U.S. patent application number 14/746751 was filed with the patent office on 2015-12-24 for determining control actions of decision modules.
The applicant listed for this patent is Atigeo Corp.. Invention is credited to Jonathan Cross, Jason Knox, Wolf Kohn, Mike Lazarus, Michael Luis Sandoval, David Talby, Vishnu Vettrivel.
Application Number | 20150370228 14/746751 |
Document ID | / |
Family ID | 54869548 |
Filed Date | 2015-12-24 |
United States Patent
Application |
20150370228 |
Kind Code |
A1 |
Kohn; Wolf ; et al. |
December 24, 2015 |
DETERMINING CONTROL ACTIONS OF DECISION MODULES
Abstract
Techniques are described for implementing automated control
systems that manipulate operations of specified target systems,
such as by modifying or otherwise manipulating inputs or other
control elements of the target system that affect its operation
(e.g., affect output of the target system). An automated control
system may have one or more decision modules that each controls at
least some of a target system, with each decision module's control
actions being automatically determined to reflect near-optimal
solutions with respect to or one more goals and in light of a
target system model having multiple inter-related constraints, such
as based on a partially optimized solution that is within a
threshold amount of a fully optimized solution. Such determination
of one or more control actions to perform may occur for a
particular time and particular decision module, as well as be
repeated over multiple times for ongoing control.
Inventors: |
Kohn; Wolf; (Seattle,
WA) ; Sandoval; Michael Luis; (Bellevue, WA) ;
Vettrivel; Vishnu; (Bothell, WA) ; Cross;
Jonathan; (Bellevue, WA) ; Knox; Jason;
(Kenmore, WA) ; Talby; David; (Mercer Island,
WA) ; Lazarus; Mike; (San Diego, CA) |
|
Applicant: |
Name |
City |
State |
Country |
Type |
Atigeo Corp. |
Bellevue |
WA |
US |
|
|
Family ID: |
54869548 |
Appl. No.: |
14/746751 |
Filed: |
June 22, 2015 |
Related U.S. Patent Documents
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Application
Number |
Filing Date |
Patent Number |
|
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62015018 |
Jun 20, 2014 |
|
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Current U.S.
Class: |
700/31 ; 700/287;
701/22; 705/44; 705/7.25 |
Current CPC
Class: |
Y02T 10/70 20130101;
B60W 2510/086 20130101; Y10S 903/93 20130101; B60L 50/60 20190201;
G06Q 10/06315 20130101; B60W 10/06 20130101; G05B 17/02 20130101;
G05B 2219/2639 20130101; G05B 19/048 20130101; B60W 10/08 20130101;
G05B 13/04 20130101; G05B 13/041 20130101; B60W 2510/0666 20130101;
G06Q 50/06 20130101; G05B 19/042 20130101 |
International
Class: |
G05B 13/04 20060101
G05B013/04; B60W 20/00 20060101 B60W020/00; G06Q 20/40 20060101
G06Q020/40; B60L 11/18 20060101 B60L011/18; G06N 5/04 20060101
G06N005/04; G06Q 10/06 20060101 G06Q010/06 |
Claims
1. A computer-implemented method comprising: obtaining, by one or
more computing systems of a collaborative distributed decision
system, coupled differential equations that represent a current
state of a physical system and that are generated from system
information and objective information and sensor information,
wherein the physical system has a plurality of inter-related
elements and has one or more outputs whose values vary based at
least in part on values of one or more manipulatable control
elements of the plurality, wherein the system information is
supplied by one or more users to describe the physical system and
includes multiple rules that each has one or more conditions to
evaluate and that specify restrictions involving the plurality of
elements, wherein the objective information identifies a goal to be
achieved during controlling of the physical system, and wherein the
sensor information identifies current state information for at
least one element of the plurality and partial initial state
information for the physical system at an earlier time; performing,
by the one or more computing systems, a piecewise linear analysis
of the coupled differential equations to identify one or more
control actions to take in the physical system that manipulate
values of the one or more manipulatable control elements and that
provide a solution for the goal within a threshold amount of an
optimal solution for the goal, wherein the performing of the
piecewise linear analysis includes: dividing a time window from the
earlier time to the specified time into a succession of a plurality
of time slices that each, other than a first time slice of the
succession, overlaps at least in part with a prior time slice of
the succession; evaluating, based on the partial initial state
information, the coupled differential equations to identify an
initial solution for the goal for the first time slice that
includes simulating effects of manipulating the one or more
manipulatable control elements to one or more initial values, and
storing a model describing a state of the physical system for the
first time slice that includes the simulated effects of the
manipulating to the one or more initial values; for each time slice
of the succession after the first time slice, updating the stored
model for the prior time slice to reflect an additional solution
for the goal for the time slice that includes simulating effects of
further manipulating the one or more manipulatable control elements
to one or more additional values; and after updating the stored
model to reflect the additional solution for the goal for a last
time slice of the succession, further updating the stored model for
the last time slice to reflect a further solution for the goal for
a next time period after the time window based at least in part on
the identified current state information, wherein the further
solution includes the identified one or more control actions to
take in the physical system for a current time; and providing
information about the identified one or more control actions, to
enable actions to be taken in the physical system for the current
time to affect the outputs based on the identified one or more
control actions.
2. The computer-implemented method of claim 1 wherein the physical
system is an electricity generating facility, wherein the plurality
of inter-related elements include multiple alternative electricity
sources within the electricity generating facility, wherein the
manipulatable control elements include one or more controls to
determine whether to accept a request to supply a specified amount
of electricity at the current time and to select which alternative
electricity source to provide the specified amount of electricity
at the current time if accepted, wherein the outputs include the
electricity being provided, and wherein the goal includes to
maximize profits for the electricity generating facility from
providing of the electricity.
3. The computer-implemented method of claim 1 wherein the physical
system is an energy generating facility, wherein the plurality of
inter-related elements include at least one energy source within
the energy generating facility and at least one energy storage
mechanism within the energy generating facility, wherein the
manipulatable control elements include one or more controls to
determine whether to accept a request to supply a specified amount
of energy at the current time and to determine to provide energy to
the at least one energy storage mechanism at the current time if
not accepted and to provide energy from the at least one energy
source at the current time if accepted, wherein the outputs include
the energy being provided, and wherein the goal includes to
maximize profits for the electricity generating facility from
providing of the energy.
4. The computer-implemented method of claim 1 wherein the physical
system is a vehicle, wherein the plurality of inter-related
elements include a motor and a battery of the vehicle, wherein the
manipulatable control elements include one or more controls to
select whether at the current time to remove energy from the
battery to power the motor or to add excess energy to the battery
and how much energy to remove from the battery, wherein the outputs
include effects of the motor to move the vehicle at the current
time, and wherein the goal includes to move the vehicle at one or
more specified speeds with a minimum of energy produced from the
battery.
5. The computer-implemented method of claim 4 wherein the plurality
of inter-related elements further includes an engine that is
manipulatable to modify energy generated from the engine, wherein
the manipulatable control elements further include one or more
additional controls to determine how much energy to generate from
the engine for use at least in part in adding the excess energy to
the battery, and wherein the goal further includes to minimize use
of fuel by the engine.
6. The computer-implemented method of claim 1 wherein the physical
system includes product inventory at one or more locations, wherein
the plurality of inter-related elements include one or more product
sources that provide products and increase the inventory at the one
or more locations and further include one or more product
recipients that receive products and decrease the inventory at the
one or more locations, wherein the manipulatable control elements
include one or more first controls to select at the current time
one or more first amounts of one or more products to request from
the one or more product sources, and further include one or more
second controls to select at the current time one or more second
amounts of at least one product to provide to the one or more
product recipients, wherein the outputs include products being
provided from the one or more locations to the one or more product
recipients, and wherein the goal includes to maximize profit of an
entity operating the one or more locations while maintaining the
inventory at one or more specified levels.
7. The computer-implemented method of claim 1 wherein the stored
model for each of the time slices of the succession is expressed
with a Hamiltonian function specific to the time slice, and wherein
each updating of the stored model for a prior time slice to reflect
an additional solution for the goal includes modifying the
Hamiltonian function expressed by the stored model for the prior
time slice.
8. The computer-implemented method of claim 1 wherein the updated
stored model for the last time slice is expressed with a
Hamiltonian function, and wherein the further updating of the
stored model for the last time slice to reflect the further
solution for the goal for the next time period includes modifying
the Hamiltonian function based at least in part on the identified
current state information.
9. The computer-implemented method of claim 1 wherein the
evaluation of the coupled differential equations to identify the
initial solution for the goal for the first time slice and the
updating for each time slice of the succession after the first time
slice of the stored model is performed to train the updated stored
model for the last time slice to reflect values of at least some of
the plurality of inter-related elements for the current time that
include one or more elements whose values are not directly
observable, and wherein training of the updated stored model for
the last time slice enables the further updating to reflect the
further solution for the goal for the next time period.
10. The computer-implemented method of claim 1 further comprising,
for each of multiple additional times after the current time,
adapting a current copy of the stored model to reflect the
additional time by: obtaining additional sensor information that
identifies state information at the additional time for one or more
elements of the plurality; determining, by the one or more
computing systems, if an updated copy of the stored model can be
generated for the additional time by attempting to identify another
solution for the goal for the additional time based at least in
part on the additional sensor information, wherein the another
solution, if identified, includes one or more further control
actions to take in the physical system for the additional time; and
if the updated copy of the stored model can be generated for the
additional time, generating and storing the updated copy of the
stored model, and providing information about one or more
additional control actions to perform in the physical system for
the additional time to further manipulate the manipulatable control
elements in a specified manner.
11. The computer-implemented method of claim 10 further comprising,
for each of the multiple additional times and if the updated copy
of the stored model for the additional time is not generated,
providing information about one or more further associated control
actions to perform in the physical system for that additional time
to further manipulate the manipulatable control elements in a
specified manner, wherein the one or more further associated
control actions are based on the current copy of the stored model
before updating for the additional time.
12. The computer-implemented method of claim 11 wherein generating
of the updated copy of the stored model for one of the multiple
additional times includes a deadline for the generating to enable
real-time control of the physical system to be performed based on
performing control actions in the physical system for the one
additional time, and wherein the generating of the updated copy of
the stored model for the one additional time fails to complete
before the deadline, such that the one or more further associated
control actions for that one additional time are performed in the
physical system for that one additional time to further manipulate
the manipulatable control elements in a specified manner.
13. The computer-implemented method of claim 10 wherein the updated
copy of the stored model for one of the multiple additional times
is not generated in a first attempt, and wherein the method further
comprises generating the updated copy of the stored model for the
one additional time during a second attempt by: determining, by the
one or more computing systems, at least one of the multiple rules
to temporarily relax by modifying at least one of the specified
restrictions corresponding to the determined at least one rule;
generating, by the one or more computing systems, additional
coupled differential equations that represent a current state of
the physical system for the one additional time based at least in
part on the modified at least one specified restrictions for the
determined at least one rule; performing, by the one or more
computing systems, a further piecewise linear analysis of the
generated additional coupled differential equations to identify the
another solution for the goal for the one additional time,
generating and storing the updated copy of the stored model for the
one additional time; and providing information about the one or
more additional control actions of the updated copy of the stored
model for the one additional time to perform in the physical system
for the one additional time to further manipulate the manipulatable
control elements in a specified manner.
14. The computer-implemented method of claim 13 wherein the
multiple rules include one or more absolute rules that specify
non-modifiable restrictions that are requirements regarding
operation of the physical system, and further include one or more
hard rules that specify restrictions regarding operation of the
physical system that can be modified in specified situations, and
wherein each determined at least one rule is one of the hard
rules.
15. The computer-implemented method of claim 13 wherein the
multiple rules include one or more soft rules whose conditions
evaluate to one of three or more possible values under differing
situations to represent varying degrees of uncertainty and further
include additional rules whose conditions evaluate to either true
or false under differing situations, and wherein one or more of the
determined at least one rules are from the soft rules.
16. The computer-implemented method of claim 10 wherein the
determining if the updated copy of the stored model can be
generated for one of the multiple additional times includes
generating values for one or more model error measurements based at
least in part on the current copy of the stored model for the one
additional time, and includes determining that at least one of the
generated values exceeds an error threshold, and wherein the method
further comprises generating the updated copy of the stored model
for the one additional time by: evaluating, by the one or more
computing systems, the generated values for the one or more model
error measurements to determine at least one of the multiple rules
that is incorrect; modifying, by the one or more computing systems,
the determined at least one rule in a manner expected to reduce
values for the one or more model error measurements; generating, by
the one or more computing systems, additional coupled differential
equations that represent a current state of the physical system for
the one additional time based at least in part on the modified
determined at least one rule; performing, by the one or more
computing systems, a further piecewise linear analysis of the
generated additional coupled differential equations to identify the
another solution for the goal for the one additional time,
generating and storing the updated copy of the stored model for the
one additional time; and providing information about the one or
more additional control actions of the updated copy of the stored
model for the one additional time to perform in the physical system
for the one additional time to further manipulate the manipulatable
control elements in a specified manner.
17. The computer-implemented method of claim 16 wherein the another
solution identified for the goal for two or more of the additional
times each has an associated error measurement within a defined
threshold relative to an optimal solution for the goal for that
additional time, and wherein the one or more model error
measurements are based on a rate of change of one or more of:
Hamiltonian functions expressed by two or more copies of the model
for two or more times; amounts of entropy included in two or more
copies of the model for two or more times; values of variables
associated with the plurality of inter-related elements in state
information for the physical system for two or more times; or a
reduction in the associated error measurements for the another
solutions for the two or more additional times.
18. The computer-implemented method of claim 10 wherein the
determining if the updated copy of the stored model can be
generated for one of the multiple additional times includes:
generating, by the one or more computing systems, the updated copy
of the stored model for the one additional time; generating, by the
one or more computing systems, values for one or more model error
measurements for the generated updated copy of the stored model for
the one additional time; determining, by the one or more computing
systems, that at least one of the generated values exceeds an error
threshold; and replacing, by the one or more computing systems, the
generated updated copy of the stored model for the one additional
time with a new generated updated copy of the stored model for the
one additional time, by causing a new copy of the model for the one
additional time to be generated without using any past copies of
the model, and storing the generated new copy of the model for the
one additional time.
19. The computer-implemented method of claim 1 further comprising
modifying the multiple rules during one of the multiple additional
times, and wherein updating of copies of the stored model after the
one additional time includes using the modified rules.
20. The computer-implemented method of claim 1 further comprising,
before the dividing of the time window into the succession of time
slices, determining, by the one or more computing systems, at least
one of a size of the time slices or a size of the time window.
21. The computer-implemented method of claim 20 wherein the
determining of the at least one of the size of the time slices or
the size of the time window includes evaluating multiple test sizes
for the at least one of the size of the time slices or the size of
the time window, and selecting, based at least in part on the
evaluating, one or more of the multiple sizes to use for the
determined at least one size of the time slices or size of the time
window.
22. The computer-implemented method of claim 20 further comprising
generating a Hamiltonian function to express a copy of the model of
the state of the physical system before the dividing of the time
window into the succession of time slices, and wherein the
determining of the at least one of the size of the time slices or
the size of the time window includes performing a symbolic
computation analysis of the Hamiltonian function to identify one or
more preferred sizes to use for the determined at least one size of
the time slices or size of the time window.
23. The computer-implemented method of claim 1 wherein the
providing of the information about the identified one or more
control actions includes performing, by the one or more computing
systems, the actions in the physical system to affect the outputs
by manipulating the manipulatable control elements in specified
manners for the identified one or more control actions.
24. A non-transitory computer-readable medium having stored
contents that cause one or more computing systems to perform a
method, the method comprising: obtaining, by the one or more
computing systems, coupled differential equations that represent a
state of a target system at a specified time and that are generated
from system information and objective information and sensor
information, wherein the target system has a plurality of elements
that are inter-related and that include one or more control
elements with modifiable values, wherein the system information is
supplied by one or more users to describe the physical system and
includes restrictions involving the plurality of elements, wherein
the objective information identifies a goal to be achieved based at
least in part on modifying the values of the control elements, and
wherein the sensor information identifies state information for the
specified time for at least one element of the plurality;
performing, by the one or more computing systems, a piecewise
linear analysis of the coupled differential equations to identify a
solution for the goal for the specified time within a threshold
amount of an optimal solution for the goal, wherein the identified
solution has one or more associated control actions that modify at
least one value of at least one of the control elements in a
specified manner, and wherein the performing of the piecewise
linear analysis includes: dividing a time window from an earlier
time to the specified time into a succession of a plurality of time
slices; evaluating, based on initial state information for the
earlier time, the coupled differential equations to identify an
initial solution for the goal for a first time slice of the
succession that includes simulating effects of modifying one or
more values of the one or more manipulatable control elements in a
specified initial manner, and storing a model describing a state of
the target system for the first time slice that includes the
simulated effects of the modifying of the one or more values; for
each time slice of the succession after the first time slice,
updating the stored model for a prior time slice to reflect an
additional solution for the goal for the time slice that includes
simulating effects of further modifying one or more values of the
one or more manipulatable control elements; and after updating the
stored model to reflect the additional solution for the goal for a
last time slice of the succession, further updating the stored
model for the last time slice to reflect a further solution for the
goal for a next time period after the time window based at least in
part on the identified state information for the specified time,
wherein the further solution includes the one or more associated
control actions; and providing information about the one or more
associated control actions, to enable modification of the at least
one value of the at least one control element for the specified
time based on the one or more associated control actions.
25. The non-transitory computer-readable medium of claim 24 wherein
the target system is a physical system having one or more outputs
whose values vary based at least in part on the values of the
control elements, wherein the one or more computing systems are
part of a collaborative distributed decision system, and wherein
the stored contents include software instructions that, when
executed, further cause the one or more computing systems to
initiate performance of the one or more associated control actions
in the physical system to modify the at least one value of the at
least one control element and to cause resulting changes in the
values of the one or more outputs for the specified time.
26. The non-transitory computer-readable medium of claim 24 wherein
the target system includes one or more computing resources being
protected from unauthorized operations, wherein the plurality of
inter-related elements include one or more sources of attempts to
perform operations, wherein the control elements include one or
more controls to determine whether a change in authorization to a
specified type of operation is needed and to select one or more
actions to take to implement the change in authorization if so
determined, and wherein the goal includes to minimize unauthorized
operations that are performed.
27. The non-transitory computer-readable medium of claim 24 wherein
the target system includes one or more information sources to be
analyzed to determine a risk level from information of the one or
more information sources, wherein the control elements include one
or more controls to determine whether the risk level exceeds a
specified threshold and to select one or more actions to take to
mitigate the risk level, and wherein the goal includes to minimize
the risk level.
28. The non-transitory computer-readable medium of claim 24 wherein
the target system includes one or more financial markets, wherein
the plurality of inter-related elements include items that can be
purchased and/or sold in the one or more financial markets, wherein
the control elements include one or more controls to determine
whether to purchase or sell particular items at particular times
and to select one or more actions to initiate transactions to
purchase or sell the particular items at the particular times, and
wherein the goal includes to maximize profit while maintaining risk
below a specified threshold.
29. The non-transitory computer-readable medium of claim 24 wherein
the target system includes functionality to perform coding for
medical procedures performed on humans, wherein the plurality of
inter-related elements include a plurality of medical codes
corresponding to a plurality of medical procedures, wherein the
control elements include one or more controls to select particular
medical codes to associate with particular medical procedures in
specified circumstances, and wherein the goal includes to minimize
errors in selected medical codes that cause revenue leakage.
30. A system comprising: one or more processors of one or more
computing systems; and one or more modules that, when executed by
at least one of the one or more processors, cause the one or more
processors to determine one or more control actions to perform as
part of controlling a physical system, the determining of the one
or more control actions including: obtaining coupled differential
equations that represent a state of a physical system for a
specified time and that are generated from system information and
objective information and sensor information, wherein the physical
system has a plurality of inter-related elements and has one or
more outputs whose values vary based at least in part on values of
one or more manipulatable control elements of the plurality,
wherein the system information is supplied by one or more users to
describe the physical system and includes restrictions involving
the plurality of elements, wherein the objective information
identifies a goal to be achieved during controlling of the physical
system, and wherein the sensor information identifies state
information for the specified time for at least one element of the
plurality; performing a first piecewise linear analysis of the
coupled differential equations to train a model that describes a
state of the physical system for the specified time and that
includes values of at least some of the plurality of elements for
the specified time, wherein the performing of the first piecewise
linear analysis includes simulating effects of manipulating the one
or more manipulatable control elements for each of one or more
prior time periods before the specified time while satisfying the
goal for the one or more prior time periods; performing a second
piecewise linear analysis of the coupled differential equations to
identify one or more control actions to take in the physical system
for the specified time that manipulate values of the one or more
manipulatable control elements and that provide a solution for the
goal for the specified time, wherein the performing of the
piecewise linear analysis includes updating the model to reflect
the solution for the goal for the specified time period based at
least in part on the identified state information for the specified
time; and providing information about the identified one or more
control actions, to enable actions to be taken in the physical
system for the specified time to affect the outputs based on the
identified one or more control actions.
31. The system of claim 30 wherein the one or more modules are part
of a collaborative distributed decision system and include software
instructions for execution by the at least one processor, wherein
the provided solution reflected in the updated model is within a
threshold amount of an optimal solution for the goal for the
specified time, and wherein the system further comprises one or
more effectuators to perform the actions in the physical system for
the specified time by manipulating the values of the one or more
manipulatable control elements in specified manners for the
identified one or more control actions to affect the outputs.
32. The system of claim 31 wherein the performing of the first
piecewise linear analysis of the coupled differential equations to
train the model further includes: for a time window from an earlier
time to the specified time that includes the one or more prior time
periods, dividing the time window into a succession of a plurality
of time slices that each, other than a first time slice of the
succession, overlaps at least in part with a prior time slice of
the succession; evaluating, based on initial state information for
the earlier time, an initial version of the coupled differential
equations to identify an initial solution for the goal for the
first time slice that includes simulating effects of manipulating
the one or more manipulatable control elements to one or more
initial values, and storing an initial version of the model that
describes the state of the physical system for the first time slice
and includes the simulated effects of the manipulating to the one
or more initial values; and for each time slice of the succession
after the first time slice, updating a version of the stored model
from the prior time slice to reflect an additional solution for the
goal for the time slice that includes simulating effects of further
manipulating the one or more manipulatable control elements to one
or more additional values, and wherein the trained model is a
version of the model after the updating to reflect the additional
solution for the goal for a last time slice of the succession.
33. The system of claim 30 wherein the one or more modules consist
of one or more means for performing the determining of the one or
more control actions to perform as part of controlling the physical
system.
Description
BACKGROUND
[0001] Various attempts have been made to implement automated
control systems for various types of physical systems that have
inputs or other control elements that the control system can
manipulate to attempt to provide desired output or other behavior
of the physical systems being controlled. Such automated control
systems have used various types of architectures and underlying
computing technologies to attempt to implement such functionality,
including to attempt to deal with issues related to uncertainty in
the state of the physical system being controlled, the need to make
control decisions in very short amounts of time to provide
real-time or near-real-time control and with only partial
information, etc.
[0002] However, various difficulties exist with existing automated
control systems and their underlying architectures and computing
technologies, including with respect to managing large numbers of
constraints (sometimes conflicting), operating in a coordinated
manner with other systems, etc.
BRIEF DESCRIPTION OF THE DRAWINGS
[0003] FIG. 1 is a network diagram illustrating an example
environment in which a system for performing cooperative
distributed control of target systems may be configured and
initiated.
[0004] FIG. 2 is a network diagram illustrating an example
environment in which a system for performing cooperative
distributed control of target systems may be implemented.
[0005] FIG. 3 is a block diagram illustrating example computing
systems suitable for executing an embodiment of a system for
performing cooperative distributed control of target systems in
configured manners.
[0006] FIG. 4 illustrates a flow diagram of an example embodiment
of a Collaborative Distributed Decision (CDD) System routine.
[0007] FIGS. 5A-5B illustrate a flow diagram of an example
embodiment of a CDD Decision Module Construction routine.
[0008] FIGS. 6A-6B illustrate a flow diagram of an example
embodiment of a decision module routine.
[0009] FIGS. 7A-7B illustrate a flow diagram of an example
embodiment of a CDD Control Action Determination routine.
[0010] FIGS. 8A-8B illustrate a flow diagram of an example
embodiment of a CDD Coordinated Control Management routine.
[0011] FIG. 9 illustrates a flow diagram of an example embodiment
of a routine for a target system being controlled.
[0012] FIG. 10 illustrates a network diagram of a portion of a
distributed architecture of an example CDI system.
[0013] FIG. 11 illustrates a network diagram of a portion of an
example Rules Builder component.
[0014] FIG. 12 illustrates a network diagram of example
sub-components of an example Rules Builder component.
[0015] FIG. 13 illustrates a network diagram of interactions of
portions of a Rules Builder component with an executing agent.
[0016] FIG. 14 illustrates a network diagram of interactions of
portions of a Rules Builder component with chattering
components.
[0017] FIG. 15 illustrates a diagram of a sliding window for use
with chattering.
[0018] FIG. 16 illustrates a diagram of using Lebesgue integration
to approximate a control trajectory for chattering.
[0019] FIG. 17 illustrates a diagram of various interactions of
different portions of a CDI system.
[0020] FIG. 18 illustrates a diagram of example different types of
rules.
[0021] FIG. 19 illustrates an example user interface related to
medical/clinical auto-coding.
[0022] FIG. 20 illustrates a diagram of some components of a CU
system.
[0023] FIG. 21 illustrates a diagram of performing Pareto
processing for use with mean field techniques.
[0024] FIG. 22 illustrates a network diagram of an example decision
module agent.
[0025] FIG. 23 illustrates a network diagram of an example of
offline workflows for knowledge capture.
[0026] FIG. 24 illustrates a network diagram of an example of
workflows for mean field computation and Pareto Optimal.
[0027] FIG. 25 illustrates a network diagram of an example of an
automated control system for a home solar micro-grid electrical
generating system.
[0028] FIG. 26 illustrates a diagram of workflow and components of
a portion of a CDI system.
[0029] FIG. 27 illustrates a diagram of a workflow for an inference
process portion of a CDI system.
[0030] FIG. 28 illustrates a diagram of an overview workflow for a
portion of a CDI system.
[0031] FIGS. 29A-29K illustrate examples of using a CDI system to
iteratively determine near-optimal solutions over time for
controlling a target system.
DETAILED DESCRIPTION
[0032] Techniques are described for implementing automated control
systems to control or otherwise manipulate at least some operations
of specified physical systems or other target systems. A target
system to be controlled or otherwise manipulated may have numerous
elements that are inter-connected in various manners, with a subset
of those elements being inputs or other control elements that a
corresponding automated control system may modify or otherwise
manipulate in order to affect the operation of the target system.
In at least some embodiments and situations, a target system may
further have one or more outputs that the manipulations of the
control elements affect, such as if the target system is producing
or modifying physical goods or otherwise producing physical
effects.
[0033] As part of implementing such an automated control system for
a particular target system, an embodiment of a Collaborative
Distributed Decision (CDD) system may use the described techniques
to perform various automated activities involved in constructing
and implementing the automated control system--a brief introduction
to some aspects of the activities of the CDD system is provided
here, with additional details included below. In particular, the
CDD system may in some embodiments implement a Decision Module
Construction component that interacts with one or more users to
obtain a description of a target system, including restrictions
related to the various elements of the target system, and one or
more goals to be achieved during control of the target system--the
Decision Module Construction component then performs various
automated actions to generate, test and deploy one or more
executable decision modules (also referred to at times as "decision
elements" and/or "agents") to use in performing the control of the
target system. When the one or more executable decision modules are
deployed and executed, the CDD system may further provide various
components within or external to the decision modules being
executed to manage their control of the target system, such as a
Control Action Determination component of each decision module to
optimize or otherwise enhance the control actions that the decision
module generates, and/or one or more Coordinated Control Management
components to coordinate the control actions of multiple decision
modules that are collectively performing the control of the target
system. Additional details related to such components of the CDD
system and their automated operations are included below.
[0034] As noted above, the described techniques may be used to
provide automated control systems for various types of physical
systems or other target systems. In one or more embodiments, an
automated control system is generated and provided and used to
control a micro-grid electricity facility, such as at a residential
location that includes one or more electricity sources (e.g., one
or more solar panel grids, one or more wind turbines, etc.) and one
or more electricity storage and source mechanisms (e.g., one or
more batteries). The automated control system may, for example,
operate at the micro-grid electricity facility (e.g., as part of a
home automation system), such as to receive requests from the
operator of a local electrical grid to provide particular amounts
of electricity at particular times, and to control operation of the
micro-grid electricity facility by determining whether to accept
each such request. If a request is accepted, the control actions
may further include selecting which electricity source (e.g., solar
panel, battery, etc.) to use to provide the requested electricity,
and otherwise the control actions may further include determine to
provide electricity being generated to at least one energy storage
mechanism (e.g., to charge a battery). Outputs of such a physical
system include the electricity being provided to the local
electrical grid, and a goal that the automated control system
implements may be, for example, is to maximize profits for the
micro-grid electricity facility from providing of the electricity.
It will be appreciated that such a physical system being controlled
and a corresponding automated control system may include a variety
of elements and use various types of information and perform
various types of activities, with additional details regarding such
an automated control system being included below.
[0035] In one or more embodiments, an automated control system is
generated and provided and used to control a vehicle with a motor
and in some cases an engine, such as an electrical bicycle in which
power may come from a user who is pedaling and/or from a motor
powered by a battery and/or an engine. The automated control system
may, for example, operate on the vehicle or on the user, such as to
control operation of the vehicle by determining whether at a
current time to remove energy from the battery to power the motor
(and if so to further determine how much energy to remove from the
battery) or to instead add excess energy to the battery (e.g., as
generated by the engine, and if so to further determine how much
energy to generate from the engine; and/or as captured from braking
or downhill coasting). Outputs of such a physical system include
the effects of the motor to move the vehicle, and a goal that the
automated control system implements may be, for example, to move
the vehicle at one or more specified speeds with a minimum of
energy produced from the battery, and/or to minimize use of fuel by
the engine. It will be appreciated that such a physical system
being controlled and a corresponding automated control system may
include a variety of elements and use various types of information
and perform various types of activities, with additional details
regarding such an automated control system being included
below.
[0036] In one or more embodiments, an automated control system is
generated and provided and used to manage product inventory for one
or more products at one or more locations, such as a retail
location that receives products from one or more product sources
(e.g., when ordered or requested by the retail location) and that
provides products to one or more product recipients (e.g., when
ordered or requested by the recipients). The automated control
system may, for example, operate at the retail location and/or at a
remote network-accessible location, such as to receive requests
from product recipients for products, and to control operation of
the product inventory at the one or more locations by selecting at
a current time one or more first amounts of one or more products to
request from the one or more product sources, and by selecting at
the current time one or more second amounts of at least one product
to provide to the one or more product recipients. Outputs of such a
physical system include products being provided from the one or
more locations to the one or more product recipients, and a goal
that the automated control system implements may be, for example,
to maximize profit of an entity operating the one or more locations
while maintaining the inventory at one or more specified levels. It
will be appreciated that such a physical system being controlled
and a corresponding automated control system may include a variety
of elements and use various types of information and perform
various types of activities, with additional details regarding such
an automated control system being included below.
[0037] In one or more embodiments, an automated control system is
generated and provided and used to manage cyber-security for
physical computing resources being protected from unauthorized
operations and/or to determine a risk level from information
provided by or available from one or more information sources. The
automated control system may, for example, operate at the location
of the computing resources or information sources and/or at a
remote network-accessible location, such as to receive information
about attempts (whether current or past) to perform operations on
computing resources being protected or about information being
provided by or available from the one or more information sources,
and to control operation of the cyber-security system by determine
whether a change in authorization to a specified type of operation
is needed and to select one or more actions to take to implement
the change in authorization if so determined, and/or to determine
whether a risk level exceeds a specified threshold and to select
one or more actions to take to mitigate the risk level. A goal that
the automated control system implements may be, for example, to
minimize unauthorized operations that are performed and/or to
minimize the risk level. It will be appreciated that such a target
system being controlled and a corresponding automated control
system may include a variety of elements and use various types of
information and perform various types of activities, with
additional details regarding such an automated control system being
included below.
[0038] In one or more embodiments, an automated control system is
generated and provided and used to manage transactions being
performed in one or more financial markets, such as to buy and/or
sell physical items or other financial items. The automated control
system may, for example, operate at the one or more final markets
or at a network-accessible location that is remote from the one or
more financial markets, such as to control operation of the
transactions performed by determining whether to purchase or sell
particular items at particular times and to select one or more
actions to initiate transactions to purchase or sell the particular
items at the particular times. A goal that the automated control
system implements may be, for example, to maximize profit while
maintaining risk below a specified threshold. It will be
appreciated that such a target system being controlled and a
corresponding automated control system may include a variety of
elements and use various types of information and perform various
types of activities, with additional details regarding such an
automated control system being included below.
[0039] In one or more embodiments, an automated control system is
generated and provided and used to perform coding for medical
procedures, such as to allow billing to occur for medical
procedures performed on humans. The automated control system may,
for example, operate at a location at which the medical procedures
are performed or at a network-accessible location that is remote
from such a medical location, such as to control operation of the
coding that is performed by selecting particular medical codes to
associate with particular medical procedures in specified
circumstances. A goal that the automated control system implements
may be, for example, to minimize errors in selected medical codes
that cause revenue leakage. It will be appreciated that such a
target system being controlled and a corresponding automated
control system may include a variety of elements and use various
types of information and perform various types of activities, with
additional details regarding such an automated control system being
included below.
[0040] It will also be appreciated that the described techniques
may be used with a wide variety of other types of target systems,
some of which are further discussed below, and that the invention
is not limited to the techniques discussed for particular target
systems and corresponding automated control systems.
[0041] As noted above, a Collaborative Distributed Decision (CDD)
system may in some embodiments use at least some of the described
techniques to perform various automated activities involved in
constructing and implementing a automated control system for a
specified target system, such as to modify or otherwise manipulate
inputs or other control elements of the target system that affect
its operation (e.g., affect one or more outputs of the target
system). An automated control system for such a target system may
in some situations have a distributed architecture that provides
cooperative distributed control of the target system, such as with
multiple decision modules that each control a portion of the target
system and that operate in a partially decoupled manner with
respect to each other. If so, the various decision modules'
operations for the automated control system may be at least
partially synchronized, such as by each reaching a consensus with
one or more other decision modules at one or more times, even if a
fully synchronized convergence of all decision modules at all times
is not guaranteed or achieved.
[0042] The CDD system may in some embodiments implement a Decision
Module Construction component that interacts with one or more users
to obtain a description of a target system, including restrictions
related to the various elements of the target system, and one or
more goals to be achieved during control of the target system--the
Decision Module Construction component then performs various
automated actions to generate, test and deploy one or more
executable decision modules to use in performing the control of the
target system. The Decision Module Construction component may thus
operate as part of a configuration or setup phase that occurs
before a later run-time phase in which the generated decision
modules are executed to perform control of the target system,
although in some embodiments and situations the Decision Module
Construction component may be further used after an initial
deployment to improve or extend or otherwise modify an automated
control system that has one or more decision modules (e.g., while
the automated control system continues to be used to control the
target system), such as to add, remove or modify decision modules
for the automated control system.
[0043] In some embodiments, some or all automated control systems
that are generated and deployed may further provide various
components within them for execution during the runtime operation
of the automated control system, such as by including such
components within decision modules in some embodiments and
situations. Such components may include, for example, a Control
Action Determination component of each decision module (or of some
decision modules) to optimize or otherwise determine and improve
the control actions that the decision module generates. For
example, such a Control Action Determination component in a
decision module may in some embodiments attempt to automatically
determine the decision module's control actions for a particular
time to reflect a near-optimal solution with respect to or one more
goals and in light of a model of the decision module for the target
system that has multiple inter-related constraints--if so, such a
near-optimal solution may be based at least in part on a partially
optimized solution that is within a threshold amount of a fully
optimized solution. Such determination of one or more control
actions to perform may occur for a particular time and for each of
one or more decision modules, as well as be repeated over multiple
times for ongoing control by at least some decision modules in some
situations. In some embodiments, the model for a decision module is
implemented as a Hamiltonian function that reflects a set of
coupled differential equations based in part on constraints
representing at least part of the target system, such as to allow
the model and its Hamiltonian function implementation to be updated
over multiple time periods by adding additional expressions within
the evolving Hamiltonian function.
[0044] In some embodiments, the components included within a
generated and deployed automated control system for execution
during the automated control system's runtime operation may further
include one or more Coordinated Control Management components to
coordinate the control actions of multiple decision modules that
are collectively performing the control of a target system for the
automated control system. For example, some or all decision modules
may each include such a Control Action Determination component in
some embodiments to attempt to synchronize that decision module's
local solutions and proposed control actions with those of one or
more other decision modules in the automated control system, such
as by determining a consensus shared model with those other
decision modules that simultaneously provides solutions from the
decision module's local model and the models of the one or more
other decision modules. Such inter-module synchronizations may
occur repeatedly to determine one or more control actions for each
decision module at a particular time, as well as to be repeated
over multiple times for ongoing control. In addition, each decision
module's model is implemented in some embodiments as a Hamiltonian
function that reflects a set of coupled differential equations
based in part on constraints representing at least part of the
target system, such as to allow each decision module's model and
its Hamiltonian function implementation to be combined with the
models of one or more other decision modules by adding additional
expressions for those other decision modules' models within the
initial Hamiltonian function for the local model of the decision
module.
[0045] Use of the described techniques may also provide various
types of benefits in particular embodiments, including
non-exclusive examples of beneficial attributes or operations as
follows: [0046] Infer interests/desired content in a cold start
environment where textual (or other unstructured) data is available
and with minimal user history; [0047] Improve inference in a
continuous way that can incorporate increasingly rich user
histories; [0048] Improve inference performance with the addition
of feedback, explicit/implicit, positive/negative and preferably in
a real-time or near-real-time manner; [0049] Derive information
from domain experts that provide business value, and embed them in
inference framework; [0050] Dynamically add new unstructured data
that may represent new states, and update existing model in a
calibrated way; [0051] Renormalize inference system to accommodate
conflicts; [0052] Immediately do inferencing in a new environment
based on a natural language model; [0053] Add new information as a
statistical model, and integrate with a natural language model to
significantly improve inference/prediction; [0054] Integrate new
data and disintegrate old data in a way that only improves
performance; [0055] Perform inferencing in a data secure way;
[0056] Integrate distinct inferencing elements in a distributed
network and improve overall performance; [0057] Easily program
rules and information into the system from a lay-user perspective;
[0058] Inexpensively perform computer inferences in a way that is
suitable for bandwidth of mobile devices; and [0059] Incorporate
constraint information.
[0060] It will be appreciated that some embodiments may not include
all some illustrative benefits, and that some embodiments may
include some benefits that are not listed.
[0061] For illustrative purposes, some embodiments are described
below in which specific types of operations are performed,
including with respect to using the described techniques with
particular types of target systems and to perform particular types
of control activities that are determined in particular manners.
These examples are provided for illustrative purposes and are
simplified for the sake of brevity, and the inventive techniques
may be used in a wide variety of other situations, including in
other environments and with other types of automated control action
determination techniques, some of which are discussed below.
[0062] FIG. 1 is a network diagram illustrating an example
environment in which a system for performing cooperative
distributed control of one or more target systems may be configured
and initiated. In particular, an embodiment of a CDD system 140 is
executing on one or more computing systems 190, including in the
illustrated embodiment to operate in an online manner and provide a
graphical user interface (GUI) (not shown) and/or other interfaces
119 to enable one or more remote users of client computing systems
110 to interact over one or more intervening computer networks 100
with the CDD system 140 to configure and create one or more
decision modules to include as part of an automated control system
to use with each of one or more target systems to be
controlled.
[0063] In particular, target system 1 160 and target system 2 170
are example target systems illustrated in this example, although it
will be appreciated that only one target system or numerous target
systems may be available in particular embodiments and situations,
and that each such target system may include a variety of
mechanical, electronic, chemical, biological, and/or other types of
components to implement operations of the target system in a manner
specific to the target system. In this example, the one or more
users (not shown) may interact with the CDD system 140 to generate
an example automated control system 122 for target system 1, with
the automated control system including multiple decision modules
124 in this example that will cooperatively interact to control
portions of the target system 1 160 when later deployed and
implemented. The process of the users interacting with the CDD
system 140 to create the automated control system 122 may involve a
variety of interactions over time, including in some cases
independent actions of different groups of users, as discussed in
greater detail elsewhere. In addition, as part of the process of
creating and/or training or testing automated control system 122,
it may perform one or more interactions with the target system 1 as
illustrated, such as to obtain partial initial state information,
although some or all training activities may in at least some
embodiments include simulating effects of control actions in the
target system 1 without actually implementing those control actions
at that time.
[0064] After the automated control system 122 is created, the
automated control system may be deployed and implemented to begin
performing operations involving controlling the target system 1
160, such as by optionally executing the automated control system
122 on the one or more computing systems 190 of the CDD system 140,
so as to interact over the computer networks 100 with the target
system 1. In other embodiments and situations, the automated
control system 122 may instead be deployed by executing local
copies of some or all of the automated control system 122 (e.g.,
one or more of the multiple decision modules 124) in a manner local
to the target system 1, as illustrated with respect to a deployed
copy 121 of some or all of automated control system 1, such as on
one or more computing systems (not shown) that are part of the
target system 1.
[0065] In a similar manner to that discussed with respect to
automated control system 122, one or more users (whether the same
users, overlapping users, or completely unrelated users to those
that were involved in creating the automated control system 122)
may similarly interact over the computer network 100 with the CDD
system 140 to create a separate automated control system 126 for
use in controlling some or all of the target system 2 170. In this
example, the automated control system 126 for target system 2
includes only a single decision module 128 that will perform all of
the control actions for the automated control system 126. The
automated control system 126 may similarly be deployed and
implemented for target system 2 in a manner similar to that
discussed with respect to automated control system 122, such as to
execute locally on the one or more computing systems 190 and/or on
one or more computing systems (not shown) that are part of the
target system 2, although a deployed copy of automated control
system 2 is not illustrated in this example. It will be further
appreciated that the automated control systems 122 and/or 126 may
further include other components and/or functionality that are
separate from the particular decision modules 124 and 128,
respectively, although such other components and/or functionality
are not illustrated in FIG. 1.
[0066] The network 100 may, for example, be a publicly accessible
network of linked networks, possibly operated by various distinct
parties, such as the Internet, with the CDD system 140 available to
any users or only certain users over the network 100. In other
embodiments, the network 100 may be a private network, such as, for
example, a corporate or university network that is wholly or
partially inaccessible to non-privileged users. In still other
embodiments, the network 100 may include one or more private
networks with access to and/or from the Internet. Thus, while the
CDD system 140 in the illustrated embodiment is implemented in an
online manner to support various users over the one or more
computer networks 100, in other embodiments a copy of the CDD
system 140 may instead be implemented in other manners, such as to
support a single user or a group of related users (e.g., a company
or other organization), such as if the one or more computer
networks 100 are instead an internal computer network of the
company or other organization, and with such a copy of the CDD
system optionally not being available to other users external to
the company or other organizations. The online version of the CDD
system 140 and/or local copy version of the CDD system 140 may in
some embodiments and situations operate in a fee-based manner, such
that the one or more users provide various fees to use various
operations of the CDD system, such as to perform interactions to
generate decision modules and corresponding automated control
systems, and/or to deploy or implement such decision modules and
corresponding automated control systems in various manners. In
addition, the CDD system 140, each of its components (including
component 142 and optional other components 117, such as one or
more CDD Control Action Determination components and/or one or more
CDD Coordinated Control Management components), each of the
decision modules, and/or each of the automated control systems may
include software instructions that execute on one or more computing
systems (not shown) by one or more processors (not shown), such as
to configure those processors and computing systems to operate as
specialized machines with respect to performing their programmed
functionality.
[0067] FIG. 2 is a network diagram illustrating an example
environment in which a system for performing cooperative
distributed control of target systems may be implemented, and in
particular continues the examples discussed with respect to FIG. 1.
In the example environment of FIG. 2, target system 1 160 is again
illustrated, with the automated control system 122 now being
deployed and implemented to use in actively controlling the target
system 1 160. In the example of FIG. 2, the decision modules 124
are represented as individual decision modules 124a, 124b, etc., to
124n, and may be executing locally to the target system 1 160
and/or in a remote manner over one or more intervening computer
networks (not shown). In the illustrated example, each of the
decision modules 124 includes a local copy of a CDD Control Action
Determination component 144, such as with component 144a supporting
its local decision module 124a, component 144b supporting its local
decision module 124b, and component 144n supporting its local
decision module 124n. Similarly, the actions of the various
decision modules 124 are coordinated and synchronized in a
peer-to-peer manner in the illustrated embodiment, with each of the
decision modules 124 including a copy of a CDD Coordinated Control
Management component 146 to perform such synchronization, with
component 146a supporting its local decision module 124a, component
146b supporting its local decision module 124b, and component 146n
supporting its local decision module 124n.
[0068] As the decision modules 124 and automated control system 122
execute, various interactions 175 between the decision modules 124
are performed, such as to share information about current models
and other state of the decision modules to enable cooperation and
coordination between various decision modules, such as for a
particular decision module to operate in a partially synchronized
consensus manner with respect to one or more other decision modules
(and in some situations in a fully synchronized manner in which the
consensus actions of all of the decision modules 124 converge).
During operation of the decision modules 124 and automated control
system 122, various state information 143 may be obtained by the
automated control system 122 from the target system 160, such as
initial state information and changing state information over time,
and including outputs or other results in the target system 1 from
control actions performed by the decision modules 124.
[0069] The target system 1 in this example includes various control
elements 161 that the automated control system 122 may manipulate,
and in this example each decision module 124 may have a separate
group of one or more control elements 161 that it manipulates (such
that decision module A 124a performs interactions 169a to perform
control actions A 147a on control elements A 161a, decision module
B 124b performs interactions 169b to perform control actions B 147b
on control elements B 161b, and decision module N 124n performs
interactions 169n to perform control actions N 147n on control
elements N 161n). Such control actions affect the internal state
163 of other elements of the target system 1, including optionally
to cause or influence one or more outputs 162. As operation of the
target system 1 is ongoing, at least some of the internal state
information 163 is provided to some or all of the decision modules
to influence their ongoing control actions, with each of the
decision modules 124a-124n possibly having a distinct set of state
information 143a-143n, respectively, in this example.
[0070] As discussed in greater detail elsewhere, each decision
module 124 may use such state information 143 and a local model 145
of the decision module for the target system to determine
particular control actions 147 to next perform, such as for each of
multiple time periods, although in other embodiments and
situations, a particular automated control system may perform
interactions with a particular target system for only one time
period or only for some time periods. For example, the local CDD
Control Action Determination component 144 for a decision module
124 may determine a near-optimal location solution for that
decision module's local model 145, and with the local CDD
Coordinated Control Management component 146 determining a
synchronized consensus solution to reflect other of the decision
modules 124, including to update the decision module's local model
145 based on such local and/or synchronized solutions that are
determined. Thus, during execution of the automated control system
122, the automated control system performs various interactions
with the target system 160, including to request state information,
and to provide instructions to modify values of or otherwise
manipulate control elements 161 of the target system 160. For
example, for each of multiple time periods, decision module 124a
may perform one or more interactions 169a with one or more control
elements 161a of the target system, while decision module 124b may
similarly perform one or more interactions 169b with one or more
separate control elements B 161b, and decision module 124n may
perform one or more interactions 169n with one or more control
elements N 161n of the target system 160. In other embodiments and
situations, at least some control elements may not perform control
actions during each time period.
[0071] While example target system 2 170 is not illustrated in FIG.
2, further details are illustrated for decision module 128 of
automated control system 126 for reference purposes, although such
a decision module 128 would not typically be implemented together
with the decision modules 124 controlling target system 1. In
particular, the deployed copy of automated control system 126
includes only the single executing decision module 128 in this
example, although in other embodiments the automated control system
126 may include other components and functionality. In addition,
since only a single decision module 128 is implemented for the
automated control system 126, the decision module 128 includes a
local CDD Control Action Determination component 244, but does not
in the illustrated embodiment include any local CDD Coordinated
Control Management component, since there are not other decision
modules with which to synchronize and interact.
[0072] While not illustrated in FIGS. 1 and 2, the distributed
nature of operations of automated control systems such as those of
122 allow partially decoupled operations of the various decision
modules, include to allow modifications to the group of decision
modules 124 to be modified over time while the automated control
system 122 is in use, such as to add new decision modules 124
and/or to remove existing decision modules 124. In a similar
manner, changes may be made to particular decision modules 124
and/or 128, such as to change rules or other restrictions specific
to a particular decision module and/or to change goals specific to
a particular decision module over time, with a new corresponding
model being generated and deployed within such a decision module,
including in some embodiments and situations while the
corresponding automated control system continues control operations
of a corresponding target system. In addition, while each automated
control system is described as controlling a single target system
in the examples of FIGS. 1 and 2, in other embodiments and
situations, other configurations may be used, such as for a single
automated control system to control multiple target systems (e.g.,
multiple inter-related target systems, multiple target systems of
the same type, etc.), and/or multiple automated control systems may
operate to control a single target system, such as by each
operating independently to control different portions of that
target control system. It will be appreciated that other
configurations may similarly be used in other embodiments and
situations.
[0073] FIG. 3 is a block diagram illustrating example computing
systems suitable for performing techniques for implementing
automated control systems to control or otherwise manipulate at
least some operations of specified physical systems or other target
systems in configured manners. In particular, FIG. 3 illustrates a
server computing system 300 suitable for providing at least some
functionality of a CDD system, although in other embodiments
multiple computing systems may be used for the execution (e.g., to
have distinct computing systems executing the CDD Decision Module
Construction component for initial configuration and setup before
run-time control occurs, and one or more copies of the CDD Control
Action Determination component 344 and/or the CDD Coordinated
Control Managements component 346 for the actual run-time control).
FIG. 3 also illustrates various client computer systems 350 that
may be used by customers or other users of the CDD system 340, as
well as one or more target systems (in this example, target system
1 360 and target system 2 370, which are accessible to the CDD
system 340 over one or more computer networks 390).
[0074] The server computing system 300 has components in the
illustrated embodiment that include one or more hardware CPU
("central processing unit") computer processors 305, various I/O
("input/output") hardware components 310, storage 320, and memory
330. The illustrated I/O components include a display 311, a
network connection 312, a computer-readable media drive 313, and
other I/O devices 315 (e.g., a keyboard, a mouse, speakers, etc.).
In addition, the illustrated client computer systems 350 may each
have components similar to those of server computing system 300,
including one or more CPUs 351, I/O components 352, storage 354,
and memory 357, although some details are not illustrated for the
computing systems 350 for the sake of brevity. The target systems
360 and 370 may also each include one or more computing systems
(not shown) having components that are similar to some or all of
the components illustrated with respect to server computing system
300, but such computing systems and components are not illustrated
in this example for the sake of brevity.
[0075] The CDD system 340 is executing in memory 330 and includes
components 342-346, and in some embodiments the system and/or
components each includes various software instructions that when
executed program one or more of the CPU processors 305 to provide
an embodiment of a CDD system as described elsewhere herein. The
CDD system 340 may interact with computing systems 350 over the
network 390 (e.g., via the Internet and/or the World Wide Web, via
a private cellular network, etc.), as well as the target systems
360 and 370 in this example. In this example embodiment, the CDD
system includes functionality related to generating and deploying
decision modules in configured manners for customers or other
users, as discussed in greater detail elsewhere herein. The other
computing systems 350 may also be executing various software as
part of interactions with the CDD system 340 and/or its components.
For example, client computing systems 350 may be executing software
in memory 357 to interact with CDD system 340 (e.g., as part of a
Web browser, a specialized client-side application program, etc.),
such as to interact with one or more interfaces (not shown) of the
CDD system 340 to configure and deploy automated control systems
(e.g., stored automated control systems 325 that were previously
created by the CDD system 340) or other decision modules 329, as
well as to perform various other types of actions, as discussed in
greater detail elsewhere. Various information related to the
functionality of the CDD system 340 may be stored in storage 320,
such as information 321 related to users of the CDD system (e.g.,
account information), and information 323 related to one or more
target systems. It will be appreciated that computing systems 300
and 350 and target systems 360 and 370 are merely illustrative and
are not intended to limit the scope of the present invention. The
computing systems may instead each include multiple interacting
computing systems or devices, and the computing systems/nodes may
be connected to other devices that are not illustrated, including
through one or more networks such as the Internet, via the Web, or
via private networks (e.g., mobile communication networks, etc.).
More generally, a computing node or other computing system or
device may comprise any combination of hardware that may interact
and perform the described types of functionality, including without
limitation desktop or other computers, database servers, network
storage devices and other network devices, PDAs, cell phones,
wireless phones, pagers, electronic organizers, Internet
appliances, television-based systems (e.g., using set-top boxes
and/or personal/digital video recorders), and various other
consumer products that include appropriate communication
capabilities. In addition, the functionality provided by the
illustrated CDD system 340 and its components may in some
embodiments be distributed in additional components. Similarly, in
some embodiments some of the functionality of the CDD system 340
and/or CDD components 342-346 may not be provided and/or other
additional functionality may be available.
[0076] It will also be appreciated that, while various items are
illustrated as being stored in memory or on storage while being
used, these items or portions of them may be transferred between
memory and other storage devices for purposes of memory management
and data integrity. Alternatively, in other embodiments some or all
of the software modules and/or systems may execute in memory on
another device and communicate with the illustrated computing
systems via inter-computer communication. Thus, in some
embodiments, some or all of the described techniques may be
performed by hardware means that include one or more processors
and/or memory and/or storage when configured by one or more
software programs (e.g., by the CDD system 340 and/or the CDD
components 342-346) and/or data structures, such as by execution of
software instructions of the one or more software programs and/or
by storage of such software instructions and/or data structures.
Furthermore, in some embodiments, some or all of the systems and/or
components may be implemented or provided in other manners, such as
by using means that are implemented at least partially or
completely in firmware and/or hardware, including, but not limited
to, one or more application-specific integrated circuits (ASICs),
standard integrated circuits, controllers (e.g., by executing
appropriate instructions, and including microcontrollers and/or
embedded controllers), field-programmable gate arrays (FPGAs),
complex programmable logic devices (CPLDs), etc. Some or all of the
components, systems and data structures may also be stored (e.g.,
as software instructions or structured data) on a non-transitory
computer-readable storage medium, such as a hard disk or flash
drive or other non-volatile storage device, volatile or
non-volatile memory (e.g., RAM), a network storage device, or a
portable media article to be read by an appropriate drive (e.g., a
DVD disk, a CD disk, an optical disk, etc.) or via an appropriate
connection. The systems, components and data structures may also in
some embodiments be transmitted as generated data signals (e.g., as
part of a carrier wave or other analog or digital propagated
signal) on a variety of computer-readable transmission mediums,
including wireless-based and wired/cable-based mediums, and may
take a variety of forms (e.g., as part of a single or multiplexed
analog signal, or as multiple discrete digital packets or frames).
Such computer program products may also take other forms in other
embodiments. Accordingly, the present invention may be practiced
with other computer system configurations.
[0077] FIG. 4 is a flow diagram of an example embodiment of a
Collaborative Distributed Decision (CDD) system routine 400. The
routine may, for example, be provided by execution of the CDD
system 340 of FIG. 3 and/or the CDD system 140 of FIG. 1, such as
to provide functionality to construct and implement automated
control systems for specified target systems.
[0078] The illustrated embodiment of the routine begins at block
410, where information or instructions are received. If it is
determined in block 420 that the information or instructions of
block 410 include an indication to create or revise one or more
decision modules for use as part of an automated control system for
a particular target system, the routine continues to block 425 to
initiate execution of a Decision Module Construction component, and
in block 430 obtains and stores one or more resulting decision
modules for the target system that are created in block 425. One
example of a routine for such a Decision Module Construction
component is discussed in greater detail with respect to FIGS.
5A-5B.
[0079] After block 430, or if it is instead determined in block 420
that the information or instructions received in block 410 are not
to create or revise one or more decision modules, the routine
continues to block 440 to determine whether the information or
instructions received in block 410 indicate to deploy one or more
created decision modules to control a specified target system, such
as for one or more decision modules that are part of an automated
control system for that target system. The one or more decision
modules to deploy may have been created immediately prior with
respect to block 425, such that the deployment occurs in a manner
that is substantially simultaneous with the creation, or in other
situations may include one or more decision modules that were
created at a previous time and stored for later use. If it is
determined to deploy one or more such decision modules for such a
target system, the routine continues to block 450 to initiate the
execution of those one or more decision modules for that target
system, such as on one or more computing systems local to an
environment of the target system, or instead on one or more remote
computing systems that communicate with the target system over one
or more intermediary computer networks (e.g., one or more computing
systems under control of a provider of the CDD system).
[0080] After block 450, the routine continues to block 460 to
determine whether to perform distributed management of multiple
decision modules being deployed in a manner external to those
decision modules, such as via one or more centralized Coordinated
Control Management components. If so, the routine continues to
block 465 to initiate execution of one or more such centralized CDD
Coordinated Control Management components for use with those
decision modules. After block 465, or if it is instead determined
in block 460 to not perform such distributed management in an
external manner (e.g., if only one decision module is executed, if
multiple decision modules are executed but coordinate their
operations in a distributed peer-to-peer manner, etc.), the routine
continues to block 470 to optionally obtain and store information
about the operations of the one or more decision modules and/or
resulting activities that occur in the target system, such as for
later analysis and/or reporting.
[0081] If it is instead determined in block 440 that the
information or instructions received in block 410 are not to deploy
one or more decision modules, the routine continues instead to
block 485 to perform one or more other indicated operations if
appropriate. For example, such other authorized operations may
include obtaining results information about the operation of a
target system in other manners (e.g., by monitoring outputs or
other state information for the target system), analyzing results
of operations of decision modules and/or activities of
corresponding target systems, generating reports or otherwise
providing information to users regarding such operations and/or
activities, etc. In addition, in some embodiments the analysis of
activities of a particular target system over time may allow
patterns to be identified in operation of the target system, such
as to allow a model of that target system to be modified
accordingly (whether manually or in an automated learning manner)
to reflect those patterns and to respond based on them. In
addition, as discussed in greater detail elsewhere, distributed
operation of multiple decision modules for an automated control
system in a partially decoupled manner allows various changes to be
made while the automated control system is in operation, such as to
add one or more new decision modules, to remove one or more
existing decision modules, to modify the operation of a particular
decision module (e.g., by changing rules or other information
describing the target system that is part of a model for the
decision module), etc. In addition, the partially decoupled nature
of multiple such decision modules in an automated control system
allows one or more such decision modules to operate individually at
times, such as if network communication issues or other problems
prevent communication between multiple decision modules that would
otherwise allow their individualized control actions to be
coordinated--in such situations, some or all such decision modules
may continue to operate in an individualized manner, such as to
provide useful ongoing control operations for a target system even
if optimal or near-optimal solutions cannot be identified from
coordination and synchronization between a group of multiple
decision modules that collectively provide the automated control
system for the target system.
[0082] After blocks 470 or 485, the routine continues to block 495
to determine whether to continue, such as until an explicit
indication to terminate is received. If it is determined to
continue, the routine returns to block 410, and otherwise continues
to block 499 and ends.
[0083] FIGS. 5A-5B illustrate a flow diagram of an example
embodiment of a CDD Decision Module Construction routine 500. The
routine may, for example, be provided by execution of the component
342 of FIG. 3 and/or the component 142 of FIG. 1, such as to
provide functionality to allow users to provide information
describing a target system of interest, and to perform
corresponding automated operations to construct one or more
decision modules to use to control the target system in specified
manners. While the illustrated embodiment of the routine interacts
with users in particular manners, such as via a displayed GUI
(graphical user interface), it will be appreciated that other
embodiments of the routine may interact with users in other
manners, such as via a defined API (application programming
interface) that an executing program invokes on behalf of a user.
In some embodiments, the routine may further be implemented as part
of an integrated development environment or other software tool
that is available for one or more users to use, such as by
implementing an online interface that is available to a variety of
remote users over a public network such as the Internet, while in
other embodiments a copy of the CDD system and/or particular CDD
components may be used to support a single organization or other
group of one or more users, such as by being executed on computing
systems under the control of the organization or group. In
addition, the CDD Decision Module Construction component may in
some embodiments and situations be separated into multiple
sub-components, such as a rules editor component that users
interact with to specify rules and other description information
for a target system, and a rules compiler engine that processes the
user-specified rules and other information to create one or more
corresponding decision modules.
[0084] The illustrated embodiment of the routine 500 begins at
block 510, where the routine provides or updates a displayed user
interface to one or more users, such as via a request received at
an online version of component that is implementing the routine, or
instead based on the routine being executed by one or more such
users on computing systems that they control. While various
operations are shown in the illustrated embodiment of the routine
as occurring in a serial manner for the purpose of illustration, it
will be appreciated that user interactions with such a user
interface may occur in an iterative manner and/or over multiple
periods of time and/or user sessions, including to update a user
interface previously displayed to a user in various manners (e.g.,
to reflect a user action, to reflect user feedback generated by
operation of the routine or from another component, etc.), as
discussed further below.
[0085] After block 510, the routine continues to block 520 to
receive information from one or more such users describing a target
system to be controlled, including information about a plurality of
elements of the target system that include one or more
manipulatable control elements and optionally one or more outputs
that the control elements affect, information about rules that
specify restrictions involving the elements, information about
state information that will be available during controlling of the
system (e.g., values of particular elements or other state
variables), and one or more goals to achieve during the controlling
of the target system. It will be appreciated that such information
may be obtained over a period of time from one or more users,
including in some embodiments for a first group of one or more
users to supply some information related to a target system and for
one or more other second groups of users to independently provide
other information about the target system, such as to reflect
different areas of expertise of the different users and/or
different parts of the target system.
[0086] After block 520, the routine continues to block 525 to
identify any errors that have been received in the user input, and
to prompt the user(s) to correct those errors, such as by updating
the display in a corresponding manner as discussed with respect to
block 510. While the identification of such errors is illustrated
as occurring after the receiving of the information in block 520,
it will be appreciated that some or all such errors may instead be
identified as the users are inputting information into the user
interface, such as to identify syntax errors in rules or other
information that the users specify. After block 525, the
illustrated embodiment of the routine continues to block 530 to
optionally decompose the information about the target system into
multiple subsets that each correspond to a portion of the target
system, such as with each subset having one or more different
control elements that are manipulatable by the automated control
system being created by the routine, and optionally have
overlapping or completely distinct goals and/or sets of rules and
other information describing the respective portions of the target
system. As discussed in greater detail elsewhere, such
decomposition, if performed, may in some situations be performed
manually by the users indicating different subgroups of information
that they enter, and/or in an automated manner by the routine based
on an analysis of the information that has been specified (e.g.,
based on the size of rules and other descriptive information
supplied for a target system, based on inter-relationships between
different rules or goals or other information, etc.). In other
embodiments, no such decomposition may be performed.
[0087] After block 530, the routine continues to block 535 to, for
each subset of target system description information (or for all
the received information if no such subsets are identified),
convert that subset (or all the information) into a set of
constraints that encapsulate the restrictions, goals, and other
specified information for that subset (or for all the information).
In block 540, the routine then identifies any errors that occur
from the converting process, and if any are identified, may prompt
the user to correct those errors, such as in a manner similar to
that described with respect to blocks 525 and 510. While not
illustrated in this example, the routine may in some situations in
blocks 525 and/or 540 return to block 510 when such errors are
identified, to display corresponding feedback to the user(s) and to
allow the user(s) to make corrections and re-perform following
operations such as those of blocks 520-540. The errors identified
in the converting process in block 540 may include, for example,
errors related to inconsistent restrictions, such as if the
restrictions as a group are impossible to satisfy.
[0088] After block 540, the routine continues to block 545 to, for
each set of constraints (or a single constraint set if no subsets
were identified in block 530), apply one or more validation rules
to the set of constraints to test overall effectiveness of the
corresponding information that the constraints represent, and to
prompt the one or more users to correct any errors that are
identified in a manner similar to that with respect to blocks 525,
540 and 510. Such validation rules may test one or more of
controllability, observability, stability, and goal completeness,
as well as any user-added validation rules, as discussed in greater
detail elsewhere. In block 550, the routine then converts each
validated set of constraints to a set of coupled differential
equations that model at least a portion of the target system to
which the underlying information corresponds.
[0089] After block 550, the routine continues to block 553 to
perform activities related to training a model for each set of
coupled differential equations, including to determine one or more
of a size of a training time window to use, size of multiple
training time slices within the time window, and/or a type of
training time slice within the time window. In some embodiments and
situations, the determination of one or more such sizes or types of
information is performed by using default or pre-specified
information, while in other embodiments and situations the users
may specify such information, or an automated determination of such
information may be performed in one or more manners (e.g., by
testing different sizes and evaluating results to find sizes with
the best performance). Different types of time slices may include,
for example, successions of time slices that overlap or do not
overlap, such that the training for a second time slice may be
dependent only on results of a first time slice (if they do not
overlap) or instead may be based at least in part on updating
information already determined for at least some of the first time
slice (if they do overlap in part or in whole). After block 553,
the routine continues to block 555 to, for each set of coupled
differential equations representing a model, train the model for
that set of coupled differential equations using partial initial
state information for the target system, including to estimate
values of variable that are not known and/or directly observable
for the target system by simulating effects of performing control
actions over the time window, such as for successive time slices
throughout the time window, and to test the simulated performance
of the trained model. Additional details related to training and
testing are included elsewhere herein.
[0090] After block 555, the routine continues to block 560 to
determine whether the training and testing was successful, and if
not returns to block 510 to display corresponding feedback
information to the users to allow them to correct errors that
caused the lack of success. If it is instead determined in block
560 that the testing and training were successful, however, the
routine continues instead to block 570 to generate an executable
decision module for each trained and tested model that includes
that model, as well as a local CCD Control Action Determination
component that the decision module will use when executed to
determine optimal or near-optimal control actions to perform for
the target system based on the information included in the model,
and in light of the one or more goals for that decision module. The
generated executable decision module may in some embodiments and
situations further include a local CCD Coordinated Control
Management component to coordinate control actions of multiple
decision modules that collectively will provide an automated
control system for the target system, such as by synchronizing
respective models of the various decision modules over time. After
block 570, the routine continues to block 580 to provide the
generated executable decision modules for use, including to
optionally store them for later execution and/or deployment.
[0091] After block 580, the routine continues to block 595 to
determine whether to continue, such as until an explicit indication
to terminate is received. If it is determined to continue, the
routine returns to block 510, and otherwise continues to block 599
and ends.
[0092] FIGS. 6A-6B illustrate a flow diagram of an example
embodiment of a routine 600 corresponding to a generic
representation of a decision module that is being executed. The
routine may, for example, be provided by execution of a decision
module 329 or as part of an automated control system 325 of FIG. 3
and/or a decision module 124 or 128 of FIG. 1 or 2, such as to
provide functionality for controlling at least a portion of a
target system in a manner specific to information and a model
encoded for the decision module, including to reflect one or more
goals to be achieved by the decision module during its controlling
activities. As discussed in greater detail elsewhere, in some
embodiments and situations, multiple decision modules may
collectively and cooperatively act to control a particular target
system, such as with each decision module controlling one or more
distinct control elements for the target system or otherwise
representing or interacting with a portion of the target system,
while in other embodiments and situations a single decision module
may act alone to control a target system. The routine 600 further
reflects actions performed by a particular example decision module
when it is deployed in controlling a portion of a target system,
although execution of at least portions of a decision module may
occur at other times, such as initially to train a model for the
decision module before the decision module is deployed, as
discussed in greater detail with respect to the CDD Decision Module
Construction routine 500 of FIGS. 5A-5B.
[0093] The illustrated embodiment of the routine 600 begins at
block 610, where an initial model for the decision module is
determined that describes at least a portion of a target system to
be controlled, one or more goals for the decision module to attempt
to achieve related to control of the target system, and optionally
initial state information for the target system. The routine
continues to block 615 to perform one or more actions to train the
initial model if needed, as discussed in greater detail with
respect to blocks 553 and 555 of FIGS. 5A-5B--in some embodiments
and situations, such training for block 615 is performed only if
initial training is not done by the routine 500 of FIGS. 5A-5B,
while in other embodiments and situations the training of block 615
is performed to capture information about a current state of the
target system at a time that the decision module begins to execute
(e.g., if not immediately deployed after initial creation and
training) and/or to re-train the model at times as discussed in
greater detail with respect to routine 700 of FIGS. 7A-7B as
initiated by block 630.
[0094] After block 615, the routine continues to block 617 to
determine a time period to use for performing each control action
decision for the decision module, such as to reflect a rate at
which control element modifications in the target system are needed
and/or to reflect a rate at which new incoming state information is
received that may alter future manipulations of the control
elements. The routine then continues to block 620 to start the next
time period, beginning with a first time period moving forward from
the startup of the execution of the decision module. Blocks 620-680
are then performed in a loop for each such time period going
forward until execution of the decision module is suspended or
terminated, although in other embodiments a particular decision
module may execute for only a single time period each time that it
is executed.
[0095] In block 625, the routine optionally obtains state
information for the time period, such as current state information
that has been received for the target system or one or more related
external sources since the last time period began, and/or by
actively retrieving current values of one or more elements of the
target system or corresponding variables as needed. In block 630,
the routine then initiates execution of a local CCD Control Action
Determination component of the decision module, with one example of
such a routine discussed in greater detail with respect to routine
700 of FIGS. 7A-7B. In block 635, the results of the execution of
the component in block 630 are received, including to either obtain
an updated model for the decision module with a local solution for
the current time period and decision module that includes one or
more proposed control action determinations that the decision
module may perform for the current time period, or to receive an
indication that no local solution was found for the decision module
in the allowed time for the execution of the component in block
630. It is then determined in block 640 whether a solution was
found, and if so continues to block 642 to store the updated model
for the decision module, and otherwise continues to block 643 to
use the prior model for the decision module to determine one or
more control action determinations that are proposed for the
current time period based on a previous model (e.g., that does not
reflect recent changes in state information and/or recent changes
in activities of other decision modules, if any), as discussed in
greater detail with respect to routine 700 of FIGS. 7A-7B.
[0096] After blocks 642 or 643, the routine continues to block 644
to determine if other decision modules are collectively controlling
portions of the current target system, such as part of the same
automated control system as the local decision module, and if so
continues to block 645. Otherwise, the routine selects the local
proposed control actions of the decision module as a final
determined control action to perform, and continues to block 675 to
implement those control actions for the current time period.
[0097] If there are other operating decision modules, the routine
in block 645 determines if the local decision module includes a
local copy of a CDD Coordinated Control Management (CCM) component
for use in synchronizing the proposed control action determinations
for the decision module's local solutions with activities of other
decision modules that are collectively controlling the same target
system. If so, the routine continues to block 647 to provide the
one or more proposed control action determinations of the decision
module and the corresponding current local model for the decision
module to the local CDD CCM component, and otherwise continues to
block 649 to provide the one or more proposed control action
determinations for the decision module and the corresponding local
model of the decision module to one or more centralized CDD CCM
components.
[0098] After blocks 647 or 649, the routine continues to block 655
to obtain results of the actions of the CDD CCM component(s) in
blocks 647 or 649, including to either obtain a further updated
model resulting from synchronization of the local model for the
current decision module with information from one or more other
decision modules, such that the further updated model indicates one
or more final control action determinations to perform for the time
period for the current decision module, or an indication that no
such synchronization was completed in the allowed time. The routine
continues to block 660 to determine whether the synchronization was
completed, and if so continues to block 665 to store the further
updated model from the synchronization, and otherwise continues to
block 670 to use the prior proposed control action determinations
locally to the decision module as the final control action
determinations for the time period.
[0099] After blocks 665 or 670, the routine continues to block 675
to implement the one or more final determined control actions for
the decision module in the target system, such as by interacting
with one or more effectuators in the target system that modify
values or otherwise manipulate one or more control elements of the
target system, or by otherwise providing input to the target system
to cause such modifications or other manipulations to occur. In
block 680, the routine optionally obtains information about the
results in the target system of the control actions performed, and
stores and/or provides information to the CDD system about such
obtained results and/or about the activities of the decision module
for the current time period.
[0100] After block 680, the routine continues to block 695 to
determine whether to continue, such as until an indication to
terminate or suspend is received (e.g., to reflect an end to
current operation of the target system or an end of use of the
decision module to control at least a portion of the target
system). If it is determined to continue, the routine returns to
block 620 to start the next time period, and otherwise continues to
block 699 and ends.
[0101] FIGS. 7A-7B are a flow diagram of a example embodiment of a
CDD Control Action Determination routine 700. The routine may, for
example, be provided by execution of the component 344 of FIG. 3
and/or components 144a-n or 244 of FIG. 2, such as to determine
control actions for a decision module to propose and/or implement
for a target system during a particular time period, including in
some embodiments to perform an optimization to determine
near-optimal actions (e.g., within a threshold of an optimal
solution) to perform with respect to one or more goals if possible.
While the illustrated embodiment of the routine is performed in a
manner local to a particular decision module, such that some or all
decision modules may each implement a local version of such a
routine, in other embodiments the routine may be implemented in a
centralized manner by one or more components with which one or more
decision modules interact over one or more networks, such as with a
particular decision module indicated to be used at a particular
time rather than acting on behalf of the local decision module.
[0102] The illustrated embodiment of the routine 700 begins at
block 703, where information or a request is received. The routine
continues to block 705 to determine a type of the information or
request, and to proceed accordingly. In particular, if a request is
received in block 703 to attempt to determine a solution for a
current time period given a current model of the local decision
module, the routine continues to block 710 to begin to perform such
activities, as discussed in greater detail with respect to block
710-790. If it is instead determined in block 705 that a request to
relax one or more rules or other restrictions for the current model
of the local decision module is received, such as discussed in
greater detail with respect to blocks 760 and 765, the routine
continues to block 765. If it is determined in block 705 that a
request is received to repair one or more rules or other
restrictions for the current model of the local decision module,
such as discussed in greater detail with respect to blocks 775 and
780, the routine continues to block 780 to obtain user input to use
during the rule repair process (e.g., to interact with a CDD
Decision Module Construction component, or to instead interact with
one or more users in another manner), such as to allow the current
model for the local decision module to later be updated and
replaced based on further resulting user actions, or if operation
of the target system can be suspended, to optionally wait to
further perform the routine 700 until such an updated model is
received. If it is instead determined in block 705 that the
information or request is of another type, the routine continues
instead to block 708 to perform one or more other indicated
operations as appropriate, and to then proceed to block 799. Such
other indicated operations may include, for example, receiving
information about current models and/or control actions proposed or
performed by one or more other decision modules that are
collectively controlling a target system with the local decision
module (such as for use in synchronizing the model of the local
decision module with such other decision modules by generating a
consensus or converged shared model, as discussed in greater detail
with respect to routine 800 of FIGS. 8A-8B), to receive updates to
a model or underlying information for the model for use in ongoing
operation of the routine 700 (e.g., from a CDD Decision Module
Construction component, such as results from interactions performed
in block 780), to receive current state information for the target
system, such as for use as discussed in routine 600 of FIGS. 6A-6B,
etc.
[0103] If it determined in block 705 that a request for a solution
was received in block 703 for a current time period and based on a
current model of the local decision module, the routine continues
to block 710 to receive a current set of coupled differential
equations that represent the current model for the local decision
module of at least a portion of the target system, optionally along
with additional state information for the target system for the
current time. The routine then continues to block 715 to determine
whether to train or re-train the model, such as if the routine is
called for a first time upon initial execution of a corresponding
decision module or if error measurements from ongoing operations
indicate a need for re-training, as discussed in greater detail
with respect to blocks 755, 770 and 730. If it is determined to
train or re-train the model, the routine continues to block 720 to
determine one or more of the size of a training time window, size
of training time slices within the time window, and/or type of
training time slices within the training time window, such as in a
manner similar to that previously discussed with respect to block
553 of routine 500 of FIGS. 5A-5B. After block 720, the routine
continues to block 725 to use partial initial state information for
the target system to train the model, including to estimate values
of state variables for the target system that are not known and/or
directly observable, by simulating effects of performing control
actions over the time window for each of the time slices, as
discussed in greater detail with respect to block 555 of routine
500 of FIGS. 5A-5B.
[0104] After block 725, or if it is instead determined in block 715
not to train or re-train the model, the routine continues to block
730 to perform a piecewise linear analysis to attempt to determine
a solution for the current model and any additional state
information that was obtained in block 710, with the solution (if
determined) including one or more proposed control action
determinations for the local decision module to take for a current
time period, as well as in some embodiments to use one or more
model error gauges to make one or more error measurements with
respect to the current model, as discussed in greater detail
elsewhere. The routine then continues to block 735 to determine if
the operations in block 730 determined a solution within a amount
of time allowed for the operation of block 730 (e.g., a defined
subset or fraction of the current time period), and if so continues
to block 740 to update the current set of coupled differential
equations and the resulting current model for the local decision
module to reflect the solution, with the resulting updated
information provided as an output of the routine 700.
[0105] If it is instead determined in block 735 that the operations
in block 730 did not determine a solution, the routine continues to
block 745 to determine if additional time is available within the
current time period for further attempts to determine a solution,
and if not continues to block 790 to provide output of the routine
700 indicating that no solution was determined for the current time
period.
[0106] If additional time is available within the current time
period, however, the routine continues to perform blocks 755-780 to
perform one or more further attempts to identify the solution--it
will be appreciated that one or more of the operations of blocks
755-780 may be repeatedly performed multiple times for a given time
period if sufficient time is available to continue further solution
determination attempts. In particular, the routine continues to
block 755 if additional time is determined to be available in block
745, where it determines whether the measurements from one or more
gauges indicate model error measurements that are over one or more
thresholds indicating modifications to the model are needed, such
as based on the model error measurements from the gauges discussed
with respect to block 730. If not, the routine continues to block
760 to determine whether there are one or more rules or other
restrictions in the current model that are available to be relaxed
for the current time period (that have not previously attempted to
be relaxed during the time period, if this is not the first pass
through this portion of the routing for the current time period),
and if so continues to block 765 to relax one or more such rules or
other restrictions and to return to block 730 to re-attempt the
piecewise linear analysis with the revised model based on those
relaxed rules or other restrictions.
[0107] If it is instead determined in block 755 that the model
error measurements from one or more of the gauges are sufficient to
satisfy one or more corresponding thresholds, the routine continues
instead to block 770 to determine whether to re-train the model
based on one or more of the gauges indicating sufficient errors to
do so, such as based on accumulated errors over one or more time
periods of updates to the model. If so, the routine returns to
block 720 to perform such re-training in blocks 720 and 725, and
then continues to block 730 to re-attempt the piecewise linear
analysis with the resulting re-trained model.
[0108] If it is instead determined in block 770 not to re-train the
model (or if the model was re-trained already for the current time
period and the resulting re-attempt in block 730 again failed to
find a solution), the routine continues to block 775 to determine
whether the model error measurements from one or more of the gauges
indicate a subset of one or more rules or other restrictions in the
model that potentially have errors that need to be repaired. If so,
the routine continues to block 780 to provide information to one or
more users via the CDD Decision Module Construction component, to
allow the users to revise the rules or other restrictions as
appropriate, although in other embodiments some or all such rule
repair activities may instead be attempted or performed in an
automated manner. After block 780, or if it is instead determined
in block 775 not to repair any rules, the routine continues to
block 790 to provide an indication that no solution was determined
for the current time period. After blocks 740, 708, or 790, the
routine continues to block 799 and ends. It will be appreciated
that if the routine 700 was instead implemented as a centralized
routine that supports one or more decision modules remote from the
executing component for the routine, the routine 700 may instead
return to block 703 to await further information or requests.
[0109] FIGS. 8A-8B are a flow diagram of an example embodiment of a
CDD Coordinated Control Management routine 800. The routine may,
for example, be provided by execution of the component 346 of FIG.
3 and/or the components 146a-n of FIG. 2, such as to attempt to
synchronize current models and their proposed control actions
between multiple decision modules that are collectively controlling
a target system. In the illustrated embodiment of the routine, the
synchronization is performed in a pairwise manner between a
particular local decision module's local current model and an
intermediate shared model for that decision module that is based on
information about the current state of one or more other decision
modules, by using a Pareto game technique to determine a Pareto
equilibrium if possible that is represented in a consensus shared
model, although in other embodiments other types of synchronization
methods may be used. In addition, in the illustrated embodiment,
the routine 800 is performed in a local manner for a particular
local decision module, such as by being included within that local
decision module, although in other embodiments the routine 800 may
be implemented in a centralized manner to support one or more
decision modules that are remote from a computing system
implementing the component for the routine and that communicate
with those decision modules over one or more computer networks,
such as with a particular decision module indicated to be used at a
particular time rather than acting on behalf of the local decision
module.
[0110] The illustrated embodiment of the routine 800 begins at
block 805, where it waits to receive information or another
indication. The routine continues to block 810 to determine if a
consensus model or other updated information for another decision
module has been received, such as from a copy of the routine 800
executing for that other decision module, and if so continues to
block 815 to use the received information to update local
intermediate shared model information for use with the local
decision module on whose behalf the current copy of the routine 800
is executing, as discussed in greater detail with respect to block
830. If it is instead determined in block 810 that the information
or request received in block 805 is not information related to one
or more other decision modules, or after block 815, the routine
continues to block 820 to determine whether to currently perform a
synchronization for the current local model of the local decision
module by using information about an intermediate shared model of
the local decision module that includes information for one or more
other decision modules, such as to do such synchronization each
time that an update to the local decision module's model is
received (e.g., based on operation of the routine 700 for a copy of
the CDD Control Action Determination component local to that
decision module) in block 805 and/or each time that information to
update the local decision module's intermediate shared model is
received in block 805 and used in block 815, or instead as
explicitly indicated in block 805--if the synchronization is to
currently be performed, the routine continues to block 825 and
begins to perform blocks 820-880 related to such synchronization
activities. Otherwise, the routine continues to block 885 to
perform one or more other indicated operations as appropriate, such
as to receive requests from the CDD system or other requestor for
current information about operation of the routine 800 and/or to
provide corresponding information to one or more entities (e.g., to
reflect prior requests), etc.
[0111] If it is determined in block 820 that synchronization is to
be currently performed, such as based on updated model-related
information that is received in block 805, the routine continues to
block 825 to obtain a current local model for the local decision
module to use in the synchronizing, with the model including one or
more proposed control actions to perform for a current time period
based on a local solution for the local decision module. The
routine then continues to block 830 to retrieve information for an
intermediate shared model of the local decision module that
represents information for one or more other decision modules
(e.g., all other decision modules) that are collectively
participating in controlling the target system, with that
intermediate shared model similarly representing one or more other
proposed control actions resulting from local solutions of those
one or more other decision modules, optionally after partial or
complete synchronization has been performed for those one or more
other decision modules between themselves.
[0112] The routine then continues to block 835 to attempt to
determine a consensus shared model that synchronizes the current
model of the local decision module and the intermediate shared
model by simultaneously providing solutions to both the local
decision module's current model and the intermediate shared model.
In some embodiments, the operations of block 835 are performed in a
manner similar to that discussed with respect to blocks 710-730 of
routine 700 of FIG. 7A-7B, such as if the local model and the
intermediate shared model are combined to create a combination
model for whom one or more solutions are to be identified. As
discussed in greater detail elsewhere, in some embodiments, the
local current model and intermediate shared model may each be
represented by a Hamiltonian function to enable a straightforward
creation of such a combined model in an additive manner for the
respective Hamiltonian functions, with the operations of routines
600 and/or 700 of FIGS. 6A-6B and 7A-7B, respectively, similarly
representing the models that they update and otherwise manipulate
using such Hamiltonian functions.
[0113] After block 835, the routine continues to block 840 to
determine whether the operations of block 835 succeeded in an
allowed amount of time, such as a fraction or other portion of the
current time period for which the synchronization is attempted to
be performed, and if so the routine continues to block 845 to
update both the local model and the intermediate shared model of
the local decision module to reflect the consensus shared model. As
earlier noted, if sufficient time is allowed for each decision
module to repeatedly determine a consensus shared model with
changing intermediate shared models representing one or more other
decision modules of a collective group, the decision modules of the
collective group may eventually converge on a single converged
shared model, although in other embodiments and situations there
may not be sufficient time for such convergence to occur, or other
issues may prevent such convergence. After block 845, the routine
continues to block 850 to optionally notify other decision modules
of the consensus shared model determined for the local decision
module (and/or of a converged shared model, if the operations of
835 were a last step in creating such a converged shared model),
such as if each of the notified decision modules is implementing
its own local version of the routine 800 and the provided
information will be used as part of an intermediate shared model of
those other decision modules that includes information from the
current local decision module's newly constructed consensus shared
model.
[0114] If it is instead determined in block 840 that a
synchronization did not occur in the allowed time, the routine
continues to perform blocks 860-875 to re-attempt the
synchronization with one or more modifications, sometimes
repeatedly if sufficient time is available, and in a manner similar
to that discussed with respect to blocks 745-780 of routine 700 of
FIGS. 7A-7B. In the illustrated example, the routine determines in
block 860 if additional time is available for one or more such
re-attempts at synchronization, and if not the routine continues to
block 880 to provide an indication that no synchronization was
performed within the allowed time. Otherwise, the routine continues
to block 870 to take one or more actions to perform one or more of
relaxing rules or other restrictions, repairing rules or other
restrictions, and/or re-training a model, with respect to one or
both of the current model of the local decision module and/or one
or more other decision modules whose information is represented in
the intermediate shared model of the local decision module. If it
is determined in block 870 to proceed in this manner, the routine
continues to block 875 to perform corresponding actions, sometimes
one at a time in a manner similar to that discussed with respect to
routine 700, including to cause resulting updates to the current
model of the local decision module and/or to the local intermediate
shared model of the local decision module, after which the routine
returns to block 835 to re-attempt to synchronize the local model
and the intermediate shared model of the local decision module.
[0115] If it is instead determined in block 870 that no further
actions are to be performed with respect to relaxation, repair
and/or re-training, the routine continues instead to block 880.
After blocks 850, 880 or 885, the routine continues to block 895 to
determine whether to continue, such as until an explicit indication
to terminate or suspend operation of the routine 800 is received,
such as to reflect an end to operation of the target system and/or
an end to use of the local decision module and/or a collective
group of multiple decision modules to control the target system. If
it is determined to continue, the routine returns to block 805, and
otherwise continues to block 899 and ends.
[0116] FIG. 9 illustrates a flow diagram of an example embodiment
of a routine 900 performed for a representative generic target
system, with respect to interactions between the target system and
one or more decision modules that are controlling at least a
portion of the target system. The routine may, for example, be
provided by execution of a target system 360 and/or 370 of FIG. 3,
and/or a target system 160 and/or 170 of FIGS. 1 and 2, such as to
implement operations specific to the target system. It will be
appreciated that the illustrated embodiment of the routine focuses
on interactions of the target system with the one or more decision
modules, and that many or all such target systems will perform many
other operations in a manner specific to those target systems that
are not illustrated here for the purpose of brevity.
[0117] The routine begins at block 910, where it optionally
provides initial state information for the target system to a CDD
system for use in an automated control system of the CDD system for
the target system, such as in response to a request from the CDD
system or its automated control system for the target system, or
instead based on configuration specific to the target system (e.g.,
to be performed upon startup of the target system). After block
910, the routine continues to block 920 to receive one or more
inputs from a collective group of one or more decision modules that
implement the automated control system for the target system,
including one or more modified values for or other manipulations of
one or more control elements of a plurality of elements of the
target system that are performed by one or more such decision
modules of the automated control system. As discussed in greater
detail elsewhere, the blocks 920, 930, 940 may be repeatedly
performed for each of multiple time periods, which may vary greatly
in time depending on the target system (e.g., a microsecond, a
millisecond, a hundredth of a second, a tenth of a second, a
second, 2 seconds, 5 seconds, 10 seconds, 15 seconds, 30 seconds, a
minute, 5 minutes, 10 minutes, 15 minutes, 30 minutes, an hour,
etc.).
[0118] After block 920, the routine continues to block 930 to
perform one or more actions in the target system based on the
inputs received, including to optionally produce one or more
resulting outputs or other results within the target system based
on the manipulations of the control elements. In block 940, the
routine then optionally provides information about the outputs or
other results within the target system and/or provides other
current state information for the target system to the automated
control system of the CDD system and/or to particular decision
modules of the automated control system. The routine then continues
to block 995 to determine whether to continue, such as until an
explicit indication to terminate or suspend operation of the target
system is received. If it is determined to continue, the routine
returns to block 920 to begin a next set of control actions for a
next time period, and otherwise continues to block 999 and ends. As
discussed in greater detail elsewhere, state information that is
provided to a particular decision module may include requests from
external systems to the target system, which the automated control
system and its decision modules may determine how to respond to in
one or more manners.
[0119] The following sections describe a variety of specific,
non-exclusive embodiments in which some or all of the described
techniques may be implemented. It will be appreciated that
particular details of particular embodiments may not be included in
or true for all embodiments, and that the described embodiments may
be implemented individually or in combination with any and all
other combinations of other embodiments.
[0120] The following discusses several non-exclusive example
embodiment, in which one or more specified embodiments of the
Collaborative Distributed Decision system are each referred to as a
Collaborative Distributed Inferencing (CDI) system or a Cooperative
Distributed Inferencing Platform (CDIP), in which one or more
specified embodiments of the Decision Module Construction component
are each referred to as the "Rules Builder" or including or being
performed by a "Rule Conversion Engine" (RCE) or a "CDL Compiler"
or a "Rule(s) Entry Interface" (REI) or a "Rules and Goals UI", in
which one or more specified embodiments of the Control Action
Determination component are each referred to as having "chattering"
subcomponents or performing "chattering" functionality or including
or being performed by a "Query Response Engine" (QRE) or an
"Optimization Element" or via "Hamiltonian-Chattering Control", in
which one or more specified embodiments of the Coordinated Control
Management component are each referred to as having or performing
"mean field" information or functionality or including or being
performed by a "Tellegen" agent or a "Pareto Multi-Criteria
Optimization Engine" (PMOE) or a "Pareto element", in which
decision modules are referred to as "agents" or "peer agents" or
"agent nodes" or "decision elements", in which an automated control
system may include a "cluster" of agents and in some cases is
referred to as a "distributed architecture", in which a target
system is referred to as an "application domain", in which a
decision module's stored information about a target system includes
an "Internal Heterogeneous Database" (IHDB) and/or a "Domain Rules
Knowledge Base", in which a decision module's consensus shared
model is referred to as an "internal optimum" generated using mean
field information, in which changes propagated from one decision
module to others is referred to as a "delta", etc.
[0121] CDI is built using a Peer-to-Peer (P2P) based distributed
architecture that allows for partitioning the overall optimization
problem into smaller sub-tasks or work-loads between peer agents.
CDI Peer Agents are equally privileged participants in the
application domain. They are configured to form a peer-to-peer
network of nodes. This allows each agent in the network to solve
the problem independently, using its own internal knowledge base.
Each agent also internally engages in Pareto game playing to reach
internal optimum before sharing changes with other agents. The
agents then communicate the mean field changes with other agents in
the network using a gossip protocol to synchronize their internal
mean field approximation in an eventually consistent fashion. FIG.
10 illustrates a network diagram 1000 of a portion of a distributed
architecture of an example CDI system.
CDI Cluster Setup
[0122] When system is initially configured certain agents are
tagged as seed-nodes in the network. The seed agent nodes can be
started in any order and it is not necessary to have all the seed
agent nodes running. When initializing the CDI cluster at least one
seed agent node is preferably running, otherwise the other CDI seed
nodes may not become initialized and other CDI agent nodes might
not join the cluster.
CDI Agent Cluster Registry
[0123] Each agent has a built-in registry of active CDI agent nodes
in the cluster. This agent registry is started on all CDI agent
nodes and is updated with every change in cluster membership
including new agents joining the cluster, agents leaving the
cluster, agent timeouts etc. Agents inform each other of active
membership through a heartbeat mechanism. The registry is
eventually consistent, i.e. changes may not immediately be visible
at other nodes, but typically they will be fully replicated to all
other nodes after a few seconds. The changed are propagated using
the change deltas and are disseminated in a scalable way to other
nodes with a gossip protocol.
CDI Cluster Agent Failure Detection
[0124] The CDI agent nodes in the cluster monitor each other by
sending heartbeats to detect if a CDI agent node is unreachable
from the rest of the CDI cluster. The heartbeat arrival times is
interpreted by an implementation of the Phi Accrual Failure
Detector (Hayashibara, Defago, Yared, & Katayama, 2004).
[0125] The suspicion level of failure is given by a value called
phi. The basic idea of the phi failure detector is to express the
value of phi on a scale that is dynamically adjusted to reflect
current network conditions.
[0126] The value of phi is calculated as:
Phi=-log 10(1-F(timeSinceLastHeartbeat))
F is the cumulative distribution function of a normal distribution
with mean and standard deviation estimated from historical
heartbeat inter-arrival times.
CDI Cluster Agent Network Partition
[0127] Once a CDI agent node becomes network partitioned it will
stop receiving mean field updates from the other agents in the
Cluster. This however, should not prevent it from continuing to
perform local optimizations based on its internal knowledge base,
local mean field and sensor inputs that it will continue to
receive.
[0128] The Rules Builder contains the following components: [0129]
Rules Entry Interface [0130] Validator [0131] CDL Compiler [0132]
Step Processor [0133] Workflow Remote Control [0134] Compiled Rules
Engine Artifact
[0135] And interacts directly with the following components: [0136]
Master Agent.
[0137] The Master Agent is responsible for interaction with all
Chattering components, including the following: [0138] System
Trainer & Bootstrapper [0139] Runner
[0140] The System also utilizes a Persistent Store to access state
and necessary models/data. FIG. 11 illustrates a network diagram
1100 of a portion of an example Rules Builder component.
Rules Entry Interface:
[0141] The Rules Entry Interface is responsible for receiving rules
in a syntax familiar to a domain expert. The entry interface is a
text entry tool where a domain expert would insert the rules, goal
and system information (rules script(s)) to provide continuous
control suggestions for a given problem. The entered rules
script(s) would be composed in the CDI Domain-Specific Language
(DSL). The CDI DSL facilitates functional translation of the
entered rules script(s) into a problem definition which houses the
control definitions (measurable variable with ranges allowed and
any associated constraints on the control variable).
[0142] For example:
[0143] In the entry interface using the CDI DSL one would specify
an upper and lower bound using the following syntax first defining
a rule, with the parameters and the rule logic that evaluates
against the system state. [0144] rule id: "max production", params:
["production"], {it <=15.0} [0145] rule id: "min production",
params: ["production"], {it >=-50.0} [0146] control var:
"production", min: "min production", max: "max production"
[0147] In the entry interface using the CDI DSL one would specify
the dynamics (variables that change with relation to other
measurable variables and the definition of this change) [0148]
delta var: "inventory", params: ["production", "demand"], {p,
d->p-d}
[0149] In the entry interface using the CDI DSL one would specify
the goal (an objective i.e., to minimize or maximize a relationship
between the control and controllable dynamics). [0150] goal
objective: "minimize", params: ["inventory", "production"],
{inventory,
production->((inventory-10)*(inventory-10))+(production*production)}
[0151] Finally, the entry interface is also responsible for
facilitating a domain expert to provide certain settings that are
used by the system as well as providing user workflow steps. The
settings provided represent key: value terms used by the system and
are available throughout by the Settings Provider encapsulated
within the Rules Engine. An domain expert might write initial
values of the following form: [0152] settings
<<[initialState: "20", initialCoState: "0", terminalCoState:
"0", numChatteringLevels: "9", horizon: "3", delta: "0.1",
iterations: "15"]
CDL Compiler:
[0153] The CDL Compiler is responsible for the conversion of said
rules into evaluatable constraints, system information and an
optimization goal. The CDI Compiler translates the entered
script(s) into a problem definition. A problem definition is
composed of labeled rules. A rule is a tuple comprised of a unique
name and a Term. A Term is an evaluatable function where the logic
was authored as previously described above, the input to which
contains a representation of the system state (StateMap).
Validator:
[0154] The Validator is responsible for the validation of converted
constraints with regards to restrictions the optimization problem
needs to satisfy as a prerequisite to its solution and/or reaching
a solution quality threshold.
[0155] The converted constraints defined in the problem definition,
along with the settings provider, is validated for controllability,
observability, stability and goal completeness. [0156]
controllability--this check ensures that every control variable has
a defined rule relating it to one or more dynamic state variables.
The above example satisfies this check since our single control
variable `production` is used in the dynamic definition of
`inventory`; [0157] observability--checks the statemap against the
defined rules in the script(s). An observability check does not
impede the solution of a problem, however if failing, represents
that our problem may not be well defined; [0158] stability--this
check ensures that our system will converge over time on a
solution, which can be tested in the testing stage; and [0159] goal
completeness--ensures that every control variable appears in the
goal.
Compiled Rules Engine:
[0160] The conversion of rule scripts into mathematical expressions
that can be yielded through an explicit contract to the chattering
agent for use in solving the optimization goal, manifests in the
compiled artifact that we label the Compiled Rules Engine
[0161] The Rules Engine defines and implements several interfaces
for acquiring the values of these mathematical expressions, as
later described with respect to "chattering" components. The Rules
Engine is compiled as an artifact available to the Chattering
component by the Compiler and made available for use in continuous
processing. FIG. 12 illustrates a network diagram 1200 of example
sub-components of an example Rules Builder component, in which
compilation is dependent upon successful Validation of the
observability, completeness, controllability, and workflow step
validation. Any errors/warnings are returned to the domain expert
for correction. Once corrections/improvements are complete, a new
Compiled Rules Engine artifact is produced along with new commands
delivered to the Step Processor.
Steps Processor and Remote Control:
[0162] Furthermore, the rules builder is also responsible for
facilitating the workflow steps an individual domain expert would
need to undertake before deploying such a system. These steps
include: [0163] training an optimization system using the converted
rules, with bootstrapped sensor data; [0164] testing the resulting
control actuator inputs (the end results of the chattering
optimization); and [0165] persisting the trained model for use in
the running of the overall system.
[0166] The workflow component allows for training and testing a
system. This is integrated into the DSL in the form of a workflow
step builder.
[0167] For example,
TABLE-US-00001 workflow { train.onComplete { persist: model,
`/tmp/model` run.withSensorData(`/tmp/simulatedData`).after(10
minutes) { persist: results, `/tmp/results` shutdown } } }
[0168] Here a domain expert instructs the default bootstrapper to
be used in loading predefined sensor data, train the system on this
data, persist the trained model output to the file `/tmp/model`,
then once trained to begin running the system (as a single agent).
The inputs are tested by analyzing the results from the run. The
artifacts and model information are persisted to a persistent
store. The Steps Processor facilitates the dispatch of interpreted
workflow steps to the Remote Control for delegation to the Master
Agent of a running system for processing.
[0169] FIG. 13 illustrates a network diagram 1300 of interactions
of portions of a Rules Builder component with an executing agent,
including the Steps processor, the Remote Control and interactions
with a running agent (the Step Processor handles interpretation of
workflow steps, and the steps are delivered for dispatch by the
Remote Control, which passes commands to the Master Agent; the
Master Agent handles communication between a running system's
controlled elements and the Remote Control, and examples of the
Controlled Elements are the Trainer and Runner).
[0170] FIG. 14 illustrates a network diagram 1400 of interactions
of portions of a Rules Builder component with chattering
components, including a Running Agent that delivers messages and
interacts with the chattering components to facilitate training and
running the system (the Trainer persists its model once training
has completed; The Trainer invokes a Bootstrapper for acquisition
of training data; The Trainer utilizes the Chattering Library; The
Trainer utilizes the Chattering Library; Chattering utilizes the
compiled rules; Chattering persists its control suggestions;
Bootstrapper persists training data).
[0171] With respect to the discussion below, FIG. 15 illustrates a
diagram 1500 of a sliding window for use with chattering, and is
referred to in the text for the example embodiment as "FIG. 1",
while FIG. 16 illustrates a diagram 1600 of using Lebesgue
integration to approximate a control trajectory for chattering, and
is referred to in the text for the example embodiment as "FIG.
2".
[0172] The following discusses an example implementation that may
be used for such chattering.
[0173] The following discusses further details regarding possible
use with such chattering.
[0174] In the chattering algorithm, there are two timing
parameters, the length of the window T, and the time slice .DELTA.
(where N.DELTA.=T). For accuracy purposes, these two parameters
should be sufficiently small, but for computational efficiency,
these two parameters should be large. Therefore, we want to keep
the parameters as large as possible, as long as the error is
tolerable.
[0175] The incremental state equation (for .delta.z) relies on the
Jacobian matrix evaluated at the beginning of the time window. To
characterize the error, we compare two linear systems, one that has
a constant matrix, denoted A(t.sub.0), and the other has a time
varying matrix, denoted A(t).
[0176] Consider both linear systems:
{dot over (x)}=A(t)x(t) (1)
{dot over (y)}=A(t.sub.0)y(t) (2)
with the same initial conditions, x(t.sub.0)=y(t.sub.0).
[0177] The solution to (1) is of the form
x(t)=.phi.(t,t.sub.0)x(t.sub.0)
where .phi.(t, t.sub.0) satisfies several properties,
{dot over (.phi.)}(t,t.sub.0)=A(t).phi.(t,t.sub.0)
.phi.(t.sub.0,t.sub.0)=1
.phi..sup.-1(t,t.sub.0)=.phi.(t.sub.0,t).
[0178] The solution to (2) is of the form
y(t)=e.sup.A(t-t.sub.0)x(t.sub.0).
[0179] The error due to approximating A(t) with A(t.sub.0) is:
E(t)=.phi.(t,t.sub.0)-e.sup.A(t-t.sup.0.sup.).
[0180] Taking the derivative yields
{dot over (E)}(t)={dot over
(.phi.)}(t,t.sub.0)-A(t.sub.0)e.sup.A(t-t.sup.0.sup.)
and replacing {dot over (.phi.)}(t, t.sub.0) with A(t).phi.(t,
t.sub.0) yields
{dot over
(E)}(t)=A(t).phi.(t,t.sub.0)-A(t.sub.0)e.sup.A(t-t.sup.0.sup.)
and substituting for .phi.(t, t.sub.0) yields
E . ( t ) = A ( t ) ( E ( t ) + A ( t - t 0 ) ) - A ( t 0 ) A ( t -
t 0 ) = A ( t ) E ( t ) + ( A ( t ) - A ( t 0 ) ) A ( t - t 0 ) .
##EQU00001##
[0181] This provides the error as a function of (A(t)-A(t.sub.0)),
and if the eigenvalues of A(t) for all t have real values less than
zero, the error is linearly proportional to (A(t)--A(t.sub.0)).
[0182] The procedure to choose T is to ask the customer what size
of error is tolerable.
[0183] The customer may say an error of 20% is tolerable, in which
case the window length T can increase until reaching the associated
threshold. The measure of the error is related to the
difference,
i = 1 I .differential. F .differential. z .alpha. i t - i = 1 I
.differential. F .differential. z .alpha. i t 0 ##EQU00002##
[0184] The size of the time slice .DELTA. is related to the
magnitude of the largest eigenvalue of
.differential. F .differential. z . ##EQU00003##
Let |.lamda.|= {right arrow over
(.lamda..sub.real.sup.2+.lamda..sub.imag.sup.2)} and suppose
|.lamda.|.sub.i has the largest value. Then
.DELTA. = 1 .lamda. i . ##EQU00004##
[0185] The following discusses further details regarding an example
of synchronizing a decision module's model and current information
with that of information from one or more other decision
modules.
[0186] In the multi-agent self-organizing architecture of CDI, all
agents synchronize in some way. This is analogous in the worst case
to the many-body problem, which Isaac Newton first formulated, and
is unsolvable for three or more bodies. However, good
approximations for systems of more than two bodies are
computationally tractable. The approach in CDI is to solve a series
of two-body problems that emulate a mean field aggregation, and
update sequentially through a Pareto game.
[0187] Consider N agents, and each agent i, i=1, . . . , N, has its
own optimization problem with criterion, state and control:
Min.sub.u.sub.iJ.sub.i(x.sub.i,u.sub.i)
st{dot over (x)}.sub.i=f(x.sub.i,u.sub.i)
[0188] The N problem in a single optimization problem is not
algorithmically solvable, but it is possible to solve
algorithmically a Pareto-optimization problem with two players (the
Pareto game), and approximate the solution to the N player problem.
The two-agent (agent 1 and agent 2) Pareto-optimization problem
is
Min.sub.u.sub.1.sub.,u.sub.2.sub.,.alpha..sub.1.sub.,.alpha..sub.2.alpha-
..sub.1J.sub.1(x.sub.1,u.sub.1)+.alpha..sub.2J.sub.2(x.sub.2,u.sub.2)
subject to
{dot over (x)}.sub.1=f(x.sub.1,u.sub.1)
x.sub.2=f(x.sub.2,u.sub.2)
.alpha..sub.1(t)+.alpha..sub.2(t)=1
[0189] The CDI approach is that each agent i plays a two-agent game
with the CDI Mean Field agent composed of the criteria from all
other agents except agent i. Instead of solving the two-agent
optimization problem directly, we convert the formulation to a
Hamiltonian problem (using Pontryagin's minimum principle).
[0190] The Hamiltonians are additive for Pareto optimization.
Therefore, we can add local Hamiltonians for individual agents to
create an aggregate mean field Hamiltonian.
[0191] Let H.sub.i be the local Hamiltonian for agent i and let
H.sub.i.sup.MF be the mean field
[0192] Hamiltonian composed of the Hamiltonians for all other
agents, excluding i. That is, the mean field Hamiltonian for agent
i is a functional form of the Hamiltonians of the other agents.
[0193] Then the two-agent optimization problem is
Min u i , u i MF , .alpha. 1 , .alpha. 2 .alpha. 1 H i ( x i , u i
) + .alpha. 2 H i MF ( x i MF , u i MF ) ##EQU00005##
subject to the state and costate equations of the combined
Hamiltonian, and
.alpha..sub.1(t)+.alpha..sub.2(t)=1.
[0194] The state consists of [x.sub.i,x.sub.i.sup.MF].sup.T, the
costate is [p.sub.i,p.sub.i.sup.MF].sup.T, and the control is
[u.sub.i,u.sub.i.sup.MF].sup.T. The initial conditions for
x.sub.i.sup.MF and p.sub.i.sup.MF come from the previous solution,
and u.sub.i is the local solution from the previous pass.
[0195] The solution to the two-agent Pareto game provides
.alpha..sub.1(t) and .alpha..sub.2(t), and the total Hamiltonian
for agent i is updated:
H.sub.i.sup.T=.alpha..sub.1H.sub.i+.alpha..sub.2H.sub.i.sup.MF
The modified Hamiltonian for agent i is constructed by projecting
the total Hamiltonian into the local state space. The modified mean
field Hamiltonian is constructed by projecting the total
Hamiltonian into the mean field state space.
[0196] Now, the another agent plays the game, solving the two-agent
Pareto game, and updating locally, and each agent updates its mean
field agent when it is ready to play its game.
[0197] The following discusses further details regarding example
embodiments.
Overview
[0198] The Cooperative Distributed Inference (CDI) platform is a
unique advanced technology offering to enable near-optimal and
near-real-time decision making on vast amount of heterogeneous and
distributed information, in complex knowledge-based decision
support systems, combining absolute, hard and soft rules to handle
various requirements from natural or governing laws, policies, and
best practices. FIG. 17 illustrates a diagram 1700 of various
interactions of different portions of a CDI system.
[0199] The CDI platform features a Distributed Architecture (DA)
for resolving queries by accessing information from both an
Internal Heterogeneous Database (IHDB) and external data sources
referred to as Sensors. CDI utilizes a network of computing devices
as its nodes--called Decision Elements (DE)--that cooperate to
resolve queries given to them.
[0200] The Decision Elements (DE) in a given DA can work together
to reach best outcome for the whole group, i.e., reaching Pareto
Efficiency (also referred to as Pareto equilibrium)--a stable state
where no change by any individual can be made to make the sum of
whole group better.
[0201] Each DE can solve the problem independently if provided with
complete knowledge and data. DE solves the query with a very unique
approach, using Optimal Control Theory, starting with a technique
called analytic continuation (transforming query and rules into
differential equations whose dependent variables represent internal
variables and parameters of the rules), then using its own internal
knowledge to solve the query, providing an outcome.
[0202] In distributed environments, group of Decision Elements
synchronize in an iterative process utilizing updates from other
DEs and provide a final result, via a Pareto multi criteria
optimization strategy.
[0203] The CDI platform needs rules, not necessarily exact
criteria, to reach the objective state, producing the near-optimal
solution. This is a very attractive feature when exact quantitative
criteria cannot be provided in advance due to uncertainty or other
reasons.
Platform Features
[0204] CDI is perfect for big data analytics, supporting many data
types: [0205] Structured: databases, ontologies [0206]
Semi-structured: spreadsheets, CSV/TSV, forms, emails [0207]
Unstructured: web pages, blogs, text documents [0208] Symbolic:
business rules, math formulas
[0209] CDI's distributed computing architecture also make it very
scalable to handle large amount (peta-size) of heterogeneous data
ingestion while performing real-time analytic results even in
microseconds (depending in part on resources available and the
complexity of queries).
Rules Support
[0210] CDI can integrate different types of business rules:
absolute, hard and soft rules, from natural or government laws,
operational or policy requirements, and practice guidelines.
Absolute rules and hard rules always take logic value 0 (false) or
1 (true) when instantiated. Soft rules, however, may take any value
in the interval [0, 1], or more generally more than two values.
Absolute rules reflects a must-satisfy situation, such as FDA/USDA
requirements; hard rules are operational requirements such as "no
serious persistent side effects", but which may be temporarily
relaxed in specific situations; and soft rules can come from
guidelines or experiences, such as "better not give drug XYZ to
diabetes patients with heart problems". FIG. 18 illustrates a
diagram 1800 of examples of different types of rules. Please note
that all chaining of the rules may happen automatically in a
dynamic manner during the optimization process. CDI handles this
complexity with said techniques above by solving a distributed
continuous-space optimization problem.
Self-Adapting and Learning
[0211] CDI platform features a self-learning design. As CDI
converts the original query solving into an optimal control
problem, it can use feedbacks from the environment (external sensor
or internal updates from peer decision elements) to refine its
internal model: a Hamilton-Jacobi-Bellman equation will be updated
to reflect new information and automatically form soft-rule like
constraints internally. The process to update internal mathematical
model is similar to reconstructing 3D images from CT or MRI imaging
by tomographic reconstruction, but used for dynamic systems.
Scalability and Performance
[0212] CDI platform enables functionality to: [0213] Integrate data
that may include 1,000,000+ variables and 100,000+ of constraints
[0214] Provide data integration over evolving distributed network
[0215] Specify queries over a broad range of languages [0216]
Specify queries of a broad range of complexity [0217] Provide best
known response to queries at the local level [0218] Operate in a
variety of environments including cloud-based or local deployments
[0219] Real-time processing of queries
Comparisons
[0220] CDI stands out in being able to do the following things:
[0221] Approximate the optimal solution of NP-hard problems (such
as planning and scheduling) by mapping criteria and constraints
onto a continuous space and solve it as an Optimal Control problem
with polynomial-time algorithms. The solutions offered by CDI may
be near optimal rather than exactly optimal but the running time
will be greatly shortened and some large-scale intractable problems
can become solvable by CDI. [0222] Blend adaptive learning together
with rules composition. Traditional rule engines might support hard
and soft rules (constraints), but the scoring mechanism and weights
need be manually adjusted to meet the goals, while in CDI, via
feedback mechanism, these weights can be optimized as internal
configurations.
[0223] The unique abilities make CDI an ideal choice for
intelligent decision support systems.
Sample Applications
[0224] CDI platform works great in areas where there are a lot of
heterogeneous data, government compliances and business
requirements, such as healthcare and energy.
Clinical Auto-Coding Application
[0225] A Clinical Auto-Coding (C.A.C.) application can be used to
detect medical under-coding, over-coding, and miscoding,
highlighting potential opportunities where higher billing is
justified. It automatically generates clinical codes, including
ICD-10 directly from clinical encounter notes such as physician
notes, lab results, and discharge records, while supporting
workflows accommodating the roles played by administrators, coders
and doctors in coding. FIG. 19 illustrates an example user
interface 1900 related to medical/clinical auto-coding. C.A.C.
incorporates user feedback to learn coding best practices as it
processes records. With its sophisticated adaptive technology,
C.A.C. improves over time, optimizing coding for each organization
to improve the efficiency, accuracy and revenue capture of the
medical coding activity
Clinical Intelligence Web Services
[0226] CDI powers the following intelligent healthcare services
that can be easily integrated: [0227] Clinical Record
Intelligence--For analysis of EMRs (electronic medical records) and
encounter notes, identification of actionable clinical terms and
concepts in those documents. The services can extract concepts,
understand issues, and analyze documents. [0228] Clinical Coding
Intelligence--For inferring clinical codes from a set of clinical
concepts, grouping codes, correlating codes, analyze documents,
audit and justification, and compliance. [0229] Patient-Centric
Intelligence--Secure delivery of end-user applications to predict,
recommend, and explain personalized actions to improve patient
outcomes. It provides a patient centric view, can provide medical
risk prediction and proactive monitoring for patients, automated
data abstraction, actionable recommendations, and more.
[0230] All these web services run in secure cloud with full
compliance measurement in place.
Fraud, Waste and Abuse Detection
[0231] In healthcare, CDI helps assess and monitor risk of medical
fraud, waste and abuse (FWA) by uncovering providers, and to a
lesser extent pharmacies, who are suspected of having committed
fraud via a variety of schemes. Similar application domains include
financial frauds.
[0232] By ingesting a wide variety of data that is difficult to
link (including live public data), a FWA service analyzes data to
find evidence and patterns of fraud, and provides a dashboard
application for agents/detectives to prioritize and act on the
discovered suspected cases of medical fraud. The service features
strict and fuzzy rules-based detection as well as automatic pattern
discovery, and runs in real-time and continuous mode to support
proactive monitoring and action.
Energy Intelligence
[0233] Smart Grid can involve uncertain bi-directional exchange in
distributed grids, so intelligent control can be used to provide
active synchronization between the network of element controllers
and the outside grid management system allows high quality of
service to be maintained in a cost-effective manner, hence the
dynamics can be learned from sensory observations. CDI enables
distributed micro-grid control, supports inductive modeling with
"soft" rules that are learned and continuously updated, to optimize
the grid.
Distributed Architecture (DA)
[0234] The DA is a network of interacting components called
decision elements (DEs). The DEs collaborate in the resolution of a
query posed by one of them. The DA's block diagram is shown in FIG.
18, where Sensors refer to external input data in general. There is
an additional translation layer to process external data to be
consumed by DEs.
Decision Element (DE)
[0235] A decision element is a higher-level functional component
solving queries locally. It can have subcomponents such as a
programmable search engine, internal heterogeneous database,
Inference engine, Inference rule base, API/user interface, and
network interface. A decision element is capable of providing a
quick and near-optimal solution to a complex query with complete
input of data.
Internal Heterogeneous Database (IHDB) and External Knowledge Base
(EKB)
[0236] IHDB is data preprocessed and stored by a specific decision
element (DE). FIG. 20 illustrates a diagram 2000 of some components
of a CDI system, including Knowledge Bases containing data sources
ranging from domain knowledge to general facts. IHDB encodes
knowledge into knowledge components (KC's). Each KC is used and
updated by a DE in the DA, and multiple KCs may share rules.
[0237] External Knowledge Base (EKB) refers to data, including
rules, as input into specific decision elements, such as patient's
body temperature, blood pressure, or instantiated rule to determine
if patient's cholesterol level is high. EKB can also contain
communicated information from other DEs. Domains for variables
include: real, complex, integer, binary numbers and symbolic token
on finite domains.
Interfacing Components
[0238] The following components act between users and the data via
API and/or GUI. [0239] Rule Entry Interface--It provides the entry
of rules into the IHDB, validates the specification of rules before
insertion, and route the rules to the appropriate DE for insertion
to their respective knowledge components. [0240] Sensor Ingestion
Interface--A sensor is a machine or service where external data
exist. Sensor Ingestion Interface (SII) enables the system to add
or remove a sensor, poll a sensor and submit data to the network.
[0241] Query Language Interface--Query Language Interface accepts
the query, submits it to the system (Distributed Architecture) and
provides response. It is an API and flexible UI can be built on top
of it.
Minimization Function Generator (MFG)
[0242] The minimization function generator converts a query to a
minimization function (i.e. analytic continuation). This is useful
because the problem is converted from search problem in a large
discrete space (like graph search problems, which are usually
NP-complete) into an optimization problem in continuous space
(polynomial algorithms exist). An analogy is NP-hard integer
programming, while continuous linear programming has a very
efficient solution.
Query Response Engine (QRE)
[0243] The Query Response Engine is the core of the whole system
responsible solving the query (locally). A mathematical model is
constructed based on these equations obtained from previous steps,
containing the current state and object state (goal state). The
continuous-space optimization automatically handles forward and
backward rule chaining by moving along the trajectory toward target
state. It also manages uncertainty by keeping a large set of
possible states and reducing the solution space only when more
information becomes available. Standard optimization techniques
(e.g. Newton-Raphson Method) can be employed to solve the
problem.
[0244] Feedbacks and updates (data from EKB or other DE in the
architecture) will be used to refine the mathematical model over
time; therefore, the core engine is self-adapting.
Pareto Multi-Criteria Optimization Engine
[0245] The Pareto Multi-Criteria Optimization Engine (PMOE) is the
aggregation step where all DE in the network settle to obtain a
good stable solution--a state where no improvements can be made to
any individual DE without reducing the whole team's performance--a
state belongs Pareto Optimal Set in Game Theory. It is like each DE
is playing a game and they communicate and interact and work
together to make the best outcome for the criteria (query). To
efficiently synchronize all decision elements, Mean Field Theory is
applied for dimension reduction using knowledge obtained. FIG. 21
illustrates a diagram 2100 of performing Pareto processing for use
with mean field techniques, including to reach Pareto
efficiency.
Data Exchange Specifications
[0246] The system can take many different data types via ingestion
API and has adapters from different public data sources. The
supported data types include common ones from XML, CSV, TSV, SQL,
Spreadsheet, JSON, and more. OData support is also available.
Output types can be API (XML or JSON), as well as exporting to CSV,
SQL or directly to services such as SOLR and Cassandra.
Running Environment
[0247] CDI can run in the cloud as Software as a Service platform.
We can provide whole end-to-end support by setting up the
infrastructure in a Virtual Private Cloud or deploy and configure
it with the cluster clients provide. The system features SaaS
architecture and provides APIs to be used by third parties. For
advanced integration requests, Java/Scala and Python libraries are
available.
[0248] To sum up, CDI platform utilizes many advanced techniques
from Mean Field Theory, Lagrangian and Hamiltonian functions,
Pareto Optimal Set, Gauge Theory, and so on derived from modern
mathematics, quantum physics, optimal control and game theory to
achieve high performance. In the detailed computation process, lots
of transformation, approximation, optimization, caching and other
advanced computing techniques are used to improve accuracy, speed
and scalability. Due to this unique approach, CDI is able to solve
complex decision making problem in a smoothly, efficient manner and
achieve near optimal results.
[0249] The Cooperative Distributed Inference (CDI) platform is a
unique advanced technology offering to enable near-optimal and near
real-time decision making on vast amount of heterogeneous and
distributed information, in complex knowledge-based decision
support systems, combining absolute, hard and soft rules to handle
various requirements from natural or governing laws, policies, and
best practices.
[0250] Each agent in the Distributed Architecture is a Decision
Element that can solve the problem independently, provided with
knowledge base and data. In a very unique manner, the agent uses
Optimal Control Theory to obtain a near optimal solution, starting
with a technique called analytic continuation (transforming query
and rules into differential equations whose dependent variables
represent internal variables and parameters of the rules), then
using its own internal knowledge to solve the problem as an
optimization problem in continuous space, to allow for efficient
solving.
[0251] The outcome will be fed back to the Rules Editor and
optimization process (the Chattering algorithm) to enable automatic
adjustment of weights of soft rules (constraints) and achieve
optimal score of the objective function.
[0252] A CDI agent is an independent decision element that can take
in sensor data (input from the environment, streaming or in
batches), as well as the knowledge-bases (rules composed by domain
experts, including absolute, hard and soft rules) to generate a set
of partial differential functions (Lagrangian constraints), through
continualization. Similarly, the objective is also converted into a
minimization function, as part of the Lagrangians which will be
solved via the Hamilton-Jacobi-Bellman equation. Each agent will
talk to other agents via a "mean field" abstraction layer to
greatly reduce the communication and computation overhead, and
incorporate additional information to reach the global optimal
state (a stable, near optimal set of states across all agents).
Finally, the agent exercises control over the system. FIG. 22
illustrates a network diagram 2200 of an example decision module
agent.
[0253] Various domain knowledge is captured from experts in the
form of Domain Specific Language. Some are constraints; logic forms
need be converted to Boolean equations; and variables in soft rules
take values [0 . . . 1]. This creates a set of equations to be
solved by the optimization algorithm. The optimization process
takes data from a time range and finds the best state configuration
to reach optimality. It starts with a "learning" process to find a
good initial configuration by taking in a short history of data,
before processing real-time streams. FIG. 23 illustrates a network
diagram 2300 of an example of offline workflows for knowledge
capture.
[0254] Each CDI agent computes a "mean field" view of the system
via its neighbors, and responds to queries with the latest updates.
The approximate mean field view of a group greatly reduces
computational dimensions. The agents synchronize with others and
understand the global state via the mean-field approximation. They
engage in games to reach Pareto Optimal (also referred to as Pareto
equilibrium)--the best output. FIG. 24 illustrates a network
diagram 2400 of an example of workflows for mean field computation
and Pareto Optimal.
[0255] An example home solar micro-grid system illustrates one
example embodiment of a CDI application or automated control
system, which takes the sensor data from the solar panel (in the
house), substation, and power network, and decides whether or not
to fulfill the utility requests in real time using a set of complex
rules. FIG. 25 illustrates a network diagram 2500 of an example of
an automated control system for a home solar micro-grid electrical
generating system.
[0256] FIGS. 26-28 provide further details regarding operations of
portions of example CDI systems and their sub-components, and FIGS.
29A-29K illustrate examples of using a CDI system to iteratively
determine near-optimal solutions over time for controlling a target
system in diagrams 2900A-2900K. In particular, FIG. 26 illustrates
a further diagram 2600 of workflow and components of a portion of a
CDI system, FIG. 28 illustrates a diagram 2800 of an overview
workflow for a portion of a CDI system, and FIG. 27 illustrates a
diagram 2700 of workflow for an inference process portion of a CDI
system.
[0257] Further details related to an example CDI system are shown
below and included in provisional U.S. Patent Application No.
62/015,018, filed Jun. 20, 2014 and entitled "Methods And Systems
For Cooperative Distributed Inferencing," which is hereby
incorporated by reference in its entirety. In addition, further
details related to example details of using gauges to perform model
error measurements are included in provisional U.S. Patent
Application No. 62/182,796, filed Jun. 22, 2015 and entitled "Gauge
Systems," which is also hereby incorporated by reference in its
entirety. Furthermore, further details related to examples of using
the CDD system in particular manners with particular types of
target systems are included in provisional U.S. Patent Application
No. 62/182,968, filed Jun. 22, 2015 and entitled "Applications Of
Cooperative Distributed Control Of Target Systems," which is also
hereby incorporated by reference in its entirety--the example types
of target systems discussed include, for example, a vehicle being
controlled, controlling inventory management, controlling a
micro-grid electrical generation facility, controlling activities
in a day trader financial setting, etc.
TABLE-US-00002 Current notation Comments t Notation for algorithmic
time. q(t) Notation of canonical coordinate vector for entire
system. q Notation of canonical coordinate vector dropping time
dimension. q.sup.(f) Notation of canonical coordinate vector for
specific function f. {dot over (q)} Notation of first time
derivative of canonical coordinate vector. {umlaut over (q)}
Notation of second time derivative of canonical coordinate vector.
h Notation of HEAD of Horn clause. .phi.(q) Notation of generic
proposition. .sigma.(q) Notation of generic proposition alternate
to .phi.. T.sub.i Notation of the TV of a soft rule. {hacek over
(r)}(q; .phi., .sigma.) Generic equational form relating two
propositions. {hacek over (.phi.)}(q) Notation of the equational
form of .phi.(q). .phi..sub.Q(q) Notation for proposition defined
by the query. {hacek over (.phi.)}.sub.Q(q) Notation for equation
defined by the query. J(q) Notation for minimization function for
the query. L Notation for static Lagrangian L.sub.k.sup.(o, T)
Notation for total static Lagrangian for DE.sub.k. q
{p.sub..alpha.} u.sup.(k) H.sub.k.sup.(o) Primary Hamiltonian for
the absolute rules for DE.sub.k. H.sub.k.sup.(A) Hamiltonian for
the Tellegen agent of the total Hamiltonian's rules.
H.sub.k.sup.(T) Total Hamiltonian for DE.sub.k.
Overview
[0258] This document introduces and specifies the architecture for
the Cooperative Distributed Inferencing Platform (CDIP). The
primary instance of this is the Distributed Architecture (DA) for
resolving queries by accessing both an Internal Heterogeneous
Database (IHDB) populated by a special class of Horn Clause rules
and external data sources referred to as sensors.
[0259] The architecture implements a network of active devices at
its nodes. These devices are called Decision Elements (DE's). The
DE's cooperate in the resolution of a query posed to one or several
of them. The DE's in a given DA are referred to as the team.
[0260] Every DE in a team is programmed to transform rules in its
domain, determined by a posed query, into an ordinary differential
equation (ODE), whose dependent variables represent internal
variables and parameters. The dependent variables include unknowns
of the query posed to the DE. The DE's in the architecture are
synchronized via a Pareto multi criteria optimization strategy.
[0261] The components of the CDIP include: [0262] Application
requirements that the system is designed to accommodate. [0263]
Functional requirements that satisfy the application requirements
and pertain directly to the construction and operation of the
system components. [0264] Subcomponents which are necessary to
implement the functional requirements [0265] Limitations which
highlight noteworthy constraints which are inherent the specified
implementation of the architecture. [0266] Architectural flow
describes key aspects of the architecture that indicate how the
system is to be constructed given the specified essence and key
behavior of the subcomponents. [0267] Software realization of the
architecture describes the key pieces of software necessary for
system implementation. [0268] Data describes the kinds of data the
system is expected to accept as input and produce as output. [0269]
Data exchange protocols reference key data types and structures
that need to be exchanged across the system and the protocols for
exchange. [0270] Environment describes the particulars of the
environments that the system will be able to operate in and
therefore should be tested in. [0271] Testing describes how the
system should be tested given the data and operating
environments.
Subcomponents
[0272] Subcomponents are fundamental parts of the architecture that
perform particular roles. This section contains descriptions for
each of the subcomponents of the architecture. The subcomponents
are: [0273] The Distributed Architecture (DA). [0274] The Internal
Heterogeneous Database (IHDB). [0275] The Rule Entry Interface
(REI). [0276] The Rule Editor (RE). [0277] The External Knowledge
Base (EKB). [0278] The Sensor Ingestion Interface (SII). [0279] The
Rule Conversion Engine (RCE). [0280] The Decision Element (DE).
[0281] The Query Language Interface (QLI). [0282] The Minimization
Function Generator (MFG). [0283] The Query Response Engine (QRE).
[0284] The Pareto Multi-Criteria Optimization Engine (PMOE).
Distributed Architecture (DA)
[0285] The DA is a platform of interacting components called DE's.
The DE's collaborate in the resolution of a query posed by one of
them. The DE's implement a distributed, dynamic optimization
process, herein referred as the optimization process (OP). OP
implements an optimization process that computes an answer to the
active queries as a function of data stored in two categories of
repositories: IHDB and EKB's. These repositories of the data needed
to implement OP given a query.
[0286] The EKB's are a collection of public or private repositories
of knowledge relevant to the DE posing a query. A DE has a list of
external repositories (LER). Each entry in an LER includes 1) a
protocol, 2) a heading sub-list, and 3) a translation grammar. Each
protocol entry prescribes the access procedure to the corresponding
knowledge repository. Each heading sub-list entry contains a
summary of the knowledge contents of the corresponding repository.
Finally, each translation grammar entry provides a procedure for
converting knowledge elements of the corresponding repository in to
the rule representation in the IHDB of the DE.
[0287] Functional characteristics of this architecture and in
particular, the DE's, IHDB, and the sensors are described,
including the following concepts: [0288] The DA [0289] A process
for resolving queries by accessing the IHDB and External Knowledge
Bases (EKB's) through sensors [0290] The constitution of DE's
[0291] A query and corresponding rules transformation into an ODE
[0292] The orchestration of a team of DE's through a Pareto multi
criteria optimization strategy
[0293] The Internal Heterogeneous Database (IHDB)
[0294] Composition of IHDB as a Set of Knowledge Components
[0295] The IHDB encodes knowledge about the implemented
application. The IHDB is divided into knowledge components (KC's).
Each KC is consulted and updated by a DE in the DA. Any pair of
KC's may have an overlapping set of rules by which they operate,
but there is no a priori constraint on intersections or inclusion.
The collection of KC's constitutes the existing knowledge of the
system.
[0296] Algorithmic Formulation of a Rule
[0297] A KC is a collection of rules, written in a restrictive Horn
clause format. The rules are logic entities. When a rule is
instantiated, it has a logic value. The logic values a rule can
have are taken from the interval [0, 1]. The entire system of rules
is evaluated using variables and parameters which are collectively
referred to as the generalized coordinates of the system and are
indexed as follows:
q(t)={q.sup.(1)(t), . . . ,q.sup.(N)(t)}. (3.2-1)
[0298] The time argument t refers to the algorithmic time of the
system which means that it has no other meaning than as a
continuous index with respect to the evolution of the system. There
is therefore no requirement that it correspond to a physical aspect
of the system although this may naturally occur. Physical time may
be represented specifically by a canonical coordinate of choice
q.sup.(i)(t). Alternatively, we may refer to the q's without
expressly stating the independent time argument and write
q(t)={q.sup.(1), . . . ,q.sup.(N)}. (3.2-2)
[0299] Then we should also note that the time derivatives are
referred to notationally as
q . = q ( t ) t , q = 2 q ( t ) t 2 ( 3.2 - 3 ) ##EQU00006##
[0300] These coordinates are referred to variously as q depending
on the context and the expected arguments of the function to which
they are applied. When it is necessary to distinguish between more
than one q in equational form we generally write q.sub.f of where f
denotes the reference function or appropriate domain. Typically, we
assume without loss of generality the entire set of canonical
coordinates q is an argument to any function, term or proposition.
In practice, we may further assume it is possible to apply the
particular required coordinates as need to mathematical construct
in question.
[0301] The rules in each knowledge component are of three types:
absolute rules, hard rules, and soft rules. Absolute rules and hard
rules take logic value 0 (false) or 1 (true) when instantiated.
Soft rules take any value in the interval [0, 1].
[0302] The format of the restrictive Horn Clauses in the IHDB is
illustrated in Equation 3.2-2. A Horn Clause is an object composed
of two objects a HEAD and a BODY connected by backward Implication
(). The logic implication transfers the logic value of the BODY to
the HEAD. If the rule is an absolute rule or a hard rule, the logic
value is 1 (if the BODY is logically true) or 0 (if the BODY is
logically false). If the rule is a soft rule, the logic value
transferred by the body is any number in [0, 1].
[0303] The HEAD is a data structure composed of two objects: A
name, h, and a list of arguments described by the argument vector
q=(q.sup.(1), q.sup.(n)). The list of arguments includes variables
and parameters. The variables take values in the domain of the rule
and the parameters are constants passed to the rule and unchanged
by the instantiation of the rule. The domain of the rule is a set
of values that each of its variables can take. In general,
variables can take values over numerical or symbolic domains. As an
example, a symbolic domain can be a list of diseases. A numeric
domain can be a set of pairs of numbers representing blood
pressure.
[0304] For the applications of CU, the domains for variables are:
real numbers, complex numbers (floating point and floating point
complex numbers), integer numbers, binary numbers and symbolic
token on finite domains.
[0305] The BODY of a clause is a data structure composed of one or
more terms, denoted .phi..sub.i(q). The composition operation is
extended- and, denoted by: . The extended- and works as a regular
and in absolute rules and hard rules and as a functional
product.sup.2 on soft rules.
[0306] A rule with a head but not a body is called a fact. A fact's
truth value is determined on the basis of the instantiation of its
variables.
[0307] Each term in the body of a rule is an extended disjunction
(or denoted by v) of sub-terms. The v operator behaves like the
standard- or for absolute and hard rules and behaves in a
functional form, described later, when connecting sub-terms
encoding heads of soft rules.
[0308] A sub-term is either the HEAD of a rule, a relation or a
truth valuation (TV). When it is a HEAD it may have the same name
as the one in the HEAD of the rule but with different arguments.
This provides a recursive mechanism for rule evaluation.
[0309] When a rule has a sub-term that is the head of another rule
it is said that the two rules are chained together by the
corresponding sub-term. Note that a rule can be chained to several
rules via corresponding sub-terms.
[0310] Constraint Domains
[0311] Constraint domains augment the BODY clause of Horn clauses
to facilitate dynamic programming. Constraints are specified as a
relationship between terms. Define the relationship between two
terms
.phi.(q)rel.sigma.(q). (3.2-4)
as a member of the following set
rel .epsilon.{=,.noteq.,.ltoreq.,.gtoreq.,statistical propagation}.
(3.2-5)
[0312] A relation can be of two types numeric or symbolic. Numeric
relations establish equational forms between two functional forms.
(For the initial phase only polynomial and affine linear functional
forms will be considered.)
[0313] In general, an equational form is a set of one or more
relations. For numeric relations, .phi.(q) rel .sigma.(q), rel
.epsilon.{=, .noteq., .ltoreq., .gtoreq., <, >, statistical
propagation}. The table below gives the relations considered and
their symbols.
TABLE-US-00003 TABLE EE Numeric Relation Symbol Code Form Equality
= .phi. = .sigma. Disequation .noteq. .phi. \= .sigma.
Less-inequality < .phi. < .sigma. Less-Equal .ltoreq. .phi.
=< .sigma. Great-equality > .phi. > .sigma. Great-
inequality .gtoreq. .phi. >= .sigma.
[0314] The adopted code forms are the ones used in constraint logic
programming.
[0315] A symbolic relation can be of two types: inclusion and
constraint. Inclusion relations are of the form:
q.epsilon.Set (3.2-6)
[0316] Where x is a variable or a parameter, .epsilon. is the
inclusion symbol and Set is a set of symbolic forms or a set of
numbers or a composite set of the form shown in the table
below.
TABLE-US-00004 TABLE FF Composite Set Symbol Code Form Intersection
.andgate. Set1/\Set2 Union .orgate. Set1\/Set2 Complement \
\Set
[0317] Constraint forms of the symbolic relational type may be one
or a set of the forms presented in the table below For numeric
relations, .phi.(q) rel .sigma.(q), rel .epsilon. {=, .noteq., .OR
right., , .OR right., }.
TABLE-US-00005 TABLE GG Symbolic Relation Symbol Code Form Equal =
.phi.# = .sigma. Not Equal .noteq. .phi.# \= .sigma. Is Contained
.OR right. .phi.# < .sigma. Contains .phi.# > .sigma. Is
Contained or Equal .OR right. .phi.# =< .sigma. Contains or
Equal .phi.# >= .sigma.
[0318] A TV is either a variable or a constant with values in the
interval [0, 1]. The TV of a Horn Clause that is an absolute rule
or a hard rule can only take two values: 1 or 0. The TV when
instantiated is 0 or 1. If the TV for an absolute or hard rule is
1, the rule is said to be in inactive state; if the TV is 0, the
rule is said to be in active state.
[0319] The TV, T.sub.i, of a soft rule satisfies
0.ltoreq.T.sub.i.ltoreq.1. (3.2-7)
[0320] If T.sub.i above satisfies,
T.sub.i.gtoreq.T.sub.threshold (3.2-8)
the soft clause is said to be in inactive state. If
T.sub.i<T.sub.threshold, (3.2-9)
the soft clause is said to be in active state, where
T.sub.threshold is a constant in [0, 1] defined for each soft
clause. The default value is 0.5.
[0321] This concludes the description of the knowledge
representation. The instantiation process of the goal in a DE, as
function of its knowledge base, is carried out by the inference
engine of the DE. This process is the central component of CU and
will be described later on the document.
Summary of Terminology
[0322] The following table summarizes the terminology we have just
reviewed.
TABLE-US-00006 TABLE HH Reference term Definition proposition
Defined as a construct as in the prepositional calculus where the
proposition takes on the value of true or false. term Recursively
according to its assigned sub-term. sub-term A sub-term may be a
Horn clause, a relation between two other sub-terms or an extended
truth valuation depending on the context of absolute, hard or soft
rules. In the case of absolute and hard rules it may be evaluated
as a proposition. In the case of soft rules it takes a value on the
interval [0, 1] and is considered to be active or true in the case
that it exceeds its specific threshold. Horn clause A disjunction
of terms with at most one positive term. definite clause A Horn
clause with exactly one positive term. goal clause A Horn clause
with no positive terms. fact A definite clause with no negative
terms. head The positive term of a definite clause. inactive state
The case when a rule will not apply for constrained optimization.
active state The case when a rule will apply for constrained
optimization. truth value, TV The value that is used to determine
whether a rule is active or inactive.
Horn Clause Example
[0323] The following example illustrates a Horn clause:
has_fever(name, temperature, white_count, heartrate,
blood_pressure)(temperature>37)((heartrate.gtoreq.70)bp(name,
temperature, blood_pressure)wc(name, white_count)) (3.2-10)
[0324] The clause establishes under which conditions the patient of
name name, has a fever. The variables in clause are: [0325] name,
temperature, white_count, heartrate, blood_pressure.
[0326] When instantiated they represent, respectively, the name of
the patient, his current body temperature, his white blood cell
count, his heart rate range, and his blood pressure.
[0327] The clause body includes other clauses: bp (blood pressure)
and wc (white count).
[0328] This completes the specification of the rule-based
framework. The next step is to specify a complete process for
converting all rules of this form to a set of equations.
[0329] Rule Entry Interface (REI)
[0330] The Rule Entry Interface provides a mechanism for: [0331]
Providing an API for the entry of rules into the IHDB. [0332]
Validating the specification of rules to be inserted into the IHDB.
[0333] Routing the rules to the appropriate DE's for insertion to
their respective KC's.
[0334] Rule Editor (RE)
[0335] The Rule Editor allows users to specify rules associated
with the systems to be interrogated.
[0336] External Knowledge Base (EKB) [0337] It may be distributed
or not [0338] It may be persisted or not [0339] It may be persisted
locally or remotely to an agent [0340] It may or may not be
architecturally co-located with the IHDB [0341] A sensor may
include any source of data used by the agent [0342] It may use
various types of buses for data communication [0343] Sensors may or
may not be co-located with agents/DEs
[0344] Rule Conversion Engine (RCE)
[0345] The rule conversion engine converts rules of the IHDB into
equations.
[0346] Method for Specification of a Simple Term as an Equation
[0347] Consider the term .phi.(q) with the following truth
assignment.
.PHI. ( q ) = { T q .di-elect cons. .PHI. F q .PHI. ( 3.6 - 1 )
##EQU00007##
[0348] Then we can define the set of arguments which yield positive
truth assignment.
{q.epsilon..sub..phi.|.phi.(q).rarw.T}. (3.6-2)
[0349] Define the corresponding equation {hacek over (.phi.)}(q) of
the term .phi.(q) as
.PHI. ( q ) = { 1 .PHI. ( q ) .rarw. T 0 .PHI. ( q ) .rarw. F ( 3.6
- 3 ) ##EQU00008##
[0350] Then extend the range of {hacek over (.phi.)}(a) to the
closed unit interval
{hacek over (.phi.)}(q).fwdarw.[0,1]. (3.6-4)
[0351] Revisiting the taxonomy of absolute, hard and soft rules, we
recognize that hard and soft rules (terms in this example) can take
values along the interval
0.ltoreq.{hacek over (.phi.)}(q).ltoreq.1. (3.6-5)
[0352] And for absolute rules we add the constraint {hacek over
(.phi.)}(q).fwdarw.{0,1}
{hacek over (.phi.)}(q)(1-{hacek over (.phi.)}(q))=0. (3.6-6)
[0353] Conversion of Fundamental Clauses of Propositional Calculus
to Equations
[0354] Define the following notation for the propositional
calculus.
TABLE-US-00007 TABLE II Symbol Function And Or Implication ~ Not
.E-backward. Exists .A-inverted. All Fuzzy rule
Theorem 3.6.1.
[0355] Given the method for the specification of equations from
propositions, we prove the following transformations.
TABLE-US-00008 TABLE JJ Proposition Equation ~.phi.(q) 1 - {hacek
over (.phi.)}(q) .phi.(q) .sigma.(q) {hacek over (.phi.)}(q) {hacek
over (.sigma.)}(q) .phi. (q) .sigma.(q) {hacek over (.phi.)}(q) +
{hacek over (.sigma.)}(q) - {hacek over (.phi.)}(q) {hacek over
(.sigma.)}(q) .phi.(q) .sigma.(q) 1 - {hacek over (.phi.)}(q) +
{hacek over (.phi.)}(q) {hacek over (.sigma.)}(q) .phi..sub.1(q)
.phi..sub.2(q) . . . .phi..sub.k-1(q) .phi.(q) .phi.(q) .phi. ~ ( n
, q ) = h ( n - 1 , q ) .sigma. ~ ( n , q ) .phi. ~ ( n - 1 , q ) -
1 ##EQU00009## (tail recursive)
[0356] Proof by enumeration for equational representation of
conjunction
[0357] Define the function {hacek over (r)}(q; .phi., .sigma.)
which represents the equation corresponding to conjunction ().
Verify by enumeration the correspondence of the mathematical
equation values corresponding to the mapping T.fwdarw.1 and
F.fwdarw.0.
TABLE-US-00009 TABLE KK .phi.(q) .sigma.(q) {hacek over (r)}(q;
.phi., .sigma.) = {hacek over (.phi.)}(q) {hacek over (.sigma.)}(q)
T T T 1 = 1 1 T F F 0 = 1 0 F F T 0 = 0 1 F F F 0 = 0 0
[0358] Proof by enumeration for equational representation of
disjunction
[0359] Define the function {hacek over (r)}(q; .phi., .sigma.)
which represents the equation corresponding to disjunction ().
Verify by enumeration the correspondence of the mathematical
equation values corresponding to the mapping T.fwdarw.1 and
F.fwdarw.0.
TABLE-US-00010 TABLE LL .phi.(q) .sigma.(q) {hacek over (r)}(q;
.phi., .sigma.) = {hacek over (.phi.)}(q) + {hacek over
(.sigma.)}(q) - {hacek over (.phi.)}(q) {hacek over (.sigma.)}(q) T
T T 1 = 1 + 1 - 1 1 T T F 1 = 1 + 0 - 1 0 F T T 1 = 0 + 1 - 0 1 F F
F 0 = 0 + 0 - 0 0
[0360] Proof by enumeration for equational representation of
negation
[0361] Define the function {hacek over (r)}(q; .phi., .sigma.)
which represents the equation corresponding to negation (). Verify
by enumeration the correspondence of the mathematical equation
values corresponding to the mapping T.fwdarw.1 and F.fwdarw.0.
TABLE-US-00011 TABLE MM .phi.(q) ~.phi.(q) {hacek over (r)}(q;
.phi., .sigma.) = 1 - {hacek over (.phi.)}(q) T F 0 = 1 - 1 F T 1 =
1 - 0
[0362] Proof by enumeration for equational representation of
implication
[0363] Define the function {hacek over (r)}(q; .phi., .sigma.)
which represents the equation corresponding to disjunction ().
First note the equivalence of
.phi.(q).sigma.(q) and .about..phi.(q).sigma.(q). (3.6-7)
[0364] Verify by enumeration the correspondence of the mathematical
equation values corresponding to the mapping T.fwdarw.1 and
F.fwdarw.0.
TABLE-US-00012 TABLE NN ~.phi.(q) .sigma.(q) {hacek over (r)}(q;
.phi., .sigma.) = 1 - {hacek over (.phi.)}(q) + {hacek over
(.phi.)}(q) {hacek over (.sigma.)}(q) T T T 1 = 1 - 1 + 1 1 T F F 0
= 1 - 1 + 1 0 F T T 1 = 1 - 0 + 0 1 F T F 1 = 1 - 0 + 0 0
[0365] Proof for equational representation of tail recursion
[0366] Tail recursion is propositionally defined as
.phi.(q).phi..sub.1(q) . . . .phi..sub.k-1(q).phi.(q) (3.6-8)
where s represent the current state. To develop an equational
representation of the recursive formulation, first define the
general function {tilde over (.phi.)}(n, q) where n represents the
n.sup.th iteration of the tail recursion ad {tilde over (.phi.)}(n,
q) is the logical consequent. Then rewrite the above formulation
using the recursive step.
{tilde over (.phi.)}(n,q){tilde over (.phi.)}.sub.2(n,q) . . .
{tilde over (.phi.)}.sub.k-1(n,q){tilde over (.phi.)}(n-1,q){tilde
over (.phi.)}(n,q) (3.6-9)
Define
{tilde over (.sigma.)}(n,q){tilde over (=)}{tilde over
(.phi.)}.sub.1(n,q){tilde over (.phi.)}.sub.2(n,q) . . . {tilde
over (.phi.)}.sub.k-1(n,q){tilde over (r)}(n-1,q){tilde over
(=)}{tilde over (.sigma.)}(n,q){tilde over (.phi.)}(n-1,q) (3.6-10)
[0367] Then the tail recursion is rewritable as
[0367] {tilde over (.sigma.)}(n,q){tilde over (.phi.)}(n-1,q){tilde
over (.phi.)}(n,q){tilde over (r)}(n-1,q){tilde over (.phi.)}(n,q).
(3.6-11)
[0368] According to the equational representation of implication,
let
{hacek over (h)}(n-1,q)=1-{hacek over ({tilde over
(.sigma.)}(n,q){hacek over ({tilde over (.phi.)}(n-1,q)+{hacek over
({tilde over (.sigma.)}(n,q){hacek over ({tilde over
(.phi.)}(n-1,q){hacek over ({tilde over (.phi.)}(n,q). (3.6-12)
[0369] Since by definition {circumflex over ({tilde over (r)}(n-1,
q)={circumflex over ({tilde over (.phi.)}(n, q){circumflex over
({tilde over (.phi.)}(n-1, q). Then
.PHI. ~ ( n , q ) = h ^ ( n - 1 , q ) + .sigma. ~ ^ ( n , q ) .PHI.
~ ^ ( n - 1 , q ) - 1 .sigma. ~ ^ ( n , q ) .PHI. ~ ^ ( n - 1 , q )
( 3.6 - 13 ) ##EQU00010##
with boundary condition n=0.
[0370] Converting Rules Based System of Inference to the Problem of
Constrained Minimization
[0371] Consider the following example. [0372] Converting rules to
constraints
[0373] The preceding discussion has established an algorithm for
convert rules of the form
h(q).phi..sub.1(q).phi..sub.2(q) . . . .phi..sub.m(q) (3.6-14)
[0374] To constraints of the form
{hacek over (h)}(q)={hacek over (.phi.)}.sub.1(q){hacek over
(.phi.)}.sub.2(q) . . . {hacek over (.phi.)}.sub.m(q). (3.6-15)
[0375] Decision Element (DE)
[0376] A diagram of the Decision Element Architecture is shown in
illustration .largecircle..largecircle.. It is composed six
elements: [0377] Programmable search engine (PSE) [0378] Internal
heterogeneous database (IHDB) [0379] Inference engine (IE) [0380]
Inference rule base (IRB) [0381] API/user interface [0382] Network
interface (NI)
[0383] List of External Repositories (LER)
[0384] A DE has a List of External Repositories (LER). Each entry
in an LER includes 1) a protocol, 2) a heading sub-list, and 3) a
translation grammar. Each protocol entry prescribes the access
procedure to the corresponding external knowledge repository. Each
heading sub-list entry contains a summary of the knowledge contents
of the corresponding repository. Finally, each translation grammar
entry provides a procedure for converting knowledge elements of the
corresponding repository in to the rule representation in the IHDB
of the DE.
[0385] Programmable Search Engine
[0386] The programmable search engine implements a standard hashing
algorithm for detecting active rules as a function of the current
instantiation of the variables in a variable buffer (VB) of the IE,
and the contents of the active rule buffer (ARB). The VB contains
the variables that form part of the query and all additional
variables incorporated to this buffer during the inference process
(IP). The VB includes all relevant data from the EKB beneficial to
perform the query. The IP will be described below. The ARB contains
all the currently active rules in the IP.
[0387] The search hashing algorithm is characterized by the search
rules in the Inference Rule Base.
[0388] Internal Heterogeneous Database
[0389] The IHDB is the repository of the application clauses
associated with the DE. These encode the domain of knowledge
characterizing the expertise of the DE. For example in a medical
application, a decision element may deal with expertise on heart
illnesses, and the corresponding clauses might encode diagnoses and
treatments for these diseases.
[0390] Inference Engine
[0391] The IE encodes an algorithm, the IP, for assigning values to
the variables appearing in the query.
[0392] Inference Rule Types
[0393] The DE incorporates inference rules (IR) that are a
collection of rules for transforming and inferring instantiations
of the goal. These rules provide the Inference Engine with
directives for processing database rules to give a satisfactory
instantiation to a given query or to request additional information
so that a satisfactory instantiation can be generated. They are
organized according to their functionality as follows. (See
Illustration PP)
[0394] Equation Rules
[0395] These rules include the formal rules for inference. This
includes all rules for natural language modeling from first
principles.
[0396] Optimizer Rules
[0397] These rules include rules for finding the interior point in
optimization.
[0398] Search Rules
[0399] These rules include rules for identifying the nature of
insufficient potential. The goal is to apply these rules to acquire
additional information required to satisfy the optimization
goal.
[0400] Adaptation Rules
[0401] Adaptation rules are used to update the soft rules to relax
them further to reduce the complexity and constrains of the
optimization problem. The adaptation also serves to update the
search rules to improve information acquisition.
[0402] Language Statistics and Pattern Rules
[0403] These rules embody the machine learning models.
[0404] Network Rules
[0405] These rules define how information is distributed over the
network and what information is available from which resources.
[0406] Hybridization Rules
[0407] The rules define how other rules may be combined.
[0408] User Interface
[0409] The UI provides the utilities for entering queries, pragma
rules, displaying query answers, status and for general interaction
with the IE.
[0410] Network Interface
[0411] The NI provides a generic mechanism for interacting with
other DE's via a procedure termed companionship. The companionship
procedure implements the active coupling for the cooperation of the
DE's in query resolution. This procedure is not hierarchical and
implements a Pareto Agreement set strategy as the mechanism for
Ca
[0412] Query Language Interface (QLI)
[0413] Information about the QLI is available elsewhere herein.
[0414] Process for Determining Active Constraints
[0415] The process for determining active constraints is available
elsewhere herein.
[0416] Minimization Function Generator (MFG) and Determining Active
Constraints
[0417] The minimization function generator converts a query to a
minimization function.
[0418] Again, we assume without loss of generality the entire set
of canonical coordinates q is an argument to any proposition
.phi..sub.i. In practice, we may further assume it is possible to
apply the particular required coordinates as need to the
proposition or function in question. Then let .phi..sup.k be the
set of propositions associated with DE.sub.k in the context of
query Q. These propositions are composed of the proposition
associated with the query .phi..sub.Q(q), and other propositions
.phi..sub.i(q), comprising the constraints of the system. The
proposition .phi..sub.Q(q) associated with a given query Q can be
converted to an equation {hacek over (.phi.)}.sub.Q(q). Queries
that are satisfiable specify a set.
{q|.phi..sub.Q(q).rarw.T} (3.10-1)
[0419] Similarly, a satisfied query represented as an equation is
also a set
{q|{hacek over (.phi.)}.sub.Q(q)=1}. (3.10-2)
[0420] Relaxing the values that {hacek over (.phi.)}.sub.Q(.cndot.)
can take to include the unit interval so that soft rules are
incorporated yields the following constrained optimization
expression. Let J(q)=({hacek over (.phi.)}.sub.Q(q)-1).sup.2. Then
specify the optimization
min q J ( q ) ( 3.10 - 3 ) ##EQU00011##
[0421] Subject to: [0422] 1. {hacek over (.phi.)}.sub.Q(q).ltoreq.1
[0423] 2. {hacek over (.phi.)}.sub.Q(q).gtoreq.0 [0424] 3. A
knowledge base on the set {{hacek over (.phi.)}.sub.1(q), . . . ,
{hacek over (.phi.)}.sub.n(q), . . . , {hacek over
(.phi.)}.sub.n+s(q)}.OR right.{hacek over (.phi.)}.sup.(k) which
represents a further set of active constraints specific to the
problem: [0425] a. {hacek over (.phi.)}.sub.i(q).gtoreq.0 for
1.ltoreq.i.ltoreq.n, [0426] b. {hacek over
(.phi.)}.sub.i(q).ltoreq.1 or, equivalently -({hacek over
(.phi.)}.sub.i(q)-1).gtoreq.0 for 1.ltoreq.i.ltoreq.n, [0427] c.
and in the case of absolute rules {hacek over
(.phi.)}.sub.l(q)(1-{hacek over (.phi.)}.sub.l(q))=0 for
n<l.ltoreq.n+s.
[0428] Introduce the indicator functions
V .PHI. i - = { 0 .PHI. i ( q ) .gtoreq. 0 .infin. .PHI. i ( q )
< 0 and ( 3.10 - 4 ) V .PHI. i + = { 0 1 - .PHI. i ( q )
.gtoreq. 0 .infin. 1 - .PHI. i ( q ) < 0 ( 3.10 - 5 )
##EQU00012##
which yields the two logarithmic barrier functions
{hacek over (V)}.sub.{hacek over (.phi.)}.sub.i.sup.-=-log({hacek
over (.phi.)}.sub.i(q)) (3.10-6)
and
{hacek over (V)}.sub.{hacek over (.phi.)}.sub.i.sup.+=-log(1-{hacek
over (.phi.)}.sub.i(q)). (3.10-7)
[0429] According to the method of Lagrange multipliers, combine
this with the equality constraints to form the static Lagrangian
function
L ( q ; .PHI. Q , .PHI. ( k ) , .omega. 1 ( + ) , , .omega. n ( + )
, .omega. n + 1 ( - ) , , .omega. 2 n ( - ) , .omega. 2 n + 1 (
.lamda. ) , , .omega. 2 n + s ( .lamda. ) , .omega. 2 n + s + 1 ( Q
) , .omega. 2 n + s + 2 ( Q ) ) = .PHI. Q ( q ) + i = 1 n [ .omega.
i ( + ) V .PHI. i + + .omega. n + i ( - ) V .PHI. i - ] + l = 1 s
.omega. 2 n + 1 ( .lamda. ) .PHI. l ( q ) ( 1 - .PHI. l ( q ) ) -
.omega. 2 n + s + 1 ( Q ) log ( .PHI. Q ( q ) ) - .omega. 2 n + s +
2 ( Q ) log ( 1 - .PHI. Q ( q ) ) , ( 3.10 - 8 ) ##EQU00013##
the roots of which can be found using a formulation of
Newton-Raphson. Since here includes absolute, hard and soft rules
we may call it the total static Lagrangian for DE.sub.k and refer
to it as .sub.k.sup.(T).
[0430] Construct Equations of Motion
[0431] Information for equations of motion is available elsewhere
herein.
[0432] Query Response Engine (QRE) which Includes Process for
Constructing Differential Equations
[0433] Application of Newton-Raphson
[0434] Consider a continuous analog of the independent variables of
(.cndot.)
q = q ( t ) = [ q ( 1 ) ( t ) q ( v ) ( t ) ] ( 3.12 - 1 )
##EQU00014##
[0435] where each of the v total independent variables of (.cndot.)
is mapped to its corresponding position in q(t), the column vector
that is represented with a lower-case q. To reiterate, the
independent variable t refers algorithmic time as opposed to
physical time which may also be represented in the system. The
corresponding unconstrained optimization goal can be written as
min q 1 , , q v L ( q ( 1 ) ( t ) , , q ( v ) ( t ) ) ( 3.12 - 2 )
##EQU00015##
so that .gradient.L(q)
.gradient. L ( q ( t ) ) = [ .differential. L .differential. q ( 1
) .differential. L .differential. q ( v ) ] = [ .gradient. L 1
.gradient. L v ] = 0 , ( 3.12 - 3 ) ##EQU00016##
with positive definite Hessian matrix
.gradient. 2 L ( q ( t ) ) = [ .differential. L .differential. q (
1 ) .differential. q ( 1 ) .differential. L .differential. q ( 1 )
.differential. q ( v ) .differential. L .differential. q ( v )
.differential. q ( 1 ) .differential. L .differential. q ( v )
.differential. q ( v ) ] = [ .gradient. L 11 .gradient. L 1 v
.gradient. L v 1 .gradient. L vv ] = > 0. ( 3.12 - 4 )
##EQU00017##
[0436] Write the recursion for Newton's method
q.sub.(k+1)(t)=q.sub.(k)(t)-(.gradient..sup.2(q.sub.(k)(t))).sup.-1.grad-
ient.(q.sub.(k)(t)). (3.12-5)
[0437] This is equivalently rewritten
q ( k + 1 ) ( t ) - q ( k ) ( t ) .delta. = - 1 .delta. (
.gradient. 2 L ( q ( k ) ( t ) ) ) - 1 .gradient. L ( q ( k ) ( t )
) . ( 3.12 - 6 ) ##EQU00018##
[0438] Via continualization we approximate the derivative
q . ( t ) = q ( t ) t = - ( .gradient. 2 L ( q ( t ) ) ) - 1
.gradient. L ( q ( k ) ( t ) ) . ( 3.12 - 7 ) ##EQU00019##
[0439] Translation of Inverted Matrix
[0440] Consider M, an invertible and positive definite matrix. Then
we make the following provable assertions. [0441] 1. A.sup.TA is
symmetric. [0442] 2. -A.sup.TA has negative eigenvalues.
[0443] Define
M ( t ) t = - A T AM ( t ) + A T ( 3.12 - 8 ) ##EQU00020##
[0444] Then as t.fwdarw..infin.,
M(t).fwdarw.A.sup.-1=.gradient..sup.2(q.sub.(k)(t)).sup.-1. Using
(3.12-3) and (3.12-4) approximate {dot over (q)}(t) by rewriting
the derivative in the context of M(t). This yields the following
two equations.
q . ( t ) = - M ( t ) .gradient. L ( q ( t ) ) = [ m 11 m 1 v m v 1
m vv ] [ .gradient. L 1 .gradient. L v ] = [ m 11 .gradient. L 1 +
+ m 1 v .gradient. L v m v 1 .gradient. L 1 + + m vv .gradient. L v
] ( 3.12 - 9 ) M ( t ) t = - ( .gradient. 2 L ( q ( t ) ) ) T (
.gradient. 2 L ( q ( t ) ) ) M ( t ) + ( .gradient. 2 L ( q ( t ) )
) T = - [ .gradient. L 11 .gradient. L v 1 .gradient. L 1 v
.gradient. L vv ] [ .gradient. L 11 .gradient. L 1 v .gradient. L v
1 .gradient. L vv ] [ m 11 m 1 v m v 1 m vv ] [ .gradient. L 11
.gradient. L v 1 .gradient. L 1 v .gradient. L vv ] = - [
.gradient. L 11 2 + + .gradient. L v 1 2 .gradient. L 11 .gradient.
L 1 v + + .gradient. L v 1 .gradient. L vv .gradient. L 11
.gradient. L 1 v + + .gradient. L v 1 .gradient. L vv .gradient. L
1 v 2 + + .gradient. L vv 2 ] [ m 11 m 1 v m v 1 m vv ] + [
.gradient. L 11 .gradient. L v 1 .gradient. L 1 v .gradient. L vv ]
= - [ ( .gradient. L 11 2 + + .gradient. L v 1 2 ) m 11 + + (
.gradient. L 11 .gradient. L 1 v + + .gradient. L v 1 .gradient. L
vv ) m v 1 + .gradient. L 11 ( .gradient. L 11 2 + + .gradient. L v
1 2 ) m 1 v + + ( .gradient. L 11 .gradient. L 1 v + + .gradient. L
v 1 .gradient. L vv ) m vv + .gradient. L v 1 ( .gradient. L 11
.gradient. L 1 v + + .gradient. L v 1 .gradient. L vv ) m 11 + + (
.gradient. L 1 v 2 + + .gradient. L vv 2 ) m v 1 + .gradient. L 1 v
( .gradient. L 11 .gradient. L 1 v + + .gradient. L v 1 .gradient.
L vv ) m 1 v + + ( .gradient. L 1 v 2 + + .gradient. L vv 2 ) m vv
+ .gradient. L vv ] = [ .gradient. m 11 .gradient. m 1 v .gradient.
m v 1 .gradient. m vv ] ( 3.12 - 10 ) ##EQU00021##
[0445] The approximation proceeds as follows, applying the Magnus
expansion to compute the integral: [0446] 1. Fix
M(0)=.gradient..sup.2(q(t)) and =.gradient..sup.2(q(t)). [0447] 2.
Use the variation of constants formula to solve
[0447]
M(T)=e.sup.-[.gradient..sup.2.sup.(q(T))].sup.2.sup.tM(0)+[.intg.-
.sub.0.sup.Te.sup.-[.gradient..sup.2.sup.(q(.tau.))].sup.2.sup.(T-.tau.)d.-
tau.].gradient..sup.2(q(T))
[0448] The following is computation flow, flowing down unless
otherwise indicated.
[0449] Process for Determining Dynamic Lagrangian Via Hemholtz
Equations
[0450] Given
G i ( q , q . , q ) = j = 1 n W i , j ( q . , q ) ( q . , q ) q ( j
) + K i ( q . , q ) = 0 j = 1 , , n ( 3.12 - 11 ) ##EQU00022##
[0451] If the three conditions
.differential. G i .differential. q ( j ) = .differential. G j
.differential. q ( i ) , .differential. G i .differential. q . ( j
) + .differential. G j .differential. q . ( i ) = t (
.differential. G i .differential. q ( j ) + .differential. G j
.differential. q ( i ) ) , .differential. G i .differential. q ( j
) - .differential. G j .differential. q ( i ) = 1 2 t (
.differential. G i .differential. q . ( j ) - .differential. G j
.differential. q . ( i ) ) , ( 3.12 - 12 ) ##EQU00023##
with i, j=1, . . . , n hold, then
j = 1 n .differential. 2 L .differential. q . ( ) .differential. q
. ( j ) q ( j ) + .differential. 2 L .differential. q ( j )
.differential. q . ( ) - .differential. L .differential. q ( ) = G
i , i = 1 , , n ( 3.12 - 13 ) ##EQU00024##
[0452] This is a second order, linear hyperbolic differential
equation on the Lagrangian L. It can be solved efficiently by the
method of characteristics.
[0453] Let
G ( q , q . , q ) = [ q ( t ) q . ( t ) M . ( t ) ] = [ q ( 1 ) q (
v ) m 11 .gradient. L 1 + + m 1 v .gradient. L v m v 1 .gradient. L
1 + + m vv .gradient. L v .gradient. m 11 .gradient. m v 1
.gradient. m 1 v .gradient. m vv ] ( 3.12 - 14 ) ##EQU00025##
[0454] Process for Determining Hessian Rank of Dynamic
Lagrangian
[0455] Information for determining Hessian rank of dynamic
Lagrangian is available elsewhere herein.
[0456] Converting the Lagrangian to the Hamiltonian Via the
Legendre transformation.
[0457] In our formulation the Lagrangian, L.sub.k.sup.(T)(q, {dot
over (q)}; .omega.), may be converted to the Hamiltonian using the
Legendre transformation, so that
H k ( T ) ( q , p ; .omega. ) = .differential. L k ( T )
.differential. q . q . - L k ( T ) ( q , q . ; .omega. ) = p T q .
- L k ( T ) ( q , q . ; .omega. ) ( 3.12 - 15 ) ##EQU00026##
[0458] Pareto Multi-Criteria Optimization Engine (PMOE)
[0459] Consider the problem of determining the relaxed Pareto
optimal solution to a given system query at a given time step.
There are N decision elements, k=1, . . . , N. A given decision
element, DE.sub.k, has the following associated parameters which
are constituent to the ARB: [0460] A generalized set of coordinates
relevant to DE.sub.k, q. [0461] A generalized set of linearly
independent momenta {p.sub.a} where the index a refers the linearly
independent momenta selected from the canonical set p. [0462] A set
of control parameters co for hard a soft rules of the system, where
0.ltoreq..omega..sub.i.ltoreq.1.
[0463] The ARB has the following components which determine the
constraints of DE.sub.k: [0464] The Hamiltonian which identifies
the fundamental dynamics of the system of the system for the k'th
decision element denoted
[0464] H.sub.k.sup.(o)(q,{p.sub.a}) (3.13-1) [0465] The summation
of the first class constraints of the system, which is
[0465] i .omega. i f i ( q ( ) , .omega. i ) ( 3.13 - 2 )
##EQU00027## [0466] The summation of the second class constraints
of the system which is
[0466] i g i ( q ( ) , .omega. i ) ( 3.13 - 3 ) ##EQU00028## [0467]
The Tellegen agent which is a function of the Hamiltonians of the
absolute rules of the other N-1 decision elements in the system
[0467] H.sub.k.sup.(A)=F.sub.k.sup.(A)(H.sub.1.sup.(T), . . .
,H.sub.k-1.sup.(T),H.sub.k+1.sup.(T), . . . ,H.sub.K.sup.(T))
(3.13-4) [0468] The total Hamiltonian of the system is denoted
H.sup.(T). [0469] Approximations to the various Hamiltonian's are
denoted H.sub.k.sup.(A), H.sup.(T) and H.sub.k.sup.(o) for the
Tellegen, total, and DE-level Hamiltonians respectively.
[0470] System Initialization
[0471] Determining the relaxed Pareto optimal point of the system
is a process which includes: [0472] Initialization of N decision
elements. [0473] Synchronization through companionship of each of
the N decision elements with its respective Tellegen agent.
[0474] System Operation
[0475] Illustration SS shows how decision elements interact with
the network, receive queries, and return results. In this example,
the distributed system effectively implements an abstract
classifier that has no real implementation. The DE's receive sensor
data from the network which includes new available information
which may benefit classification. The user submits a query that is
received by a DE which then returns a result.
[0476] Gauge Systems in a Hamiltonian Domain
[0477] The time integral of the Lagrangian L(q, {dot over (q)}) is
the action S.sub.L defined as
S.sub.L=.intg..sub.t.sub.1.sup.t.sup.2L(q,{dot over (q)})dt
where
q . = q ( t ) t . ##EQU00029##
The Lagrangian conditions for stationarity are first that
t L q . ( n ) - L q ( n ) = 0 ( 3.14 - 1 ) ##EQU00030##
where n=1, . . . , N,
L q . ( n ) = .differential. L .differential. q . ( n ) , and
##EQU00031## L q ( n ) = .differential. L .differential. q ( n ) .
##EQU00031.2##
And, secondarily
[ n ' = 1 N q ( n ' ) ] L q . ( n ) q . ( n ) = L q ( n ) - q . ( n
) L q . ( n ) q ( n ) ( 3.14 - 2 ) ##EQU00032##
where
q ( n ' ) = 2 q ( n ' ) t 2 ##EQU00033## and ##EQU00033.2## L q . (
n ) q ( n ) = .differential. 2 L .differential. ( q . ( n ) ) 2 .
##EQU00033.3##
The generalized accelerations {umlaut over (q)}.sup.(n) are
immediately determined if L.sub.{dot over (q)}(n).sub.{dot over
(q)}(n) is invertible, or equivalently
det(L.sub.{dot over (q)}(n){dot over (q)}(n)).noteq.0 (3.14-3)
for i=1, . . . , N. If for some n, det(L.sub.{dot over
(q)}(n).sub.{dot over (q)}(n))=0, the acceleration vector {umlaut
over (q)}.sup.(n) will not be uniquely determined.
[0478] The departing point for the Hamiltonian approach is the
definition of conjugate momentum
p.sub.n=L.sub.{dot over (q)}(n) (3.14-4)
where n=1, . . . , N. We will see that (3.14-3) is the condition of
non-invertibility of
L q . q . = [ L q . ( 1 ) q . ( 1 ) L q . ( n ) q . ( N ) L q . ( N
) q . ( 1 ) L q . ( N ) q . ( N ) ] ##EQU00034##
of the velocities of the functions of the coordinates q and momenta
p. In other words, in this case, the momenta defined in (3.14-4)
are not all independent. Define the relations that follow from
(3.14-4) as
.phi..sub.m(q,p) (3.14-5)
where m=1, . . . , M. Write (3.14-4) in vector notation as
p=L.sub.{dot over (q)}(q,{dot over (q)}).
[0479] Then compatibility demands
.phi..sub.m(q,L.sub.{dot over (q)}(q,{dot over (q)}))=0
is an identity with m=1, . . . , M.
[0480] Relations specified in (3.14-5) are called primary
constraints. For simplicity let's assume that rank(L.sub.{dot over
(q)}{dot over (q)}) is constant throughout the phase space, (q,
{dot over (q)}), so that (3.14-5) defines a submanifold smoothly
embedded in the phase space. This manifold is known as the primary
constraint surface. Let
rank(L.sub.{dot over (q)}{dot over (q)})=N-M' (3.14-6)
[0481] Then there are M' independent constraints among (3.14-5) and
the primary constraint surface is a phase space submanifold of
dimension 2N-M'.
[0482] We do not assume that all the constraints are linearly
independent so that
M'.ltoreq.M. (3.14-7)
[0483] It follows from (3.14-5) that the inverse transformation
from the p's to the q's is multivalued. That is, given q, p that
satisfies (3.14-5), the inverse image (q, that satisfies
p = ( .differential. L .differential. q . ) T ( 3.14 - 8 )
##EQU00035##
is not unique, since (3.14-8) defines a map from a 2N-dimensional
manifold (q, {dot over (q)}) to the smaller (2N-M')-dimensional
manifold. Thus the inverse image of the points of (3.14-5) form a
manifold of dimension M'.
[0484] Conditions on the Constraint Function
[0485] There exist many equivalent ways to represent a given
surface by means of equations of the form of (3.14-5). For example
the surface p.sub.1=0 can be represented equivalently by
p.sub.1.sup.2=0, {square root over (|p.sub.1|)}=0, or redundantly
by p.sub.1=0 and p.sub.1.sup.2=0. To use the Hamiltonian formalism,
it is necessary to impose some restrictions which the regularity
conditions for the constraints.
[0486] Regularity Conditions
[0487] The (2N-M')-dimensional constraint surface .phi..sub.m(q, p)
should be covered of open region: in each region the constraints
can be split into independent constraints
{.phi..sub.m'|m'=1, . . . ,M'}.
[0488] Their Jacobian matrix
{ .differential. .phi. m ' .differential. p n , q ( n ) } = [
.differential. .phi. 1 .differential. p 1 , q ( 1 ) .differential.
.phi. 1 .differential. p n , q ( n ) .differential. .phi. m '
.differential. p 1 , q ( 1 ) .differential. .phi. m '
.differential. p n , q ( n ) ] ##EQU00036##
with m'=1, . . . , M' and n=1, . . . , N, is of rank M'.
[0489] The dependent constraints .phi..sub.m, m=M'+1, . . . , M of
the other .phi..sub.m'=0.phi..sub.m=0. Alternatively the condition
on the Jacobian. [0490] 1. The function .phi..sub.m' can be taken
locally as the first M' coordinates of a new regular system in the
vicinity of the constraint surface or the differentials
d.phi..sub.1, . . . , d.phi..sub.M' are locally linearly
independent:
[0490] d.phi..sub.1 . . . d.phi..sub.M'.noteq.0 (3.14-9) [0491] 2.
The variations .delta..phi..sub.m' are of order .epsilon. for
arbitrary variations .delta.q.sup.(i), .delta.p.sub.i of order
.epsilon. (Dirac's approach).
[0492] Theorem 3.14.1. If a smooth, phase space function G vanishes
on {.phi..sub.m=0} then
G = m = 1 M g ( m ) .phi. m ( 3.14 - 10 ) ##EQU00037##
[0493] Proof: (local proof). Set .phi..sub.m', m'=1, . . . , M' as
coordinates (y.sub.m',x.sub..alpha.) with y.sub.m'=.phi..sub.m'. In
these coordinates G(0, x)=0 and
G ( y , x ) = .intg. 0 1 t G ( ty , x ) t m ' = 1 M ' y m ' .intg.
0 1 .differential. .differential. y m ' G ( ty , x ) t m ' = 1 M '
g ( m ' ) ( y , x ) .phi. m ' ( y , x ) ( 3.14 - 11 ) with g ( m '
) ( y , x ) = .intg. 0 1 .differential. .differential. y m ' G ( ty
, x ) t . ##EQU00038##
[0494] Theorem 3.14.2. If the sum
.SIGMA.(.lamda..sup.(n).delta.q.sup.(n)+.mu..sub.n.delta.p.sub.n)=0
for arbitrary variations .delta.q.sup.(i), .delta.p.sub.i tangent
to the constraint surface {.phi..sub.m(q, p)=0|m=1, . . . , M},
then
.lamda. ( n ) = m = 1 M u ( m ) .differential. .phi. m
.differential. q ( n ) ( 3.14 - 12 ) .lamda. n = m = 1 M u ( m )
.differential. .phi. m .differential. q n ( 3.14 - 13 )
##EQU00039##
[0495] Proof. The dimension of {.phi..sub.m} is 2N-M'. Thus the
variations at a point (p, q) forms a 2N-M' dimensional space
n = 1 N ( .lamda. ( n ) .delta. q ( n ) + .mu. n .delta. p n ) = 0
( 3.14 - 14 ) ##EQU00040##
[0496] By the singularity assumption, there exists exactly M'
solutions to (3.14-14). Clearly, the gradients
{ .differential. .phi. m ' .differential. q ( n ) } and {
.differential. .phi. m ' .differential. p n } ##EQU00041##
are linearly independent. They are the basis for solutions to
(3.14-14).
[0497] Note that in the presence of redundant constraints, the
functions u.sup.(m) exist but are not unique.
[0498] Canonical Hamiltonian
[0499] The Hamiltonian in canonical coordinates is
H ( q , p ) = n = 1 N q . ( n ) p n - L ( q , q . ) ( 3.14 - 15 )
##EQU00042##
[0500] The rate {dot over (q)} enters through the combination
through conjugate momenta defined for each coordinate
p.sub.n(q,{dot over (q)})=L.sub.{dot over (q)}.sub.(n)(q,{dot over
(q)}) (3.14-16)
[0501] This remarkable property is essential for the Hamiltonian
approach. It is verified by evaluating the change 6H involved by
arbitrary independent variations of position and velocities.
.delta. H = n = 1 N ( q . ( n ) .delta. p n + .delta. q . ( n ) p n
) - .delta. L = n = 1 N ( q . ( n ) .delta. p n + .delta. q . ( n )
p n ) - n = 1 N ( L q ( n ) .delta. q ( n ) + L q . ( n ) .delta. q
. ( n ) ) ( 3.14 - 17 ) ##EQU00043##
[0502] Utilizing (3.14-16) in (3.14-17) yields
.delta. H = n = 1 N ( q . ( n ) .delta. p n - L q ( n ) .delta. q (
n ) ) ( 3.14 - 18 ) ##EQU00044##
[0503] The Hamiltonian defined by (3.14-15) is not unique as a
function of p, q. This can be inferred from (3.14-18) by noticing
that {.delta.p.sub.n|n=1, . . . , N} are not all independent. They
are restricted to preserve the primary constraints
.phi..sub.m.apprxeq.0 which are identities when the p's are
expressed as functions of q's via (3.14-16).
[0504] Using the definition of the differential in several
variables applied to .delta.H=.delta.H({q.sup.(n)}, {p.sub.n}),
(3.14-18) can be rewritten
n = 1 N ( .differential. H .differential. q ( n ) .delta. q ( n ) +
.differential. H .differential. p n .delta. p n ) = n = 1 N ( q . (
n ) .delta. p n - .delta. q ( n ) .differential. L .differential. q
( n ) ) or n = 1 N ( .differential. H .differential. q ( n ) +
.differential. H .differential. q ( n ) ) .delta. q ( n ) + n = 1 N
( .differential. H .differential. p n - q . ( n ) ) .delta. p n = 0
( 3.14 - 19 ) ##EQU00045##
[0505] From theorem 2 we then conclude for each n that.
.differential. H .differential. q ( n ) + .differential. L
.differential. q ( n ) = m = 1 M u ( m ) .differential. .phi. m
.differential. q ( n ) and .differential. H .differential. p n - q
. ( n ) = m = 1 M u ( m ) .differential. .phi. m .differential. p n
. ( 3.14 - 20 ) ##EQU00046##
[0506] So for each n:
q . ( n ) = .differential. H .differential. p n + m = 1 M u ( m )
.differential. .phi. m .differential. p n , n = 1 , , N and ( 3.14
- 21 ) - .differential. L .differential. q ( n ) = .differential. H
.differential. q ( n ) + m = 1 M u ( m ) .differential. .phi. m
.differential. q ( n ) , n = 1 , , N . ( 3.14 - 22 )
##EQU00047##
[0507] Note that if the constraints are independent, the
vectors
n = 1 N .differential. .phi. m .differential. p n ,
##EQU00048##
m=1, . . . , M are also independent because of the regularity
conditions (this is proved later). Hence no two sets of
{u.sup.(m)|m=1, . . . , M} can yield the same velocities via
(3.14-21).
[0508] Thus, using
q . ( n ) = .differential. H .differential. p n + m = 1 M u ( m ) (
q , q . ) .differential. .phi. m .differential. p n ( q , p ( q , q
. ) ) ##EQU00049##
we can find u.sup.(m)(p,{dot over (q)}). If we define the
transformation from (q,{dot over (q)}) to the manifold
{.phi..sub.m(q, p)=0|m=1, . . . , M}, from q, {dot over (q)},
u.fwdarw.q, p, u by
q=q, n=1, . . . ,N
p.sub.n=L.sub.q.sub.(n)(q,{dot over (q)}), n=1, . . . ,N-M'
u.sup.(m)=u.sup.(m)(q,{dot over (q)}), m=1, . . . ,M'
[0509] We see that this transformation is invertible since one has
from q, p, u.fwdarw.q, {dot over (q)}, u
q = q ##EQU00050## q . ( n ) = .differential. H .differential. p n
+ m = 1 M u ( m ) .differential. .phi. m .differential. p n
##EQU00050.2## .phi. n ( q , p ) = 0 ##EQU00050.3##
[0510] Thus invertibility of the Legendre transformation when
det(L.sub.{dot over (q)}{dot over (q)})=0
can be regained at the prices of adding extra variables.
Action Principle of the Hamiltonian Form
[0511] With (3.14-21) and (3.14-22) we can rewrite (3.14-1) in the
equivalent Hamiltonian form
q . ( n ) = .differential. H .differential. p n + m = 1 M u ( m )
.differential. .phi. m .differential. p n p . n = - .differential.
H .differential. p n - m = 1 M u ( m ) .differential. .phi. m
.differential. p n .phi. m ( q , p ) = 0 , m = 1 , , M ' ( 3.14 -
23 ) ##EQU00051##
[0512] The Hamiltonian Equations (3.14-23) can be derived from the
following variational principle:
.delta. .intg. t 1 t 2 [ n = 1 N q ( n ) p n - H - m = 1 M u ( m )
.phi. m ] = 0 ( 3.14 - 24 ) ##EQU00052##
for arbitrary variations of .delta.q.sup.(n), .delta.p.sub.n, and
.delta.u.sup.(m) subject to
.delta.q(t.sub.1)=.delta.q(t.sub.2)=0
where the u.sup.(m) appear now as Lagrange multipliers enforcing
the primary constraints
.phi..sub.m(q,p)=0, m=1, . . . ,M.
[0513] Let F(p, q) be an arbitrary function of the canonical
variables, then
F t = n = 1 N .differential. F .differential. q ( n ) q . n + n = 1
N .differential. F .differential. p n p . n = n = 1 N
.differential. F .differential. q ( n ) [ .differential. H
.differential. p n + m = 1 M u ( m ) .differential. .phi. m
.differential. p n ] + n = 1 N .differential. F .differential. p n
[ - .differential. H .differential. p n - m = 1 M u ( m )
.differential. .phi. m .differential. p n ] = [ F , H ] + m = 1 M u
( m ) [ F , .phi. m ] ( 3.14 - 25 ) ##EQU00053##
[0514] The equation (3.14-25) introduces the new binary operator
[.cndot.,.cndot.] which is the Poisson bracket and has the form
[ F , G ] = n = 1 N [ .differential. F .differential. q ( n )
.differential. G .differential. p n + .differential. F
.differential. p n .differential. G .differential. q ( n ) ] = n =
1 N [ F q ( n ) G p n + F p n G q ( n ) ] ( 3.14 - 26 )
##EQU00054##
[0515] Secondary Constraints
[0516] The basic consistency condition is that the primary
constraints be preserved in time. So for
F(p,q)=.phi..sub.m(q,p)
we should have that {dot over (.phi.)}.sub.m=0. {.phi..sub.m(q,
p)=0}. So this means
[ .phi. m , H ] + m ' = 1 M ' u ( m ' ) [ .phi. m , .phi. m ' ] = 0
( 3.14 - 27 ) ##EQU00055##
[0517] This equation can either reduce to a relation independent of
the u.sup.(m'), or, it may impose a restriction on the u's.
U=-{[.phi..sub.m,.phi..sub.m']}[.phi..sub.m,H](q,p) (3.14-28)
[0518] In the case (3.14-27) is independent of the u's (3.14-27) is
called a secondary constraint. The fundamental difference of
secondary constraints with respect to primary constraints is that
primary constraints is that primary constraints are the consequence
of the definition (3.14-8) while secondary constraints depend on
the dynamics.
[0519] If X(q, p)=0 is an external constraint, we most impose a
compatibility condition
[ X , H ] + m = 1 M ' u ( m ) [ X , .phi. m ] = 0 ( 3.14 - 29 )
##EQU00056##
[0520] Next we need to test whether this constraint:
.PHI. ( p , q ) = [ X , H ] + m = 1 M ' u ( m ) [ X , .phi. m ] = 0
( 3.14 - 30 ) ? ? indicates text missing or illegible when filed (
3.14 - 31 ) ##EQU00057##
[0521] Implies new secondary constraints or whether it only
restricts the u's. After the process is finished we are left with a
number of secondary constraints which will be denoted by
.phi..sub.k=0, k=M+1, . . . ,M+K
where K is the total number of secondary constraints. In general,
it will be useful to denote all the constraints (primary and
secondary) in a uniform way as
.phi..sub.j(q,p)=0, j=1, . . . ,M+K=J (3.14-32)
[0522] We make the same regularity assumptions on the full set of
constraints.
[0523] Weak and Strong Equations
[0524] Equation (3.14-32) can be written as
.phi..sub.j(.cndot.).apprxeq.0 (3.14-33)
[0525] To emphasize, the quantity .phi..sub.j is numerically
restricted to be zero but does not vanish throughout the space.
What this is means is that .phi..sub.j has non-zero Poisson
brackets with the canonical variables.
[0526] Let F, G be functions that coincide on the manifold
{.phi..sub.j.apprxeq.0|j=1, . . . , J} are said the be weakly equal
and denoted by F.apprxeq.G. On the other hand, an equation that
holds throughout the entire phase space and not just on the
submanifold {.phi..sub.j.apprxeq.0} is called strong. Hence, by
theorem 1
F .apprxeq. G .revreaction. F - G = j = 1 J c ( j ) ( p , q ) .phi.
j ( 3.14 - 34 ) ##EQU00058##
[0527] Restrictions on the Lagrange Multipliers
[0528] Assume that we have found a complete set of constraints
{ .phi. j .apprxeq. 0 | j = 1 , , J } ( 3.14 - 35 ) [ .phi. j , H ]
+ m = 1 M u ( m ) [ .phi. j , .phi. m ] .apprxeq. 0 ( 3.14 - 36 )
##EQU00059##
[0529] We consider (3.14-36) as a set of non-homogeneous linear
equations with M.ltoreq.J unknowns with coefficients that are
functions of the q's and p's.
[0530] The general solution of (3.14-36) for each j is of the
form
u.sup.(m)=U.sup.(m)+V.sup.(m), m=1, . . . ,M (3.14-37)
with V.sup.(m) the solution of the homogeneous equation
m = 1 M V ( m ) [ .phi. j , .phi. m ] .apprxeq. 0 ( 3.14 - 38 )
##EQU00060##
[0531] The most general solution of (3.14-38) is a linear
combination of linearly independent solutions of
V.sub..alpha..sup.(m) where .alpha.=1, . . . , A with A.ltoreq.M.
Under the assumption that the matrix
[ [ .phi. 1 , .phi. 1 ] [ .phi. 1 , .phi. M ] [ .phi. J , .phi. 1 ]
[ .phi. J , .phi. M ] ] ( 3.14 - 39 ) ##EQU00061##
is of constant rank, the number of independent solutions A is the
same for all p, q. Thus the general solution to (3.14-36) can be
written as
u ( m ) .apprxeq. U ( m ) + .alpha. = 1 A v ( .alpha. ) V .alpha. (
m ) , m = 1 , , M ( 3.14 - 40 ) ##EQU00062##
[0532] Irreducible and Reducible Cases
[0533] If the equations {.phi..sub.j=0|j=1, . . . , J} are not
independent, one says that the constraints are reducible. The
system is irreducible when the constraints are independent. However
the separation of constraints into dependent and independent ones
might be difficult to perform. It also may disturb invariance
properties under some important symmetry. In some cases it may be
impossible to separate irreducible from irreducible contexts.
Reducible cases arise for example when the dynamical coordinates
include p-form gauge fields.
[0534] Any irreducible set of constraints can always be replaced by
a reducible set by introducing constraints of the ones already at
hand. The formalism should be invariant under such
replacements.
[0535] Total Hamiltonian
[0536] We now discuss details of the dynamic equation (3.14-25)
F . .apprxeq. [ F , H ' + .alpha. = 1 A v ( .alpha. ) .phi. .alpha.
] ( 3.14 - 41 ) ##EQU00063##
where from (3.14-40)
H ' = H + m = 1 M U ( m ) .phi. m and .phi. .alpha. = m = 1 M V
.alpha. ( m ) .phi. m , .alpha. = 1 , , A ( 3.14 - 42 )
##EQU00064##
[0537] This is the result of theorem 3 (see below).
[0538] Theorem 3.
[ F , m = 1 M U ( m ) .phi. m ] m = 1 M U ( m ) [ F , .phi. m ] (
3.14 - 43 ) [ F , .alpha. = 1 A V .alpha. ( m ) .phi. m ] .alpha. =
1 A V .alpha. ( m ) [ F , .phi. m ] ( 3.14 - 44 ) [ F , m = 1 M U (
m ) .phi. m ] = i = 1 N { .differential. F .differential. q ( i )
.differential. .differential. p i m = 1 M U ( m ) .phi. m -
.differential. F .differential. p i .differential. .differential. q
( i ) m = 1 M U ( m ) .phi. m } = i = 1 N { .differential. F
.differential. q ( i ) [ m = 1 M .differential. U ( m )
.differential. p i .phi. m + m = 1 M U ( m ) .differential. .phi. m
.differential. p i ] } - i = 1 N { .differential. F .differential.
p i [ m = 1 M .differential. U ( m ) .differential. q ( i ) .phi. m
+ m = 1 M U ( m ) .differential. .phi. m .differential. q ( i ) ] }
= m = 1 M { [ F , U ( m ) ] .phi. m + U ( m ) [ F , .phi. m ] } So
[ F , m = 1 M U ( m ) .phi. m ] - m = 1 M U ( m ) [ F , .phi. m ] =
m = 1 M [ F , U ( m ) ] .phi. m ( 3.14 - 45 ) ##EQU00065##
Proof.
[0539] and from (3.14-34) in (3.14-45), (3.14-43) follows. By a
similar process we show (3.14-44). We now prove the validity of
(3.14-41).
[0540] Theorem 4. Let F(q,p) be a regular function, then F(p,q)
propagates in time according to the approximate equation
(3.14-41).
[0541] Proof. From (3.14-25),
F t = [ F , H ] + m = 1 M u ( m ) [ F , .phi. m ] . ( 3.14 - 46 )
##EQU00066##
[0542] From (3.14-40) into (3.14-46) we obtain,
F t .apprxeq. [ F , H ] + m = 1 M { U ( m ) + .alpha. = 1 A v (
.alpha. ) V .alpha. ( m ) } [ F , .phi. m ] or F t .apprxeq. [ F ,
H ] + m = 1 M U ( m ) [ F , .phi. m ] + m = 1 M .alpha. = 1 A v (
.alpha. ) V .alpha. ( m ) [ F , .phi. m ] ( 3.14 - 47 )
##EQU00067##
[0543] Thus from (3.14-43) and (3.14-44) of theorem 3, we get
F t .apprxeq. [ F , H ] + m = 1 M [ F , U ( m ) .phi. m ] + .alpha.
= 1 A v ( .alpha. ) [ F , m = 1 M V .alpha. ( m ) .phi. m ]
.apprxeq. [ F , H + m = 1 M U ( m ) .phi. m + .alpha. = 1 A v (
.alpha. ) m = 1 M V .alpha. ( m ) .phi. m ] .apprxeq. [ F , H ' + m
= 1 M U ( m ) .phi. m + .alpha. = 1 A v ( .alpha. ) .phi. .alpha. ]
( 3.14 - 48 ) with H ' = H + m = 1 M U ( m ) .phi. m ( 3.14 - 49 )
.phi. .alpha. = m = 1 M V .alpha. ( m ) .phi. m ( 3.14 - 50 )
##EQU00068##
[0544] Now define
H T = H ' + .alpha. = 1 A V ( .alpha. ) .phi. .alpha. . ( 3.14 - 51
) ##EQU00069##
[0545] So we obtain
F t .apprxeq. [ F , H T ] ( 3.14 - 52 ) ##EQU00070##
[0546] First and Second Class Functions
[0547] The distinction between primary and secondary constraints is
of little importance.
[0548] We now consider a fundamental classification. It depends on
the concept of first class and second class functions.
[0549] Definition 1. A function F(q, p) is said to be first class
if its Poisson bracket with every constraint vanishes weakly,
[F,.phi..sub.j].apprxeq.0, j=1, . . . , J. A function of the
canonical variables that is not first class is called second class.
Thus F is second class if [F, .phi..sub.k].apprxeq.0 for at least
one k, k=1, . . . , M.
[0550] Theorem 5. If F and G are first class functions, then their
Poisson bracket is also a first class function.
[0551] Proof: By Hypothesis,
[ F , .phi. j ] = k = 1 M f j ( k ) .phi. k ( 3.14 - 53 ) [ G ,
.phi. j ] = l = 1 M g j ( l ) .phi. l ( 3.14 - 54 )
##EQU00071##
[0552] Applying the Jacobi identity, we get
[ [ F , G ] , .phi. j ] = [ F , [ G , .phi. j ] ] - [ G , [ F ,
.phi. j ] ] = [ F , l = 1 M g j ( l ) .phi. l ] - [ G , k = 1 M f j
( k ) .phi. k ] = i { .differential. F .differential. q ( i )
.differential. .differential. p i l = 1 M g j ( l ) .phi. l -
.differential. F .differential. p i .differential. .differential. q
( i ) l = 1 M g j ( l ) .phi. l } - n { .differential. G
.differential. q ( n ) .differential. .differential. p n k = 1 M f
j ( k ) .phi. k - .differential. G .differential. p n
.differential. .differential. q ( n ) k = 1 M f j ( k ) .phi. k } =
i { .differential. F .differential. q ( i ) l = 1 M {
.differential. g j ( l ) .differential. p i .phi. l + g j ( l )
.differential. .phi. l .differential. p i } - .differential. F
.differential. p i l = 1 M { .differential. g j ( l )
.differential. q ( i ) .phi. l + g j ( l ) .differential. .phi. l
.differential. q ( i ) } } - n { .differential. G .differential. q
( n ) k = 1 M { .differential. f j ( k ) .differential. p n .phi. k
+ f j ( k ) .differential. .phi. k .differential. p n } -
.differential. G .differential. p n k = 1 M { .differential. f j (
k ) .differential. q ( n ) .phi. k + f j ( k ) .differential. .phi.
k .differential. q ( n ) } } = l = 1 M { .phi. l i { .differential.
F .differential. q ( i ) .differential. g j ( l ) .differential. p
i - .differential. F .differential. p i .differential. g j ( l )
.differential. q ( i ) } + g j ( l ) i { .differential. F
.differential. q ( i ) .differential. .phi. l .differential. q ( i
) - .differential. F .differential. p i .differential. .phi. l
.differential. p i } } - k = 1 M { .phi. k n { .differential. G
.differential. q ( n ) .differential. f j ( k ) .differential. p n
- .differential. G .differential. p n .differential. f j ( k )
.differential. q ( n ) } + f j ( k ) n { .differential. G
.differential. q ( n ) .differential. .phi. k .differential. p n -
.differential. G .differential. p n .differential. .phi. k
.differential. q ( n ) } } = l = 1 M { .phi. l [ F , g j ( l ) ] +
g j ( l ) [ F , .phi. l ] } - k = 1 M { .phi. k [ G , f j ( k ) ] +
f j ( k ) [ G , .phi. k ] } = l = 1 M [ F , g j ( l ) ] .phi. l - k
= 1 M [ G , f j ( k ) ] .phi. k + l ' = 1 M { l = 1 M g j ( l ) f l
( l ' ) } .phi. l ' - k ' = 1 M { k = 1 M f j ( k ) g k ' k } .phi.
k ' .apprxeq. 0 ##EQU00072##
[0553] We now use theorem 5 to show the following.
[0554] Theorem 6. H' defined by (3.14-49) and .phi..sub..alpha.
defined by (3.14-50) are first class functions.
[0555] Proof: This follows directly from (3.14-36) and
(3.14-38).
[0556] We learn from theorem 6 that the total Hamiltonian defined
by (3.14-51) is the sum of the first class Hamiltonian H' and the
first class primary constraints .phi..sub..alpha. multiplied by
arbitrary coefficients.
[0557] First Class Constraints as Generators of Gauge
Transformations
[0558] Gauge transformations are transformations that do not change
the physical state.
[0559] The presence of arbitrary functions of time v.sup.(.alpha.),
.alpha.=1, . . . , A in the total Hamiltonian, H.sub.T (see
(3.14-51)) imply that not all the q's and p's are observable given
a set of q's and p's where the state of the physical system is
uniquely determined. However the converse is not true: there is
more than one set of values of the canonical variables that defines
a state. To illustrate this, we see that if we give an initial set
of values of physical state at time t, we expect the equations of
motion to fully determine the state at other times. Thus any
ambiguity in the value of the canonical variables at
t.sub.2.noteq.t.sub.1 should be irrelevant from the physical point
of view.
A Derivation Example
[0560] We propose here an alternate formulation of Dirac's
formalism.
[0561] Primary Constraints
[0562] Recall that the momenta, canonically conjugate to the
generalized "coordinates" q.sup.(j), j=1, . . . , N is given by
p j = .differential. L ( q , q . ) .differential. q . ( j ) , j = 1
, , N . ( E -- 1 ) ##EQU00073##
[0563] For non-singular systems the equations allows us to express
{dot over (q)}.sup.(j), j=1, . . . , N in terms of the canonical
variables,
{dot over (q)}.sup.(i)=f.sub.i(q,p), i=1, . . . ,N (E--2)
[0564] By performing a Legendre transformation
H c ( p , q ) = i = 1 N p i f ( q , p ) + L ( q , f ( p , q ) )
##EQU00074##
[0565] We obtain the Hamiltonian of the system H.sub.c. And from
this function we obtain the standard equations of motion of the
system.
q . = .differential. H c .differential. p p . = - .differential. H
c .differential. q ( E -- 3 ) ##EQU00075##
[0566] For (E--2) to be well-defined we need to have the Hessian W
of satisfy
det W.noteq.0 (E--4)
[0567] In this case the accelerations {umlaut over (q)}.sup.(i) are
uniquely determined by the q.sup.(j) and {dot over
(q)}.sup.(i).
[0568] When det W.noteq.0, the Hamiltonian equations of motion do
not take the standard form, and we speak of a singular Lagrangian.
For illustration purposes, consider a Lagrangian of the form
L ( q , q . ) = 1 2 i = 1 N j = 1 N W ij ( q ) q . ( i ) q . ( j )
+ i = 1 N .eta. i ( q ) q . ( i ) - V ( q ) ( E -- 5 )
##EQU00076##
with W a symmetric matrix. From (E--1), the canonical momentum for
(E--5) is given by
p i = 1 2 j = 1 N W ij ( q ) q . ( j ) + .eta. i ( q ) , i = 1 , ,
n . ( E -- 6 ) ##EQU00077##
[0569] If W is singular of rank R.sub.W, then it possesses
N--R.sub.W eigenvectors with corresponding zero eigenvalues. Then
for eigenvectors v.sub.j.sup.(.alpha.)
j = 1 N W ij ( q ) v j ( .alpha. ) ( q ) = 0 , .alpha. = 1 , , N -
R W ##EQU00078##
[0570] So pre-multiplying (E--6) by v.sub.j.sup.(.alpha.) and
summing over i we get
i = 1 N v i ( .alpha. ) ( q ) p i = i = 1 N [ j = 1 N ( v i (
.alpha. ) ( q ) W ij ( q ) q . ( j ) ) + v i ( .alpha. ) ( q )
.eta. i ( q ) ] = i = 1 N .nu. i ( .alpha. ) ( q ) .eta. i ( q ) ,
.alpha. = 1 , , N - R W So i = 1 N v i ( .alpha. ) ( q ) ( p i -
.eta. i ( q ) ) = 0 , .alpha. = 1 , , N - R W . ( E -- 7 )
##EQU00079##
[0571] Let {p.sub..alpha.}, .alpha.=1, . . . , N--R.sub.W, denote
the linearly dependent elements of p. Let {p.sub..alpha.}, a=1, . .
. , R.sub.a be the momenta satisfying (E--1). Then the constraint
equations are of the form
.beta. = 1 N - R W N .alpha..beta. ( q ) p .beta. - F .alpha. ( q ,
{ p a } ) = 0 , .alpha. = 1 , , N - R W M .alpha..beta. ( q ) = v
.beta. ( .alpha. ) and ( E -- 8 ) F .alpha. ( q , { p .beta. } ) =
i = 1 N v i ( .alpha. ) ( q ) .eta. i ( q ) + b = 1 R W v b (
.alpha. ) ( q ) p b ( E -- 9 ) ##EQU00080##
[0572] The matrix {M.sub..alpha..beta.} is necessarily invertible
because otherwise M would possess eigenvectors with zero
eigenvalues, implying existence of additional constraints.
[0573] Note that (E--8) can be written as
p .alpha. - g .alpha. ( q , { p a } ) = 0 , .alpha. = 1 , , N - R W
with g .alpha. ( q , { p a } ) = .beta. = 1 N - R W M .alpha..beta.
- 1 F .beta. ( q , { p a } ) ( E -- 10 ) ##EQU00081##
with dim{p.sub.a}=R.sub.W. So we can define,
.phi..sub..alpha.(q,p)=p.sub..alpha.-g.sub..alpha.(q,{p.sub.a})=0,
.alpha.=1, . . . ,N--R.sub.W (E--11)
[0574] In Dirac's terminology, constraints of the form of (E--11)
are referred to as primary constraints. Although the derivation
above is based on a Lagrangian, quadratic in the velocity terms, it
is generally valid for Lagrangians which depend on q and {dot over
(q)} but not on higher derivatives.
[0575] Note: Primary constraints follow exclusively from the
definition of canonical momenta.
[0576] The derivation above is valid for general Lagrangians and
their Hessian. Let's assume {W.sub.ij(q, {dot over (a)})} is the
Hessian of a given Lagrangian L. Let {W.sub.ab|a, b=1, . . . ,
R.sub.W} be the largest sub-matrix of {W.sub.ij} with suitable
rearrangement if necessary. We then solve (E--1) for R.sub.W
velocities {dot over (q)}.sup.(a) in terms of {q.sup.(i)|i=1, . . .
, n}, {p.sub.a|a=1, . . . R.sub.W} and {q.sup.(.alpha.)|.alpha.=1,
. . . , N-R.sub.W}. That is
{dot over (q)}.sup.(a)=f.sub.a(q,{p.sub.b},{{dot over
(a)}.sup.(.beta.)}) (E--12)
with a, b=1, . . . , R.sub.W and .beta.=R.sub.W+1, . . . , N.
[0577] Inserting these relations into (E--1), we get relations of
the form
p.sub.j=h.sub.j(q,{p.sub.a},{{dot over (q)}.sup.(.alpha.)})
(E--13)
with a, j=1, . . . , R.sub.W and .alpha.=R.sub.W+1, . . . , N. This
relation reduces to an identity by construction. The remaining
equations are of the form
p.sub..alpha.=h.sub..alpha.(q,{p.sub.a},{{dot over
(q)}.sup.(.beta.)}) (E--14)
with .alpha.=1, . . . , N-R.sub.W. However, the right hand side
cannot depend on {{dot over (q)}.sup.(.beta.)} since otherwise we
could express more velocities in terms of the momenta of the
coordinates of the momenta and the remaining velocities.
[0578] Hamiltonian Equations of Motion for Constrained Systems
[0579] Theorem 3.16.1. In the space .GAMMA..sub.p define by
.GAMMA..sub.p={.phi..sub..alpha.(p, q)|.alpha.=1, . . . ,
N-R.sub.W} where .phi..sub..alpha. is defined as (E--11). The
Hamiltonian is only a function of {q.sup.(i)|i=1, . . . , N} and
momenta {p.sub.a|a=1, . . . , R.sub.W} and does not depend on {{dot
over (q)}.sup.(.alpha.)|.alpha.=1, . . . , N-R.sub.W}
[0580] Proof. On .GAMMA..sub.p the Hamiltonian is given by
H o = H c | .GAMMA. p = a = 1 R W p a f a - .alpha. = 1 N - R W g
.alpha. q . ( .alpha. ) - L ( q , { f b } , { q . ( .beta. ) } ) (
E -- 15 ) ##EQU00082##
where f.sub.a, a=1, . . . , N-R.sub.W is given by (E--12) and
g.sub..alpha., .alpha.=1, . . . , R.sub.W is given by (E--10). We
want to show that H.sub.o does not depend on {dot over
(q)}.sup.(.beta.), .beta.=1, . . . , N-R.sub.W. We compute
( E -- 16 ) ##EQU00083## .differential. H o .differential. q . (
.beta. ) = a = 1 R W p a .differential. f a .differential. q . (
.beta. ) - g .beta. - a = 1 R W .differential. L .differential. q .
( a ) | q . ( a ) = f a .differential. f a .differential. q . (
.beta. ) - .differential. L .differential. q . ( .beta. ) | q . ( a
) = f a = .alpha. = 1 R w ( p .alpha. - .differential. L
.differential. q . ( .alpha. ) | q . ( .alpha. ) = f .alpha. )
.differential. f .alpha. .differential. q . ( .beta. ) - g .beta. -
.differential. L .differential. q . ( .beta. ) | q . ( .alpha. ) =
f .alpha. ##EQU00083.2##
[0581] Since by definition
p a = .differential. L .differential. q . ( a ) , a = 1 , , R W
##EQU00084##
[0582] And from (E--11)
g .beta. = p .beta. = .differential. L .differential. q . ( .beta.
) q . a = f a . So .differential. H o .differential. q . ( .beta. )
= 0 , .beta. = 1 , , N - R W . ( E - 17 ) ##EQU00085##
and therefore
H.sub.o(q,{p.sub.a},{{dot over
(q)}.sup.(a)})=H.sub.o(q,{p.sub.a}).
[0583] Theorem 3.16.2. In the presence of primary constraints
(E--11), the Hamilton equations of motion are given by
q . ( i ) = .differential. H o .differential. p i + .beta. = 1 N q
. ( .beta. ) .differential. .phi. .beta. .differential. p i , i = 1
, , N p . i = - .differential. H o .differential. q ( i ) + .beta.
= 1 n q . ( .beta. ) .differential. .phi. .beta. .differential. q (
i ) , i = 1 , , N .phi. .alpha. ( p , q ) = 0 , .alpha. = 1 , , N -
R W ( E - 18 ) ##EQU00086##
where {dot over (q)}.sup.(.beta.) are a priori underdetermined
velocities.
[0584] Proof: From (E--15) we obtain and the application of Theorem
3.16.1
.differential. H o .differential. p a = f a + b = 1 R W p b
.differential. f b .differential. p a + .beta. = 1 N - R W
.differential. g .beta. .differential. p a q . ( .beta. ) - b = 1 R
W .differential. L .differential. q . ( b ) .differential. f b
.differential. p a = q . ( a ) + b = 1 R W ( p b - .differential. L
.differential. q . ( b ) ) .differential. f b .differential. p a +
.beta. = 1 N - R W .differential. g .beta. .differential. p a q . (
.beta. ) = q . ( a ) + .beta. = 1 N - R W .differential. g .beta.
.differential. p a q . ( .beta. ) ( E - 19 ) ##EQU00087##
with a=1, . . . , n-R.sub.W. Further
.differential. H o .differential. q ( i ) = b = 1 R W p b
.differential. f b .differential. q ( i ) + .beta. = 1 N - R W q .
( .beta. ) .differential. g .beta. .differential. q ( i ) -
.differential. L .differential. q ( i ) q . a = f a - b = 1 R W
.differential. L .differential. q . b q . b = f b .differential. f
b .differential. q ( i ) = b = 1 R W ( p b - .differential. L
.differential. q . ( b ) q . ( b ) = f b ) .differential. f b
.differential. q ( i ) + .beta. = 1 N - R W q . ( .beta. )
.differential. g .beta. .differential. q ( i ) - .differential. L
.differential. q ( i ) q . ( a ) = f a = .beta. = 1 N - R W q . (
.beta. ) .differential. g .beta. .differential. q ( i ) -
.differential. L .differential. q ( i ) q . ( a ) = f a = .beta. =
1 N - R W q . ( .beta. ) .differential. g .beta. .differential. q (
i ) - t ( .differential. L .differential. q . ( i ) ) q . ( a ) = f
a ( E - 20 ) ##EQU00088##
from (add reference).
.differential. H o .differential. q ( i ) = - p . i + .beta. = 1 N
- R W q . ( .beta. ) .differential. g .beta. .differential. q ( i )
( E - 21 ) ##EQU00089##
[0585] From (E--19) and (E--20) we get:
q . ( a ) = .differential. H o .differential. p a - .beta. = 1 N -
R W .differential. g .beta. .differential. p a q . ( .beta. ) , a =
1 , , R W p . i = - .differential. H o .differential. q ( i ) +
.beta. = 1 n - R W q . ( .beta. ) .differential. g .beta.
.differential. q ( i ) , i = 1 , , N ( E - 22 ) ##EQU00090##
[0586] Since
.differential. H o .differential. p a = 0 and .differential. .phi.
.beta. .differential. p a = .delta..beta. .alpha. ##EQU00091##
we can supplement these equations with
q . ( .alpha. ) = .differential. H o .differential. p .alpha. -
.beta. = 1 N - R W .differential. g .beta. .differential. p .alpha.
q . ( .beta. ) , .alpha. = 1 , , N - R W ( E - 23 )
##EQU00092##
[0587] So we can write
q . ( i ) = .differential. H o .differential. p i + .beta. = 1 N -
R W .differential. g .beta. .differential. p i q . ( .beta. ) , i =
1 , , N p . i = - .differential. H o .differential. q ( i ) -
.beta. = 1 N - R W q . ( .beta. ) .differential. g .beta.
.differential. q ( i ) , i = 1 , , N ( E - 24 ) ##EQU00093##
[0588] For consistency with (E--11) we should write
q . ( .alpha. ) = t - g .alpha. ( q , { p a } ) , .alpha. = 1 , , N
- R W ( E - 25 ) ##EQU00094##
where {dot over (p)}.sub..alpha. is given by the right hand side of
(E--22).
Streamlining the Hamiltonian Equation of Motion (EOM)
[0589] Definition 3.16-1. A function f is weakly equal to g denoted
by f.apprxeq.g, if f and g are equal on the subspace defined by the
primary constraints,
.phi..sub..beta.=0 when
f|.sub..GAMMA..sub.p=g|.sub..GAMMA..sub.p
and
f(q,p).apprxeq.g(q,p)f(q,p)=g(q,p) when
{.phi..sub..alpha.(q,p)=0}
[0590] Theorem 3.16.3. Assume f, g are defined over the entire
space spanned by {q.sup.(i)}, {p.sub.i}. Then if
f ( q , p ) .GAMMA. p = g ( q , p ) .GAMMA. p Then .differential.
.differential. q ( i ) ( f - .beta. .phi. .beta. .differential. f
.differential. p .beta. ) .differential. .differential. q ( i ) ( h
- .beta. .phi. .beta. .differential. h .differential. p .beta. )
and ( E - 26 ) .differential. .differential. p i ( f - .beta. .phi.
.beta. .differential. f .differential. p .beta. ) .differential.
.differential. p ( h - .beta. .phi. .beta. .differential. h
.differential. p .beta. ) ( E - 27 ) ##EQU00095##
for i=1, . . . N.
[0591] Proof: Consider the two functions f(q, {p.sub.a},
{p.sub..beta.}) and h(q, {p.sub.a}, {p.sub..beta.}). Using (E--11)
and from the hypothesis of the theorem,
f(q,{p.sub.a},{g.sub..alpha.})=h(q,{p.sub.a},{g.sub..alpha.})
(E--28)
[0592] Thus is follows
( .differential. f .differential. q ( i ) + a .differential. f
.differential. p a .differential. p a .differential. q ( i ) +
.beta. .differential. f .differential. p .beta. .differential. g
.beta. .differential. q ( i ) ) .GAMMA. p = ( .differential. h
.differential. q ( i ) + a .differential. h .differential. p a
.differential. p a .differential. q ( i ) + .beta. .differential. h
.differential. p .beta. .differential. g .beta. .differential. q (
i ) ) .GAMMA. p and ( E - 29 ) ( .differential. f .differential. p
i + a .noteq. i .differential. f .differential. p a .differential.
p a .differential. p i + .beta. .differential. f .differential. p
.beta. .differential. g .beta. .differential. p i ) .GAMMA. .beta.
= ( .differential. h .differential. p i + a .noteq. i
.differential. h .differential. p a .differential. p a
.differential. p i + .beta. .differential. h .differential. p
.beta. .differential. g .beta. .differential. p i ) .GAMMA. p ( E -
30 ) ##EQU00096##
[0593] Note since
.phi..sub..alpha.(q,p)=p.sub..alpha.-g.sub..alpha.(q,{p.sub.a}), we
have
.differential. g .beta. .differential. q ( i ) = - .differential.
.phi. .beta. ( q , p ) .differential. q ( i ) ##EQU00097## and
##EQU00097.2## .differential. g .beta. .differential. p i =
.differential. .phi. .beta. ( q , p ) .differential. p i
##EQU00097.3## and ##EQU00097.4## .differential. .phi. .alpha. ( q
, p ) = 0 ##EQU00097.5##
for .alpha.=1, . . . , N-R.sub.W. We have
( .differential. f .differential. q ( i ) - .beta. .differential. f
.differential. p .beta. .differential. .phi. .beta. .differential.
q ( i ) ) .GAMMA. p = ( .differential. h .differential. q ( i ) -
.beta. .differential. h .differential. p .beta. .differential.
.phi. .beta. .differential. q ( i ) ) .GAMMA. .beta.
##EQU00098##
which can be written as
.differential. .differential. q ( i ) ( f - .beta. .phi. .beta.
.differential. f .differential. p .beta. ) .differential.
.differential. q ( i ) ( h - .beta. .phi. .beta. .differential. h
.differential. p .beta. ) ##EQU00099##
since
.phi..sub..beta..sup..differential..sup.2f/.differential.p.sub..bet-
a..sup.2=0 because .phi..sub..beta.=0. Similarly,
.differential. .differential. p i ( f - .beta. .phi. .beta.
.differential. f .differential. p .beta. ) .differential.
.differential. p i ( h - .beta. .phi. .beta. .differential. h
.differential. p .beta. ) ##EQU00100##
[0594] Corrolary 3.16-1.
q . ( i ) = .differential. H .differential. p i + .beta. v ( .beta.
) .differential. .phi. .beta. .differential. p i ##EQU00101## p . i
= - .differential. H .differential. q ( i ) - .beta. v ( .beta. )
.differential. .phi. .beta. .differential. q ( i )
##EQU00101.2##
for i=1, . . . , N.
[0595] Proof. We consider two Hamiltonians H({q.sup.(i)},{p.sub.i})
and H.sub.o({q.sup.(i)},{p.sub.a}). Define H({q.sup.(i)},{p.sub.i})
as follows
H({(q.sup.(i)},{p.sub.i}).apprxeq.H.sub.o({q.sup.(i)},{p.sub.a}).
[0596] Then using the result of Theorem 3.16.1, from (E--29) with
f=H and h=H.sub.o
.differential. H o .differential. q ( i ) .apprxeq. .differential.
.differential. q ( i ) ( H - .beta. = 1 N - R w .phi. .beta.
.differential. H .differential. p .beta. ) ( E - 31 )
.differential. H o .differential. p i .apprxeq. .differential.
.differential. p i ( H - .beta. = 1 N - R w .phi. .beta.
.differential. H .differential. p .beta. ) ( E - 32 )
##EQU00102##
[0597] Using (E--31) and (E--32) in (E--24), we get
q . ( i ) .apprxeq. .differential. .differential. p i ( H - .beta.
.phi. .beta. .differential. H .differential. p .beta. ) + .beta. q
. ( .beta. ) .differential. .phi. .beta. .differential. p i ( E -
33 ) and p . i .apprxeq. - .differential. .differential. q ( i ) (
H - .beta. .phi. .beta. .differential. H .differential. p .beta. )
- .beta. q . ( .beta. ) .differential. .phi. .beta. .differential.
q ( i ) or q . ( i ) .apprxeq. .differential. .differential. p i (
H - .beta. .phi. .beta. ( .differential. H .differential. p .beta.
- q . ( .beta. ) ) ) and p . i .apprxeq. - .differential.
.differential. q ( i ) ( H - .beta. .phi. .beta. ( .differential. H
.differential. p .beta. - q . ( .beta. ) ) ) Define v .beta.
.ident. q . ( .beta. ) - .differential. H .differential. p .beta. H
T .ident. H + .beta. v ( .beta. ) .phi. .beta. ##EQU00103##
[0598] So (E--33) becomes
q . ( i ) .apprxeq. .differential. H T .differential. p i p . i
.apprxeq. - .differential. H T .differential. q ( i ) ( E - 34 )
##EQU00104##
[0599] Constrained Hamiltonian Systems
[0600] Local symmetries on a Lagrangian based model. Consider
q.sup.(i).fwdarw.q.sup.(i)(t)+.delta.q.sup.(i)(t)
{dot over (q)}.sup.(i).fwdarw.{dot over (q)}.sup.(i)(t)+.delta.{dot
over (q)}.sup.(i)(t)
with i=1, . . . , N. The action of the system is given by
S(q,{dot over (q)})=.intg.L(q,{dot over (q)})dt
where q and {dot over (q)} are n-dimensional column vectors. The
action differential
.delta. S = .intg. L ( q + .delta. q , q . + .delta. q . ) t -
.intg. L ( q , q . ) t = .intg. L ( q + .delta. q , q . + .delta. q
. ) t - .intg. L ( q , q . ) t = .intg. [ i .differential. L
.differential. q ( i ) .delta. q ( i ) + i .differential. L
.differential. q . ( i ) .delta. q . ( i ) ] t = - .intg. i [ t
.differential. L .differential. q . ( i ) - .differential. L
.differential. q ( i ) ] .delta. q ( i ) t = - t i E i ( o ) ( q ,
q . , q ) .delta. q ( i ) ##EQU00105##
where we define the Euler-Lagrange differential operator
E i ( o ) ( q , q . , q ) = t .differential. L .differential. q . (
i ) - .differential. L .differential. q ( i ) . ##EQU00106##
[0601] Note that
.intg. i = 1 N E i ( o ) ( q , q . , q ) .delta. q ( i ) t .ident.
0 ( 3.17 - 1 ) ##EQU00107##
on shell. Expanding E.sub.i.sup.(o)
E i ( o ) ( q , q . , q ) = j [ .differential. 2 L ( q , q . )
.differential. q . ( i ) .differential. q ( j ) q ( i ) +
.differential. 2 L ( q , q . ) .differential. q . ( i )
.differential. q ( j ) q . ( i ) ] - .differential. L ( q , q . )
.differential. q ( i ) = j W ij ( q , q . ) q ( j ) + j
.differential. 2 L ( q , q . ) .differential. q . ( i )
.differential. q ( j ) q . ( i ) - .differential. L ( q , q . )
.differential. q ( i ) = j W ij ( q , q . ) q ( j ) + k i ( q , q .
) ##EQU00108##
[0602] If L is singular, W.sub.(N.times.N) is not invertible so
(3.17-1) cannot be solved for {umlaut over (q)}.sub.i, =1, . . . ,
N. If Rank(W(q,{dot over (q)}))=R.sub.W on shell, then there exist
N-R.sub.W in the theory. There exist N-R.sub.W independent left (or
right) zero mode eigenvectors w.sub.i.sup.(o,k), i=1, . . . ,
N-R.sub.W such that
i w i ( o , k ) ( q , q . ) W ij ( q , q . ) = 0 , k = 1 , , N - R
W Thus .phi. ( o , k ) = i = 1 N w i ( o , k ) ( q , q . ) E i ( o
) ( q , q . , q ) ( 3.17 - 2 ) ##EQU00109##
depend on q and {dot over (q)} only. The .phi..sup.(o,k) also
vanish on shell:
.phi..sup.(o,k)(q,{dot over (q)})=0, k=1, . . . ,N-R.sub.W
[0603] The set {.phi..sup.(o,k)|k=1, . . . , N-R.sub.W} are the
zero generation constraints. It is possible that not all the
{.phi..sup.(o,k)} are linearly independent. So we may find linear
combinations of the zero mode eigenvectors
v i ( o , n o ) = k c k ( n o ) w i ( o , k ) ##EQU00110##
such that we have
G.sup.(o,n.sup.o.sup.)=v.sup.(o,n.sup.o.sup.)E.sup.(o).ident.0,
n.sub.o, . . . ,N.sub.o (3.17-3)
[0604] These are called gauge identities.
[0605] Any variation .delta.q.sub.i, i=1, . . . , N, of the
form
.delta. q i = n o n o v i ( o , n o ) ##EQU00111##
[0606] Is action invariant by (3.17-1).
[0607] Given this definition of .delta.q.sub.i and (3.17-3), we
conclude
.delta. S = .intg. t i = 1 N E i ( o ) ( q , q . , q ) n o n o ( t
) v i ( o , n o ) = .intg. t i = 1 N n o n o E i ( o ) ( q , q . ,
q ) v i ( o , n o ) ( q , q . ) = .intg. t i = 1 N n o G ( o , n o
) .ident. 0 ##EQU00112##
everywhere. The remaining zero generating modes which we denote by
u.sup.(o,n.sup.0.sup.) lead to genuine constraints. They are of the
form .phi..sup.(o,n.sup.o.sup.)(q,{dot over (q)})=0 on shell,
where
.phi..sup.(o,n.sup.o.sup.)=u.sup.(o,n.sup.o.sup.)E.sup.(o).
(3.17-4)
[0608] The algorithm now proceeds as follows. We separate the gauge
identities (3.17-3) from the nontrivial constraints (3.17-4) and
will list them separately. They will be used for determining local
symmetry transformations.
[0609] Next we want to search for additional constraints. We do
this by searching for further functions of the coordinates and
velocities which vanish in the space of physical trajectories. To
this effect consider the following N+N.sub.o vector constructed
from E.sup.(o) and the time derivative of the constraints
(3.17-4)
[ E ( 1 ) ] = [ E ( o ) t ( u ( o , 1 ) E ( o ) ) t ( u ( o , n o )
E ( o ) ) ] = [ E ( o ) t .phi. ( o ) ] ( 3.17 - 5 )
##EQU00113##
by construction. The constraint .phi..sup.(o) is valid for all time
and therefore
t .phi. ( o ) = 0 ##EQU00114##
on shell, but
.phi. ( o , i ) t = .gradient. q . ( u ( o , i ) E ( o ) ) q +
.gradient. q ( u ( o , i ) E ( o ) ) q . So [ E i 1 ( 1 ) ] = j = 1
n W i 1 j ( 1 ) ( q , q . ) q . ( j ) + k i 1 ( 1 ) ( q , q . ) (
3.17 - 6 ) ##EQU00115##
where i.sub.1=1, . . . , N+N.sub.o, and
[ W i 1 i ( 1 ) ] = [ W ( o ) .gradient. q . ( u ( o , i ) E ( o )
) .gradient. q . ( u ( o , N o ) E ( o ) ) ] [ k i 1 ( 1 ) ] = [ k
( o ) i .differential. .differential. q ( j ) ( u ( o , i ) E ( o )
) q . ( j ) i .differential. .differential. q ( j ) ( u ( o , N o )
E ( o ) ) q . ( j ) ] ( 3.17 - 7 ) ##EQU00116##
[0610] We next look for the zero modes of W.sup.(1). By
construction, these zero modes include the o modes of the previous
level. The gauge identities at level 1 are.
G ( 1 , n 1 ) = v ( 1 , n 1 ) E 1 - n o = 1 N o M n 1 n o ( 1 , 0 )
( u ( o , n o ) E ( o ) ) .ident. 0 ( 3.17 - 8 ) ##EQU00117##
where n.sub.1=1, . . . , N.sub.1 and the genuine constraints are of
the form
.phi..sup.(1,n.sup.1.sup.)=.phi..sup.(1,n.sup.1.sup.)E.sup.1=0
(3.17-9)
with n.sub.1=1, . . . , N.sub.1 on shell.
[0611] We next adjoin the new identities (3.17-8) to the ones
determined earlier (3.17-3) with the remaining constraints (3.17-9)
we proceed as before, adjoining their time derivatives to (3.17-5)
and construct W.sub.i.sub.1.sub.i.sup.(1) and
k.sub.i.sub.1.sup.(1).
[0612] The iterative process will terminate at some level M if
either i) there is not further zero modes, or ii) the new
constraints can be expressed as linear combinations of previous
constraints.
[0613] The Maximal Set of Linearly Independent Gauge Identities
Generated by the Algorithm
[0614] Note that the algorithm steps are of the form
G ( o , n o ) = u ( o , n o ) E ( o ) .ident. 0 ( 3.17 - 10 ) G ( l
, n l ) = u ( l , n l ) E ( l ) - l ' = 0 l - 1 n l ' = 0 N l ' M n
l n l ' ( l , l ' ) .phi. ( l ' , n l ' ) ( 3.17 - 11 )
##EQU00118##
with L=1, . . . , N.sub.l. The
M.sub.n.sub.l.sub.n.sub.l'.sup.(l,l'), are only functions of q and
{dot over (q)}. And
.phi. ( l , n l ) = u ( l , n l ) E ( l ) , n 1 = 1 , , N l , (
3.17 - 12 ) E ( l ) = [ E ( o ) .phi. ( o ) t .phi. ( l - 1 ) t ] (
3.17 - 13 ) ##EQU00119##
where .phi..sup.(l) is a column vector with N.sub.1 components
.phi..sup.(l,n.sup.l.sup.). Thus we conclude from (3.17-13) and
(3.17-11) that the general form of the gauge identity given by
(3.17-11) is of the form
G ( l , n l ) = i = 1 N l l = 1 M m = 1 l mi ( l , m l ) m t m E i
( o ) .ident. 0 ( 3.17 - 14 ) ##EQU00120##
where .delta..sub.mi.sup.(l,m.sup.1.sup.)(q,{dot over (q)}) and
N.sub.l<M. From (3.17-14) it also follows that
l = 1 M n l = 1 l ( l , n l ) G ( l , n l ) .ident. 0 ( 3.17 - 15 )
##EQU00121##
[0615] This identity can also be written as
.delta. q ( i ) E i ( o ) - t F where .delta. q ( i ) = l = 1 M n l
= 1 N l m = q l ( - 1 ) m m t m m ( l , n l ) ( l , m l ) ( t ) (
3.17 - 16 ) ##EQU00122##
and F is a complicated function of q and {dot over (q)}. By
collecting indices l, n.sub.l together
.delta. q i = l = 1 M n l = 1 N l m = q l ( - 1 ) m m i ( a ) ( a )
( t ) ##EQU00123##
[0616] Example of Constrained Hamiltonian System in Lagrangian
Form
[0617] Let
L ( q , q . ) = 1 2 q . 2 ( 1 ) + q . ( 1 ) q ( 2 ) + 1 2 ( q ( 1 )
- q ( 2 ) ) 2 ( 3.17 - 17 ) E ( o ) = [ t .differential.
.differential. q . ( 1 ) - .differential. L .differential. q ( 1 )
t .differential. .differential. q . ( 2 ) - .differential. L
.differential. q ( 2 ) ] = [ q ( 1 ) + 2 q ( 2 ) - q ( 1 ) q ( 1 )
- q ( 2 ) ] ( 3.17 - 18 ) W = [ 1 0 0 0 ] ( 3.17 - 19 ) k = [ q . (
2 ) - q ( 1 ) + q ( 2 ) - q . ( 1 ) - q ( 2 ) + q ( 1 ) ] ( 3.17 -
20 ) ##EQU00124##
[0618] The only o mode is
u ( o ) = [ 0 , 1 ] Then E ( o ) = W ( o ) q + k ( o ) = [ 1 0 0 0
] [ q ( 1 ) q ( 2 ) ] + [ q . ( 2 ) - q ( 1 ) + q ( 2 ) - q . ( 1 )
- q ( 2 ) + q ( 1 ) ] Then u ( o ) E ( o ) = [ 0 1 ] [ [ 1 0 0 0 ]
[ q ( 1 ) q ( 2 ) ] + [ q . ( 2 ) - q ( 1 ) + q ( 2 ) - q . ( 1 ) -
q ( 2 ) + q ( 1 ) ] ] = - q . ( 1 ) - q ( 2 ) + q ( 1 ) = 0 ( 3.17
- 21 ) ##EQU00125##
on shell. Then there are no gauge identities for E.sup.(o). Now
construct E.sup.(1).
E ( 1 ) = [ E ( o ) t t u ( o ) E ( o ) ] = [ q . ( 2 ) - q ( 1 ) +
q ( 2 ) - q . ( 1 ) - q ( 2 ) + q ( 1 ) - q ( 1 ) - q . ( 2 ) + q .
( 1 ) ] ##EQU00126##
which can be written
E ( 1 ) = W ( 1 ) q + k ( 1 ) = [ 0 0 0 0 - 1 0 ] [ q ( 1 ) q ( 2 )
] + [ q . ( 2 ) - q ( 1 ) + q ( 2 ) - q . ( 1 ) - q ( 2 ) + q ( 1 )
- q ( 2 ) + q . ( 1 ) ] ##EQU00127##
[0619] There zero modes of w.sup.(1) are
W ( 1 ) { [ 0 1 0 ] [ 1 0 1 ] ##EQU00128##
[0620] The first zero mode is the previous one augmented by one
dimension and reproduces the previous constraint. The second mode
reproduces the negative of the constraint (3.17-21). That is,
v.sup.(1)E.sup.(1)=-u.sup.(o)E.sup.(o)
with v.sup.(1)=[1 0 1]. This leads to the gauge identity
G.sup.(1)=v.sup.(1)E.sup.(1)+u.sup.(o)E.sup.(o).ident.0
Companionship: Reconciling Agents in the Network.
[0621] The outline of the companionship process is as follows for a
system of N agents. [0622] Determine the state action space of the
system for N-1 agents to create a Tellegen decision element. [0623]
Update the remaining agent with the Tellegen DE. [0624] Repeat
process so that all N agents are updated with respect to their
Tellegen DEs. [0625] User submits query. [0626] System used KB to
establish equations of motion for system in Lagrangian or
Hamiltonian form. [0627] System determines optimal trajectory via
optimization algorithm of the equations of motion that conform to
the principle of least action. [0628] System returns solution which
is a point in the phase space and also serves as an answer to the
query.
[0629] It will also be appreciated that in some embodiments the
functionality provided by the routines discussed above may be
provided in alternative ways, such as being split among more
routines or consolidated into fewer routines. Similarly, in some
embodiments illustrated routines may provide more or less
functionality than is described, such as when other illustrated
routines instead lack or include such functionality respectively,
or when the amount of functionality that is provided is altered. In
addition, while various operations may be illustrated as being
performed in a particular manner (e.g., in serial or in parallel,
synchronously or asynchronously, etc.) and/or in a particular
order, those skilled in the art will appreciate that in other
embodiments the operations may be performed in other orders and in
other manners. Those skilled in the art will also appreciate that
the data structures discussed above may be structured in different
manners, such as by having a single data structure split into
multiple data structures or by having multiple data structures
consolidated into a single data structure. Similarly, in some
embodiments illustrated data structures may store more or less
information than is described, such as when other illustrated data
structures instead lack or include such information respectively,
or when the amount or types of information that is stored is
altered.
[0630] From the foregoing it will be appreciated that, although
specific embodiments have been described herein for purposes of
illustration, various modifications may be made without deviating
from the spirit and scope of the invention. Accordingly, the
invention is not limited except as by the appended claims and the
elements recited therein. In addition, while certain aspects of the
invention are presented below in certain claim forms, the inventors
contemplate the various aspects of the invention in any available
claim form. For example, while only some aspects of the invention
may currently be recited as being embodied in a computer-readable
medium, other aspects may likewise be so embodied.
* * * * *