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A game theoretic approach to robust controller synthesis
Computers them. Engng. Vol. 14. No. 4/S, pp. 381-389, Printed in Great Britain. All rights reserved
A
GAME
1990 Copyright
THEORETIC CONTROLLER
APPROACH TO SYNTHESIS
009%1354/90 $3.00 + 0.00 0 1990 Pergamon Press plc
ROBUST
V. MANOUSIOUTHAKIS Department of Chemical Engineering, University of California Los Angeles, Los Angeles, CA 90024, U.S.A. (Received
6 Nouemher
1989;
received
for publicorion
28 November
1989)
Abstract-This paper presents a novel game theoretic approach to robust controller synthesis. The main advantage of this problem formulation is that it provides simultaneous controller synthesis and process model selection. The approach is illustrated with several numerical examples.
1. INTRODUCTION
the last few years, control system design paradigms continuously incorporate an ever increasing number of issues in an effort to reduce the gap between conceptual controller design and implementable controller design. The current status of this evolution calls for the satisfaction of control objectives, constraints and specifications in the presence of uncertainty (i.e. for all process models that belong to a process uncertainty description given a priori). Current control system design methodologies cannot easily address this issue. In that respect, the principal disadvantage of model predictive control (MPC) stems from its attempt to specify (in an optimal way) the system input u (see Fig. 1). In doing so, MPC fails to recognize that within a closed-loop setting the input u is no longer an independent variable. Indeed, as the disturbance d and/or the process description P vary, the only independent parameter at the designer’s disposal is the controller algorithm C. Robust control system design methodologies, on the other hand, have avoided this flaw by focusing on the design of the controller C. However, their dependence on the notion of a nominal model has limited their success to linear systems and to frequency-dependent uncertainty descriptions and performance specifications. Indeed, the theoretical basis of these methods are, nominal model related, necessary and sufficient conditions for robust stability. We feel that as the scope of robust control expands to nonfrequency-dependent uncertainty descriptions and to nonlinear systems the derivation of similar conditions will become increasingly difficult and will require the use of so general uncertainty descriptions that the resulting controllers will be overly conservative. We also feel that as control system performance specifications tighten, it will be of paramount importance to identify the critical modeling information for improved control system performance. Current robust control methods do not address this important task. In
381
In this paper we present a novel control system design methodology that addresses these issues. The main advantage of this novel game theoretic problem formulation is that it avoids the notion of a nominal process model and provides simultaneous controller synthesis and process model selection. The remaining paper is structured as follows: in Section 2 the proposed design methodology is presented. In Section 3 numerical examples illustrating its use are provided. In Section 4 the implications of the gametheoretic approach for nonlinear robust control are discussed. Finally, in Section 5 we draw conclusions and discuss some of our current research efforts. 2.
THE
GAME
THEORETIC
APPROACH
Consider the F-P-L-C control scheme (Fig. 2) (Manousiouthakis, 1985, 1987a, b) which represents the most general way to interconnect a process P and its controller C. For given filters, F, L, let N = n(P, C) denote the map relating the internal (2 , , Zz, &, &, ii+, 5,) and external (U,, fi2) signals of this control scheme. It is well known (Francis, 1986) that closed-loop performance requirements can often be. expressed in terms of the “smallness” of norm of the II w,nw* IllPrvthe inducedextended-p weighted closed-loop map H (IV,, W, represent pstable performance weights). As pointed out earlier, H = H(P, C) is a function of P and C. The controller C belongs to a known set C while the plant P belongs to a set P which is also known (e.g. C may consist of all rational transfer functions while P may be defined through the notions of a nominal plant and a bounded additive perturbation). The task of the designer can then be stated as: “Identify a controller C E C that achieves the best possible bound on the worst closed-loop performance level that can occur as P varies over P”. Given the aforementioned notion of closed-loop performance the above task can be mathematically quantified as:
(1)
V. MANOUSIOIITHAKIS
382
d
Fig. 1. Classical feedback scheme.
Therefore, robust controller synthesis and model selection has been formulated as a zero-sum two-person game with payoff function: &(P? C) = IIW,H(P,
C)W,Il,~,
DeJinition
The closed-loop robustly achieves the performance level specified by the weights W, I W, ifE
(2)
with one player (the controller) choosing strategies in C to minimize r#~ (P, C> while the second player (the plant) chooses strategies in P to maximize +(P, C). The game (1) is said to have the value v if: L’I G infsup+(P,C)=v=supinfti(P,C)e:z. PEPccc CECPSP
0, < 1.
Formulating the robust performance issue as a game has resulted in the optimization problem (1) whose solution is not straightforward and will be discussed in a later section. It is always true though, that:
(3)
VI 3 02,
If for some (PO, CO), q5(PO. CO) = v, then (PO, C’,) is an optimal strategy pair. P,, is the “nominal process model” and C, is the controller to be implemented. If in addition:
VPEP,
(4)
(5)
[I inf+(P,C)<+(P,C) CeC
;:T-& ::7f, 4J(P. C) S ;yg
*
;ye
VPEP, 4J(P. Cl,
VCEC, VC E c,
VCEC,
then the game has a saddlepoint. Based on the above problem definition and assuming that the performance weights W, , W, are appropriately chosen we can define robust performance as:
;$fc 4J(P9 Cl s >:7’, ;:g
Furthermore, there are cases in which U, = v2 (see Balakrishnan, 1976 and references therein), although
F
Fig. 2. The F-P-L-C
4 (PT Cl Il-
control scheme.
Approach
to robust controller synthesis
no conditions on the class P are known that can guarantee the equality of U, , u2. Let us therefore focus now on identifying v2. In this case P can be thought of as a fixed (but arbitrary) element of P. Since 4 (P, C) is + 00 if (F, P, L, C) is not L,,-stable it becomes apparent that C must belong to S(P) 17 C, where S(P) is the set of p-stabilizing controllers of P. For linear systems, using the results in Manousiouthakis (1985) one can express C as a function of P and Q and bring W,H(P, C) W, in a Q-affine form, that is: W,N(P,
C) w, = T,(P)
-
where Q is any &stable map are &-stable maps. Then: 4 (P. C) = IIT,(P)
the
above
the
and
i = 1,2,
T,(P),
C rational:
=
Q
C =
, _ P(A)Q,
Q stable, For a weighted sensitivity problem we have:
Q) = II T,(P)
+‘(P>
3,
-
2-induced
T,(P)QT,(P)
II i,, Q>.
(6)
problem
O.lw;‘s
-
Iliz
0;‘s
W,J’@)Q Then:
O.lw,‘s
S+5
w;‘s
+ 1 k Q
+ 1 >
Following standard RH,-optimization (see Francis, 1986) we identify that: Q>,
d’(P,
s.t. C(Q)
(7)
E C.
IIm
procedures 1 k 1 > ’
O.lw;‘(2+A)+
(?i$ &‘(P, Q) = E z
II m
+ 1 >
s - (2+A)
u2
_
+ 1 k
IK
inf sup Act-I.11 PERHr
rational
minimization
the RN,-norm.
where II . II ocdenotes v2 =
maximum
vz = sup pFp ai&
w;‘(2+A)+
Therefore
Optimization problem (6) is mathematically tractable and its solution identifies an optimum pair (P,. Co). This pair can then be used either to verify u, = u2 or to provide upper and lower bounds for u,. Let us now illustrate the procedure with numerical examples.
2j2= The
solution
max *Et-I, II
2
w;‘(2+A)+
O.lw;‘+
w;‘+
corresponding
1 *
O.lw;‘(2+A)+
to this problem
v = The
3. EXAMPLES
Example
S(P)
= II W, -
e +‘(P, Based on becomes:
Then
T,(P)QT,(P),
TAP)QT,V’)
-
383
1 k
1 >
1
> .
is:
for
A,,=
-1.
PO, Co are:
S-l
I
PO = -
Consider an open-loop system function belongs to the set:
whose
S +5’
transfer
c, = P = {P rational:
P (s, A) = ’ -s(:;A),A~[-l,
Let also, the C consist tions F=I,L=l,N=l/(l+PC),
of all rational
11). transfer
func-
=
1 k
o.‘w,‘s+
s
+ l)(O.lo,’
(w;‘s (O.lw;’
Let us now verify whether
11 1 + P(A)C,
=$%) = O.lw;’ CD;‘+
/I
(O.lw;’ S + l)(O.lo,’ + 1) O.lw;‘(w;‘+ 1)s +(1 + 1.9w;‘+O.9w;‘A) II + 1 1
+ 1) + 1)
v, = v2 or not (for k = 1):
W, A:[:?.
S + l)(w;’
(it is stable!).
1.
VI
s+5
Qo= S-l
I 1o;‘s+l ’ 1 w*=
w
-0.9w;‘(s + 5) QO 1 - P,,Q, = (0.1~;’ + l)(w;‘s + 1)’
for
A--l.
II~
V. MANOUSIOUTHAKIS
384
But this is equal to u2(remember k = 1) and therefore v, = vr. It is easy to verify that u, is less than one, irrespective of the value of 0,. Therefore one can conclude that the controller C, can robustly meet the performance requirements imposed by the weight 0 W . Example
Since
the
1’
~__
AE[-1,
The task at hand is to design, in a systematic way, a control system that maintains the value of y in the interval [-a,a]Vt 20 VAe[-1, 11. This objective must be met for all disturbances d with values in the interval [- 1, 1] Vt > 0. Using the notation of Fig. 1 we have: 3
P = P, P,],
_J5= bl,
,
Q =
6=k-1,
H 1
w,=
0 .
For any arbitrary (but fixed) element P E P the set of stabilizing controllers S(P) is (Manousiouthakis, 1985): = {C: C = Q(Z -
Q stable}.
FPLQ)-‘,
For that same element P E P, 4 (P, C) becomes: #(P,
Q, = II W,(J’ - J’LQFP)
C) g $‘(J’,
v2 =
sup psp ci&
v* =
Apia,, sup ,, .)I&
tt w,PW,
- @‘,J’LQFPW, ll;o,-,
II T,(A) - ~AA)Qz II ioj3
where
W,
-(2+A)lQ a(s + 4)2
To mathematically proceed as follows:
formulate
this
problem
we
Then Q, stable -
Vt >Oosupld(t))< 150
Similarly the control requiring that:
T(A)
4’
Is= 2+ A = 0 and ur becomes:
objective
s-t. 7-4WIs=2+.4=
1
IAl
Ildll,
C 1.
can be quantified
by
objective
IA]<
2.a
is therefore
#(P,
C)G
1,
=z 1.
Based on Dahleh and Pearson (1987) we replace the minimization problem with its dual and then have:
s.t. [be-(*+*“l
where
w,
0,
uz= max[bT,@)l,=2+Al, A.b
II W,Y IIz G 1,
where
~
4
-
The control
r,(A)
2
uz = syp ;ml II T,(A) - T,(A) II izc1
It holds that 1
W,II,,.
Based on the discussion in Section 2 we now concentrate on the maximin problem v2:
Js+2--)b
is
m’PW211i;aer
II w,(Z+PZ-JCF)
T2(A) - Q, G W, PLQFP
-1
ii,
where
S(P)
I]}.
and
then + (P. C) becomes: dJ(P.C)=
P = {P rational] P = [P, P2]
Fz
i&=[+[:]d,
2
Consider an open-loop system with one measurable output ( y ), one manipulatable input (u) and one nonmanipulatable input (d), where y = P,d + P,u. The system’s transfer function P = [P, P2] belongs to the set P defined as:
between
map
H(P,C)=(Z+PLCF)-‘Pandy,=y,
met if:
The first constraint and as a result:
G 1
vr 20,
1.
achieves its maximum
u2=i%x[a(04~A)]’ s.t. lb I c 1,
IAl < 1.
at t = 0
Approach to robust controllersynthesis After some algebra it can be shown that for any positive value of CL, the maximum is achieved at B = 1, A = - 1. The resulting value of v2 is: vz=-.
0.8 I%
For the control objective to be robustly met it is necessary that v2 < 1. Therefore, the previous expression suggests that for any linear controller C the peak of the closed-loop system response cannot become smaller than 0.8. The transfer function QZOfor which u2 is achieved can be evaluated as follows: The difference T,( - 1) - T4( - 1) must be equal to a constant 6 (since the first constraint of the dual problem was activated only at t = 0). Evaluating this difference at s = 1 we get: &!?.
c?
The nominal and uncertain closed-loop transfer functions between y and d then become respectively: (1 -
f’,Q,,V’,,,
= 0.8,
0.8(s + 2 -A) (1 - P20Q20)P1 1 + (P2 - P2,,)QZ0= s + 2.8 - 0.2A’
(S + 3)
3
Consider the open-loop stable plant: [(3+A)z-11 P(z) =
-0.12~~
(
&z-l
- 0.4~ + 1
>
’
Act-1,
11,
where P(z)
=
5
i=ll
P,z’ (and not zWi)_
Consider also the performance weight W(z) = l/[u (1 - zk) + bzk] which approximately reflects the requirement that the output lie in [-a, a] at all times and in [--b, b] after k sample intervals. Using the controller parametrization C = Q /( 1 - PQ) the maximin problem v2 becomes:
.
0.8(s + 2 - A) 4~[P(A), Co1 = a (S + 2.8 - 0.2A) ilo’ /I which can be bounded above and below as:
/I
3.36 -= 2.6~
Example
0.2(s + 4)
It should be noted that r$[P(A), C,] then becomes:
1.29 -= a
For any finite parametrization of the set P this yields a nonlinear programming problem since the above inequality constraints are becoming less restrictive as i increases. This solution methodology is illustrated in the following example.
The resulting maximization problem can then be stated as follows:
where Qzo is Q20 =
385
s.t. -1
+&Jt,(yc
1,
i=o,
1,
-lcA
max &[P(A).C~]Z=V~BV~=~. AS,_,. Ij
0
The mathematical technique used in arriving at the solution of the above maximin problem is of general value. By replacing the internal minimization problem with its dual, one can transiate the maximin problem into a maximization problem. For the case of [,-robust-optimal control of discrete time systems the maximin problem can then be formulated as a nonlinear programming problem as demonstrated next:
For a = 1, b = 0.1 the solution of this problem is next tabulated as a function of k (the number of samples after which the system output is required to be within 1-b, bl): k
3 4 5 8 10
u2 1.2727 1.1284 1.0623 1.0076 1.0019
A -1 -1 - 1 - 1 -1
The resulting closed-loop response can be evaluated through solution of the equation: W -
WP(-
1)Q = 5
bjz'9
j=O
c2 = sup inf IIT,(P) - T,(P) 11,co, PEP T-d stable s.t. T,(P)I;=oicp, = 0,
where 1 + IV = 2 is the number of activated constraints and b, must satisfy:
i = 1, m,
where a,(p) are the zeros of T,(P) inside the unit circle. Based on Dahleh and Pearson (1986), v2 then becomes: u2= sup max 5 a,T,(P)iz_~,~,,, Pep I, ,-, s.t. - 1c
CACE I4IC%-D
2 a,u:(P)
j=o,1,2
,...
C bp-j(1) = W[a,(I)] i = 1,2. i=O The behavior of the closed-loop system is captured in Fig. 3 for A = - 1 and for two different values of the filter parameter k (k = 3, k = 10). The system output, as expected, is below 1.2727 (1.0019) at all times and below 0.12727 (0.10019) after 3(10) sample intervals. q
386
V. MANOUSIOUTXAKI~
cc
\
I 0
I ~_______________ 20
10
30
Samples
Fig. 3. Unit step response to an output disturbance. -0.8 4.
IMPLICATIONS ROBUST
FOR NONLINEAR CONTROL
One of the principal advantages of the game approach proposed above is its applicability to both linear and nonlinear systems. Indeed, relation (4), u, < 1, remains the necessary and sufficient condition for robust performance even if the classes C, P contain nonlinear systems. Naturally, identifying the value of q becomes even more complex for nonlinear systems but at least conceptually the evaluation of u, for both linear and nonlinear systems is possible. However, before outlining a conceptual approach to the evaluation of vL we first justify, with an example, our use of the induced-extended-p-norm, II.11,peras a control system performance measure and then discuss an approach to its computation. Example
I 0
20
Tlme Fig. 4. Step and bang-bang
simulations.
The induced-extended-p-gain of the unbiased operator N: 15: + Lg over the set W 5 LF can be defined as:
HN II Let W Then:
be
chosen
vws as
II Nu Ilp. llu(lp
3
W={UEL;~O<]~UI~~,(~).
11 Nu 11 P
sup ,,”ya ’ IIN tlipw= 0<6S6 6’
(9)
and for n = 1, p = 00
4
II N IIiaow= sup
Consider a closed-loop system whose transfer function between the disturbance d and the output y is: G(s)
=
--s
+0.1
(s+lY
-
We are interested in identifying the maximum deviation of y from 0 as the disturbance d varies between - 1 and 1. The response of G (s) to a unit step change would suggest that the maximum deviation is -0.34 (Fig. 4). If, however, the disturbance d is of a bang-bang type then the maximum deviation can be -0.79 (Fig. 4). Therefore, if one of the closed-loop system performance specifications is that the output deviation never exceeds 0.4, then the step-change simulation would suggest that this requirement is met while in reality it may be violated. The maximum deviation, 0.79, is also equal to the semiinfinite integral (from 0 to ~13) of the absolute value of the impulse response of G(s) (Fig. 5). This in turn is the induced-extended-co norm of G(S). Having illustrated the usefulness of the inducedextended-co norm as a closed-loop performance measure we now address issues related to its computation for nonlinear systems. This has been successfully addressed in Nikolaou and Manousiouthakis (1989) and NikoIaou (1989) and is only outlined here.
“l”“==;,
.
0<.5<6
(10)
The above relation suggests a procedure for the computation of IIN II_,,,. For given 6’, one can solve a nonsmooth optimal control problem and can identify the signal u and the time t for which (Nu(i)) becomes the largest. Repeated application of this
Unit
-1.2
1 0
impulse
I
Tfme Fig. 5. Impulse
simulation.
10
Approach to robust controllersynthesis procedure for different b’ provides IiN ((_++,. As an example consider a heater with manipulated input output y G AT/T, and disturbance u G AFIF,, d G Ac/Kb. A PI controller, K(s) = (s + a)/bas, is used to regulate the system and the resulting closedloop system is realized by the equations:
x 7;,[1 + d(f)1 - TJ1 +Y (r)l TX + UAITc - TsP +y(r)l) 3
PCP TS
>
G(r) = -w(t). where V = 1 m3,
T,$= 300 K,
T, = 375 K,
Based on (1) and (10) and for p = co, 0, can be written as: v, = inf sup sup sup CEC PCP O
where 1.1denotes any vector norm in R”. It becomes clear that even for nonlinear systems the evaluation of V, lies in the solution of a minimax game. To solve this game one should first characterize the classes P, C. P can often be described in terms of a general model which contains uncertainty blocks A (e.g. uncertain real parameters) of bounded magnitude. C on the other hand can be conveniently described using the Q-parameterization for nonlinear systems (Desoer and Liu, 1982). In turn, the nonlinear stable operator Q can admit (Boyd and Chua, 1985) a state-space realization of the form:
z=28m3h-‘. P
i=Ax+Bu,
Steady state:
Y’?
T, = 370 K,
F,=2m’h-‘,
c = 0.03 h.
387
a,(&, c “u-1 11.. im= 1
. . . , i,)Xf,
. . . xjm,
where Re [A,(A)] -c 0. Consider now, for illustration purposes, W, H W, : ii, + jj2, F, L to be identity and P to be &-incrementally-stable and depend on the uncertainty block A. Then V, is equivalent to:
IlW-_=~“W+
a
The minimum principle yields: H[x(r),fJ(t),
d(r)1
+ E[x(r),p(t)]:
--6’<
v, =
C,,,;~pg,‘Y(Z)’ sup %O
w Q
)
6’
> which in turn dictates that
The above relation suggests that unless the nonsmooth optimal control problem has a singular arc the extremizing disturbance signal d will have a bang-bang form. This suggestion is confirmed computationally and helps identify the extremizing signal d. The solution reveals that a 5% deviation of the disturbance from its steady-state value causes 3.8% deviation in the output of the linearized system and 4.9% deviation (30% larger) in the output of the nonlinear system. Having demonstrated the utility and computability of II. II,/Jrrwe now proceed to the description of a conceptual approach for the evaluation of 0,.
inf A,.“, 8.1,
sup
A.b'.r.liii~Ilr
s.t -I;_(t) =f[xlt),
h(t),
A],
5’0)=f[C;(th
vi(t),
A,],
i(t)
= AZ(~)
+ RF,(t),
h(t)
= g[x(r)>
Mt),
5
2
m=l i,.....i,=l
&(t) =72(t)
-72O(th
Fz(t) = St)
+-J,(Z),
lE,(f)l /IAl~
< 0,
A],
A,13
720(f) =g[<(thB,(t)v
J,(t) =
Re[Aj(A)]
a,(i,,
__, i,)q,
. . . z,,(t),
G 4 <
1,
II4,Il
G 1,
which is a differential game. Solution of differential games has been a popular endeavor in the past and a wealth of literature is available on the subject. Therefore, we expect that the transformation of the robust controller synthesis problem into a differential game will lead to the problem’s solution. The above unification of linear and nonlinear robust control not only facilitates controller design but also permits the quantification of performance improvement for nonlinear over linear control. In
V.
388
MANOUSIOUTHAKIS
that respect, a philosophically interesting question is the following: “Does there exist a nontrivial class of plants for which nonlinear robust control is nof superior to linear robust control in the 11.[I,F sense?” As shown next, the answer to this question is positive (at least for p = 2). Theorem Let P consist only of linear time-invariant plants. Then, for p = 2, vu 2 VIff >, V2L.
Proof
since the set of nonlinear controllers strictly contains the set of linear controllers: (b) UIN2
“2NI
this has been shown in Section 2, relation (5);
(cl
VZN
=
sup
inf
II W,H(P,
PEP cnonlinsar
C) W,
II i2p-
In regard to the internal minimization problem it has been shown (Khargonekar and Poolla, 1986) that if there is no uncertainty then induced-2 nonlinear control of a linear plant is equivalent to induced-2 linear control of that linear plant. Therefore, for any fixed (but unknown) P E P it holds that: inf II W H(P, C) W2 II i2e cnon,inenr
=
inf II W,H(P, Clinear
0
W2
II izc
Taking supremum of both sides over all P E P we conclude that tj2N= v2Land then combining a, b, c we obtain uIL3 u,,v2, uZL. QED 0 Using the above theorem one may evaluate bounds on nonlinear robust control system performance based on linear control system performance calculations. In fact, for linear systems, the minimax to maximin gap for linear control provides a range for any improvement that nonlinear control may offer over linear control. Corollary
There exists a nontrivial class of plants for which nonlinear robust control is not superior to linear robust control in the 11.II i2Fsense. Proof In Example I we identified a class P and weights w, 9 W, for which vIL= Vet_ Based on the above theorem, t’,L=V,N=t&=V2L. QED 0
5. CONCLUSIONS
AND
DlSCUSSlON
In this paper we have presented a novel game theoretic formulation of the robust controller sythesis and model selection problem. Two games have been formulated in regard to this problem. A minimax game u , , whose solution delivers the robust controller CO and a representative process model P,, and a maximin game vr whose solution provides a lower (possibly tight) bound on vr . The maximin problem u2, which can be readily solved, has several ramifications for the robust controller synthesis and model selection problem.
(a) it provides a solution to the v, problem when v, = 02; (b) it allows the use of controller design methodologies that focus on nominal system performance; (c) it provides a lower bound on v, _ As a result, if v2 is larger than one then the desirable control system performance specifications cannot be met; (d) it provides a pair (P,,, C,,) which can serve as an initial guess, if such a guess is needed, for any solution procedure of v,; (e) for linear systems, the linear minimax and maximin games, vIL and vZL provide upper and lower bounds, respectively, for the nonlinear minimax game vIN. The minimax problem v, is the subject of our current research investigations which focus on the development of solution methodologies for minimax games. would like to acknowledge the NSF-PYI Program for partial support of our research through award No. CBT 88 57867. We would also like to thank Dr Michael Nikolaou and Mr Dennis Sourlas for participation in many intelligent discussions and for carrying out most of the computations. Acknowkdgements-We
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Circuits
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CAS32,
11 (1985).
Dahleh M. and J. B. Pearson, I’-Optimal feedback controllers for discrete-time systems. ACC Proc., pp. 19641968 (1986). Dahleh M. and J. B. Pearson, L’-Optimal compensators for continuous-time systems. IEEE TAC AC-32 (1987). Dahleh M. and J. B. Pearson, Optimal rejection of persistent disturbances, robust stability, and mixed sensitivity minimization. IEEE TAC AC-33 (1988). Desoer C. and R. W. Liu, Global properties of feedback systems with nonlinear plants systems. Control Lerrs 1, 249 (1982). Francis B., A Course in H, Control Theory. Springer Verlag, New York (1986). Khargonekar P. P. and K. Poolla, Uniformly optimal control of linear time-invariant plants: nonlinear timevarying controllers. Systems Control Letrs 6, 303-308 (1986).
Approach
to robust controller synthesis
Manousiouthakis V., A parametrization of all stabilizing compensators for a general control scheme. UCLA Internal Report (1985). Manousiouthakis V., On the stability properties of a general control scheme and their implications for robustly reliable control. IFAC Proc., Munich, West Germany (1987a). Manousiouthakis V., Robust control. An overview and some new directions. Shell Process Control Workshop (D.
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Prett and M. Morari. Eds). Buttetworths, London (1987b). Nikolaou M., A hybrid approach to stability and performance of nonlinear control systems. Ph.D. Thesis, UCLA (1989). Nikolaou M. and V. Manousiouthakis, A hybrid approach to nonlinear system stability and performance. AIChE JI 35, 559-563 (1989).
A
GAME
1990 Copyright
THEORETIC CONTROLLER
APPROACH TO SYNTHESIS
009%1354/90 $3.00 + 0.00 0 1990 Pergamon Press plc
ROBUST
V. MANOUSIOUTHAKIS Department of Chemical Engineering, University of California Los Angeles, Los Angeles, CA 90024, U.S.A. (Received
6 Nouemher
1989;
received
for publicorion
28 November
1989)
Abstract-This paper presents a novel game theoretic approach to robust controller synthesis. The main advantage of this problem formulation is that it provides simultaneous controller synthesis and process model selection. The approach is illustrated with several numerical examples.
1. INTRODUCTION
the last few years, control system design paradigms continuously incorporate an ever increasing number of issues in an effort to reduce the gap between conceptual controller design and implementable controller design. The current status of this evolution calls for the satisfaction of control objectives, constraints and specifications in the presence of uncertainty (i.e. for all process models that belong to a process uncertainty description given a priori). Current control system design methodologies cannot easily address this issue. In that respect, the principal disadvantage of model predictive control (MPC) stems from its attempt to specify (in an optimal way) the system input u (see Fig. 1). In doing so, MPC fails to recognize that within a closed-loop setting the input u is no longer an independent variable. Indeed, as the disturbance d and/or the process description P vary, the only independent parameter at the designer’s disposal is the controller algorithm C. Robust control system design methodologies, on the other hand, have avoided this flaw by focusing on the design of the controller C. However, their dependence on the notion of a nominal model has limited their success to linear systems and to frequency-dependent uncertainty descriptions and performance specifications. Indeed, the theoretical basis of these methods are, nominal model related, necessary and sufficient conditions for robust stability. We feel that as the scope of robust control expands to nonfrequency-dependent uncertainty descriptions and to nonlinear systems the derivation of similar conditions will become increasingly difficult and will require the use of so general uncertainty descriptions that the resulting controllers will be overly conservative. We also feel that as control system performance specifications tighten, it will be of paramount importance to identify the critical modeling information for improved control system performance. Current robust control methods do not address this important task. In
381
In this paper we present a novel control system design methodology that addresses these issues. The main advantage of this novel game theoretic problem formulation is that it avoids the notion of a nominal process model and provides simultaneous controller synthesis and process model selection. The remaining paper is structured as follows: in Section 2 the proposed design methodology is presented. In Section 3 numerical examples illustrating its use are provided. In Section 4 the implications of the gametheoretic approach for nonlinear robust control are discussed. Finally, in Section 5 we draw conclusions and discuss some of our current research efforts. 2.
THE
GAME
THEORETIC
APPROACH
Consider the F-P-L-C control scheme (Fig. 2) (Manousiouthakis, 1985, 1987a, b) which represents the most general way to interconnect a process P and its controller C. For given filters, F, L, let N = n(P, C) denote the map relating the internal (2 , , Zz, &, &, ii+, 5,) and external (U,, fi2) signals of this control scheme. It is well known (Francis, 1986) that closed-loop performance requirements can often be. expressed in terms of the “smallness” of norm of the II w,nw* IllPrvthe inducedextended-p weighted closed-loop map H (IV,, W, represent pstable performance weights). As pointed out earlier, H = H(P, C) is a function of P and C. The controller C belongs to a known set C while the plant P belongs to a set P which is also known (e.g. C may consist of all rational transfer functions while P may be defined through the notions of a nominal plant and a bounded additive perturbation). The task of the designer can then be stated as: “Identify a controller C E C that achieves the best possible bound on the worst closed-loop performance level that can occur as P varies over P”. Given the aforementioned notion of closed-loop performance the above task can be mathematically quantified as:
(1)
V. MANOUSIOIITHAKIS
382
d
Fig. 1. Classical feedback scheme.
Therefore, robust controller synthesis and model selection has been formulated as a zero-sum two-person game with payoff function: &(P? C) = IIW,H(P,
C)W,Il,~,
DeJinition
The closed-loop robustly achieves the performance level specified by the weights W, I W, ifE
(2)
with one player (the controller) choosing strategies in C to minimize r#~ (P, C> while the second player (the plant) chooses strategies in P to maximize +(P, C). The game (1) is said to have the value v if: L’I G infsup+(P,C)=v=supinfti(P,C)e:z. PEPccc CECPSP
0, < 1.
Formulating the robust performance issue as a game has resulted in the optimization problem (1) whose solution is not straightforward and will be discussed in a later section. It is always true though, that:
(3)
VI 3 02,
If for some (PO, CO), q5(PO. CO) = v, then (PO, C’,) is an optimal strategy pair. P,, is the “nominal process model” and C, is the controller to be implemented. If in addition:
VPEP,
(4)
(5)
[I inf+(P,C)<+(P,C) CeC
;:T-& ::7f, 4J(P. C) S ;yg
*
;ye
VPEP, 4J(P. Cl,
VCEC, VC E c,
VCEC,
then the game has a saddlepoint. Based on the above problem definition and assuming that the performance weights W, , W, are appropriately chosen we can define robust performance as:
;$fc 4J(P9 Cl s >:7’, ;:g
Furthermore, there are cases in which U, = v2 (see Balakrishnan, 1976 and references therein), although
F
Fig. 2. The F-P-L-C
4 (PT Cl Il-
control scheme.
Approach
to robust controller synthesis
no conditions on the class P are known that can guarantee the equality of U, , u2. Let us therefore focus now on identifying v2. In this case P can be thought of as a fixed (but arbitrary) element of P. Since 4 (P, C) is + 00 if (F, P, L, C) is not L,,-stable it becomes apparent that C must belong to S(P) 17 C, where S(P) is the set of p-stabilizing controllers of P. For linear systems, using the results in Manousiouthakis (1985) one can express C as a function of P and Q and bring W,H(P, C) W, in a Q-affine form, that is: W,N(P,
C) w, = T,(P)
-
where Q is any &stable map are &-stable maps. Then: 4 (P. C) = IIT,(P)
the
above
the
and
i = 1,2,
T,(P),
C rational:
=
Q
C =
, _ P(A)Q,
Q stable, For a weighted sensitivity problem we have:
Q) = II T,(P)
+‘(P>
3,
-
2-induced
T,(P)QT,(P)
II i,, Q>.
(6)
problem
O.lw;‘s
-
Iliz
0;‘s
W,J’@)Q Then:
O.lw,‘s
S+5
w;‘s
+ 1 k Q
+ 1 >
Following standard RH,-optimization (see Francis, 1986) we identify that: Q>,
d’(P,
s.t. C(Q)
(7)
E C.
IIm
procedures 1 k 1 > ’
O.lw;‘(2+A)+
(?i$ &‘(P, Q) = E z
II m
+ 1 >
s - (2+A)
u2
_
+ 1 k
IK
inf sup Act-I.11 PERHr
rational
minimization
the RN,-norm.
where II . II ocdenotes v2 =
maximum
vz = sup pFp ai&
w;‘(2+A)+
Therefore
Optimization problem (6) is mathematically tractable and its solution identifies an optimum pair (P,. Co). This pair can then be used either to verify u, = u2 or to provide upper and lower bounds for u,. Let us now illustrate the procedure with numerical examples.
2j2= The
solution
max *Et-I, II
2
w;‘(2+A)+
O.lw;‘+
w;‘+
corresponding
1 *
O.lw;‘(2+A)+
to this problem
v = The
3. EXAMPLES
Example
S(P)
= II W, -
e +‘(P, Based on becomes:
Then
T,(P)QT,(P),
TAP)QT,V’)
-
383
1 k
1 >
1
> .
is:
for
A,,=
-1.
PO, Co are:
S-l
I
PO = -
Consider an open-loop system function belongs to the set:
whose
S +5’
transfer
c, = P = {P rational:
P (s, A) = ’ -s(:;A),A~[-l,
Let also, the C consist tions F=I,L=l,N=l/(l+PC),
of all rational
11). transfer
func-
=
1 k
o.‘w,‘s+
s
+ l)(O.lo,’
(w;‘s (O.lw;’
Let us now verify whether
11 1 + P(A)C,
=$%) = O.lw;’ CD;‘+
/I
(O.lw;’ S + l)(O.lo,’ + 1) O.lw;‘(w;‘+ 1)s +(1 + 1.9w;‘+O.9w;‘A) II + 1 1
+ 1) + 1)
v, = v2 or not (for k = 1):
W, A:[:?.
S + l)(w;’
(it is stable!).
1.
VI
s+5
Qo= S-l
I 1o;‘s+l ’ 1 w*=
w
-0.9w;‘(s + 5) QO 1 - P,,Q, = (0.1~;’ + l)(w;‘s + 1)’
for
A--l.
II~
V. MANOUSIOUTHAKIS
384
But this is equal to u2(remember k = 1) and therefore v, = vr. It is easy to verify that u, is less than one, irrespective of the value of 0,. Therefore one can conclude that the controller C, can robustly meet the performance requirements imposed by the weight 0 W . Example
Since
the
1’
~__
AE[-1,
The task at hand is to design, in a systematic way, a control system that maintains the value of y in the interval [-a,a]Vt 20 VAe[-1, 11. This objective must be met for all disturbances d with values in the interval [- 1, 1] Vt > 0. Using the notation of Fig. 1 we have: 3
P = P, P,],
_J5= bl,
,
Q =
6=k-1,
H 1
w,=
0 .
For any arbitrary (but fixed) element P E P the set of stabilizing controllers S(P) is (Manousiouthakis, 1985): = {C: C = Q(Z -
Q stable}.
FPLQ)-‘,
For that same element P E P, 4 (P, C) becomes: #(P,
Q, = II W,(J’ - J’LQFP)
C) g $‘(J’,
v2 =
sup psp ci&
v* =
Apia,, sup ,, .)I&
tt w,PW,
- @‘,J’LQFPW, ll;o,-,
II T,(A) - ~AA)Qz II ioj3
where
W,
-(2+A)lQ a(s + 4)2
To mathematically proceed as follows:
formulate
this
problem
we
Then Q, stable -
Vt >Oosupld(t))< 150
Similarly the control requiring that:
T(A)
4’
Is= 2+ A = 0 and ur becomes:
objective
s-t. 7-4WIs=2+.4=
1
IAl
Ildll,
C 1.
can be quantified
by
objective
IA]<
2.a
is therefore
#(P,
C)G
1,
=z 1.
Based on Dahleh and Pearson (1987) we replace the minimization problem with its dual and then have:
s.t. [be-(*+*“l
where
w,
0,
uz= max[bT,@)l,=2+Al, A.b
II W,Y IIz G 1,
where
~
4
-
The control
r,(A)
2
uz = syp ;ml II T,(A) - T,(A) II izc1
It holds that 1
W,II,,.
Based on the discussion in Section 2 we now concentrate on the maximin problem v2:
Js+2--)b
is
m’PW211i;aer
II w,(Z+PZ-JCF)
T2(A) - Q, G W, PLQFP
-1
ii,
where
S(P)
I]}.
and
then + (P. C) becomes: dJ(P.C)=
P = {P rational] P = [P, P2]
Fz
i&=[+[:]d,
2
Consider an open-loop system with one measurable output ( y ), one manipulatable input (u) and one nonmanipulatable input (d), where y = P,d + P,u. The system’s transfer function P = [P, P2] belongs to the set P defined as:
between
map
H(P,C)=(Z+PLCF)-‘Pandy,=y,
met if:
The first constraint and as a result:
G 1
vr 20,
1.
achieves its maximum
u2=i%x[a(04~A)]’ s.t. lb I c 1,
IAl < 1.
at t = 0
Approach to robust controllersynthesis After some algebra it can be shown that for any positive value of CL, the maximum is achieved at B = 1, A = - 1. The resulting value of v2 is: vz=-.
0.8 I%
For the control objective to be robustly met it is necessary that v2 < 1. Therefore, the previous expression suggests that for any linear controller C the peak of the closed-loop system response cannot become smaller than 0.8. The transfer function QZOfor which u2 is achieved can be evaluated as follows: The difference T,( - 1) - T4( - 1) must be equal to a constant 6 (since the first constraint of the dual problem was activated only at t = 0). Evaluating this difference at s = 1 we get: &!?.
c?
The nominal and uncertain closed-loop transfer functions between y and d then become respectively: (1 -
f’,Q,,V’,,,
= 0.8,
0.8(s + 2 -A) (1 - P20Q20)P1 1 + (P2 - P2,,)QZ0= s + 2.8 - 0.2A’
(S + 3)
3
Consider the open-loop stable plant: [(3+A)z-11 P(z) =
-0.12~~
(
&z-l
- 0.4~ + 1
>
’
Act-1,
11,
where P(z)
=
5
i=ll
P,z’ (and not zWi)_
Consider also the performance weight W(z) = l/[u (1 - zk) + bzk] which approximately reflects the requirement that the output lie in [-a, a] at all times and in [--b, b] after k sample intervals. Using the controller parametrization C = Q /( 1 - PQ) the maximin problem v2 becomes:
.
0.8(s + 2 - A) 4~[P(A), Co1 = a (S + 2.8 - 0.2A) ilo’ /I which can be bounded above and below as:
/I
3.36 -= 2.6~
Example
0.2(s + 4)
It should be noted that r$[P(A), C,] then becomes:
1.29 -= a
For any finite parametrization of the set P this yields a nonlinear programming problem since the above inequality constraints are becoming less restrictive as i increases. This solution methodology is illustrated in the following example.
The resulting maximization problem can then be stated as follows:
where Qzo is Q20 =
385
s.t. -1
+&Jt,(yc
1,
i=o,
1,
-lcA
max &[P(A).C~]Z=V~BV~=~. AS,_,. Ij
0
The mathematical technique used in arriving at the solution of the above maximin problem is of general value. By replacing the internal minimization problem with its dual, one can transiate the maximin problem into a maximization problem. For the case of [,-robust-optimal control of discrete time systems the maximin problem can then be formulated as a nonlinear programming problem as demonstrated next:
For a = 1, b = 0.1 the solution of this problem is next tabulated as a function of k (the number of samples after which the system output is required to be within 1-b, bl): k
3 4 5 8 10
u2 1.2727 1.1284 1.0623 1.0076 1.0019
A -1 -1 - 1 - 1 -1
The resulting closed-loop response can be evaluated through solution of the equation: W -
WP(-
1)Q = 5
bjz'9
j=O
c2 = sup inf IIT,(P) - T,(P) 11,co, PEP T-d stable s.t. T,(P)I;=oicp, = 0,
where 1 + IV = 2 is the number of activated constraints and b, must satisfy:
i = 1, m,
where a,(p) are the zeros of T,(P) inside the unit circle. Based on Dahleh and Pearson (1986), v2 then becomes: u2= sup max 5 a,T,(P)iz_~,~,,, Pep I, ,-, s.t. - 1c
CACE I4IC%-D
2 a,u:(P)
j=o,1,2
,...
C bp-j(1) = W[a,(I)] i = 1,2. i=O The behavior of the closed-loop system is captured in Fig. 3 for A = - 1 and for two different values of the filter parameter k (k = 3, k = 10). The system output, as expected, is below 1.2727 (1.0019) at all times and below 0.12727 (0.10019) after 3(10) sample intervals. q
386
V. MANOUSIOUTXAKI~
cc
\
I 0
I ~_______________ 20
10
30
Samples
Fig. 3. Unit step response to an output disturbance. -0.8 4.
IMPLICATIONS ROBUST
FOR NONLINEAR CONTROL
One of the principal advantages of the game approach proposed above is its applicability to both linear and nonlinear systems. Indeed, relation (4), u, < 1, remains the necessary and sufficient condition for robust performance even if the classes C, P contain nonlinear systems. Naturally, identifying the value of q becomes even more complex for nonlinear systems but at least conceptually the evaluation of u, for both linear and nonlinear systems is possible. However, before outlining a conceptual approach to the evaluation of vL we first justify, with an example, our use of the induced-extended-p-norm, II.11,peras a control system performance measure and then discuss an approach to its computation. Example
I 0
20
Tlme Fig. 4. Step and bang-bang
simulations.
The induced-extended-p-gain of the unbiased operator N: 15: + Lg over the set W 5 LF can be defined as:
HN II Let W Then:
be
chosen
vws as
II Nu Ilp. llu(lp
3
W={UEL;~O<]~UI~~,(~).
11 Nu 11 P
sup ,,”ya ’ IIN tlipw= 0<6S6 6’
(9)
and for n = 1, p = 00
4
II N IIiaow= sup
Consider a closed-loop system whose transfer function between the disturbance d and the output y is: G(s)
=
--s
+0.1
(s+lY
-
We are interested in identifying the maximum deviation of y from 0 as the disturbance d varies between - 1 and 1. The response of G (s) to a unit step change would suggest that the maximum deviation is -0.34 (Fig. 4). If, however, the disturbance d is of a bang-bang type then the maximum deviation can be -0.79 (Fig. 4). Therefore, if one of the closed-loop system performance specifications is that the output deviation never exceeds 0.4, then the step-change simulation would suggest that this requirement is met while in reality it may be violated. The maximum deviation, 0.79, is also equal to the semiinfinite integral (from 0 to ~13) of the absolute value of the impulse response of G(s) (Fig. 5). This in turn is the induced-extended-co norm of G(S). Having illustrated the usefulness of the inducedextended-co norm as a closed-loop performance measure we now address issues related to its computation for nonlinear systems. This has been successfully addressed in Nikolaou and Manousiouthakis (1989) and NikoIaou (1989) and is only outlined here.
“l”“==;,
.
0<.5<6
(10)
The above relation suggests a procedure for the computation of IIN II_,,,. For given 6’, one can solve a nonsmooth optimal control problem and can identify the signal u and the time t for which (Nu(i)) becomes the largest. Repeated application of this
Unit
-1.2
1 0
impulse
I
Tfme Fig. 5. Impulse
simulation.
10
Approach to robust controllersynthesis procedure for different b’ provides IiN ((_++,. As an example consider a heater with manipulated input output y G AT/T, and disturbance u G AFIF,, d G Ac/Kb. A PI controller, K(s) = (s + a)/bas, is used to regulate the system and the resulting closedloop system is realized by the equations:
x 7;,[1 + d(f)1 - TJ1 +Y (r)l TX + UAITc - TsP +y(r)l) 3
PCP TS
>
G(r) = -w(t). where V = 1 m3,
T,$= 300 K,
T, = 375 K,
Based on (1) and (10) and for p = co, 0, can be written as: v, = inf sup sup sup CEC PCP O
where 1.1denotes any vector norm in R”. It becomes clear that even for nonlinear systems the evaluation of V, lies in the solution of a minimax game. To solve this game one should first characterize the classes P, C. P can often be described in terms of a general model which contains uncertainty blocks A (e.g. uncertain real parameters) of bounded magnitude. C on the other hand can be conveniently described using the Q-parameterization for nonlinear systems (Desoer and Liu, 1982). In turn, the nonlinear stable operator Q can admit (Boyd and Chua, 1985) a state-space realization of the form:
z=28m3h-‘. P
i=Ax+Bu,
Steady state:
Y’?
T, = 370 K,
F,=2m’h-‘,
c = 0.03 h.
387
a,(&, c “u-1 11.. im= 1
. . . , i,)Xf,
. . . xjm,
where Re [A,(A)] -c 0. Consider now, for illustration purposes, W, H W, : ii, + jj2, F, L to be identity and P to be &-incrementally-stable and depend on the uncertainty block A. Then V, is equivalent to:
IlW-_=~“W+
a
The minimum principle yields: H[x(r),fJ(t),
d(r)1
+ E[x(r),p(t)]:
--6’<
v, =
C,,,;~pg,‘Y(Z)’ sup %O
w Q
)
6’
> which in turn dictates that
The above relation suggests that unless the nonsmooth optimal control problem has a singular arc the extremizing disturbance signal d will have a bang-bang form. This suggestion is confirmed computationally and helps identify the extremizing signal d. The solution reveals that a 5% deviation of the disturbance from its steady-state value causes 3.8% deviation in the output of the linearized system and 4.9% deviation (30% larger) in the output of the nonlinear system. Having demonstrated the utility and computability of II. II,/Jrrwe now proceed to the description of a conceptual approach for the evaluation of 0,.
inf A,.“, 8.1,
sup
A.b'.r.liii~Ilr
s.t -I;_(t) =f[xlt),
h(t),
A],
5’0)=f[C;(th
vi(t),
A,],
i(t)
= AZ(~)
+ RF,(t),
h(t)
= g[x(r)>
Mt),
5
2
m=l i,.....i,=l
&(t) =72(t)
-72O(th
Fz(t) = St)
+-J,(Z),
lE,(f)l /IAl~
< 0,
A],
A,13
720(f) =g[<(thB,(t)v
J,(t) =
Re[Aj(A)]
a,(i,,
__, i,)q,
. . . z,,(t),
G 4 <
1,
II4,Il
G 1,
which is a differential game. Solution of differential games has been a popular endeavor in the past and a wealth of literature is available on the subject. Therefore, we expect that the transformation of the robust controller synthesis problem into a differential game will lead to the problem’s solution. The above unification of linear and nonlinear robust control not only facilitates controller design but also permits the quantification of performance improvement for nonlinear over linear control. In
V.
388
MANOUSIOUTHAKIS
that respect, a philosophically interesting question is the following: “Does there exist a nontrivial class of plants for which nonlinear robust control is nof superior to linear robust control in the 11.[I,F sense?” As shown next, the answer to this question is positive (at least for p = 2). Theorem Let P consist only of linear time-invariant plants. Then, for p = 2, vu 2 VIff >, V2L.
Proof
since the set of nonlinear controllers strictly contains the set of linear controllers: (b) UIN2
“2NI
this has been shown in Section 2, relation (5);
(cl
VZN
=
sup
inf
II W,H(P,
PEP cnonlinsar
C) W,
II i2p-
In regard to the internal minimization problem it has been shown (Khargonekar and Poolla, 1986) that if there is no uncertainty then induced-2 nonlinear control of a linear plant is equivalent to induced-2 linear control of that linear plant. Therefore, for any fixed (but unknown) P E P it holds that: inf II W H(P, C) W2 II i2e cnon,inenr
=
inf II W,H(P, Clinear
0
W2
II izc
Taking supremum of both sides over all P E P we conclude that tj2N= v2Land then combining a, b, c we obtain uIL3 u,,v2, uZL. QED 0 Using the above theorem one may evaluate bounds on nonlinear robust control system performance based on linear control system performance calculations. In fact, for linear systems, the minimax to maximin gap for linear control provides a range for any improvement that nonlinear control may offer over linear control. Corollary
There exists a nontrivial class of plants for which nonlinear robust control is not superior to linear robust control in the 11.II i2Fsense. Proof In Example I we identified a class P and weights w, 9 W, for which vIL= Vet_ Based on the above theorem, t’,L=V,N=t&=V2L. QED 0
5. CONCLUSIONS
AND
DlSCUSSlON
In this paper we have presented a novel game theoretic formulation of the robust controller sythesis and model selection problem. Two games have been formulated in regard to this problem. A minimax game u , , whose solution delivers the robust controller CO and a representative process model P,, and a maximin game vr whose solution provides a lower (possibly tight) bound on vr . The maximin problem u2, which can be readily solved, has several ramifications for the robust controller synthesis and model selection problem.
(a) it provides a solution to the v, problem when v, = 02; (b) it allows the use of controller design methodologies that focus on nominal system performance; (c) it provides a lower bound on v, _ As a result, if v2 is larger than one then the desirable control system performance specifications cannot be met; (d) it provides a pair (P,,, C,,) which can serve as an initial guess, if such a guess is needed, for any solution procedure of v,; (e) for linear systems, the linear minimax and maximin games, vIL and vZL provide upper and lower bounds, respectively, for the nonlinear minimax game vIN. The minimax problem v, is the subject of our current research investigations which focus on the development of solution methodologies for minimax games. would like to acknowledge the NSF-PYI Program for partial support of our research through award No. CBT 88 57867. We would also like to thank Dr Michael Nikolaou and Mr Dennis Sourlas for participation in many intelligent discussions and for carrying out most of the computations. Acknowkdgements-We
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Approach
to robust controller synthesis
Manousiouthakis V., A parametrization of all stabilizing compensators for a general control scheme. UCLA Internal Report (1985). Manousiouthakis V., On the stability properties of a general control scheme and their implications for robustly reliable control. IFAC Proc., Munich, West Germany (1987a). Manousiouthakis V., Robust control. An overview and some new directions. Shell Process Control Workshop (D.
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Prett and M. Morari. Eds). Buttetworths, London (1987b). Nikolaou M., A hybrid approach to stability and performance of nonlinear control systems. Ph.D. Thesis, UCLA (1989). Nikolaou M. and V. Manousiouthakis, A hybrid approach to nonlinear system stability and performance. AIChE JI 35, 559-563 (1989).