U.S. patent application number 10/672681 was filed with the patent office on 2005-03-31 for servo loop pid compensator with embedded rate limit.
Invention is credited to Wand, Martin A..
Application Number | 20050067997 10/672681 |
Document ID | / |
Family ID | 34274779 |
Filed Date | 2005-03-31 |
United States Patent
Application |
20050067997 |
Kind Code |
A1 |
Wand, Martin A. |
March 31, 2005 |
SERVO LOOP PID COMPENSATOR WITH EMBEDDED RATE LIMIT
Abstract
This invention describes a reconfigured form of the PID
compensator such that the rate of change of the position error is
inherently limited without affecting the performance of the servo
loop when the position error is small. The technique described here
maintains the performance of the conventional PID compensator when
the position error is close to zero, which is the operating point
of primary interest.
Inventors: |
Wand, Martin A.; (Plano,
TX) |
Correspondence
Address: |
TEXAS INSTRUMENTS INCORPORATED
P O BOX 655474, M/S 3999
DALLAS
TX
75265
|
Family ID: |
34274779 |
Appl. No.: |
10/672681 |
Filed: |
September 26, 2003 |
Current U.S.
Class: |
318/610 |
Current CPC
Class: |
G05B 11/42 20130101;
G05B 5/01 20130101 |
Class at
Publication: |
318/610 |
International
Class: |
G05B 011/42 |
Claims
What is claimed is:
1. A servomechanism having proportional, integral, and differential
control (PID) receiving an error signal and generating a command
signal comprising: a first gain block having an input receiving the
error signal, an output and a first gain; a first rate-limiter
block having an input connected to said output of said first gain
block, an output and a first rate limit; a derivative block having
an input receiving the error signal and an output; a first
summer-block having a first input connected to said output of said
first rate-limiter block and a second input connected to said
output of said derivative block and a first sum output; a second
rate-limiter block having an input connected to said output of said
first gain block, and output and a second rate limit less than said
first rate limit; an integrator block having an input connected to
said first sum output of said first summer-block and an output; a
second gain block having an input connected to said output of said
integrator block, an output and a second gain; a third gain block
having an input connected to said output of said derivative block,
an output and a third gain; and second summer-block having a first
input connected to said output of said second gain block, a second
input connected to said output of said third gain block, a third
input connected to said output of said second rate-limiter block,
and a second sum output, said second sum output being said command
signal.
2. The servomechanism of claim 1 further including: a fourth gain
block having an input connected to said second sum output of said
second summer-block, an output being said command signal and a
fourth gain.
3. The servomechanism of claim 1 further including: at least one
low-pass filter connected between said output of said derivative
block and said input of said first summer-block and said input of
said third gain block.
4. The servomechanism of claim 3 wherein: said at least one
low-pass filter includes a first low-pass filter having an input
connected to said output of said derivative block, an output and s
first cutoff frequency, and a second low-pass filter block having
an input connected to said output of said first low-pass filter,
and output and a second cutoff frequency higher than said first
cutoff frequency.
5. The servomechanism of claim 4 wherein: said first cutoff
frequency and said second cutoff frequency are both higher than an
effective cutoff frequency of said integrator block and an
effective cutoff frequency of said derivative block.
6. The servomechanism of claim 5 wherein: said first gain K.sub.R,
said second gain K.sub.I, said third gain K.sub.D and said fourth
gain K.sub.C are set whereby 8 K i = 2 .times. w 1 K r = 2 .times.
w 1 K d = ( 4 .times. w 1 ) w 2 K c = 1 ( 4 .times. w 1 ) where
W.sub.1 is said effective cutoff frequency of said integrator block
and W.sub.2 is said effective cutoff frequency of said derivative
block.
7. A method of servo control receiving an error signal and
generating a command signal comprising the steps of: amplifying the
error signal by a first gain; limiting the amplified error signal
by a first rate limit; forming a derivative of the error signal;
summing the a first rate limited amplified error signal and the
derivative of the error signal thereby forming a first sum signal;
limiting the amplified error signal by block second rate limit, the
second rate limit less than the first rate limit; integrating block
the first sum signal; amplifying the derivative of the error signal
by a second gain; amplifying the integrated first sum signal by a
third gain; and summing amplified first sum signal, the amplified
derivative signal and the second rate limited error signal thereby
forming a second sum signal being said command signal.
8. The method of claim 7 further including: amplifying the second
sum signal by a fourth gain.
9. The method of claim 7 further including: low-pass filtering the
derivative signal.
10. The method of claim 9 wherein: said step of low-pass filtering
the derivative signal includes a first low-pass filtering having a
first cutoff frequency, and a second low-pass filtering having a
second cutoff frequency higher than said first cutoff
frequency.
11. The method of claim 10 wherein: said first cutoff frequency and
said second cutoff frequency are both higher than an effective
cutoff frequency of said integrator block and an effective cutoff
frequency of said derivative block.
12. The method of claim 11 wherein: said first gain K.sub.R, said
second gain K.sub.I, said third gain K.sub.D and said fourth gain
K.sub.C are set whereby 9 K i = 2 .times. w 1 K r = 2 .times. w 1 K
d = ( 4 .times. w 1 ) w 2 K c = 1 ( 4 .times. w 1 ) where W.sub.1
is said effective cutoff frequency of said integrator block and
W.sub.2 is said effective cutoff frequency of said derivative
block.
Description
TECHNICAL FIELD OF THE INVENTION
[0001] The technical field of this invention is feedback control
systems employing proportional, integral and differential
compensation.
BACKGROUND OF THE INVENTION
[0002] Feedback control systems, also known as servomechanisms or
servo devices have been developed using a wide variety of
technologies and techniques. These systems have a broad spectrum of
applications. Many special types of servos are used in high
performance equipment. A special type of servo loop acting to
achieve proportional, integral and differential (PID) compensation
is often used to processes an error signal and generate a command.
The goal of this loop is to generate the proper command to
ultimately drive the error signal to zero. The command generated by
the PID compensator consists of three components.
[0003] 1. The component proportional to the error (proportional
P).
[0004] 2. The component proportional to the cumulative sum of the
error (integral I).
[0005] 3. The component proportional to the rate of change of the
error (derivative D).
[0006] Consider the case where the error is a position error. The
task of the PID compensator is to drive the servomechanism to a
commanded position, thus reducing the position error to zero. The
design of the proper PID compensator is well-known to practitioners
in the field of control systems and the details of such design
approaches are not the major focus of this invention. Conventional
design techniques assure that the design is stable with appropriate
stability margins. Digital servos are increasingly common because
they are very effective due to development in recent years.
[0007] When a design is implemented considerations must be given to
factors such as mechanical, electrical and timing limits. These
limits may be exceeded if the mechanism moves too quickly. An
example is when the compensator generates a command to a very fast
actuator and the loop must have a large bandwidth to hold the
position in the presence of high frequency disturbances. Such
devices work well when the position error is small and all
techniques have been brought to bear to overcome these
disturbances. Consider, what happens when a new position command is
issued. The position error is very large resulting in an extremely
large component of proportional correction signal. This results in
a large correction command given to the actuator driver that tends
to cause correction. The integral component also begins changing
but its effect is not as immediate. The derivative component is an
impulse because the rate of change of the position error is large.
In a prior art design a high performance actuator can quickly reach
high speeds.
[0008] FIG. 1 illustrates the components of a prior art PID
compensator. The control equations are easily recognized in FIG. 1.
The input signal is position error 100. The output signal is torque
command 110. The proportional signal couples position error 100 to
summing junction 103. The integrate signal comes from integration
block 101 and the modifying gain constant factor W.sub.1 in
amplifier block 102. The signal operation comes from derivative
block 105, the modifying gain constant factor 1/W.sub.2 in
amplifier block 106, low pass filter 107 having a cutoff frequency
of W.sub.3 and low pass filter 108 having a cutoff frequency of
W.sub.4. The transfer function frequency characteristics are shaped
at low frequencies through parameters W.sub.1 and W.sub.2 and by
parameters W.sub.3 and W.sub.4 at high frequency.
[0009] There is need to introduce a moderating factor in the servo
operation to prevent unwarranted over-drive of the high performance
actuator. The PID compensator of FIG. 1 may be described
mathematically by transfer function equation [1] which omits
W.sub.3 and W.sub.4 for simplicity: 1 H 0 ( s ) = 1 + w 1 s + s w 2
[ 1 ]
[0010] FIG. 2 illustrates a piece-wise graphical representation of
the PID compensator of FIG. 1 and Equation 1 with an arbitrary
overall gain. The integral portion 200 removes low frequency
offsets (DC bias) and the derivative portion 207 provides fast
response to high frequency disturbances. The proportional portion
205 bridges the integral and derivative regions. FIG. 2 also
illustrates the derivative poles, W.sub.3 203 and W.sub.4 204 for
the typical PID compensator illustrated in FIG. 1. In FIG. 2, the
lower frequency break points W.sub.1 201 and W.sub.2 202 of the
transfer function versus frequency are introduced by the frequency
responses of respective amplification stages 102 and 106. Since the
differentiator is the dominant component at frequencies above
W.sub.2 202, the poles W.sub.3 203 and W.sub.4 204 can be included
in the implementation without affecting the validity of the
approaches to be described here.
SUMMARY OF THE INVENTION
[0011] This invention describes a reconfigured form of the well
known proportional, integral, differential (PID) servo compensator.
The reconfiguration provides inherent limits on the rate of change
of the position error without affecting the performance of the
servo loop when the position error is small. The technique
maintains the high-performance of current PID servo compensators
when the position error is close to zero, the operating point of
primary interest.
BRIEF DESCRIPTION OF THE DRAWINGS
[0012] These and other aspects of this invention are illustrated in
the drawings, in which:
[0013] FIG. 1 illustrates a functional block diagram of a PID servo
compensator of the prior art;
[0014] FIG. 2 illustrates the log gain versus frequency
characteristic of the PID servo compensator illustrated in FIG.
1;
[0015] FIG. 3 illustrates the functional block diagram of a
modified PID servo compensator with explicit rate limit of this
invention;
[0016] FIG. 4 illustrates the functional block diagram of the PID
servo compensator of this invention with embedded rate limit;
and
[0017] FIG. 5 illustrates the log gain versus frequency
characteristic of the servo PID compensator with embedded rate
limit of this invention.
DETAILED DESCRIPTION OF PREFERRED EMBODIMENTS
[0018] This invention recognizes an important principle. An
alternate form of PID servo compensator providing rate limiting is
desirable. This invention permits computations of the rate command
based on the position error so that the rate command could be
limited. FIG. 3 illustrates one possible approach. Equation 2 below
is the corresponding transfer function. The three forward paths of
FIG. 1 are modified in FIG. 3 to form the four forward paths of
FIG. 3. This gives the same functional features while including a
rate limiting block 306. Derivative block 301 forms the derivative
term. Gain block 302 controls the derivative gain. Similarly, gain
block 305, integrator 308 and integral gain block 309 implement the
integral term. The proportional factor is implemented through two
paths in FIG. 3. These two paths use: derivative block 301,
integrator 308, and integral block 309 for one path; and gain
blocks 305 and 302 for the other path. These two paths sum to equal
the cumulative proportional effect. Summing junction 303 forms the
torque command 310.
[0019] Employing the principle of superposition since these are
linear networks, the overall transfer function can be derived by
inspection in terms of the four forward paths. In operational
mathematics notation, Equation 2 represents mathematically the
transfer functional of the block diagram of FIG. 3. For simplicity
and conciseness in describing the techniques of the invention, the
differentiator poles, W.sub.3 203 and W.sub.4 204, are omitted at
this point from FIG. 3 though these differentiator poles must be
accounted for in the final implementation. 2 H 2 ( s ) = ( ( K R
.times. K P + K I + ( K R .times. K I ) s + K P .times. s ) ) [ 2
]
[0020] The block diagram of FIG. 3 includes a computed rate command
factor that can be limited and an integrator that can be limited
and reset to prevent windup.
[0021] Equating the coefficients of the PID terms in Equation 1 and
Equation 2 yields three equations in three unknowns, illustrated in
Equations 3. 3 K R .times. K P + K I = 1 K R .times. K I = w 1 K P
= ( 1 w 2 ) [ 3 ]
[0022] Solving Equation 3 for K.sub.R, K.sub.P, and K.sub.I in
terms of W.sub.1 and W.sub.2 can yield complex numbers. This form
is thus not always realizable in hardware.
[0023] Another path or paths must be added in the controller of
FIG. 3 to insure it is realizable. FIG. 4 illustrates such an
embodiment of the servo compensator. This is similar to the servo
compensator of FIG. 3 with changes to the path for the computed
rate and an additional proportional path with limit to the output.
FIG. 4 also includes slight changes in the gain blocks. The gain
after the last summing junction Kc normalizes the compensator gain
to match the gain of the original compensator of FIG. 3.
[0024] Equation 4 shows the transfer function for the compensator
of FIG. 4. The limit blocks are ignored for now. The differentiator
poles, W.sub.3 203 and W.sub.4 204, are omitted from FIG. 4 though
these differentiator poles must be included in the final
implementation. 4 K C .times. [ ( K I + K R ) + K R .times. K I s +
K D .times. s ] [ 4 ]
[0025] Equating the coefficients as before yields three equations
with four unknowns: 5 K C .times. ( K I + K R ) = 1 K C .times. K R
.times. K I = w 1 K C .times. K D = ( 1 w 2 ) [ 5 ]
[0026] Equations 5 have more than one solution because there are
four unknowns. Solving for K.sub.I results in a quadratic. 6 K I =
1 1 - 4 .times. K C .times. w 1 2 .times. K C [ 6 ]
[0027] The value of K.sub.C can be selected such that the quadratic
term of Equation 6 is zero, guaranteeing a single real value for
KI. This becomes the fourth equation in the solution. The results
of the four equations are shown in Equation 7. These coefficients
yield the same closed loop results as the original form of Equation
1. 7 K i = 2 .times. w 1 K r = 2 .times. w 1 K d = ( 4 .times. w 1
) w 2 K c = 1 ( 4 .times. w 1 ) [ 7 ]
[0028] Now the rate limit can be applied. Note that the commanded
rate goes to two different limit blocks 406 and 411. When the
position error is large and the measured rate equals the desired
rate in steady state operation (constant slew rate), a constant
output that counteracts friction or any torque offsets such as
gravity or springs is desirable. Of course the output is not truly
constant, since the spring torque may change as the actuator moves
but that is why the loop is closed. This constant output should
come from the integrator. By setting rate_limit.sub.--2 to the
desired rate, the integrator input becomes zero. Thus the
integrator output is instantaneously constant. Setting
rate_limit.sub.--1 equal to the desired rate times K.sub.D, the
other paths contribute nothing to the output in this particular
steady state condition.
[0029] What do the two limits do to the stability and performance
of the servo loop? The condition where neither limit is reached is
identical to the original design that was stable by design. When
the position error is mid-range where the position error exceeds
rate_limit.sub.--2 but not rate_limit.sub.--1, the limit has the
same effect as reducing K.sub.R to K.sub.R' such that
[K.sub.R.times.position_error]=rate_limit.sub.--2. In Equation 5,
this reduction of KR only affects the integral coefficient so that
now K.sub.C.times.K.sub.R'.times.K.sub.I=W.sub.1. So reducing KR
has the effect of reducing W.sub.1. This reduces the effect of the
integrator by reducing the frequency where the integrator ends. In
the process, it reduces the gain of all frequencies up to the
original W.sub.1. FIG. 5 shows this graphically. Below
rate_limit.sub.--2 the curve 500 is applicable. Above
rate_limit.sub.--2 but below rate_limit.sub.--1, the curve 511 is
applicable. Finally above rate_limit.sub.--1, the curve 512 is
applicable.
[0030] Assuming that the crossover frequencies where the gain and
phase margins are recorded are sufficiently higher than W.sub.1,
the reduction of W.sub.1 has little effect on the stability of the
controller. This is in fact the very type of servo that this
invention is useful for, where the servo has a high bandwidth.
[0031] When the position error is large and exceeds both
rate_limit.sub.--2 and rate_limit.sub.--1, the effect is similar to
a reduction in K.sub.R for both paths. As before, refer to the
reduced gain of the integral path as K.sub.R'. Because
rate_limit.sub.--1 equals rate_limit.sub.--2.times.K.sub.D, the
equivalent K.sub.R for that path is K.sub.R'.times.K.sub.D.
[0032] Note from equation 5 that the derivative term is still not
affected by the limits. Both the proportional and integral paths
are affected. The frequency where the integral region ends
continues to decrease and the gain of the proportional region
decreases as the position error increases. FIG. 5 shows this
graphically. In the region 506 the derivative term remain
unaffected by rate limits. In the region 505 the proportional term
gain is decreased with lower corner frequency also reduced to 509
or 513, upon exceeding rate_limit.sub.--2 and rate_limit.sub.--1
respectively. In the region 500 the integral term remains on the
original curve 500 for small error but shifts to 511 when the
position error exceeds rate_limit.sub.--2 and shift to curve 512
when the position error exceeds rate_limit.sub.--1.
[0033] FIG. 5 illustrates curves 500, 511 and 512 separately
whereas they actually represent three distinct operating points.
The integral portion of the response transitions smoothly from
curve 500 to the left as the position error increases. For example,
when position error equals rate_limit.sub.--2, curves 500 and 511
are identical.
[0034] As before, this has little effect on the higher frequencies
so it has little effect on the stability of the controller. The
exact effect of the limits on the stability of the servo can be
analyzed if desired.
[0035] The two derivative low pass filters with respective cut off
frequencies of W.sub.3 and W.sub.4 are provided in the derivative
path following derivative block 401 in a manner similar to that
illustrated in FIG. 1.
[0036] There are many ways to configure a PID compensator but no
implementation that is the same as described in this invention.
Other common ways to accomplish rate control or a position servo
are listed below.
[0037] 1. The position command can be ramped to the desired
position as the actuator moves causing the rate to be controlled.
The ramp rate of the position command will determine the achieved
rate. This is not always straightforward, since in some servo
systems, the position command does not explicitly exist. Only the
position error may be available, as is often the case when all
positioning is relative instead of absolute.
[0038] 2. The position error can be limited prior to the PID
compensator. The problem with this type of implementation is that
the derivative term is effectively removed from the compensator
while the position error exceeds the limit because the position
error into the PID compensator remains constant until it is less
than the limit. There is no apparent motion. Also, the integral
term must be limited because it will windup during the entire
move.
[0039] 3. A separate rate loop can be designed to handle the move
then the controller can switch to the PID compensator as the
position error approaches zero. This requires design of two
separate loops and care must be exercised during the switchover to
prevent glitches in the command. Such a rate loop also needs some
sort of profile control to slow down the actuator prior to the
switchover. This method might be preferable over the method
described in this disclosure when very precise control of the rate
is required.
[0040] Advantages of the Invention
[0041] 1. Maintains the stability performance of the traditional
PID compensator design.
[0042] 2. Controls the rate of the servo while maintaining the
effect of a PID compensator during the entire move.
[0043] 3. Does not require the design of multiple compensators for
different operating modes.
[0044] 4. If one considers the two limits within the loop as
representing different operating modes, the switch between modes is
automatic, inherently glitch-free and no special considerations are
required to handle the transitions.
[0045] 5. This technique can be applied in both digital and analog
implementations since it is not a multi-mode controller requiring
sophisticated mode control.
[0046] This invention is usable in any servo that meets the
criteria of high bandwidth, fast actuator and required rate limit
during moves.
* * * * *