U.S. patent application number 14/360109 was filed with the patent office on 2014-10-30 for method and system for controlling vibrations in a drilling system.
The applicant listed for this patent is SHELL INTERNATIONALE RESEARCH MAATSCHAPPIJ B.V., SHELL OIL COMPANY. Invention is credited to Apostolos Doris.
Application Number | 20140318865 14/360109 |
Document ID | / |
Family ID | 47216292 |
Filed Date | 2014-10-30 |
United States Patent
Application |
20140318865 |
Kind Code |
A1 |
Doris; Apostolos |
October 30, 2014 |
METHOD AND SYSTEM FOR CONTROLLING VIBRATIONS IN A DRILLING
SYSTEM
Abstract
A control system and method for limiting vibrations in a
drilling system, including a drill string and a drive system for
providing drive torque for rotating the drill string at a reference
frequency. The control system includes a sensor module for
determining an uphole parameter of the drilling system, a model
module provided with a model of the drilling system and adapted to
provide modeled parameters of the drilling system using the drive
torque as an input, a model gain module for providing a model gain
vector to the model module in response to one or more of the
modeled parameters and the drive torque. The model gain vector
enables the model module to update the model thereby obtaining an
updated model, and a control module provides a torque correction
factor to the drive system depending on the modeled parameters, the
uphole parameter, the reference frequency, and the drive
torque.
Inventors: |
Doris; Apostolos; (Rijswijk,
NL) |
|
Applicant: |
Name |
City |
State |
Country |
Type |
SHELL INTERNATIONALE RESEARCH MAATSCHAPPIJ B.V.
SHELL OIL COMPANY |
THE HAGUE
Houston |
TX |
NL
US |
|
|
Family ID: |
47216292 |
Appl. No.: |
14/360109 |
Filed: |
November 22, 2012 |
PCT Filed: |
November 22, 2012 |
PCT NO: |
PCT/EP2012/073322 |
371 Date: |
May 22, 2014 |
Current U.S.
Class: |
175/40 |
Current CPC
Class: |
E21B 44/00 20130101 |
Class at
Publication: |
175/40 |
International
Class: |
E21B 44/00 20060101
E21B044/00 |
Foreign Application Data
Date |
Code |
Application Number |
Nov 25, 2011 |
EP |
11190673.1 |
Claims
1. A control system for controlling vibrations in a drilling
system, the drilling system including an elongate body extending
from surface into a borehole formed in an earth formation and a
drive system for providing a drive torque (Tm) to the elongate body
for rotating said elongate body at a reference frequency
(.OMEGA.ref), the control system comprising: a sensor module for
determining at least one uphole parameter of the drilling system; a
model module provided with a model of the drilling system, the
model module being adapted to provide modeled parameters of the
drilling system using the drive torque (Tm) as an input; a model
gain module for providing a model gain vector (L) to the model
module in response to one or more of the modeled parameters and the
drive torque (Tm), the model gain vector enabling the module to
update the model thereby obtaining an updated model; and a control
module for providing a torque correction factor (u) to the drive
system depending on the modeled parameters, the uphole parameter,
the reference frequency (.OMEGA.ref), and the drive torque
(Tm).
2. The system of claim 1, wherein the modeled parameters include:
modeled uphole angular position ({circumflex over
(.theta.)}.sub.u); modeled downhole angular position ({circumflex
over (.theta.)}.sub.l); and modeled downhole rotary velocity
(.omega..sub.l,m).
3. The system of claim 1, wherein the uphole parameter of the
drilling system as determined by the sensor module comprises the
uphole rotary velocity (.omega..sub.u).
4. The system of claim 1, wherein the control module is adapted to
determine a difference in uphole angular position and downhole
angular position (.theta..sub.u-.theta..sub.l) using the drive
torque (Tm).
5. The system of claim 4, wherein the difference in uphole angular
position and downhole angular position
(.theta..sub.u-.theta..sub.l) is determined by formula: ( .theta. u
- .theta. l ) = T m k .theta. ##EQU00009## wherein k.sub..theta. is
a constant.
6. The system of claim 1, wherein the control module (20) is
provided with the following formula to calculate the torque
correction factor (u): u = - k 1 [ .theta. ^ u - .theta. ^ l - ( T
m k .theta. ) ] - k 2 [ .omega. u - .OMEGA. ref ) ] - k 3 [ .omega.
l , m - .OMEGA. ref ] ##EQU00010## wherein k.sub.1, k.sub.2,
k.sub.3 and k.sub..theta. are constants.
7. A method of controlling vibrations in a drilling system, the
drilling system including an elongate body extending from surface
into a borehole formed in an earth formation and a drive system for
providing a drive torque (Tm) to the elongate body for rotating
said elongate body at a reference frequency (.OMEGA.ref), the
method comprising the steps of: providing the reference frequency
(.OMEGA.ref) to the drive system; the drive system providing the
drive torque (Tm) to the elongate body of the drilling system;
determining at least one uphole parameter of the drilling system;
providing the drive torque (Tm) to a model module which is provided
with a model of the drilling system, the model module providing
modeled parameters of the drilling system; providing one or more of
the modeled parameters and the drive torque (Tm) to a model gain
module for providing a model gain vector (L) to the model module in
response thereto; obtaining an updated model by the module using
the model gain vector; and providing the modeled parameters, the
uphole parameter, the reference frequency (.OMEGA.ref), and the
drive torque (Tm) to a control module; the control module providing
a torque correction factor (u) to the drive system to correct the
drive torque (Tm).
8. The method of claim 7, wherein the modeled parameters include:
modeled uphole angular position ({circumflex over
(.theta.)}.sub.u); modeled downhole angular position ({circumflex
over (.theta.)}.sub.l); and modeled downhole rotary velocity
(.omega..sub.l,m).
9. The method of claim 7, wherein the uphole parameter of the
drilling system comprises the uphole rotary velocity
(.omega..sub.u).
10. The method of claim 7, wherein the control module determines a
difference in uphole angular position and downhole angular position
(.theta..sub.u-.theta..sub.l) using the drive torque (Tm).
11. The method of claim 10, wherein the difference in uphole
angular position and downhole angular position
(.theta..sub.u-.theta..sub.l) is determined by formula: ( .theta. u
- .theta. l ) = T m k .theta. ##EQU00011## wherein k.sub..theta. is
a constant.
12. The method of claim 7, wherein the control module calculates
the torque correction factor (u) using formula: u = - k 1 [ .theta.
^ u - .theta. ^ l - ( T m k .theta. ) ] - k 2 [ .omega. u - .OMEGA.
ref ) ] - k 3 [ .omega. l , m - .OMEGA. ref ] ##EQU00012## wherein
k.sub.1, k.sub.2, k.sub.3 and k.sub..theta. are constants.
13. The method of claim 7, including the step of: Replacing the
drive torque (Tm) with a corrected drive torque (Tc), using
formula: T.sub.c=T.sub.m-u.
14. The method of claim 7, including the step of using only
parameters which can be measured or modeled uphole.
15. A method of controlling vibrations in a drilling system, the
drilling system including an elongate body extending from surface
into a borehole formed in an earth formation and a drive system for
rotating the elongate body by providing a drive torque to the
elongate body, the method comprising: a) operating the drive system
to provide the drive torque to the elongate body, and determining a
system parameter that relates to an uphole parameter of the
drilling system; b) obtaining a model of the drilling system; c)
applying the model to determine a modeled system parameter that
corresponds to said system parameter; d) determining a difference
between the system parameter and the modeled system parameter; e)
updating the model in dependence of said difference, thereby
obtaining an updated model; f) determining from the updated model
at least one modeled parameter of rotational motion, and adjusting
the drive torque in dependence of each modeled parameter of
rotational motion to control vibrations of the elongate body.
16. The method of claim 15, wherein said uphole parameter of the
drilling system relates to an uphole torque in the drilling
system.
17. The method of claim 15, wherein said uphole parameter of the
drilling system relates to torque (T) in the elongate body at or
near the earth's surface.
18. The method of claim 16, wherein said model of the drilling
system includes a modeled torsional stiffness (k.sub..theta.m) of
the elongate body, and wherein said drilling parameter comprises a
ratio of said torque (T) over said modeled torsional stiffness
(k.sub..theta.m).
19. The method of claim 15, wherein said modeled system parameter
relates to a modeled difference between an uphole rotational
position of the elongate body and a downhole rotational position of
the elongate body.
20. The method of claim 15, wherein said uphole parameter of the
drilling system is a first uphole parameter, and wherein step (c)
comprises applying the model using an input parameter relating to a
second uphole parameter of the drilling system.
21. The method of claim 20, wherein the drive system comprises a
rotary drive coupled to an uphole end of the elongate body, and
wherein said second uphole parameter is or comprises torque
provided by the rotary drive to said uphole end of the elongate
body.
22. The method of claim 15, wherein the model includes at least one
modeled state parameter and wherein step (e) comprises adding to
each modeled state parameter the product of said difference and a
respective gain factor pertaining to the modeled state
parameter.
23. The method of claim 22, wherein each modeled state parameter
relates to a modeled parameter of rotational motion of the elongate
body.
24. The method of claim 22, wherein said at least one modeled state
parameter is selected from a modeled difference between an uphole
angular velocity and a downhole angular velocity of the elongate
body, a modeled uphole angular acceleration of the elongate body,
and a modeled downhole angular acceleration of the elongate
body.
25. The method of claim 22, wherein step (b) comprises obtaining a
state observer in which the model is included, the state observer
further including a gain module for calculating each said gain
factor.
26. The method of claim 15, wherein said at least one modeled
parameter of rotational motion includes at least one of a modeled
difference between an uphole rotational position and a downhole
rotational position of the elongate body, a modeled uphole angular
velocity of the elongate body, and a modeled downhole angular
velocity of the elongate body.
Description
[0001] The present invention relates to a method and to a system
for controlling vibrations in a drilling system.
[0002] Numerous vibrations can occur in an elongate body extending
into a borehole formed in a subsurface formation, such as in a
drill string operated to drill the borehole for the production of
hydrocarbon fluid from the subsurface formation.
[0003] Drilling of an oil or gas wellbore is typically done by
rotary drilling. Herein the wellbore may include vertical sections
and/or sections deviating from vertical, e.g. horizontal sections.
Rotary drilling generally employs a drill string including a drill
bit at its downhole end. The drill string typically includes drill
pipe sections which are mutually connected by threaded couplings.
In operation, a drive system located at or near surface may provide
torque to the drill string to rotate the drill string to extend the
borehole. The drive system may include, for example, a top drive or
a rotary table. The drill string transmits the rotational motion to
the drill bit. Generally the drill string also provides weight on
bit and may transmit drilling fluid to the drill bit.
[0004] As a drill string may be several kilometres long, e.g.
exceeding 5 or 10 km, the drill string may have a very large length
to diameter ratio. As a result, the drill string behaves as a
rotational spring and can be twisted several turns during drilling.
Different modes of vibration may occur during drilling, e.g.
rotational, lateral and/or longitudinal (axial) vibrations,
possibly causing alternating slip-stick motions of the drill string
or the drill bit relative to the borehole wall. Such vibrations are
due to, for example, fluctuating bit-rock interactions and pressure
pulses in the drilling fluid generated by the mud pumps.
[0005] In a model description, a drill string can often be regarded
as a torsional pendulum wherein the top of the drill string rotates
with a substantially constant angular velocity, whereas the drill
bit performs a rotation with varying angular velocity. The varying
angular velocity can have a constant part and a superimposed
torsional vibration part. In extreme cases, the bit periodically
comes to a complete standstill. Maintaining rotation of the drill
string at surface builds up torque and eventually causes the drill
bit to come loose and to suddenly rotate again, typically leading
to a downhole angular velocity being much higher than the angular
velocity at surface, typically more than twice the speed of the
nominal speed of the motor at surface, e.g. a top drive or rotary
table. The downhole angular velocity is dampened again whereafter
the process is repeated, causing an oscillating behaviour of the
lower part of the drill string. This phenomenon is known as
stick-slip.
[0006] It is desirable to reduce or prevent these vibrations in
order to reduce one or more of shock loads to the drilling
equipment, excessive bit wear, premature tool failures and
relatively poor drilling rate. High peak speeds occurring during
the slip phase can lead to secondary effects like extreme axial and
lateral accelerations and forces. Proper handling of downhole
vibrations can significantly increase reliability of the drilling
equipment.
[0007] To suppress the stick-slip phenomenon, control methods and
systems have been applied in the art to control the speed of the
drive system such that the rotational speed variations of the drill
bit are dampened or prevented.
[0008] One such method and system is disclosed in EP-B-443689,
whereby the energy flow through the drive system of the drilling
assembly is controlled to be between selected limits, the energy
flow being definable as the product of an across-variable and a
through-variable. The speed fluctuations are reduced by measuring
at least one of the variables and adjusting the other variable in
response to the measurement.
[0009] In EP-B-1114240 it is pointed out that the control system
disclosed in EP-B-443689 can be represented by a combination of a
rotational spring and a rotational damper associated with the drive
system. To obtain optimal damping, the spring constant of the
spring and the damping constant of the damper are to be tuned to
optimal values, whereby the rotational stiffness of the drill
string plays an important role in tuning to such optimal values. To
aid this tuning, EP-B-1114240 discloses a method and system for
determining the rotational stiffness of a drill string for drilling
of a borehole in an earth formation.
[0010] WO 2010/063982 discloses a method and system for dampening
stick-slip operations, wherein the rotational speed is controlled
using a PI controller that is tuned such that the drilling
mechanism absorbs torsional energy at or near the stick-slip
frequency. The method can also comprise the step of estimating a
bit speed, which is the instantaneous rotational speed of a
bottom-hole assembly. The bit speed is displayed at a driller's
graphical interface and is regarded as a useful optional feature to
help the driller visualize what is happening downhole.
[0011] A basic control theory for a non-smooth mechanical systems
is described in A. Doris, Output-feedback design for non-smooth
mechanical systems: Control synthesis and experiments, Ph.D.
thesis, Eindhoven University of Technology, September 2007
(hereinafter referred to as the Doris publication).
[0012] The Doris publication uses a dynamic rotor system, including
an upper disc connected to the motor (the top drive) and a lower
disc connected to the bit. Inputs to the model are the angular
position (phase) and the speed (first derivative of the phase) of
both the upper disc and the lower disc. For the model to provide
accurate results, the speed and phase of the lower disc will have
to be measured using a downhole sensor.
[0013] Jens Rudat ET AL: "Development of an innovative model-based
stick/slip control system", SPE/IADC Drilling Conference and
Exhibition, 139996, 1 Mar. 2011, discloses a system using surface
measured rotary speed .OMEGA. and hook load H as inputs to both the
real drilling process and to a model thereof. The output of the
model y.sub.m, comprising downhole rotary speed .omega.,
weight-on-bit W and torque on bit T, is compared to the measured
output vector y of the drilling process. The comparison is used to
adapt parameters of the model to the real drilling system. The
model itself however remains the same.
[0014] Disadvantage of the system is the requirement to measure
downhole. As disclosed in Rudat et al., in the bottom hole assembly
a dynamics measurement tool was positioned close to the bit
enabling the measurement of the downhole parameters downhole rotary
speed .omega., weight-on-bit W and torque on bit T. The model
parameters related to the measured values have to be transmitted to
surface, using limited bandwidth telemetry systems.
[0015] US-2009/229882 discloses a system for active vibration
damping which relies on downhole sensors to measure motions of the
drill string.
[0016] In practice, it is difficult to accurately measure the
angular position and speed of the downhole disc, i.e. typically the
drill bit. For instance, measurement of angular position and
rotational speed may for instance be measured using a
two-dimensional gravity sensor. During a slip-phase, wherein the
bit suddenly accelerates from a complete standstill to a rotational
speed exceeding the speed of the top drive, the phase accuracy is
often lost. And not knowing the exact angular position will render
further output of the model inaccurate. Herein, please note that
data transmission rates between a downhole location and surface,
using currently available techniques, are very low, typically less
than 1 Hz. These low transmission rates allow for instance only one
sample per 15 seconds or less.
[0017] In addition, the exact initial angular position of the drill
bit is not known, which implies there is always an uncertainty or
error in the measurement of the angular displacement of the bit.
Since the drill string system is a non-linear system, for instance
due to the friction, and exhibits multiple steady state solutions
for the same excitation input, this error can drive the system in
one or the other solution. A steady state solution, for instance,
is constant rotational velocity at the top drive and stick-slip
behaviour at the bit. Another steady state solution, for instance,
is constant rotational velocity at the top drive and at the bit.
Again, this is also due to the low data transmission rate as
explained above.
[0018] The known methods and systems assume a specific frequency of
stick-slip oscillations (vibrations), and tune the control system
to that effect. Such control strategy is inadequate in case the
stick-slip vibrations occur at a different frequency than the
expected frequency, or when there are multiple vibration
frequencies which may change with operating conditions.
[0019] There is a need for an improved method of controlling
vibrations in a drilling system, which overcomes the drawbacks of
the prior art.
[0020] In accordance with the invention there is provided a method
of controlling vibrations in a drilling system, the drilling system
including an elongate body extending from surface into a borehole
formed in an earth formation and a drive system for rotating the
elongate body by providing a drive torque to the elongate body, the
method comprising:
[0021] a) operating the drive system to provide a drive torque to
the elongate body, and determining a system parameter that relates
to an uphole parameter of the drilling system;
[0022] b) obtaining a model of the drilling system;
[0023] c) applying the model to determine a modeled system
parameter that corresponds to said system parameter;
[0024] d) determining a difference between the system parameter and
the modeled system parameter;
[0025] e) updating the model in dependence of said difference,
thereby obtaining an updated model;
[0026] f) determining from the updated model at least one modeled
parameter of rotational motion, and adjusting the drive torque in
dependence of each modeled parameter of rotational motion so as to
control vibrations of the elongate body.
[0027] The invention also relates to a control system for
controlling vibrations in a drilling system, the drilling system
including an elongate body extending from surface into a borehole
formed in an earth formation and a drive system for rotating the
elongate body by providing a drive torque to the elongate body, the
control system comprising:
[0028] an operating device for operating the drilling system to
provide a drive torque to the elongate body, and for determining a
system parameter that relates to an uphole parameter of the
drilling system;
[0029] a model of the drilling system; and
[0030] computer means for applying the model to determine a modeled
system parameter that corresponds to said system parameter, for
determining a difference between the system parameter and the
modeled system parameter, for updating the model in dependence of
said difference thereby obtaining an updated model, for determining
from the updated model at least one modeled parameter of rotational
motion, and for adjusting the drive torque in dependence of each
modeled parameter of rotational motion so as to control vibrations
of the elongate body.
[0031] By using a model that predicts the responses of the drilling
system, e.g. displacement, angular velocity, acceleration,
frictional torque between drill string and rock formation, it is
achieved that stick-slip vibrations of the drill string are
eliminated for a range of angular velocities from angular
velocities close to zero up to very high angular velocities.
Furthermore, by updating the model in dependence of a difference
between the system parameter and the modeled system parameter, it
is achieved that the parameters of the model converge rapidly to
the parameters of the real drilling system so that the model
accurately represents the state of the real drilling system. Also,
with the method and control system of the invention, a constructive
approach is used to adjust the drive torque delivered to the drill
string so that there is no need for trial-and-error approach that
can be time consuming. The model can include high system modes as
many as required to simulate the drilling system accurately for the
control purposes. It is robust in terms of model inaccuracies due
to changes in the interaction between the rock formation and the
drill bit or the drill string (frictional changes, damping changes,
etc). The controller provides information to the drive system to
adjust the drive torque in order to avoid undesirable drill string
vibrations. The adjusted drive torque results in a
winding/unwinding of the drill string able to eliminate the
stick-slip vibrations of the bottom-hole assembly.
[0032] Suitable, said uphole parameter of the drilling system
relates to an uphole torque in the drilling system. An example of a
parameter related to uphole torque can be a torque parameter
provided by a rotary drive coupled to an uphole end of the elongate
body, for example as available in modern top drives. Alternatively
or in addition a parameter related to uphole torque can be a torque
parameter, such as torque, measured at an uphole position of the
elongate body.
[0033] For example, said uphole parameter of the drilling system
suitably relates to torque (T) in the elongate body at or near the
earth's surface.
[0034] In one embodiment, said model of the drilling system
includes a modeled torsional stiffness (k.sub..theta.m) of the
elongate body, and wherein said drilling parameter comprises a
ratio of said torque (T) over said modeled torsional stiffness
(k.sub..theta.m).
[0035] Suitably, said modeled system parameter relates to a modeled
difference between an uphole rotational position of the elongate
body and a downhole rotational position of the elongate body.
[0036] In one embodiment, said uphole parameter of the drilling
system is a first uphole parameter, and wherein step (c) comprises
applying the model using an input parameter relating to a second
uphole parameter of the drilling system.
[0037] Suitably, the drive system comprises a rotary drive coupled
to an uphole end of the elongate body, and wherein said second
uphole parameter is or comprises torque (T.sub.m) provided by the
rotary drive to said uphole end of the elongate body.
[0038] To accurately model the drilling system, the model suitably
includes at least one modeled state parameter and wherein step (e)
comprises adding to each modeled state parameter the product of
said difference and a respective gain factor pertaining to the
modeled state parameter.
[0039] In one embodiment, each modeled state parameter relates to a
modeled parameter of rotational motion of the elongate body.
[0040] Suitably, said at least one modeled state parameter is
selected from a modeled difference between an uphole angular
velocity and a downhole angular velocity of the elongate body, a
modeled uphole angular acceleration of the elongate body, and a
modeled downhole angular acceleration of the elongate body.
[0041] In one embodiment, step (b) comprises obtaining a state
observer in which the model is included, the state observer further
including a gain module for calculating each said gain factor.
[0042] Suitably, said at least one modeled parameter of rotational
motion includes at least one of a modeled difference between an
uphole rotational position and a downhole rotational position of
the elongate body, a modeled uphole angular velocity of the
elongate body, and a modeled downhole angular velocity of the
elongate body.
[0043] The term "uphole" may refer to locations within, for
example, 200 m from the earth surface or from a drilling rig used
in the method of the invention. In case of an offshore operation,
the earth surface is formed by the seabed. The term "downhole" may
refer to locations within, for example, 200 m from the lower end of
the elongate body. Suitably, the elongate body comprises a drill
string having a drill bit at its downhole end.
[0044] The invention will be described hereinafter in more detail
by way of example, with reference to the drawings, in which:
[0045] FIG. 1 schematically shows a drilling system to be
controlled by a preferred embodiment of the method and control
system of the invention;
[0046] FIG. 2A schematically shows an embodiment of the control
system in modular form;
[0047] FIG. 2B shows a schematic representation of an embodiment of
the control system of the invention;
[0048] FIGS. 3a, 3b, 3c, 4a, 4b and 4c schematically show various
results achieved using the method and control system of the
invention.
[0049] In the description, like reference numerals relate to like
components.
[0050] FIG. 1 shows a drilling system 1 including a drill string 2
extending from surface into a borehole (not shown) formed in an
earth formation. The drill string 2 can be several thousand meters
in length, and therefore behaves as a torsional spring. A drive
system 4 is arranged at surface to rotate the drill string 2 in the
borehole by providing a drive torque to the drill string 2. The
drive system generally includes a motor arranged to drive a rotary
table or a top drive (not shown). The drill string 2 typically
includes a downhole end part 6. Said downhole end part may include
a bottom hole assembly (BHA) 6 including a drill collar having an
increased weight which provides the necessary weight on bit during
drilling. Top drive may imply a drive system which rotates an upper
end of the drill string. Upper end implies the end at surface, i.e.
near the location where the drill string is suspended from a
drilling rig.
[0051] Reference sign 7 represents torque resistance T.sub.u of the
upper part of the drill string, e.g. due to electrostatic forces in
the motor, friction in the ball bearings, etc. Reference sign 8
represents torque resistance T.sub.l of the lower part of the drill
string due to interaction of the BHA with the rock formation and
the drilling mud.
[0052] The following parameters are used in the discussion
below:
[0053] T.sub.m: drive torque provided by the drive system 4 to the
drill string 2;
[0054] V: voltage input to a motor (not shown) of the drive system
4;
[0055] T: torque in the drill string 2 as determined at or near the
earth surface;
[0056] u: an update value for controlling drive torque;
[0057] .theta..sub.u, .theta..sub.l: rotational position of the
drill string 2 at respective uphole and downhole locations;
[0058] {dot over (.theta.)}.sub.u, {dot over (.theta.)}.sub.l:
rotational velocity of the drill string 2 at respective uphole and
downhole locations;
[0059] {umlaut over (.theta.)}.sub.u, {umlaut over
(.theta.)}.sub.l: rotational acceleration of the drill string 2 at
respective uphole and downhole locations;
[0060] .theta..sub.u, {dot over (.theta.)}.sub.u, {umlaut over
(.theta.)}.sub.u: model estimates of respective parameters
.theta..sub.u, {dot over (.theta.)}.sub.u, {umlaut over
(.theta.)}.sub.u;
[0061] {circumflex over (.theta.)}.sub.u, {circumflex over
(.theta.)}.sub.l, {circumflex over ({dot over (.theta.)})}.sub.l,
{circumflex over ({umlaut over (.theta.)})}.sub.l: observer
estimates of respective parameters .theta..sub.u, .theta..sub.l,
{dot over (.theta.)}.sub.l, {umlaut over (.theta.)}.sub.l;
[0062] {dot over (.theta.)}.sub.u,eq, {dot over
(.theta.)}.sub.l,eq={dot over (.theta.)}.sub.eq: equilibrium values
of {dot over (.theta.)}.sub.u and {dot over (.theta.)}.sub.l;
[0063] .theta..sub.u,eq, .theta..sub.l,eq: equilibrium values of
.theta..sub.u and .theta..sub.l.
[0064] Furthermore, arrow 9 (FIG. 1) refers to the parameters
.theta..sub.u, {dot over (.theta.)}.sub.u, {umlaut over
(.theta.)}.sub.u, arrow 10 refers to the parameter T.sub.m, and
arrow 12 refers to the parameters .theta..sub.l, {dot over
(.theta.)}.sub.l, {umlaut over (.theta.)}.sub.l.
[0065] Generally, the subscript "u" ("upper") refers to an uphole
position, preferably at or near the surface of the earth, and the
subscript "l" refers to a downhole position, preferably at or near
the downhole end of the elongate body. A bar above a symbol
indicates a modeled parameter. A dot above a symbol refers to a
single time derivative, i.e. a single dot indicates a velocity, and
a double dot indicates acceleration. A "hat" above a symbol (such
as {circumflex over (.theta.)}.sub.l) refers to a parameter of a
state observer. The subscript "eq" refers to an equilibrium value,
that is a value for a state in which the system is free of
vibrations. When the system is in equilibrium, the bit will
generally rotate at the same frequency as the drill string at the
connection to the motor. Angular velocity is also referred to as
rotational velocity.
[0066] Uphole parameters of the drilling system 1 are determined at
or near surface for use in the method of the invention. At or near
surface implies that accurate measurements can be obtained using
high-frequency sensors. High-frequency is for instance exceeding 1
kHz, i.e. more than 1000 samples per second. One such uphole
parameter relates to uphole torque T in the upper part of the drill
string 2. In the practice of the invention, the torque T.sub.m
applied in a modern drive system or a parameter directly related to
T.sub.m is often available as a digital parameter. Generally T
differs slightly from T.sub.m due to, for example, friction in the
drive system itself and/or higher-frequency contributions that may
not be transmitted between the drive system and the drill string.
In case the drive system 4 includes a rotary table, such difference
also can be due to transmission losses. In any event, uphole torque
T or a parameter directly related to T can be determined for
example by measuring, e.g. by a torque sensor at a location at or
near the earth surface. Further uphole parameters can be measured
by suitable sensors.
[0067] Uphole rotary velocity {dot over (.theta.)}.sub.u or a
related parameter may also be measured by a sensor at or near
surface. Such related parameter is for example a period of one
rotation of the drill string 2 at an uphole position. The period of
rotation is directly related to and representative of angular
velocity.
[0068] FIG. 2A shows a block diagram of a control system for
controlling vibrations in the drilling system 1. The control system
comprises a state observer 14 for providing an estimate of the
state of the drilling system 1. The state observer 14 may use
measurements of the input and the output of the drilling system 1.
The state observer 14 includes a mathematical model 16 of the
drilling system and a gain module 18 for updating the model 16. The
gain module may use input and output measurements of the drilling
system 1. The model 16 may typically be implemented in a computer
system running software, e.g. written in Matlab. It is known in the
art how to build a model for a given drill string, and for the
drill string in the borehole. The model 16 can be a simple two
degree-of-freedom (DOF) model, e.g. similar to the one used in
section 6.2.2. of the Doris publication. The model can also be a
more complex multi degree-of-freedom model. It is also possible to
derive a two degree-of-freedom model from a multi degree-of-freedom
model using model reduction techniques known per se. The skilled
person knows how to build a model that describes the dynamics of a
specific drilling system accurately enough for the controller
needs, by including sufficient eigen-modes of the drilling system.
The control system further comprises a controller 20 arranged to
control a motor 22 that drives the drill string 2.
[0069] The state observer 14 may receive an input signal 24
representing Tm. In practice, said motor torque Tm is available to
the driller, as it may be derived from the current drawn by the top
drive. Input signal 24 may also include T. The model 16 provides
output signals 28, 30, 32 representing respective parameters
{circumflex over (.theta.)}.sub.u, {circumflex over
(.theta.)}.sub.l, {circumflex over ({dot over (.theta.)})}.sub.l.
{circumflex over (.theta.)}.sub.u, {circumflex over
(.theta.)}.sub.l are supplied to both the gain module 18 and the
controller 20. {circumflex over ({dot over (.theta.)})}.sub.l is
supplied to the controller 20. The controller 20 also receives an
input signal 34 representing parameter {dot over (.theta.)}.sub.u
and an input signal 36 representing parameters {dot over
(.theta.)}.sub.eq, .theta..sub.u,eq-.theta..sub.l,eq. Furthermore,
reference sign 38 represents voltage V supplied to motor 22,
reference sign 40 represents drive torque T.sub.m supplied by motor
22 to drill string 2, and reference sign 42 represents a gain
vector L supplied by gain module 18 to model 16.
[0070] During normal use of the drilling system 1, voltage V is
supplied to the motor 22 and as a result the motor produces torque
T.sub.m that drives the drill string 2 in rotation. T and {dot over
(.theta.)}.sub.u are measured at surface. T may be input to the
observer. The observer output comprises {circumflex over
(.theta.)}.sub.u, {circumflex over (.theta.)}.sub.l and {circumflex
over ({dot over (.theta.)})}.sub.l. These parameters together with
{dot over (.theta.)}.sub.u and {dot over (.theta.)}.sub.eq,
.theta..sub.u,eq-.theta..sub.l,eq are input to the controller where
they are multiplied by a controller gain. The controller output is
-u which is input back into the motor. The motor adapts T.sub.m by
-u and supplies the adjusted torque to the drill string 2.
[0071] A more detailed description of the way in which the observer
14, the model 16 and the controller 20 are used, is presented
below.
[0072] The equations of motions of the drilling system 1 are
governed by two inertias J.sub.u, J.sub.l, a spring flexibility
k.sub..theta., two frictional torques T.sub.fu, T.sub.fl, and
torque input from the motor T.sub.m. J.sub.u is rotational inertia
of the top drive and part of the drill string, J.sub.l is
rotational inertia of the Bottom-Hole-Assembly (BHA) and the
remaining part of the drill-pipe. k.sub..theta. is rotational
stiffness of the drill string. T.sub.fu torque resistance in
torsional motion of the upper part of the drill string (e.g.
electrostatic forces in the motor, friction in the ball bearings,
etc.) and T.sub.fl represents frictional interaction of the BHA
with the formation and the drilling mud. The differential equations
that describe the torsional dynamics of this system are given in
formulas (1)-(8):
J u .theta. u + k .theta. ( .theta. u - .theta. l ) + T fu (
.theta. . u ) - T m = 0 ( 1 ) J l .theta. l - k .theta. ( .theta. u
- .theta. l ) + T fl ( .theta. . l ) = 0 ( 2 ) T fu ( .theta. . u )
.di-elect cons. { T cu ( .theta. . u ) sgn ( .theta. . u ) for
.theta. . u .noteq. 0 [ - T su + .DELTA. T su , T su + .DELTA. T su
] for .theta. . u = 0 ( 3 ) ( 4 ) T cu ( .theta. . u ) = T su +
.DELTA. T su sgn ( .theta. . u ) + b u .theta. . u + .DELTA. b u
.theta. . u ( 5 ) T fl ( .theta. . l ) .di-elect cons. { T cl (
.theta. . l ) sgn ( .theta. . l ) for .theta. . l .noteq. 0 [ - T
sl , T sl ] for .theta. . l = 0 ( 6 ) ( 7 ) T cl ( .theta. . l ) =
T sl + ( T sl - T cl ) - .theta. . l .omega. st .delta. st + b l
.theta. . l . ( 8 ) ##EQU00001##
wherein:
[0073] T.sub.cu, T.sub.su, .DELTA.T.sub.su, b.sub.u,
.DELTA.b.sub.u, T.sub.cl, T.sub.sl, b.sub.l are constant parameters
governed by frictional torque in the upper part of the drill
string, such as in the drive system, and in the
Bottom-Hole-Assembly. Example values of these parameters are
presented in Table 1 at the end of this section. The torque T in
the drill string 2 at or near surface is:
T=k.sub..theta.(.theta..sub.u-.theta..sub.l).
[0074] The torque T can be derived from the current in the motor
22, and in practice it is always available to the drill-string
operator.
[0075] Equations (9), (10) below are a copy of equations (1), (2)
however with some disturbances included in the parameters
k.sub..theta., J.sub.u, J.sub.l, T.sub.fu and T.sub.fl. These
disturbances are used to simulate modeling inaccuracies when
deriving a model for an oil-field drilling system. This set of
equations (called: model of the drilling system) will be used to
build the observer shown in the cascade of FIG. 2A.
J.sub.um {umlaut over (.theta.)}.sub.u+k.sub..theta.m(
.theta..sub.u- .theta..sub.l)+T.sub.fum( {dot over
(.theta.)}.sub.u)-T.sub.m=0 (9)
J.sub.lm {umlaut over (.theta.)}.sub.l-k.sub..theta.m(
.theta..sub.u- .theta..sub.l)+T.sub.flm( {dot over
(.theta.)}.sub.l)=0 (10)
[0076] where k.sub..theta.m, J.sub.um, J.sub.lm, T.sub.fum and
T.sub.flm are model values of respective parameters k.sub..theta.,
J.sub.u, J.sub.l, T.sub.fu, T.sub.fl. T.sub.fum and T.sub.flm have
the same structure as T.sub.fu and T.sub.fl respectively. The
parameters of the frictional forces of the model that are different
from those of the drilling system are T.sub.sum (instead of
T.sub.su), b.sub.um (instead of b.sub.u), T.sub.clm (instead of
T.sub.cl) and b.sub.lm (instead of b.sub.l).
[0077] In state-space form the dynamics of the drilling system can
be written as:
x . 1 = x 2 - x 3 x . 2 = 1 J u [ - k .theta. x 1 - T fu ( x 2 ) +
T m ] x . 3 = 1 J l [ k .theta. x 1 - T fl ( x 3 ) ] ( 11 )
##EQU00002##
[0078] where x.sub.1=.theta..sub.u-.theta..sub.l, x.sub.2={dot over
(.theta.)}.sub.u and x.sub.3={dot over (.theta.)}.sub.l.
[0079] In state-space form the dynamics of the model of the
drilling system can be written as:
x _ . 1 = x _ 2 - x _ 3 x _ . 2 = 1 J um [ - k .theta. m x _ 1 - T
fum ( x _ 2 ) + T m ] x _ . 3 = 1 J l m [ k .theta. m x _ 1 - T flm
( x _ 3 ) ] ( 12 ) ##EQU00003##
[0080] where x.sub.1= .theta..sub.u- .theta..sub.l, x.sub.2= {dot
over (.theta.)}.sub.u and x.sub.3= {dot over (.theta.)}.sub.l.
[0081] In state-space form the observer is represented by the
following expression:
x ^ . 1 = x ^ 2 - x ^ 3 + l 1 ( T k .theta. m - x ^ 1 ) x ^ . 2 = 1
J um [ - k .theta. m x ^ 1 - T fum ( x ^ 2 ) + T m + l 2 ( T k
.theta. m - x ^ 1 ) ] x ^ . 3 = 1 J l m [ k .theta. m x ^ 1 - T flm
( x ^ 3 ) + l 3 ( T k .theta. m - x ^ 1 ) ] ( 13 ) ##EQU00004##
[0082] where l.sub.1, l.sub.2 and l.sub.3 are observer gains and
{circumflex over (x)}.sub.1={circumflex over
(.theta.)}.sub.u-{circumflex over (.theta.)}.sub.l, {circumflex
over (x)}.sub.2={circumflex over ({dot over (.theta.)})}.sub.u and
{circumflex over (x)}.sub.3={circumflex over ({dot over
(.theta.)})}.sub.l. The gain vector L=[l.sub.1, l.sub.2,
l.sub.3].
[0083] By applying the observer design techniques disclosed in
Chapters 5 and 6 of the Doris publication, the values l.sub.1,
l.sub.2, l.sub.3 can be computed. The goal of the observer is to
derive estimates of the drilling system states that are as close as
possible to the real drill-string system's states. To derive the
observer gains l.sub.1, l.sub.2, l.sub.3, a set of linear matrix
inequalities (LMIs) is to be solved, for example using the software
Matlab and in particular the Matlab toolbox LMI-tool. The procedure
to derive these LMIs is described in the Doris publication.
[0084] In a further step, an adjustment to the drive torque is
applied for torque control so as to control vibrations. The
adjustment takes the following form in this example:
u=-k.sub.1[{circumflex over (.theta.)}.sub.u-{circumflex over
(.theta.)}.sub.l-(.theta..sub.u,eq-.theta..sub.l,eq)]-k.sub.2[{circumflex
over ({dot over (.theta.)})}.sub.u-{dot over
(.theta.)}.sub.u,eq)]-k.sub.3[{circumflex over ({dot over
(.theta.)})}.sub.l-{dot over (.theta.)}.sub.l,eq] (14)
where the subscript eq refers herein to equilibrium values of the
model and the drill-string system. The adjustment u to the drive
torque is calculated using modeled downhole parameters of
rotational motion. {dot over (.theta.)}.sub.u,eq and {dot over
(.theta.)}.sub.l,eq are equal to each other, as these are the
desired values of the drilling system while drilling because no
stick-slip vibrations occur when they are equal. Hence the bit will
rotate at the same rotational speed as the drill string at the
connection to the drive system.
[0085] In order to calculate .theta..sub.u,eq, .theta..sub.l,eq,
the acceleration component in equations (9), (10) is nullified
(i.e. {umlaut over (.theta.)}.sub.u,eq=0, {umlaut over
(.theta.)}.sub.l,eq=0), substitutions {dot over
(.theta.)}.sub.u={dot over (.theta.)}.sub.u,eq={dot over
(.theta.)}.sub.eq, {dot over (.theta.)}.sub.l={dot over
(.theta.)}.sub.l,eq={dot over (.theta.)}.sub.eq are performed, and
equations (9), (10) are solved:
k.sub..theta.m( .theta..sub.u,eq- .theta..sub.l,eq)+T.sub.fum({dot
over (.theta.)}.sub.eq)-T.sub.m=0
-k.sub..theta.m( .theta..sub.u,eq- .theta..sub.l,eq)+T.sub.flm({dot
over (.theta.)}.sub.eq)=0
so that:
k.sub..theta.m( .theta..sub.u,eq- .theta..sub.l,eq)+T.sub.fum({dot
over (.theta.)}.sub.eq)-T.sub.m=-k.sub..theta.m( .theta..sub.u,eq-
.theta..sub.l,eq)+T.sub.flm({dot over (.theta.)}.sub.eq)
and:
( .theta. _ u , eq - .theta. _ l , eq ) = T flm - T fum ( .theta. .
eq ) + T m 2 k .theta. m ##EQU00005##
[0086] Herein, ( .theta..sub.u,eq- .theta..sub.l,eq) is constant.
Formulas (9) and (10) may be used to derive either (
.theta..sub.u,eq- .theta..sub.l,eq), i.e. the output of the model
16 or ({circumflex over (.theta.)}.sub.u,eq-{circumflex over
(.theta.)}.sub.l,eq), i.e. the output of the observer 14. Also,
(.theta..sub.u,eq-.theta..sub.l,eq) may be derived. Regarding the
latter, optionally a measured value for .theta..sub.u,eq may be
included, for instance provided by sensor 54. In formula (14),
k.sub.1, k.sub.2, k.sub.3 are constants calculated according to the
control theory of the Doris publication using the model (9), (10).
Example values of these parameters are presented in Table 1.
[0087] Formula (14) provides a correction factor to the torque. The
corrected torque Tc applied to the drill string, after the
adjustment, is for instance:
T.sub.c=T.sub.m-u
This torque is then substituted in equation (1) replacing
T.sub.m.
[0088] FIG. 2B shows a schematic flow scheme, representing the
above described control system of the invention in an alternative
form.
[0089] A driller 50 operates a drilling rig (not shown) comprising
the drive system 22. The driller 50 sets the voltage input V by
signal 38 to the drive system 22. In response to the voltage signal
38, the drive system 22 will try to rotate the drill string 2 of
the drilling system 1 at a reference rotation .OMEGA..sub.ref.
[0090] Thus, the reference rotation .OMEGA..sub.ref is set by the
driller, by signal 38. When the drill string system would rotate in
equilibrium, the equilibrium rotary speed at surface {dot over
(.theta.)}.sub.u,eq and downhole {dot over (.theta.)}.sub.l,eq
would be equal to the set reference rotation .OMEGA..sub.ref:
{dot over (.theta.)}.sub.u,eq={dot over
(.theta.)}.sub.l,eq=.OMEGA..sub.ref
[0091] These values are therefore readily available, and may be
provided for instance by the drive system 22, see signal 52.
[0092] To rotate the drill string, the drive system 22 provides a
motor torque Tm to the drilling system 1. In response to the
received motor torque Tm, the drill string and drill bit of the
drilling system 1 will rotate. A resulting output vector y may
include rotary position and rotary speed both at surface and
downhole. However, in the system of the invention, downhole
components of the output vector y of the drilling system may be
disregarded. Only one or more uphole components, which can be
accurately measured, are required. Downhole measurements are
obviated.
[0093] It is for instance sufficient to measure the rotary speed
.omega..sub.u={dot over (.theta.)}.sub.u at the connection between
the drive system and the drilling system, for instance using sensor
54. Sensor 54 may be a separate module, or may be included in the
drive system 22. Said surface rotary speed .omega..sub.u={dot over
(.theta.)}.sub.u may be provided to the controller 20. See signal
34.
[0094] A measured torque value, for instance the motor torque Tm,
may be provided to the model 16 and to the model gain module 18.
See signals 24.
[0095] In response to receiving the motor torque Tm, the model
provides a model output vector y.sub.m. Said model output vector
y.sub.m may comprise angular position and rotary speed both at
surface and downhole respectively.
[0096] The signal 52 may also comprise the value of
(.theta..sub.u-.theta..sub.l), which is available due to the
relation thereof to the torque T in the drill string 2 at or near
surface:
T = k .theta. ( .theta. u - .theta. l ) or ( .theta. u - .theta. l
) = T k .theta. ##EQU00006##
The torque T can be derived from the current in the drive system
22. In practice the value T is available to the operator 50.
Otherwise, T can be measured accurately at or near the connection
between the drive system 22 and the drill string 2.
[0097] When the drilling system rotates in equilibrium or steady
state, the above also provides:
( .theta. u , eq - .theta. l , eq ) = T eq k .theta.
##EQU00007##
The value of (.theta..sub.u,eq-.theta..sub.l,eq) may thus be
derived from the torque value when the drilling system operates at
equilibrium.
[0098] The value (.theta..sub.u-.theta..sub.l) may be provided to
the control module 20 via signal 52. Alternatively, the control
module 20 may calculate the value (.theta..sub.u-.theta..sub.l)
using the torque T as provided by the drive system.
[0099] The model module 16 may be provided with any suitable model
of the drilling system 1. Using the input signal 24, which
comprises the motor torque Tm, the model module provides the model
output vector y.sub.m.
[0100] The model output vector y.sub.m comprises for instance the
modeled rotary positions {circumflex over (.theta.)}.sub.u,
{circumflex over (.theta.)}.sub.l at surface and downhole
respectively. Also, y.sub.m may comprise modeled downhole rotary
speed {dot over ({circumflex over
(.theta.)})}.sub.l=.omega..sub.l,m. Herein, .omega..sub.l,m and
{dot over ({circumflex over (.theta.)})}.sub.l are both
representations of the modeled downhole rotary speed.
[0101] The modeled rotary positions {circumflex over
(.theta.)}.sub.u, {circumflex over (.theta.)}.sub.l at surface and
downhole respectively may be provided to a model gain module 18.
See signal 56. The model gain module 18 calculates the gain vector
L=[l.sub.1, l.sub.2, l.sub.3], as described above relating to
Formula (13). The model gain module provides the gain vector L to
the model module 16. See signal 58. The model module 16 uses the
gain vector L to improve the parameters of the model, and
consequently to improve the output vector y.sub.m.
[0102] The values of {circumflex over (.theta.)}.sub.u, {circumflex
over (.theta.)}.sub.l, .omega..sub.l,m as provided by the model
module 16, which preferably have been adjusted and improved using
the input of the gain module 18, are provided to control module 20.
See signal 60.
[0103] The control module 20 uses the available inputs (included in
signals 34, 52, and 60) in formula (14) to provide a torque
correction factor u:
u=-k.sub.1[{circumflex over (.theta.)}.sub.u-{circumflex over
(.theta.)}.sub.l-(.theta..sub.u,eq-.theta..sub.l,eq)]-k.sub.2[{circumflex
over ({dot over (.theta.)})}.sub.u-{dot over
(.theta.)}.sub.u,eq)]-k.sub.3[{circumflex over ({dot over
(.theta.)})}.sub.l-{dot over (.theta.)}.sub.l,eq]
[0104] Until the drilling system is in equilibrium, and given the
available inputs, said formula can also be written as:
u = - k 1 [ .theta. ^ u - .theta. ^ l - ( T m k .theta. ) ] - k 2 [
.omega. u - .OMEGA. ref ) ] - k 3 [ .omega. l , m - .OMEGA. ref ]
##EQU00008##
[0105] The torque correction factor u is provided to the drive
system 22, which subtracts said factor u from the motor torque
T.sub.m, to arrive at a corrected torque value T.sub.c:
T.sub.c=T.sub.m-u
The corrected torque T.sub.c is then substituted in equation (1)
replacing T.sub.m.
[0106] Herein, please note that the reference frequency
.OMEGA..sub.ref as set by the driller 50 is not affected by the
above correction. Rather, the correction is applied to adjust the
torque that the drive system applies to the drill string to arrive
at said .OMEGA..sub.ref.
[0107] Reference is further made to FIGS. 3a-c showing results for
the drill string 2, the model 16 and the observer 14, whereby the
controller 20 is de-activated in order to illustrate convergence of
the drill string states as determined by the model 16 and the
observer 14 to the real drill string states. FIG. 3a shows
.alpha.=.theta..sub.u-.theta..sub.l as a function of time t. FIG.
3b shows .omega..sub.u={dot over (.theta.)}.sub.u as a function of
time t. FIG. 3c shows .OMEGA..sub.l={dot over (.theta.)}.sub.l as a
function of time t. These figures indicate that the drilling
system's state as determined by the observer 16 rapidly converges
to the drilling system's real state.
[0108] Reference is further made to FIGS. 4a-c showing results
whereby the controller 20 is activated. Herein the control system
operates in closed-loop with the drilling system so as to dampen
stick-slip behaviour of the drill string 2. FIG. 4a shows
.alpha.=.theta..sub.u-.theta..sub.l as a function of time t, FIG.
4b shows .omega..sub.u={dot over (.theta.)}.sub.u as a function of
time t, and FIG. 4c shows .omega..sub.l={dot over (.theta.)}.sub.l
as a function of time t. These figures demonstrate that the control
loop is able to rapidly eliminate the stick-slip behaviour of the
drilling system. Rapidly herein implies for instance in less than a
minute.
[0109] Example values of the various parameters discussed
hereinbefore are presented in Table 1 below.
[0110] The present invention is not limited by the above-described
embodiments thereof, wherein many modifications are conceivable
within the scope of the appended claims. Features of respective
embodiments may for instance be combined.
TABLE-US-00001 TABLE 1 Parameter Value Unit Drilling system J.sub.u
0.4765 kg m.sup.2 J.sub.l 0.0414 kg m.sup.2 T.sub.su 0.37975 N m
.DELTA.T.sub.su -0.00575 N m b.sub.u 2.4245 kg m.sup.2/rad s
.DELTA.b.sub.u -0.0084 kg m.sup.2/rad s k.sub..theta. 0.0775 N
m/rad T.sub.sl 0.2781 N m T.sub.cl 0.0473 N m .omega..sub.st 1.4302
rad/sec .delta..sub.st 2.0575 [--] b.sub.l 0.0105 kg m.sup.2/rad s
Model k.sub..theta.m 0.0787 Nm/rad J.sub.um 0.5003 kg m.sup.2
T.sub.sum 0.3987 N m b.sub.um 2.5457 kg m.sup.2/rad s J.sub.lm
0.0455 kg m.sup.2 T.sub.clm 0.052 N m b.sub.lm 0.0116 kg
m.sup.2/rad s Controller k.sub.1 14.5 N m/rad k.sub.2 1.5 N m
sec/rad k.sub.3 30 N m sec/rad Observer l.sub.1 13.5751 -- l.sub.2
-4.458 -- l.sub.3 -152.204 --
* * * * *