a comparation betweeen diferente control laws
TRANSCRIPT
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Appendix C
A Com par ison of th e
Self-Tun ing Regulator
o
blstrom and Wit tenm ark w ith
th e Techniques
o
Adapt ive
Inverse Con tro l
The best-known adaptive control methods are based on the self-tuning regulator of Astrom
and Wittenmark. Their
1973
paper
[2]
has had great influence worldwide
in
the field of
adaptive control. Chapter
3
of their book entitled Adaptive Control [13 summ arizes their
work on the self-tuning regulator. Figure
C.
1 is a generic diagram of the self-tuning regu-
lator, based on F ig.
3.1
of Adaptive Con trol.
Process parameters
Design Estimator
Regulator Process
Figure C.l
Conrrol
(Reading, MA: Addison-Wesley, 1989 .
The self-tuning regulator based on Fig.
3.1
of
K.J.
STROM, and B. W I T T E N M A R K , A ~ ~ ~ ~ ~
The system of Fig. C.l is linear and SISO, and
it
works
in
the following way. The
process
or
plant is excited by its input
u
ts output is
y .
This output contains a response
363
Adaptive Inverse Control: Signal Processing App roach Reissue Edition
by Bernard Widrow and Eugene Walach
Copyright 8 the Institute of Electrical and Electronics Engineers
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364
Comparing t he Self-Tuning Regulatorwith Adaptive
Inverse
Control
App. C
u c
to
u
plus plant disturbance. An estimator, receiving both the signal input and signal output
of the plant, estimates the plant parameters. These estimates are fed to an automatic design
algorithm that sets the parameters of the regulator. This regulator could be an input con-
troller, or a controller within the feedback loop, or both. For convenience, we have redrawn
the diagram of Fig. C.
1
in Fig. C.2.
m y
nput
controller
Plant
disturbance
Figure
C.2
An alternative representation of the self-tuning regulator.
In Chapter
8,
we demonstrated that the adaptive disturbance canceler of Fig. 8.1 min-
imizes the plant output disturbance power. In fact, we have shown that no other linear sys-
tem, regardless of its configuration, can reduce the variance of the plant output disturbance
to a level lower than that of Fig.
8.I
Comparing the self-tuning regulator of Fig.
C.2
with
the adaptive disturbance canceler of Fig. 8.1, the question is: Can the feedback controller of
the self-tuning regulator be designed to cancel the plant disturbance as well as the adaptive
disturbance canceler? Another question that arises is: Can an input controller
be
designed
for the self-tuning regulator so that when
it
is cascaded with the plant and its feedback con-
troller, the entire control system will have a transfer function equal to the transfer function of
a selected reference model? It is not obvious that the self-tuning regulator and the adaptive
inverse control system will deliver performances that are equivalent to each other.
C . l
DESIGNING A SELF-TUNING REGULATOR T O BEHAVE L IKE
A N
A DA PTIVE INVERSE CONTROL SYSTEM
To address these issues, we redraw Figs. C.2 and
8.1
as Figs. C.3 and C.4, respectively, in
order to simplify and bring2ut essential features of these block diagrams. For simplicity, we
have drawn Fig. 8.1 with P z ) approximated by P z ) . Also, we included a necessary unit
delay
z-
within
the
feedback loop of the self-tuning regulator that will be necessary only if
there is no delay either in the plant or in the feedback controller. We will assume that
P z )
is stable. If the plant is not really stable, let it be stabilized by a separate feedback stabilizer
and let P z ) represent the stabilized plant. This creates no theoretical problems for the self-
tuning regulator, and
as
demonstrated in Appendix D, creates no theoretical problems for
adaptive inverse control. To compare the two approaches, we need to first show, if possible,
that the transfer function from the plant disturbance injection point to the plant output is
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Sec.
C.l
Designing
a
Self-Tuning Regulator 365
the same for the self-tuning regulator as for the adaptive disturbance canceler. For the self-
tuning regulator of Fig. C.3 his transfer function is
Plant disturbance
W z
Unit delay
C z )
Input
controller
Feedback
controller
Figure C.3
Another representation of the self-tuning regulator.
Plant disturbance
N z )
Input
Plant
output
l- ~--j7+-1
Figure C.4
Another representation of the adaptive plant disturbance canceler
For the adaptive disturbance canceler of Fig.
C.4,
this transfer function is
In order for these transfer functions to be equal, it is necessary that
To
obtain
F C z ) ,
we need
Q z )
and
P z ) .
In practice, both wouldbe readily available
to
a close approximation from adaptive processes already described. So there would be no
problem in getting a good expression for F C z ) .
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66
Comoarina
the
Self Tunina Regulator with AdaDtive Inverse Control
ADD.C
If the feedback controller given by
Eq. C.3)
were used in the self-tuning regulator,
would the resulting system
be
stable? The answ er is yes. Referring to Fig.
C.3,
we may
note that the transfer func tion through the feedb ack loop to the plant output is
This transfer func tion is stable since both
P z )
and
Q z )
are stable.
SO
ar SO good.
To
control the dynam ic response of the sy stem, to make it behave like the dynam ic
response of a selected reference model, we multiply the transfer function 2.4) by ~ C Z )
and set the product equal to
M z ) :
C.5)z ) = ZC Z)
.
I P z) 1 z- P z )
.
Q z ) ) .
Accordingly,
-
M z ) .F C z )
-
Z I P z) Q z )
For the entire self-tuning regulator to be stable, it is necessary that Z C z )be stable. Stability
has already been established for the rest of the system .
M z )
s stable. Since
P z )
and
Q z )
are stable, they would not cancel any unstable poles of F C z ) hat may occur. It is necessary ,
therefore, for
F C z )
o
be
stable in order for
Z C z )
o be stable, although this
is
not sufficient
for stability. I C z ) will
be
unstable if either P z ) , Q z ) , r both are nonminimum-phase.
How would one build a self-tuning regulator if its feedback controller
F C z )
were
unstable? There are two possibilities. One possibility would be to choose an
F C z )
hat is
stable but not optima l for plant disturbance canceling. The other possibility would be to use
the optimal, unstable F C z ) inside the feedback loop and build the input controller ZC(z)
having the ind icated poles and zeros but allowing components of
Z C z )
to be noncausal
as required for stability, The noncausal filter could be realized app roximately with an ap-
propriate delay. The entire system response would be a delayed version of the response of
M z ) .
These difficulties are not encoun tered when the optimal
F C z )
s used and the input
controller
ZC(z)
is stable.
C.2
SOME EXAMPLES
Specific exam ples will help to clarify some of the issues. Suppose that the plant disturbance
is constant, that it is a
Dc
offset or bias. The question is: How w ell do the adaptive dis-
turbance canceler and the se lf-tuning regulator handle this disturbance? For the adaptive
disturbance canceler, the transfer function from the plant disturbance injection point to the
plant outpu t is given by
C.2).
We would like this transfer function to have a va lue of zero
at zero frequency, that is, at z = 1. This is easily accom plished as follows:
1
1
- P 1 ) .
Q 1 )
= 0, or Q 1) =
)
(C.7)
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Sec. C.3
Summary
367
Any form of Q z ) would allow perfect canceling of the constan t disturbance as long as the
value of
Q
at
z
= 1 is the reciprocal of the value of P at z = 1.
So
the
adaptive plant disturbance canceler will perfectly eliminate constant plant dis-
turbance. What will the self-tuning regulator do with this disturbance? Its transfer function
from plant disturbance injection point to plant output point is g iven by
C.1).
This transfer
function shou ld equal zero at zero frequency to eliminate the DC plant disturbance. Accord-
ingly,
Assuming that the plant transfer function is well behaved at
z = 1, i t
is clear that F C z )
must be infinite at z = 1. One way to accomplish this would be to let F C z ) be a digital
integrator,
F C z )
=
(C.9)
1 - - I
giving it a simple pole at
z =
1. T he input controller has the transfer function given by
C.6):
1
C .10)
P z ) and Q z ) are both stable and finite at z = 1. If in addition M z ) has a finite value
at z
= 1, I C z )
will have a pole at
z = 1
and will thereby be unstable making the entire
system unstable. A noncausal realization of I C z ) would not be useful in this case. The
only possibility would be to choose an
F C z )
having its pole slightly inside the unit circle,
sacrificing some d isturbance canceling capability for a stable
I C z ) .
The sam e kind of result would be obtained
if
the plant disturbance were a constant-
amplitude sine wave. The adap tive disturbance canceler would adap t and learn to elimi-
nate it perfectly. The self-tuning regulator would either be unstable or, if stable, would give
somew hat less than optima l disturbance reducing performanc e.
C.3 SUMMARY
The self-tuning regulator has an easier job of disturbance reduction and dynamic control
than adaptive inverse control when the plant is unstable with a pole or poles on the unit circle
or
outside the unit circle. Feedback used by the self-tuning regulator has the capability of
moving the plant poles inside the
unit
circle to stabilize the plant, reduce disturbance, and
control its dynamics. For adaptive inverse control, the first step would be to stabilize the
plant with feedback. The choice of feedback transfer function for stabilization would not be
critical and would not need to be optimized. The only requirement would be to somehow
stabilize the plant. Then adaptive inverse control could be applied
in
the usual manner. A
discussion of initial stabilization for adaptive inverse control systems is given i n Appendix
D.
A difficult case for both approaches occurs when the plant has one
or
more zeros on
the
unit
circle. T he inv erse controller tries to put the
FIR
equivalent of a pole
or
poles on top
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368
Comparing the Self-Tuning Regulator with Adaptive Inverse Control
App. C
of the zeros. The inverse impulse response does not die exponentially but instead persists
forever. An optimal
FIR
inverse filter cannot be constructed. The self-tuning regulator of
Fig. C.3 also canno t cope w ith such a p roblem . Its feedback canno t move around the zeros.
It too would try to choose ZC(z) to put poles on top
of
these zeros, but this would make
ZC(z)
unstable and thereby make the entire system unstable. A remedy that w ould work
for both approaches would be to cascade with the plant a digital filter having poles m atching
the unit circle zeros of the plant and to envelop the cascade within a feedback loop. If, for
exam ple, the plant had a single zero on the
unit
circle at z = 1, the cascade filter would be
a digital integrator. The loop around the cascade would need to be designed to be stable for
adaptive inverse control.
When the plant is nonminimum-phasewith zeros outside the unit circle, we have seen
in Chapters
5,6
and 7 how adaptive inverse control can readily cope w ith such a plant. Deal-
ing with this kind of plant w ith a self-tuning regulator is difficult, much more difficult than
dealing with a m inimum -phase plant. The literature is not clear on how this can be done.
Com paring adaptive inverse control with the self-tuningregulator reveals cases where
one approach is advan tageous, and cases where the other approach is advantageous. Also,
there are m any cases where both approaches give equivalen t performance although the sys-
tem configurations and m ethods of adaptation are totally different.
Bibliography for Appendix C
[ l ] K.J.ASTROM, nd B .
W I T T E N MA R K,daptive
control, 2nd ed. Menlo Park, CA:
[2]
K.J.
ASTROM, nd
B.
W I T T E N M A R K ,
On self-tuningregulators,
Aufomafica ,
Vol. 9,
Addison Wesley, 1995).
No. 2 1973).
Th is situation is not so clear. Refer to simulation examples
in
Chapter 12