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Robust Control of a General Servomechanism Problem: The Servo Compensator
ROBUST CONTROL OF A GENERAL SERVOMECHANISM PROBLEM: THE SERVO COMPENSATOR
E.J. Davison A. Goldenberg Department of Electrical Engineering University of Toronto Toronto, Canada
ABSTRACT The robust control of a completely general servomechanism problem, is considered in this paper. Necessary and sufficient conditions, together with a characterization of all robust controllers which enables asymptotic tracking to occur, independent of disturbances in the plant and perturbations in the plant parameters and gains of the system is obtained. (It new type of compensator, introduced in reference ,called a servo-compensator which is quite distinct from an observer is shown to play an essential role in the robust servomechanism problem. It is shown that this compensator, which corresponds to an integral controller in classical control theory, must be used in any servomechanism problem to assure that the controlled system is stabilizable and achieves robust control; in particular, it is shown that a robust controller of a general servomechanism problem must consist of two devices (i) a servo-compensator and (ii) a stabilizing compensator. A study of the stabilizing compensator is made; in parti cular, i t is shown that a new type of stabi l i zing compensator called a complementary controller, may be used together with the servo-compensator to form a robust controller for the servomechanism problem. INTRODUCTION This paper deals with an extension of reference (1) for the case of finding a robust controller for a system so that asymptotic tracking, in the presence of disturbances, occurs independent of perturb at ions in the plant's or gain parameters of the system. The following problem is considered: A linear, time-invariant system (plant) is described by the following equations: x = Ax
+
Bu
+
Ew
Cx
+
Du
+
Fw
y
e ~ Y - Yref
(1)
m where x ~ Rn is the state, u ~ R is the input, y ~Rr is the output which is to be regulated and w ~RSi is a disturbance vector which may be either measurable or unmeasurable. e ~ Rr is the error in
the system, which is the difference between the ou~ put y and the specified reference input Yref' The disturbance vector is assumed to satisfy the follo~ ing equation: (2 ) n
where zl ~ R I and where (Cl' AI) is observable and zl (0) mayor may not be known. The specified reference input vector Yref ~ Rr is assumed to satisfy the following equation: y ref = Go, z2
(3)
o
C2 z2
n2 where z2 ~ R and where (C 2 , A2 ) is observable and Z2(0) is known. It is desired to find a robust (or weak robust) controller, in a precise sense to be defined later, for (1) so that the errornle ~ 0 as t rt = 'tJ x(O) ~Rn , 'tJ zl (0) ~ R ,\f z2(0) ~ R 2 and such that the resultant controlled system is stable. It is assumed that the outputs Ym where: Ym = Cmx
+
Dm u
+
Fm w
(4)
are the only outputs which are available for measurement. It is assumed without loss of gpnerality that rank C = r, rank B = m, rank L~ = rank Cl = Si , rank G = rank C = dim (0). ~n addition, it is 2 assumed for non-tri viali ty, that all the eigenva lues of AI' A2 are in the closed right hand part of the comp lex plane, and that max {rank rank (G)} ~ 1.
[n'
Davison (1) has considered the above problem when (1) is written as follows:
i = Ax e
= Cx
+
Bu
+
~
(Ei' 0)
i=l +
Du
+
f
(5)
(F i' - Gi )
i=l This work has been supported by the National Research Council of Canada under Grant No. A4396.
231
where
r
L F.w. is the output, t~~lsp~c~fied reference input,
y = Cx + Du +
Definition (2)
~ G.y. is i=l I I r and Ym = Cmx + D u + L Fm w. are the only outputs m i=l I I which are available for measurement and wi ' y. i = 1,2, ... ,r each satisfy a different differential equation. References (9) , (10) have considered the problem when D = 0 by using geometric methods. Yref =
The following definitions are required: Definition
(7)
Remark 3
2
1
Let the coefficients 0., i = 1,2, ... ,q be given by the coefficients of theIpolynomial ~ (A - Ai) where i=l A. are given by (8), i . e.
Definition Gi ven a real matrix denote
n+ min (r,m)
<
Let all the zeros of the minimal polynomial of Al be given by {A!} and let all the zeros of the minimal polynoffilal,of A2 be given by {A } . Then 2 define
9iven the system (1) assume a feedback controller n = E n + Elx ,u = Klx + K2n has been applied; 2 then the controlled system is said to be{[A 0 ] controllable (stabilizable) if the pair E E ' controllable (stabilizable).
BJ D
rank [ A -C AI
It may be noted that the set of transmission zeros may contain zero elements, a finite number of symmetric complex numbers or include the whole complex plane.
PRELIMINARIES
[~J}iS
Given the system x = Ax + Bu, y = Cx + Du, denoted by (C,A,B,D), the transmission zeros of (C,A,B,D) are defined to be the set of complex numbers A which satisfy the following inequality:
m.. (. fl IJ
fl
E:
E:
=
M = {m .. } IJ
let
I
M (. fl
q q 1 q-2 q + ... +02A+ol~ IT (A- A.) A + OqA - +Oq_lA i=l I
E:
where:
{m·.1 Im·IJ·1 IJ
<
d
and let
(6)
C be a
qxq 1
Defini tion (1)
o
Given the system (1), suppose that there exists a controller so that asymptotic regulation takes place and so that the resultant controlled system is stable. Let the plant parameters A, Band C now be perturbed, i.e. A 7 A + aA, B-7 B + oB~ C 7 C + OC where oA (. flE: ' oB (. fl , oC (. fl E: . Then E if there exists an E > 0 so that the controlled system is still stable and such that asymptotic regulation still take s place for VoA (. fl , VoB (. 11 , VaC (. fl (VOA (. fl , VoB (. fl ), the contro1rer is E E E E said to be a robust controller (weak robust controll e r) Remark 1 Note that there is no assumption in the above definition that £ > 0 is arbitrary small, i . e . finite changes, and not just arbitrarily small perturbations, are allowed in the plant I s parameters
(9)
companion matrix given by:
0 1
l]
(10)
Defini tion The actual output y is defined to be equal to the physical outputs of the plant which are to be regulated corresponding to y in (1). The following lemma is required in the development to follow : Lemma 1 (3) : I f (C,A) is obs.e rvable, and i f A has k invariant polynomials, then there exists a coordinate transformation T so that (CT- l , TAT-I) is observabl e with :
A, B, C.
where Remark 2 The motivation for the above definition comes from the fact that in practice, the equations describing a plant of th e form x = Ax + Bu, y = Cx + Du are always approximate, i.e. they really arise from a linearization, about an operating point, of the nonlinear equations = f(x,z,u), ~z = g(x,z,u), y = h(x,z,u) where z corresponds to the "high frequency modes" of the plant which have been negl e cted and ~ 7 O. See reference (l) for details of this motivation.
F. I
o0 ·· [· x
1
0
x
0 ... 0J 1 ... 0
x
.. .
(12)
x
and x designates a number not necessarily zero, and where rank(Cr l ) > k . Here F . , i = 1,2, ... ,k are the companion matrices of the invariant polynomials a (of degree ~i) of A with a the i l minimal pol ynomial of A and with a dIviding i a i_I' i = 2,3, ... ,k .
x
232
{block diag(C, C, ... ,C) , B}
DEVELOPMENT : THE SERVO-COMPENSATOR The following main result is obtained:
( 17)
r matrices
Theorem 1 is controllable . x is the output of a stabilizing compens ator S* with inputs y, u, t; where S*, K , K are found using standardmmethods, e.g. o observer theory, to stabilize the following augmented system:
A necessary and sufficient condition that there exists either a robust or weak robust linear timeinvariant controller for (1) such that e ~ 0 as t ~ for all measurable or unmeasurable disturbances w described by (2), and for all specified reference-inputs Yref described by (3), and such that the controlled system is stabilizable, is that the following conditions ' all hold : 00
(a)
(18)
(A, B) Le stabilizable.
(b)
(Cm' A) be detectable .
(c)
m > r
(d)
The transmission zeros of (C, A, B, D) do not coincide with Ai' i = 1,2, ... ,q given by (8).
(e)
In the case of robust control, Ym must contain the actual output y, i .e. there exists a nonsingular transformation T so that TYm = [~-l, where y is the actual
where the p air l[:*c
(I r , 0) T* (Ym- Dmu)
+
Du
In the case of weak robust control, the most general controller structure which regulates (1) has, except for a pathological case, exactly the same structure above except that the output y used to form the error input to (15) may be re-constructed and need not be actually measured. The pathological case occurs only if D F 0 and can only occur provided there exists a gain matrix K* which lies on a hypersurface(4) in the parameter space of (C, D), given by the condition rank(C + DK*C ) < r In this pathological case, there may exist m controllers of smaller order than rq. This pathological case is not very important however, since (i) the hypersurface condition is "almost never"(4) satisfied for any gain matrices of the system, (ii) the resultant controller would be hyper-sensitive with respect to feedback gain perturbations in the system.
Conditions (c), (d) reduce to the following condition: A - A.I B] = n+r, i=1,2, .... ,q(l3) rank [ C ~
D
Controller Structure The most general robust controller which regulates (1) must have the following structure:
x+
Kt;
(14)
where t;, a rq vector, is the output of the general servo-compensator(l) given by :
* C t;
+
* Be, e
fi
y - Yref
(15)
where T
block diag
(C, C, ... ,C)
T
The robust (or weak robust) controller obtained by putting T=Iand letting B=bl ock di ag (y y ,." " y r ) r 2 in (16) where Yl = Y2 ="'=Y r = (00 .. . 01) gi ve s :
-1
r matrices
u = Kx 0
(16) B*
T
where
B
t;
is a
rq
Kt;
+
+
where
T
C* = block diag(C, C, . .. ,C) r matrices
233
(19)
vector given by:
t; = C* t;
is a nonsingular real matrix, C is defined in (10) and B E:: Rrq x r is a real matrix of rank r with the property that:
where
~*J}iS
is observable (detectable) if (Cm' A) is observable (detectable). In the above robust controller, the error input e to the general servo-compensator (15) must be obtained by using the actual output y obtained from Ym' Note that the above robust controller always has order > rq. Figure 1 gives a schematic diagram of this robust controller .
Remark 4
u = K o
controllable
(stabilizable) i f (A, B) is controllabte (stabilizable) and the pair{[~m ~J' ~*C
output y. InYmthe case of weak robust control, Ym must contain enough information to be able to re-construct y from it, i . e. there exists a*nonsingular tra[~Jsformaf~onJ T* so that T (Cmx + Fm w) = C x + LFm w ; in this case, y
~*J, r:*DJ~iS
B* e
(20)
provided only that tne resultant closed loop system remains stable. (Zl)
4.
and is particularly simple to implement. The compensator (ZO) is called a servo-compensator and was introciuced in references(1),(5) It is to be emphasized that the servo-compensator
(ZO) or more generally the general servo-compensator (15) is not some type of observer. It is a compensator in its own right, quite distinct from an observer. It corresponds in classical control theory, for the case of a system subj ect to constant disturbances, to an integral controller. Remark 5 It is to be noted that the general servo-compensator (15) has minimal order rq and is a feedback controller, consisting of r unstable compensators with identical dynamics corresponding to the class of disturbance/reference-inputs, connected to the error outputs of the system. Note that all error outputs of the system must be connected ~the servo-compensator. This feedback structure is an essential feature of any robust controller.
The proof of Theorem 1 follows almost immediately from reference (1) , after (1), (4) are rewritten in a form similar to (5). The following lemma is required:
Properties of the Robust (or Weak Robust) Controller 1.
Lemma Z
In the above robust controller, it is not necessary to know E, F, G, Cl' CZ' AI' AZ . It is essential however to know the zeros of the minimal polynomials of AI' A .
The system (1)-(4) is equivalent to the following representation:
r
z
Z.
x = Ax + Bu +
L i=l
The robust controller is robust for any finite changes (and not just arbitrarily small perturbations) in the plant parameters A, B, C, D, feedback gain parameters K, K, stabilizing compensator parameter~ of S* and for any changes in the order of the assumed mathematical model describing the plant and also for any changes in the order of the stabilizing compensator S*, provided only that the resultant closed loop system remains stable. It is also robust for finite perturbations in some of the parameters of the general servocompensator (15), in particular for perturbations in the elements of T , and parameters of the matrix B provided that the resultant closed loop system remains stable. Note that , 0Z' " .,0 it is essential that the elements of the servo-compensator (15) to be 1 fixed q precisely t ,and that the error e 6 (y - Y f) = re input to (15) to be measured precisely, i.e. it is assumed that the actual output y is obtained from Ym in an exact way.
w.
(E. ,0)
1
[J 1
+
Yi
r
[j
vi
L L (.E.
, 0)
i=lj=l J 1
w.] 1
jYi (ZZ)
[w.~
tW~
r (F.,-G.) 1+ Ir vi e=Cx+DJ+ L L (.F.,- . G. J 1 i=l 1 1 Yi i=l j=l J 1 J 1 / i i r v Fm w. + L 1 1 L L i=l i=l j=l
r
y = C x + D u m m III E.
where rank [ F:
+
0 ]
G i
~ 1, i
.F
J
m
.w.
J 1
1,Z, ... ,r and where the
elements of the vectors wi' Yi' i satisfy the differential equation:
°
3.
Note that a robust controller exists for "almost all" (C, A, B, D) systems' provided that (i) m > r, (ii) the outputs to be regulated can be physically measured; i.e. the class of (C, A, B, D) systems in which Theorem 1 does not hold, lies on a hypersurface in the parameter space of (C, A, B, D). (See reference(4) for a definition and properties of a hypersurf~c~.) This immediately follows from reference ll ). Thus if a system to be regulated has at least the same number of independent inputs as there are independent outputs to be regulated and if these outputs are physically measurable, there "almost alway!l' is a solution to the robust servomechanism problem. If either (i) or (ii) above fail to hold, then there never is a solution to the robust servomechanism problem. Similarly a weak robust controller exists for "almost all" (C, A, B, D) systems provided that (i) m > r, (ii) the outputs to be regulated can be reconstructed; if either (i) or (ii) fails to hold, there is never a solution to the weak robust servomechanism problem.
1, Z, . . . ,r
(Z 3)
The weak robust controller is robust for any finite changes in the plant parameters A, B, feedback gain parameters Ko' K, stabilizing compensator parameters of S* and servocompensator parameters T, B
where the initial conditions of (Z3) are known for the elements of Yi and known (unknown) for the elements of w. if the disturbances ware r measurable (un~easurable), and where i~l {Si}= t This is usually the case in practice ,especially when the disturbance-reference inputs are polynomial or sinusoidal functions of time.
234
{AI' A ,···,A} where {Si}, i = 1,2, ... ,r denotes
2
the character~atic roots of (23). The elements of the vectors .W., .y., i=1,2, ... ,r satisfy a differential] ~quati~n, with initial conditions which depend on the initial conditions of (23) and, with characteristic roots equal to those of (23), except that the multiplicity of all roots is now reduced by at least one. Proof of Lemma 2
= Ax
+ Bu + (E, 0)
e
= ex
+
Du
+
(F, -G)
[~J (24 )
[~J
where
[~~J =[ ~l ~J [:J (25 )
(30)
where the elements of w ' Yi satisfy the i differential equation (23), and where the elements of . W . , .y. satisfy a differential equation, with 1
]
1
initial conditions which depend on the initial conditions of (23), with characteristic roots equal to those of (23), except that the multiplicity of all roots is now reduced by at least one. Here Si' Ti are Q x dim(w ) , dim(o) x dim(Yi) matrices i respectively with the property that rank Si= dim(w ), i rank Ti = dim(Yi), i = l, .. . ,r . (This follows since the
[:J=[~l ~J[:~J and where
E [Si [W]= ° i=l 0
]
The system (1) may be written as : x
.. (~ 1 '-l) , the property that 0'1 2=0'1, 1 1 0'3= 1 0'1 1, ... ,0 1. '1." = °'1 (2 S) may be expanded into the fOllowing: 1
pair{[~l ~2J' [~l ~2J}iS o
Applying now lemma 1, let a transformation applied to (25) to give:
T be
0J~[EF
[ Ei F. -G .
rank
Note that the minimal zeros equal to
On expanding out the disturbance, reference-inputs
[~J of (27), the following is obtained : (2S)
°i
(29)
0'. 1
~[:l :l]. .1
.2
Tk
Tk
Oil]
On noting that 0i
0 - G.
j1 J~W'J y.
j1j1'1
Q,
°J,i
T.
rank
1,2, ... , r
(32)
1
[~J =
dim
0,
it
~~~ -G~J~ ~:
1, i
_~J~
1,2, .. . , r
1 , i
= 1,2, ... , r .
(33)
(34 )
Proof of Theorem 1 Proof of Sufficiency
0i satisfies the differential equation:
where T
E. F.
On substituting (31) into (1) and (4), the lemma 1S proved. Q.E.D.
2 , · ·· ,A q } defined by (S) .
where
[~J =
rank
{AI' A
[~J JJ~:J
0J[Si -G 0
1
which implies thar~.
Fl is a companion matrix of the minimal
[~l ~2Jhas
1
follows that
(27)
polynomial of
1j
(31)
Now since rank
[~l ~2J.
i
where
(26 )
polynomial of
1 I j1 J[W~ r Ii[ y. = =
1
1
where
This
implies that:
- G.
[~l ~J, [~l ~J is observable.
observable.)
~ ~i2 has [ °i ~ .
The proof of sufficiency of theorem 1 and properties of the contI~1lT5)now immediately f?lloW from references' on uS1ng the equ1valent representation for (1) given by (22). Proof of Necessity The proof of necessity of the theorem and the characterization result of the servo compensator now immediate ly follows from reference (l) on using (22)b) In particular, the proof is identical to reference , except there may be additional disturbance terms 6 ' 6 which depend on w. , y. added to the - 1 - 2 -1 -1 (1) equation appearing before (14) in reference , i.e .
1
235
I~* E.
w.
]
. -1-1 1=1 [
r
now becomes
(F. ;;;-. - G. y. )
.r~i ~i [ 1=1 ~*
L .
. -1 -1 -1-1 1=1
1=1
_ _
+
The Complementary Controller
§l __
Assume that A in (1) is stable and let the following robust (or weak robust) controller be applied to (1):
(F. w. - G. y . ) +
-1-1 -1-1
(35 ) where the elements of §l' §} are a linear c?mbination of time functions ~i (t , i = 1,2, ... ,~ ,where
::(~': .'\':'
o
f Il
~Win:::';;:.'0;;,:.
1
(~+ ~t +. .. +G"" t
) e
i ' l ,2,
(38) where C* 1;
+ B* e
(39)
is the general servo-compensator described by (15), (16) and where ~ is the output of the complemental)' controller for (1) defined by:
, ,'
1£ q1~2
q{l
~
(36) Equation (16) and (17) in reference(l)now will also have the additional disturbance terms §l' §2 appearing in them corresponding to (35). No other changes are made. The argument now follows exactly as in reference(l)with no changes, on noting that in the case of robust control, it is necessary that the controller satisfactorily re gulate (1) \foCC,,£ ; this is possible to do, only if the error e is physically measured and not re-constructed. The pathological case, which may arise for a weak robust controllef ~n the case D f 0 is discussed in reference 6 . It does not arise in the case of a robust controller, because it is impossible to have rank (C+oC+DKo) < r VoCC"!:' Q.E.D . CHARACTERIZATION OF THE STABILIZING COMPENSATOR: THE COMPLEMENTARY CONTROLLER
=~
+
Bu
(40)
where K, K are found so that the controllable o (stabilizable) system
( 41)
has a satisfactory stable dynamic response, by using conventional methods e.g., pole assignment, modal techniques, linear optimal control etc. This controller has the following important property:
In the robust or weak robust controller given by (14)-(18), the sole purpose of the stabilizing compensator S* is to stabilize the controllable (stabilizable), observable (detectable) system (18) given by (37) which is con tro ll.able (stabi lizab le), observable (detectable) if (Cm' A, B) is controllable (stabilizable) and observable (detectable), assuming that theorem 1 holds. It is clear that this can always be done by an observer . In the case (37) is observable, then the order of such an observer is equal to (n - r - r) where rank Cm = r +re' re ~ 0 . e Alternately, it could be done by the BraschPearson Compensator(7). In this case, if (A,B,C ) m is controllable and observable, complete pole assignment of the resultant closed loop system can be achieved (unlike the case when an observer is used) . The purpose of this section is to introduce a new compensator, the complementary compensator, which may be used for the stabilizing compensator S* It is a degenerate type of observer, which has certain engineering advantages over the conventional observer, when applied to the robust servomechanism problem. It is assumed that theorem 1 holds in this development.
Theorem 2 Let the robust (or weak robust) controller (38)-(40) be applied to (1); then the eigenvalues of the resultant closed loop system matrix are equal to: A. (A) ,A. [A+BKo 1 1 B*C + B*DKo where
BK]
(42)
C*+ B*DK
Ai (.) denotes the eigenvalues of (').
Proof The closed loop system, obtained by applying the robust controller (38)-(40) to (1) is ¥iven as follows:
y
=
(C
which can
~~.]=[::~:~*DKO ~~B*DK k-x 0
236
0
(44)
Ax
x
from which the result follows.
+
Bu ( 49)
Figure 2 gives a schematic diagram of this robust controller. Remark 6 It is seen that the complementary controller allows the response of the closed loop robust controller to be "just as fast as" the response of the open loop plant (assuming it is stable), but no faster. It is also seen that the complementary controller is a model of the plant. Note, that in this regard, the servo-compensator is, in a way, a duplicated model of the disturbance/reference-inputs of the system. Remark 7 The complementary controller described here is a generalization of the classical single-input, single-output complementary controller, e.g. see Shinskey(8J.
where A is chosen so that (A - AC ) is stable. This is always possible to do sincem(A, B) is at least stabilizable and (C , A) is at least detectable. Note that if (C ,mA, B) is controllable and m observable, then the observer can always be chosen so that the resultant closed loop system has arbitrarily fast dynamics. In this case, the following robust (or weak robust) controller is used: u
O
= K
o
x + Kt;
(50)
where t; is the output of the general servocompensator
t
C*E; + B*e
(51 )
The Complementary Controller with Output Feedback
described by (15), (16), and K , K are found so that the following stabilizable sys£em:
If t~e plant has output feedback applied to it,e.g. u = K(y - D u) + U O to stabilize or improve the m m plant's dynamic response, then the robust controller (38) becomes:
[~
U
O
= K x + Kt;
(45 )
o
1
t;
B (C+DK) -B DK C (C+DK
y
-DK
E;
u
(46)
ym =Cx+Du m m
and t; is the output of (39) . Ko' K are now found so that the controllable (stabilizable) system: ;.] [A+BKC [
~
=
t3*(C+~C m)
~J
o
[;}[:*D}o ,
u = (Ko' K)
[:]
(52) 0
"
has a satisfactory dynamic response. The stabilizability of (52) follows_from theorem I on noting that the pair and that A+Bl(:.A. I
°
rank [
1
C+DK
~[A+:K ~~~cmJ, [~J}iS -BK
~AC m-A.l 1
B1 :
• mk
stabilizable
[~U
(C+DKC
m
0) [:] +
:'
° :ACm (53)
-DK
1
= 2n + r
Du
has a satisfactory stable response. The eigenvalues_of the closed loop system are now equal to Ai (A+BKC ) together with the eigenvalues of the m resulting closed loop system matrix of (47).
x Ym p
m
m
o
m
be applied to (1) so as to provide a suitable satisfactory dynamic response to the following system:
A~ + BU
Kp +
u
u
0
C x + Du m m
(54 )
(A- AC ) p + A(y - D u) + Bu m m m
In this case, on following the argument used to establish (42), the eigenvalues of the resultant closed loop system will be equal to:
Let the full order observer: p= (A-AC )p+A(y -D u)+Bu, u = Kp+u
i=1,2, ... ,q
x is the output of the general complementary controller defined by:
The General Complementary Controller In the case A is unstable, then the following modification can be made:
J
B D
(47)
y
j
[X
u, u=(Ko,O,K) x-p BD t;
[;-j .
0)
where x is the output of the complementary controller: x = Ax + Bu
J
-BK 0J[x ~-p] = [A+BK *0 _ A-~C~ 0* x-p + [B0*
+B(K+K )
(48) A.(A+BK),A.(A-AC ),A.(A-ACI,A. 1.
1
m
1
IT!
[
1.*
*
BK
0
C+B D(K+K ) C*+B * DK o
which in the case (Cm' A, B) is controllable and
237
J
as there are outputs to be regulated, and if the output errors car. actually be physically measured, there "almost always" (in a precise technical sense) is a solution to the robust servomechanism'problem; if ei ther of these two assumptions are' not satisfied, then there never is a solution to the robust servomechanism problem.
observable, can be made arbitrarily fast. Properties of the Complementary Controller The complementary controller has three significant engineering advantages: (1) In the case of the complementary controller (40), there are no unspecified parameters in the controller, after the gains K, K have been determined. This is quite dif~erent from the observer in which one must specify the observer poles in some "ad hoc" manner. (2) In the case of the complementary controller (46), existing conventional output feedback controls which have been found to satisfactorily control the plant can be used in the final control configuration, i.e. one does not have to "throwaway" satisfactory controllers already existing on the plant in the final control configuration. (3) In the case of the general complementary controller (54), one can apply the robust controller (50) to the plant in a 3-stage sequential manner. First the plant's dynamics are "speeded up", without any servo-compensator being applied, by using the local observer (48). Then the gains (K , K) are found so that the dynamics of (52) ~re "at least as fast" as the dynamics of the controlled plant. Finally, the servo-compensator and complementary controller together are then applied to the controlled plant, without changing the existing control configuration found in step one. A disadvantage of the complementary controller is that it has higher dimensions than required by the minimum order observer. CONCLUSIONS Necessary and sufficient conditions have been obtained for the existence of a solution to the robust servomechanism problem, so that asymptotic track i ng, in the presence of either measurable or unmeasurable disturbances, occurs for all plant parameter variations. A characterization of all robust controllers has also been obtained; it is shown that any robust controller must consist of two separate devi ces: (i) a servo- compensator, which is quite distinct from an observer and which corresponds to a generalization of the integral controller in classical control theory, (ii) a stabilizing compensator, whose sole purpose is to stabilize the augmented system obtained when the servo-compensator is connected to the plant. The servo-compensator consists of r (where r is the number of outputs to be regulated) unstable compensators with identical dynamics, corresponding to the class of disturbance/reference-inputs, connected via feedback, to the error outputs of the plant. This servo-compensator structure is an essential feature of any robust controller. The above characterization result is extremely important, since it justifies the widespread use of error feedback in practical control configurations. It is also shown that, if there are at least the same number of independent control inputs
238
A new type of stabilizing compensator, called a complementary controller, has also been introduced which has engineering advantages over the conven~ional observer, when used in the robust servomechanism problem. It is a degenerate type of observer and corresponds to a generalization of the single-input, single-output complementary contro!ler of classical control theory, e.g. see Shinskey(8J. REFERENCES (1) Davison E.J. ,"The Feedforward,Feedback and Robust Control of a General Servomechanism Problem-Part Il" ,Department of Electrical Engineering, University of Toronto, Control System Report No. 7305 April 1973 (Revised Nov. 1973), presented at 11 th Allerton Conference on Ci rcui t and Sys tem Theory, October 1973. (2) Davison E.J.,Wang S.H., "Properties and Calculation of Transmission Zeros of Linear Multivariable Systems", Automatica vol. 10, No. 6, 1974, pp.643-6s8. (3) Heyrnann M., "On the Input and Output Reducibili ty of Multivariable Linear Systems", IEEE Trans. on Automatic Control, vol. AC-ls, No.s, October, 1970, pp. 563-569. (4) Davison, E. J., Wang, S. H., "Properties of Linear Multivariable Systems Subject to Arbitrary Output and State Feedback", IEEE Trans. on Automatic Control, Vol. AC-18, No.l, February 1973, pp.2432. (5) Davison E. J., "The Output Control of Linear TimeInvariant Multivariable Systems with Unmeasurable Arbitrary Dis turbances", IEEE Trans. on Automatic Control, Vol.AC-17, No.s,October 1972,pp . 621-630. (6) Davison E.J., "A Generalization of the Output Control of Linear Multivariable Systems with Unmeasurable Arbitrary Disturbances",Department of Electrical Engineering, University of Toront~ Control Systems Report No.74l4 June 1974, Submitted for publication. (7) Brasch F.M., Pearson J .B., "Pole Placement using
Dynamic Compensators", IEEE Trans.on Automatic Control, Vol.AC-ls, No.l, February 1970, pp.3443. (8) Shinskey F.G., Process Control Systems, McGraw Hill, 1967, p.l04. (9) Francis B., Sebakhy O.A . , Wonham W.M.,"Synthesis of Multivariable Regulators: The Internal Model Principle", Applied Math. & Optimization, Vol.l, No.l, 1974, pp.64-86. (lO)Staat s P.W., Pearson J.B., "Robust Solution of the Linear Servomechanism Problem", Tech. Report No. 7401, Electrical Engineering Department, Rice University, Houston, Texas, September 1974.
Servo Compensotor
(15)
Figure 1
The Genera l Robus t Servomechanism Controll e r. y is th e Actua l Output of the Pl ant obtained from Y ' m
W
Plont U
Compensator
(15)
x= Ax + Bu + Ew Y = Cx + Du +Fw Ym
Figure 2
= Cmx +
Dm u + Fm W
General Rob us t Controll e r with Comp l ementary Contro ll er CA is assumed stable) . y i s tile Actual Out put obtained from Yn ·
239
Y
E.J. Davison A. Goldenberg Department of Electrical Engineering University of Toronto Toronto, Canada
ABSTRACT The robust control of a completely general servomechanism problem, is considered in this paper. Necessary and sufficient conditions, together with a characterization of all robust controllers which enables asymptotic tracking to occur, independent of disturbances in the plant and perturbations in the plant parameters and gains of the system is obtained. (It new type of compensator, introduced in reference ,called a servo-compensator which is quite distinct from an observer is shown to play an essential role in the robust servomechanism problem. It is shown that this compensator, which corresponds to an integral controller in classical control theory, must be used in any servomechanism problem to assure that the controlled system is stabilizable and achieves robust control; in particular, it is shown that a robust controller of a general servomechanism problem must consist of two devices (i) a servo-compensator and (ii) a stabilizing compensator. A study of the stabilizing compensator is made; in parti cular, i t is shown that a new type of stabi l i zing compensator called a complementary controller, may be used together with the servo-compensator to form a robust controller for the servomechanism problem. INTRODUCTION This paper deals with an extension of reference (1) for the case of finding a robust controller for a system so that asymptotic tracking, in the presence of disturbances, occurs independent of perturb at ions in the plant's or gain parameters of the system. The following problem is considered: A linear, time-invariant system (plant) is described by the following equations: x = Ax
+
Bu
+
Ew
Cx
+
Du
+
Fw
y
e ~ Y - Yref
(1)
m where x ~ Rn is the state, u ~ R is the input, y ~Rr is the output which is to be regulated and w ~RSi is a disturbance vector which may be either measurable or unmeasurable. e ~ Rr is the error in
the system, which is the difference between the ou~ put y and the specified reference input Yref' The disturbance vector is assumed to satisfy the follo~ ing equation: (2 ) n
where zl ~ R I and where (Cl' AI) is observable and zl (0) mayor may not be known. The specified reference input vector Yref ~ Rr is assumed to satisfy the following equation: y ref = Go, z2
(3)
o
C2 z2
n2 where z2 ~ R and where (C 2 , A2 ) is observable and Z2(0) is known. It is desired to find a robust (or weak robust) controller, in a precise sense to be defined later, for (1) so that the errornle ~ 0 as t rt = 'tJ x(O) ~Rn , 'tJ zl (0) ~ R ,\f z2(0) ~ R 2 and such that the resultant controlled system is stable. It is assumed that the outputs Ym where: Ym = Cmx
+
Dm u
+
Fm w
(4)
are the only outputs which are available for measurement. It is assumed without loss of gpnerality that rank C = r, rank B = m, rank L~ = rank Cl = Si , rank G = rank C = dim (0). ~n addition, it is 2 assumed for non-tri viali ty, that all the eigenva lues of AI' A2 are in the closed right hand part of the comp lex plane, and that max {rank rank (G)} ~ 1.
[n'
Davison (1) has considered the above problem when (1) is written as follows:
i = Ax e
= Cx
+
Bu
+
~
(Ei' 0)
i=l +
Du
+
f
(5)
(F i' - Gi )
i=l This work has been supported by the National Research Council of Canada under Grant No. A4396.
231
where
r
L F.w. is the output, t~~lsp~c~fied reference input,
y = Cx + Du +
Definition (2)
~ G.y. is i=l I I r and Ym = Cmx + D u + L Fm w. are the only outputs m i=l I I which are available for measurement and wi ' y. i = 1,2, ... ,r each satisfy a different differential equation. References (9) , (10) have considered the problem when D = 0 by using geometric methods. Yref =
The following definitions are required: Definition
(7)
Remark 3
2
1
Let the coefficients 0., i = 1,2, ... ,q be given by the coefficients of theIpolynomial ~ (A - Ai) where i=l A. are given by (8), i . e.
Definition Gi ven a real matrix denote
n+ min (r,m)
<
Let all the zeros of the minimal polynomial of Al be given by {A!} and let all the zeros of the minimal polynoffilal,of A2 be given by {A } . Then 2 define
9iven the system (1) assume a feedback controller n = E n + Elx ,u = Klx + K2n has been applied; 2 then the controlled system is said to be{[A 0 ] controllable (stabilizable) if the pair E E ' controllable (stabilizable).
BJ D
rank [ A -C AI
It may be noted that the set of transmission zeros may contain zero elements, a finite number of symmetric complex numbers or include the whole complex plane.
PRELIMINARIES
[~J}iS
Given the system x = Ax + Bu, y = Cx + Du, denoted by (C,A,B,D), the transmission zeros of (C,A,B,D) are defined to be the set of complex numbers A which satisfy the following inequality:
m.. (. fl IJ
fl
E:
E:
=
M = {m .. } IJ
let
I
M (. fl
q q 1 q-2 q + ... +02A+ol~ IT (A- A.) A + OqA - +Oq_lA i=l I
E:
where:
{m·.1 Im·IJ·1 IJ
<
d
and let
(6)
C be a
qxq 1
Defini tion (1)
o
Given the system (1), suppose that there exists a controller so that asymptotic regulation takes place and so that the resultant controlled system is stable. Let the plant parameters A, Band C now be perturbed, i.e. A 7 A + aA, B-7 B + oB~ C 7 C + OC where oA (. flE: ' oB (. fl , oC (. fl E: . Then E if there exists an E > 0 so that the controlled system is still stable and such that asymptotic regulation still take s place for VoA (. fl , VoB (. 11 , VaC (. fl (VOA (. fl , VoB (. fl ), the contro1rer is E E E E said to be a robust controller (weak robust controll e r) Remark 1 Note that there is no assumption in the above definition that £ > 0 is arbitrary small, i . e . finite changes, and not just arbitrarily small perturbations, are allowed in the plant I s parameters
(9)
companion matrix given by:
0 1
l]
(10)
Defini tion The actual output y is defined to be equal to the physical outputs of the plant which are to be regulated corresponding to y in (1). The following lemma is required in the development to follow : Lemma 1 (3) : I f (C,A) is obs.e rvable, and i f A has k invariant polynomials, then there exists a coordinate transformation T so that (CT- l , TAT-I) is observabl e with :
A, B, C.
where Remark 2 The motivation for the above definition comes from the fact that in practice, the equations describing a plant of th e form x = Ax + Bu, y = Cx + Du are always approximate, i.e. they really arise from a linearization, about an operating point, of the nonlinear equations = f(x,z,u), ~z = g(x,z,u), y = h(x,z,u) where z corresponds to the "high frequency modes" of the plant which have been negl e cted and ~ 7 O. See reference (l) for details of this motivation.
F. I
o0 ·· [· x
1
0
x
0 ... 0J 1 ... 0
x
.. .
(12)
x
and x designates a number not necessarily zero, and where rank(Cr l ) > k . Here F . , i = 1,2, ... ,k are the companion matrices of the invariant polynomials a (of degree ~i) of A with a the i l minimal pol ynomial of A and with a dIviding i a i_I' i = 2,3, ... ,k .
x
232
{block diag(C, C, ... ,C) , B}
DEVELOPMENT : THE SERVO-COMPENSATOR The following main result is obtained:
( 17)
r matrices
Theorem 1 is controllable . x is the output of a stabilizing compens ator S* with inputs y, u, t; where S*, K , K are found using standardmmethods, e.g. o observer theory, to stabilize the following augmented system:
A necessary and sufficient condition that there exists either a robust or weak robust linear timeinvariant controller for (1) such that e ~ 0 as t ~ for all measurable or unmeasurable disturbances w described by (2), and for all specified reference-inputs Yref described by (3), and such that the controlled system is stabilizable, is that the following conditions ' all hold : 00
(a)
(18)
(A, B) Le stabilizable.
(b)
(Cm' A) be detectable .
(c)
m > r
(d)
The transmission zeros of (C, A, B, D) do not coincide with Ai' i = 1,2, ... ,q given by (8).
(e)
In the case of robust control, Ym must contain the actual output y, i .e. there exists a nonsingular transformation T so that TYm = [~-l, where y is the actual
where the p air l[:*c
(I r , 0) T* (Ym- Dmu)
+
Du
In the case of weak robust control, the most general controller structure which regulates (1) has, except for a pathological case, exactly the same structure above except that the output y used to form the error input to (15) may be re-constructed and need not be actually measured. The pathological case occurs only if D F 0 and can only occur provided there exists a gain matrix K* which lies on a hypersurface(4) in the parameter space of (C, D), given by the condition rank(C + DK*C ) < r In this pathological case, there may exist m controllers of smaller order than rq. This pathological case is not very important however, since (i) the hypersurface condition is "almost never"(4) satisfied for any gain matrices of the system, (ii) the resultant controller would be hyper-sensitive with respect to feedback gain perturbations in the system.
Conditions (c), (d) reduce to the following condition: A - A.I B] = n+r, i=1,2, .... ,q(l3) rank [ C ~
D
Controller Structure The most general robust controller which regulates (1) must have the following structure:
x+
Kt;
(14)
where t;, a rq vector, is the output of the general servo-compensator(l) given by :
* C t;
+
* Be, e
fi
y - Yref
(15)
where T
block diag
(C, C, ... ,C)
T
The robust (or weak robust) controller obtained by putting T=Iand letting B=bl ock di ag (y y ,." " y r ) r 2 in (16) where Yl = Y2 ="'=Y r = (00 .. . 01) gi ve s :
-1
r matrices
u = Kx 0
(16) B*
T
where
B
t;
is a
rq
Kt;
+
+
where
T
C* = block diag(C, C, . .. ,C) r matrices
233
(19)
vector given by:
t; = C* t;
is a nonsingular real matrix, C is defined in (10) and B E:: Rrq x r is a real matrix of rank r with the property that:
where
~*J}iS
is observable (detectable) if (Cm' A) is observable (detectable). In the above robust controller, the error input e to the general servo-compensator (15) must be obtained by using the actual output y obtained from Ym' Note that the above robust controller always has order > rq. Figure 1 gives a schematic diagram of this robust controller .
Remark 4
u = K o
controllable
(stabilizable) i f (A, B) is controllabte (stabilizable) and the pair{[~m ~J' ~*C
output y. InYmthe case of weak robust control, Ym must contain enough information to be able to re-construct y from it, i . e. there exists a*nonsingular tra[~Jsformaf~onJ T* so that T (Cmx + Fm w) = C x + LFm w ; in this case, y
~*J, r:*DJ~iS
B* e
(20)
provided only that tne resultant closed loop system remains stable. (Zl)
4.
and is particularly simple to implement. The compensator (ZO) is called a servo-compensator and was introciuced in references(1),(5) It is to be emphasized that the servo-compensator
(ZO) or more generally the general servo-compensator (15) is not some type of observer. It is a compensator in its own right, quite distinct from an observer. It corresponds in classical control theory, for the case of a system subj ect to constant disturbances, to an integral controller. Remark 5 It is to be noted that the general servo-compensator (15) has minimal order rq and is a feedback controller, consisting of r unstable compensators with identical dynamics corresponding to the class of disturbance/reference-inputs, connected to the error outputs of the system. Note that all error outputs of the system must be connected ~the servo-compensator. This feedback structure is an essential feature of any robust controller.
The proof of Theorem 1 follows almost immediately from reference (1) , after (1), (4) are rewritten in a form similar to (5). The following lemma is required:
Properties of the Robust (or Weak Robust) Controller 1.
Lemma Z
In the above robust controller, it is not necessary to know E, F, G, Cl' CZ' AI' AZ . It is essential however to know the zeros of the minimal polynomials of AI' A .
The system (1)-(4) is equivalent to the following representation:
r
z
Z.
x = Ax + Bu +
L i=l
The robust controller is robust for any finite changes (and not just arbitrarily small perturbations) in the plant parameters A, B, C, D, feedback gain parameters K, K, stabilizing compensator parameter~ of S* and for any changes in the order of the assumed mathematical model describing the plant and also for any changes in the order of the stabilizing compensator S*, provided only that the resultant closed loop system remains stable. It is also robust for finite perturbations in some of the parameters of the general servocompensator (15), in particular for perturbations in the elements of T , and parameters of the matrix B provided that the resultant closed loop system remains stable. Note that , 0Z' " .,0 it is essential that the elements of the servo-compensator (15) to be 1 fixed q precisely t ,and that the error e 6 (y - Y f) = re input to (15) to be measured precisely, i.e. it is assumed that the actual output y is obtained from Ym in an exact way.
w.
(E. ,0)
1
[J 1
+
Yi
r
[j
vi
L L (.E.
, 0)
i=lj=l J 1
w.] 1
jYi (ZZ)
[w.~
tW~
r (F.,-G.) 1+ Ir vi e=Cx+DJ+ L L (.F.,- . G. J 1 i=l 1 1 Yi i=l j=l J 1 J 1 / i i r v Fm w. + L 1 1 L L i=l i=l j=l
r
y = C x + D u m m III E.
where rank [ F:
+
0 ]
G i
~ 1, i
.F
J
m
.w.
J 1
1,Z, ... ,r and where the
elements of the vectors wi' Yi' i satisfy the differential equation:
°
3.
Note that a robust controller exists for "almost all" (C, A, B, D) systems' provided that (i) m > r, (ii) the outputs to be regulated can be physically measured; i.e. the class of (C, A, B, D) systems in which Theorem 1 does not hold, lies on a hypersurface in the parameter space of (C, A, B, D). (See reference(4) for a definition and properties of a hypersurf~c~.) This immediately follows from reference ll ). Thus if a system to be regulated has at least the same number of independent inputs as there are independent outputs to be regulated and if these outputs are physically measurable, there "almost alway!l' is a solution to the robust servomechanism problem. If either (i) or (ii) above fail to hold, then there never is a solution to the robust servomechanism problem. Similarly a weak robust controller exists for "almost all" (C, A, B, D) systems provided that (i) m > r, (ii) the outputs to be regulated can be reconstructed; if either (i) or (ii) fails to hold, there is never a solution to the weak robust servomechanism problem.
1, Z, . . . ,r
(Z 3)
The weak robust controller is robust for any finite changes in the plant parameters A, B, feedback gain parameters Ko' K, stabilizing compensator parameters of S* and servocompensator parameters T, B
where the initial conditions of (Z3) are known for the elements of Yi and known (unknown) for the elements of w. if the disturbances ware r measurable (un~easurable), and where i~l {Si}= t This is usually the case in practice ,especially when the disturbance-reference inputs are polynomial or sinusoidal functions of time.
234
{AI' A ,···,A} where {Si}, i = 1,2, ... ,r denotes
2
the character~atic roots of (23). The elements of the vectors .W., .y., i=1,2, ... ,r satisfy a differential] ~quati~n, with initial conditions which depend on the initial conditions of (23) and, with characteristic roots equal to those of (23), except that the multiplicity of all roots is now reduced by at least one. Proof of Lemma 2
= Ax
+ Bu + (E, 0)
e
= ex
+
Du
+
(F, -G)
[~J (24 )
[~J
where
[~~J =[ ~l ~J [:J (25 )
(30)
where the elements of w ' Yi satisfy the i differential equation (23), and where the elements of . W . , .y. satisfy a differential equation, with 1
]
1
initial conditions which depend on the initial conditions of (23), with characteristic roots equal to those of (23), except that the multiplicity of all roots is now reduced by at least one. Here Si' Ti are Q x dim(w ) , dim(o) x dim(Yi) matrices i respectively with the property that rank Si= dim(w ), i rank Ti = dim(Yi), i = l, .. . ,r . (This follows since the
[:J=[~l ~J[:~J and where
E [Si [W]= ° i=l 0
]
The system (1) may be written as : x
.. (~ 1 '-l) , the property that 0'1 2=0'1, 1 1 0'3= 1 0'1 1, ... ,0 1. '1." = °'1 (2 S) may be expanded into the fOllowing: 1
pair{[~l ~2J' [~l ~2J}iS o
Applying now lemma 1, let a transformation applied to (25) to give:
T be
0J~[EF
[ Ei F. -G .
rank
Note that the minimal zeros equal to
On expanding out the disturbance, reference-inputs
[~J of (27), the following is obtained : (2S)
°i
(29)
0'. 1
~[:l :l]. .1
.2
Tk
Tk
Oil]
On noting that 0i
0 - G.
j1 J~W'J y.
j1j1'1
Q,
°J,i
T.
rank
1,2, ... , r
(32)
1
[~J =
dim
0,
it
~~~ -G~J~ ~:
1, i
_~J~
1,2, .. . , r
1 , i
= 1,2, ... , r .
(33)
(34 )
Proof of Theorem 1 Proof of Sufficiency
0i satisfies the differential equation:
where T
E. F.
On substituting (31) into (1) and (4), the lemma 1S proved. Q.E.D.
2 , · ·· ,A q } defined by (S) .
where
[~J =
rank
{AI' A
[~J JJ~:J
0J[Si -G 0
1
which implies thar~.
Fl is a companion matrix of the minimal
[~l ~2Jhas
1
follows that
(27)
polynomial of
1j
(31)
Now since rank
[~l ~2J.
i
where
(26 )
polynomial of
1 I j1 J[W~ r Ii[ y. = =
1
1
where
This
implies that:
- G.
[~l ~J, [~l ~J is observable.
observable.)
~ ~i2 has [ °i ~ .
The proof of sufficiency of theorem 1 and properties of the contI~1lT5)now immediately f?lloW from references' on uS1ng the equ1valent representation for (1) given by (22). Proof of Necessity The proof of necessity of the theorem and the characterization result of the servo compensator now immediate ly follows from reference (l) on using (22)b) In particular, the proof is identical to reference , except there may be additional disturbance terms 6 ' 6 which depend on w. , y. added to the - 1 - 2 -1 -1 (1) equation appearing before (14) in reference , i.e .
1
235
I~* E.
w.
]
. -1-1 1=1 [
r
now becomes
(F. ;;;-. - G. y. )
.r~i ~i [ 1=1 ~*
L .
. -1 -1 -1-1 1=1
1=1
_ _
+
The Complementary Controller
§l __
Assume that A in (1) is stable and let the following robust (or weak robust) controller be applied to (1):
(F. w. - G. y . ) +
-1-1 -1-1
(35 ) where the elements of §l' §} are a linear c?mbination of time functions ~i (t , i = 1,2, ... ,~ ,where
::(~': .'\':'
o
f Il
~Win:::';;:.'0;;,:.
1
(~+ ~t +. .. +G"" t
) e
i ' l ,2,
(38) where C* 1;
+ B* e
(39)
is the general servo-compensator described by (15), (16) and where ~ is the output of the complemental)' controller for (1) defined by:
, ,'
1£ q1~2
q{l
~
(36) Equation (16) and (17) in reference(l)now will also have the additional disturbance terms §l' §2 appearing in them corresponding to (35). No other changes are made. The argument now follows exactly as in reference(l)with no changes, on noting that in the case of robust control, it is necessary that the controller satisfactorily re gulate (1) \foCC,,£ ; this is possible to do, only if the error e is physically measured and not re-constructed. The pathological case, which may arise for a weak robust controllef ~n the case D f 0 is discussed in reference 6 . It does not arise in the case of a robust controller, because it is impossible to have rank (C+oC+DKo) < r VoCC"!:' Q.E.D . CHARACTERIZATION OF THE STABILIZING COMPENSATOR: THE COMPLEMENTARY CONTROLLER
=~
+
Bu
(40)
where K, K are found so that the controllable o (stabilizable) system
( 41)
has a satisfactory stable dynamic response, by using conventional methods e.g., pole assignment, modal techniques, linear optimal control etc. This controller has the following important property:
In the robust or weak robust controller given by (14)-(18), the sole purpose of the stabilizing compensator S* is to stabilize the controllable (stabilizable), observable (detectable) system (18) given by (37) which is con tro ll.able (stabi lizab le), observable (detectable) if (Cm' A, B) is controllable (stabilizable) and observable (detectable), assuming that theorem 1 holds. It is clear that this can always be done by an observer . In the case (37) is observable, then the order of such an observer is equal to (n - r - r) where rank Cm = r +re' re ~ 0 . e Alternately, it could be done by the BraschPearson Compensator(7). In this case, if (A,B,C ) m is controllable and observable, complete pole assignment of the resultant closed loop system can be achieved (unlike the case when an observer is used) . The purpose of this section is to introduce a new compensator, the complementary compensator, which may be used for the stabilizing compensator S* It is a degenerate type of observer, which has certain engineering advantages over the conventional observer, when applied to the robust servomechanism problem. It is assumed that theorem 1 holds in this development.
Theorem 2 Let the robust (or weak robust) controller (38)-(40) be applied to (1); then the eigenvalues of the resultant closed loop system matrix are equal to: A. (A) ,A. [A+BKo 1 1 B*C + B*DKo where
BK]
(42)
C*+ B*DK
Ai (.) denotes the eigenvalues of (').
Proof The closed loop system, obtained by applying the robust controller (38)-(40) to (1) is ¥iven as follows:
y
=
(C
which can
~~.]=[::~:~*DKO ~~B*DK k-x 0
236
0
(44)
Ax
x
from which the result follows.
+
Bu ( 49)
Figure 2 gives a schematic diagram of this robust controller. Remark 6 It is seen that the complementary controller allows the response of the closed loop robust controller to be "just as fast as" the response of the open loop plant (assuming it is stable), but no faster. It is also seen that the complementary controller is a model of the plant. Note, that in this regard, the servo-compensator is, in a way, a duplicated model of the disturbance/reference-inputs of the system. Remark 7 The complementary controller described here is a generalization of the classical single-input, single-output complementary controller, e.g. see Shinskey(8J.
where A is chosen so that (A - AC ) is stable. This is always possible to do sincem(A, B) is at least stabilizable and (C , A) is at least detectable. Note that if (C ,mA, B) is controllable and m observable, then the observer can always be chosen so that the resultant closed loop system has arbitrarily fast dynamics. In this case, the following robust (or weak robust) controller is used: u
O
= K
o
x + Kt;
(50)
where t; is the output of the general servocompensator
t
C*E; + B*e
(51 )
The Complementary Controller with Output Feedback
described by (15), (16), and K , K are found so that the following stabilizable sys£em:
If t~e plant has output feedback applied to it,e.g. u = K(y - D u) + U O to stabilize or improve the m m plant's dynamic response, then the robust controller (38) becomes:
[~
U
O
= K x + Kt;
(45 )
o
1
t;
B (C+DK) -B DK C (C+DK
y
-DK
E;
u
(46)
ym =Cx+Du m m
and t; is the output of (39) . Ko' K are now found so that the controllable (stabilizable) system: ;.] [A+BKC [
~
=
t3*(C+~C m)
~J
o
[;}[:*D}o ,
u = (Ko' K)
[:]
(52) 0
"
has a satisfactory dynamic response. The stabilizability of (52) follows_from theorem I on noting that the pair and that A+Bl(:.A. I
°
rank [
1
C+DK
~[A+:K ~~~cmJ, [~J}iS -BK
~AC m-A.l 1
B1 :
• mk
stabilizable
[~U
(C+DKC
m
0) [:] +
:'
° :ACm (53)
-DK
1
= 2n + r
Du
has a satisfactory stable response. The eigenvalues_of the closed loop system are now equal to Ai (A+BKC ) together with the eigenvalues of the m resulting closed loop system matrix of (47).
x Ym p
m
m
o
m
be applied to (1) so as to provide a suitable satisfactory dynamic response to the following system:
A~ + BU
Kp +
u
u
0
C x + Du m m
(54 )
(A- AC ) p + A(y - D u) + Bu m m m
In this case, on following the argument used to establish (42), the eigenvalues of the resultant closed loop system will be equal to:
Let the full order observer: p= (A-AC )p+A(y -D u)+Bu, u = Kp+u
i=1,2, ... ,q
x is the output of the general complementary controller defined by:
The General Complementary Controller In the case A is unstable, then the following modification can be made:
J
B D
(47)
y
j
[X
u, u=(Ko,O,K) x-p BD t;
[;-j .
0)
where x is the output of the complementary controller: x = Ax + Bu
J
-BK 0J[x ~-p] = [A+BK *0 _ A-~C~ 0* x-p + [B0*
+B(K+K )
(48) A.(A+BK),A.(A-AC ),A.(A-ACI,A. 1.
1
m
1
IT!
[
1.*
*
BK
0
C+B D(K+K ) C*+B * DK o
which in the case (Cm' A, B) is controllable and
237
J
as there are outputs to be regulated, and if the output errors car. actually be physically measured, there "almost always" (in a precise technical sense) is a solution to the robust servomechanism'problem; if ei ther of these two assumptions are' not satisfied, then there never is a solution to the robust servomechanism problem.
observable, can be made arbitrarily fast. Properties of the Complementary Controller The complementary controller has three significant engineering advantages: (1) In the case of the complementary controller (40), there are no unspecified parameters in the controller, after the gains K, K have been determined. This is quite dif~erent from the observer in which one must specify the observer poles in some "ad hoc" manner. (2) In the case of the complementary controller (46), existing conventional output feedback controls which have been found to satisfactorily control the plant can be used in the final control configuration, i.e. one does not have to "throwaway" satisfactory controllers already existing on the plant in the final control configuration. (3) In the case of the general complementary controller (54), one can apply the robust controller (50) to the plant in a 3-stage sequential manner. First the plant's dynamics are "speeded up", without any servo-compensator being applied, by using the local observer (48). Then the gains (K , K) are found so that the dynamics of (52) ~re "at least as fast" as the dynamics of the controlled plant. Finally, the servo-compensator and complementary controller together are then applied to the controlled plant, without changing the existing control configuration found in step one. A disadvantage of the complementary controller is that it has higher dimensions than required by the minimum order observer. CONCLUSIONS Necessary and sufficient conditions have been obtained for the existence of a solution to the robust servomechanism problem, so that asymptotic track i ng, in the presence of either measurable or unmeasurable disturbances, occurs for all plant parameter variations. A characterization of all robust controllers has also been obtained; it is shown that any robust controller must consist of two separate devi ces: (i) a servo- compensator, which is quite distinct from an observer and which corresponds to a generalization of the integral controller in classical control theory, (ii) a stabilizing compensator, whose sole purpose is to stabilize the augmented system obtained when the servo-compensator is connected to the plant. The servo-compensator consists of r (where r is the number of outputs to be regulated) unstable compensators with identical dynamics, corresponding to the class of disturbance/reference-inputs, connected via feedback, to the error outputs of the plant. This servo-compensator structure is an essential feature of any robust controller. The above characterization result is extremely important, since it justifies the widespread use of error feedback in practical control configurations. It is also shown that, if there are at least the same number of independent control inputs
238
A new type of stabilizing compensator, called a complementary controller, has also been introduced which has engineering advantages over the conven~ional observer, when used in the robust servomechanism problem. It is a degenerate type of observer and corresponds to a generalization of the single-input, single-output complementary contro!ler of classical control theory, e.g. see Shinskey(8J. REFERENCES (1) Davison E.J. ,"The Feedforward,Feedback and Robust Control of a General Servomechanism Problem-Part Il" ,Department of Electrical Engineering, University of Toronto, Control System Report No. 7305 April 1973 (Revised Nov. 1973), presented at 11 th Allerton Conference on Ci rcui t and Sys tem Theory, October 1973. (2) Davison E.J.,Wang S.H., "Properties and Calculation of Transmission Zeros of Linear Multivariable Systems", Automatica vol. 10, No. 6, 1974, pp.643-6s8. (3) Heyrnann M., "On the Input and Output Reducibili ty of Multivariable Linear Systems", IEEE Trans. on Automatic Control, vol. AC-ls, No.s, October, 1970, pp. 563-569. (4) Davison, E. J., Wang, S. H., "Properties of Linear Multivariable Systems Subject to Arbitrary Output and State Feedback", IEEE Trans. on Automatic Control, Vol. AC-18, No.l, February 1973, pp.2432. (5) Davison E. J., "The Output Control of Linear TimeInvariant Multivariable Systems with Unmeasurable Arbitrary Dis turbances", IEEE Trans. on Automatic Control, Vol.AC-17, No.s,October 1972,pp . 621-630. (6) Davison E.J., "A Generalization of the Output Control of Linear Multivariable Systems with Unmeasurable Arbitrary Disturbances",Department of Electrical Engineering, University of Toront~ Control Systems Report No.74l4 June 1974, Submitted for publication. (7) Brasch F.M., Pearson J .B., "Pole Placement using
Dynamic Compensators", IEEE Trans.on Automatic Control, Vol.AC-ls, No.l, February 1970, pp.3443. (8) Shinskey F.G., Process Control Systems, McGraw Hill, 1967, p.l04. (9) Francis B., Sebakhy O.A . , Wonham W.M.,"Synthesis of Multivariable Regulators: The Internal Model Principle", Applied Math. & Optimization, Vol.l, No.l, 1974, pp.64-86. (lO)Staat s P.W., Pearson J.B., "Robust Solution of the Linear Servomechanism Problem", Tech. Report No. 7401, Electrical Engineering Department, Rice University, Houston, Texas, September 1974.
Servo Compensotor
(15)
Figure 1
The Genera l Robus t Servomechanism Controll e r. y is th e Actua l Output of the Pl ant obtained from Y ' m
W
Plont U
Compensator
(15)
x= Ax + Bu + Ew Y = Cx + Du +Fw Ym
Figure 2
= Cmx +
Dm u + Fm W
General Rob us t Controll e r with Comp l ementary Contro ll er CA is assumed stable) . y i s tile Actual Out put obtained from Yn ·
239
Y