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On the Design of Linear Control Systems with Plant Uncertainty Via Nondifferentiable Optimization*
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ON THE DESIGN OF LINEAR CONTROL SYSTEMS WITH PLANT UNCERTAINTY VIA NONDIFFERENTIABLE OPTIMIZA TION* E. Polak and D. M. Stimler Depa rt ml'1lt of t:lectncal E nf,{l1u'l'r1nK and Compll tl'r SCIeNce" aud Ih t' E Il'Ctro llln [" 1II1'er,,1.' 0/ CtlIi/onl/tI. B erkfil~. CA 'J./ 720. ["SA
ABSI'RAcr We present a three phase semi-infinite optimization algorithm which is motivated by worst case, single-input single-output control system design problems. When suitably interpreted, the algorithm can also be used for multivariable control system design with performance requirements that lead to semi-infinite inequality constraints.
R l'~e(l r(h
Labomlun,
meetinl!> this goal is to carry out a worst ca.se design which ensures that specifications will be met for all mathematical plant models in a prescribed class. It was shown in [Sti.1] that worst case design of single-input single-output (SI SO) control systems, in the frequency domain, can be expressed as a somewhat unusual semi-infinite optimization problem. In this section we shall summarize the findings presented in [Sti.1] so as to justify the structure of the algorithm to be presented in the next section. We shall use the system configuration of Figure 1 for illustration. We assume that the plant transfer function is of the form,
1. Introduction Over the last decade, the control system designer's ability to make intell~ent decisions has been tremendously enhanced by the introduction of interactive computing packages which enable the designer to define problems by means of graphical editors, simulate specified system responses and display graphically the results. As the power of these definition. simulation and display systems grows, designers are becoming more and more aware of the fact that a major obstacle to obtaining significantly better p erformance than is currently possible is traceable to the limited ability humans have for searching through alternatives in a multidimensional design parameter space. In response to this phenomenon in control engineering as well as in other engineering areas, there has arisen a new area of research in optimization: semi-1nfinite optimization, i.e., optimization over a finite dimensional desi~n vector which must satisfy a continuum of inequalities (see, e.~., [Bec.!. Gon.!. Gon.2, May. 1, PoLl, Pol.2, Pol.3, Pol.4, Pol.5 J). These inequalities are referred to as semi-infinite. Such inequalities arise when system responses are constrained to lie in envelopes over frequency or time . They also arise when the plant depends on uncertain parameters which leads to a worst case design situation. An important issue in optimization-bas ed computeraided design is that of problem formulation . In particuler, care must be taken to formulate design requirements as constraints which involve functions that are compatible with the requirements of the optimization algorithm and are computationally tractable . 1n order that we may use nondifferentiable optimization techniques [Cla.1], the constraint functions must be locally Upschitz continuous. In Section 2, we show that the worst case design of single-input single-output (SISO) control systems can be reduced to a tractable, tbough ratber unusual semi-infinite optimization problem. The unusual properties of this problem are due to the fact that a c losed loop system whose component subsystems have rational transfer functions may have transfer functions which are dis continuous as a function of frequency. Since there are no algorithms in the literature that solve this type of semi-infinite optimization problem, we have constructed a special algorithm for its solution. The algorithm is presented in Section 3. (When suitably modified. the algorithm can also be used for multiinput mUlti-output control system design).
P(s,a,l(s)) ~ P.(s,a)l(s)
(2.1)
Po(s,a) ~ KTI(s + z');TI(s + pi)
(2.2)
where ,
i
with a a vector whose components are the gain K, the zeros z' and the poles pi and l (-) EL is a perturbation function which is used to express unstructured uncertainty attributable to unmodelled dynamics, high frequency measurement errors, etc. We introduce the possibility of structured v:ncertainty in our model by assuming that the gain K, the zeros z' and the poles pi lie in confidence int er vals of the formt
The product of all the uncertainty intervals in (2.3) will be denoted by A; it is a compact subset of !Rn.. The set of unstructured perturbation functions L is assumed to consist of all rational transfer functions, l (s), which have equal numerator and denominator degree and which are bounded in magnitude and phase as follows:
l,v(r.»";; Il(jr.»
1 ,.;;
lAl(r.»
V r.>E!R+
(2.4a) (2.4b)
where l,v, rAl,1.t,~ :R+"'R are bound functions which satisfy
o
continuously differentiable
,.;; 1,.;; [Al(r.»
V r.> E R.-
(2.4c) (2.4d)
It is convenient to use the faclored form, below, for the compensator transfer functions :
C(%,s)
=
Kc{s + de) I](s2+2ats +(bt-)2) (s +dg) TI(s2+2ats +(bC)2) i
(25) .
A similar form will be used for F(% ,s). The design vector EIR", to be determined by optimization, is composed of the free coefficients in these representations of C and F. It was shown in [Pol.6,Sti.1] that many design requirements (including disturbance rejection, saturation avoidance, closed loop stability, etc.) may be state d as inequality constraints on the design vector. These in equalities are of the form
%
2. Worst Case Design as a Semi-Infinite Optimization Problem A major goal in control system design is to ensure satisfactory system performance in the pres ence of uncertainty in the mathematical model of the plant. One approach to
ct%,r.>,a,l(jr.»)";; 0
V r.>EO, V aEAand V l EL ,(2.6)
• Research spmwored by the National Science Foundation under grant t For complex zeros and poles which must occur in complex cO::1jugate EC8-S121l49, the otfice of Naval Research \IDder contract NOOOI4-83-K-0602 the Jdr Farce otfice of Scienililc Research grant AFOSR-B3-0361, the Sem: pairs we assume that the confidence "intervals" are in fac t rect angles in C lconductor Re""arch Comortium grant SRC-82-1 HlOB, and a Universi t y of !(u,v) ER2 IRe,(w')";; u,.;; Re(wi).Im,(wi),.;; v ,.;; Sydney TraveJlin& Scholandrip.
~l~w'] ~
16 15
Ir~'{(wi)l
E. Polak and D. M. Stimler
1616
where 0 c JR is compact. Verification that (2.6) is satisfied is equivalent to evaluating max ~(x.(.).a.l(j(.)) . ~~~ 'EL
(2.7)
description to ( in (2.9a) . Finally. taking all these considerations into account. when a cost function f :Rn .. lR is introduced (which we will assume to be continuously differentiable). the worst case design problem assumes the form of the semi-infinite optimization problem below: minI! (x) Ihi(x) S O.i
S'mce this is computationally extremely difficult. we have developed in [Pol.6.Sti.1] a set of techniques for replacing inequalities of the form (2.6). by slightly more conservallve. but computation ally much more tractable inequalities of the form
where the differentiable.
(2.13)
= 1.2•. ..1;gi(x) S O.j = 1.2 . . ..m;x EX! f .h' :Rn"lR,i = 1.2•... .1 are continuously gl (x) ~ max(i (x .(.).v)
(2.14)
"EO, VEN
where N is a compact subset of R4. By (2.8) being more conservative than (2.6) we mean that any x ElRn which satisfies (2.8) also satisfies (2.6). The function { in (2.8). which arises from tl1e process descri~d in [StL1J. is upper semi-continuous on Xx lR, x R4. where X is the set of design parameters which stabilize all the closed loop systems within the uncertainty range. The fun,!,ltion ( may become unbounded outside of this domain. On Xx OxJR'. (has the form (.) E [(.)0.(.)1) (x .(.).v)
=
where for i
with O · a compact interval for each j EI1.2 ... .. ml and X define~ as in (2.12a). The functions (I are assumed to have the same structure as the function (in (2.9a).
3. Development of the Optimal Design Algorithm We shall present a Phase (}-Phase I-Phase 11 algorithm for solving design problems of the form (2.13). Under a mild hypothesis. in the phase 0 stage. starting from xoEJRn . an arbitrary initial point. this algorithm constructs a finite I(X'(.)'V) sequence. Ixd:~1 such that XlooEX Next. in phase Ithe algo(10 (x.(.).v) (.) E ( (.)10-1.(.)10 ) .k E I 2.3 ..... s l 10 ( rithm computes a sequence Ix.; ldlo o' such that Xi EX for all (.H(X . (.). v) (.)~ (.),.(.).. 1] i;;" k o. In general there is a finite index. k l • such that XIo is I axl(1o (x .(.). v). (10+ I(X .(.).v)j (.) - (.)~.k E 11.2 ..... s l a feasible point for (2.13). In phase II. the algorithm constructs a sequence Ix.dt= 10,'1 such that x.; is feasible for the (2.9a) problem (2.13) for all i Elk l +1.k l +2.k l +3 .... l and the cost
~
1.2 •.. ..s. (.)i
DO
E 0 ~ [c.>o.(.).. I]
(2.9b)
and for k Ell. 2 •...• s + lJ. (10 is continuously differentiable in x at (z.(.).v) for all XEX for all (.)EO. for all vER'. We shall use the notation
sequence.
I! (x.;)li =k 1+ 1 decreases monotonically.
To simplify notation we shall restrict our discussion to the simplest version of problem (2.13). Le .• minI! (x) 19 (x)
~ TE~ (Cx .(.). v)
S
o.X EX!
(3.1a)
vEM
(2.9c)
where
X~ Ix ElRn Ig(x) ~ max (x.(.).v) S Ol . "EO
It follows (see [Cia. 1]) that
vEM
9 (z) ~ max ~(x .(.).v) "EO
(2. 10)
vEM
is a regular function and that its generalized gradient is given by
co I Vz (I; (X. (.). v) 1 «(.).v) Ero(x)l if (.) E «(.)1;-10(.)1; ).k E 11.2.... .s + II /lg (z)
= coIVzMx.(.).v) lj Elk.k+1l. (i(z.(.).v)
=(Cx,(.).v).
«(.).v) Ero(x)jif (.) = (.)1;. k EI1.2 •.... s+1l.
(2.11a)
R
where ro:R" .. 2 ,xJl< is defined by ro(x) ~ 1«(.). v) EOxN I (.) is a global maximizer of (Cx ... ·) over 0 x Nl . (2.11 b) The set se does not have a particularly convenient description for use ~ an optimization algorithm. Hence we define a subset X of X which is determined by a set of inequalities which constitute a sufficient condition for closed loop stability (see [Pol.4.Sti.1]). Hence we obtain that
X~ Ix ElR" lil'(x) SO. i
= 1.2... .,[; gi(x) S
O. j
= 1.2.... .m:l (2.12a)
where the il' :R" .. lR, i E 11.2 ..... fj are continuously differentiable. and for each j EI1.2 •. ..•m:l the gl are of the form. (2. 12b) It was shown in [Sti.1] that for each j E 11.2 ..... m:l. the functions (1:R"XR.XR~"lR are regular on R.:'xlR.-OwlxR4. where OdU is as in (2.9c) and that the have a similar
e
(3.1b)
The extension of the algorithm to the more general case of problem (2. 13) is straightforward. In the presentation to follow. we shall use many concepts and results in nondifferentiable optimization which may be found in [Cla1]. We shall omit proofs and indicate where in the literature these results may be found. 3.1 An Algorithm Model The algorithm we are going to present is based on a general algorithm theory. presented in [Pol.5]' A key aspect of this theory is the characterization of search direction finding rules. Our development of algorithms has been greatly assisted by the use of certain abstract models of the direction-finding process. We begin by defining such a model for computing descent directions for a function -t:R""lR which is locally Lipschitz continuous (hereafter abbreviated to l.L.c.) [Pol.5]' Definition 3.1: [ ? ] Let -y,:lRn .. R be l.L.c. A family of maps Rft I G,¥'Ol. ~ 0 such that G.-y,:R" .. 2 is said to be a family of convergent directianfinding (c.d.f.) maps for -y, if: (i) For all x ElR". Ga-y,(x) 8-y,(x). the Clarke generalized radient [CIa 1] of -y, at x. ii) For all x E lR" . if £ < t·. then G.-y,(x) c G.<1/I(x). ill) (a) For any t ;;" 0 and for all x E lRn. G.-y,(x) is cOnYel[. (b) G..1/I(x) is u~per semicontinuous (u.s.c.) in the sense of Berge [Ber. 1J in (t.x) at (O.x) for all x ERn. (iv) Given any E Rn . t > 0 and '3 > O. there exists a p > 0 such that for any x E B(x .p) and any fj E B-y,(x). there exists an '7 E ~-y,(x) such that 1'7-fj ~ So. • It is easy to show that if xERn is such that O~ ~i) and £ ;;" 0 is such that 0 It G.-y,(x). then
=
~
x
h.(x) ~ argmin!}~ 11 h 11 2+ max (v . h) l vEG.t(f)
(3.2)
is a descent direction for -y, at X. Setting t = 0 for all iterations results in a steepest descent like algorithm for solving min-y,(x) which may fail to zER"
converge (When -y, is not differentiable) because of the
Design of Linear Control Syst ems possibly discontinuous behavior of the search direction defined by (3.2) . The function of the parameter t is to "smooth" out this discontinuous behavior to the extent that convergence can be established. Now consider the problem min!t{x) 11/I(x),; O!
(3.3)
where /:R" -o1R is continuously differentiable and 1/I:lRn -oR is l.L.c. We show how the c.d.f. map for 1/1 may be used to construct a convergent algorithm for (3.3). Let t .., 0 and let 1/1+ {x ) ~ max!0.1/I{x)!
(3.4)
d/{x;h) =
(3.5)
d ./,{ 'h) tl
.yx.
-
maJi;
r E
(v.h)
G,'I'{z) -00
if 1/I{x) .., -t if1/l{x)
For "I > 0.11 ~ (ho. hi) e:lRn +' with h' e:Rn. we define the phase I - phase IT t-search direction for problem (3.3) by ri.(x)
\6\ 7
Remark 3 .1: The hypothesis that for all x e: Rn such that '!/I(x) > O. d 01/l(x;h),t0. in Theorem 3.1. ensures that the alg~ rithm does not jam up at infeasible point. _ 3.2 Convergent Direction FIDding M:aps for Problem (2. 13) We shall now develop an algorithm for the problem (2.13) by developing a pair of c.d.f. maps for the l.L.c. functions ff and '!/I where 1/I(x) ~ max!ff(x).g(zH .
(3.12)
The convergence properties of the algorithm that we shall state will follow directly from Theorem 3.1. First we develop_a c.d.f. map for ff under the simplifying assumption that ~(-. ... ) is continuously ditferentiable on Rn X ~ XR4. We shall need the following hypothesis. Assumption S. l : For each x e: IR". the function f(x .. .. ) has a finite number of local exlrema in OxN . -
Let r:~xR"-o;!l+xR' be defined by
f.(x) ~ !('->.II) e: OX N I ('->.11)
=(h~ (x ).hi (x»
is a local maximizer of f(x. · .. ) over OxN and
~ argminB~ II11 112 + max
g(x) - f(x .'->.11) ,; t!
Ii'ER"+l
(3 .13)
We now let !d/(x;h') -hoV11f+(x). d.1/I(x;h')!! .
(3.7)
It can be shown using the Von Neumann minimax theorem [Ber.1] that
11.(x)
=
~f~p!*11V11 2Iv
e:co![Vv1r;»].
[~'ll
).
(3.8)
lEG,-;(Z)
Let a.(x) denote the minimum value of the quadratic program on the right hand side of (3.8) . Note that 9a(x) = 0 is a standard first order optimality condition for (3.3) (see. e.g .. [Cia. 1. Pol.5]). Next. we must introduce a mechanism for reducing the "smoothing" parameter. t. as we approach a solution point of problem (3.3) (see [Pol.5]). since otherwise the resulting algorithm would converge to points that satisfy an unnecessarily weak optimality condition. For 6 e: (0.1). let
E~ !0!v!6"" Ik e:7l+!
(3.9)
(3. 14) where co denotes the convex hull. The next result was established in [Sti.1]. Proposition 3.2: Suppose Assumption 3.1 holds. Then the sets !G'.gO!.;o.o are a family ofc .d.f. maps for ff . When the function f(x .. •11) is of the form (2.9a) we must modify (3. 14) to the following form:
G.§(x) ~
(3.10) Combining the above search direction rule with the Armijo stepsize rule [Arm. 1] and an update rule. we obtain the following Phase I - Phase D Algorithm M:odel 3.1 Parameters: a .{J.6 e: (0.1). "I > O. Data: Xo e: Rn . . Step 0 : Set i O. Step 1: Compute t(",) and the direction. f4 ~ hi,,"!) (",). Step 2: If 1/1(",) > O. compute the largest stepsize Si e:! 1.(3.(32 •... ! such that
=
1/I(:r;
+ Sif4) -1/I(Xi) ,; -Siae(:r;) .
(3.lla)
If 1/1(",) ,; O. compute the largest stepsize Si e: !1.(3.~ • ... ! such that
(3.11b) and 1/I(Xi + sif4) ,; O.
(3.11c)
(3.15)
Theorem 3.3: Suppose Assumption 3.1 holds. Then the sets !G.oO!.;o.o are a family of c.d.f. maps for ~ 9(x) - mrf.II) .
-
"E ~E
G.,,(:z:)
~
O. we let
~ CO{lllz (X.'->.II) I('->.11) e: ri(x)! v!llz ~(x .'->.11) I (3.16)
where
rl(x) ~ !('->.II) e:OxN I ('->.11) is a local maximizer off(x ... ·) aver OXN.and1/l(x) - f(x.'->.II)'; e!
(3 . 17)
and r:(:z:) is obtained by replacing f by t in the above definition. It has been shown in [Sti.1] that. if t(x .... ) has a finite number of local extrema in OXN. for each x e: IR". then the family I G.",' H. ~ is a family of c.d.f. maps for 1/1.
°
3.3 APhaae I>-Pbase I-Pbase D Algorithm
We shall now construct a phase 0 - phase I - phase 11 algorithm. based on Algorithm Model 3.1. for solving problem (2.13) . We define the phase 0 e-search direction at :z:e:R" by (3. 18)
Step 3: Set :r;+, i
= :r; + Sif4
=i
+
1
and go to Step 1. _ The following convergence result was established in [Pol.5]' Theorem 3 .1: Suppose that for all x e: Rn such that '!/I(x) > O. d 01/l(x;h),t0. If £ is any accumulation point of a sequence !:r; !i-=O. constructed by Algorithm Model 3.1. then '!/I(£) ,; 0 and 9 0 (£)
Ilzt(x.'->.v) .
In [Sti.l] we find the following result.
Sunilarly. for t
and let
co
''' .~)E r"Z)
= O.
-
and the phase I - phase 11 e-search direction at x e:R" by 11.{x)
={h~ (x ).hi(x» (3. 19)
We denote by G.{x) and 9.{x) the minimum value of the quadratic program in (3. 18) and (3. 19). resp ec tively. Next. we
E. Polak and D. M. Stiml er
1618
introduce a pair of smoothing functions,
&(x) ~ max!I:E:E I8.(x),;; -I:l
(3.20)
l:(x)~max!I:E:E I 6.(x),;; -I:l .
(3.21)
(3.22)
[Gon.2) Gonzaga. C., and Polak, E., "On constraint dropping schemes and optirnality functions for a class of outer approximations algorithms", J. SIAM Control and Optimiza.tion Vol. 17, 1979.
(3 .23)
[May.1) Mayne, D. Q. , and Polak, E., "A qua dra tic ally c onvergent algorithm for solving infinite-dim e nsional ine qualities" , Jour . 0/ /!ppl . Ma.th . a.nd. Optimiza.tion, VoL 9, pp. 25-4-0, 1982.
F'mally we define the phase 0 search direction at x by h(x) ~ hc(z)(x)
and the phase I - phase II search direction at x by
h(x)~h.l(z)(X) .
Remark 3.2: We note that the computation of h(x) and h (x) involves a loop. • We now state the algorithm. Optimal Design Algorithm 3.2: Parameters: Cl, p, 6 E: (0,1), '"1 > O. Data: Xo . Step 0 : Set i = O. _ Step 1: If o9(X,»OA com{>ute &(x;) and ht ~ h(xi)' Else, computel:(x,) and ht = h(x,). Step 2: If o9(x,»O, compute the largest Si E:!l,P,p2, ... l such that
(3. 23) If 09 (x;) ,;; 0 and ..p(x,) such that
> 0, compute the largest Si E: !l,P,p2,,,.l o9(x, + s,ht) ,;; 0
(3. 24-a)
..p(x, + s,ht) - ..p(Xi) ,;; -SiClI:(x;) .
(3. 24-b)
If ..p(x,) ,;; 0, compute the largest Si E: !1,p,{l2,
[Gon.1) Gonzaga. C., Polak, E., and Trahan, R., "An improved algorithm for optimization problems with function a l inequality constraints", IEEE 1'rans., Vol. AC-25, No . 1, 1980.
...l
such
that
/ (x, + s,ht) - / (Xi)';; -SiClI:(X,)
(3.25a)
,,(X, + s,ht),;; 0 .
(3.25b)
and
Step 3: Set (3.26) replace i by i + 1 and go to Step 1. • The following convergence result was established in [Sti.1]. Theorem 3 .4: (a) Suppose that for all x E:lRn such that o9(x);;" 0, 90 (x) < O. Then for each s e quenc e !x,l c onstructe d by Algorithm 3.2 the re exists a finite positive integer, ko, such that g(x/e~ ,;; O. (b) Suppose that for all x E:IRn such that ..p(x);;" 0, 8 0 (x)<0 . If fi is any accumulation point of a s e quence !xi It=o constructed by Algorithm 3.2, then ..p(fi) ,;; 0 and 6 0 (fi) O. •
=
4. Conclusion We have shown that the worst case design of SISO control systems leads to rather unusual semi-infinite optimization problems which are outside of the cla ss solvable b y existing optimization algorithms . We have also shown that the theory of nonditIerentiable algorithms pre sent e d in [Pol. 5) provides a map which can be used to construct an optimal design algorithm for these problems. 5. References
[Arm.1) Armijo, 1.., "Minimization of functio ns having continuous partial derivatives", Pacific J. Math ., vol. 16, pp. 1-3, 1966. [Bec .1) Becker, R. G., Heunis, A. J., and Mayne, D. Q., "Computer-aided design of control systems via optimization", Proe . lEE, vol. 126, no. 6, 1979. [Ber.1) Berge, C., Topological Spaces , Macmillan, New York, N.Y., 1963. [Cla. 1) Clarke, F. H., Optimizaton and Nons'l7LlJoth Analysis, Wiley-Interscience, New York, N.Y., 1963.
[PoLl) Polak, E. and Mayne, D. Q. , "An Algorithm for optimization problems with fun c tional ine quality constraints" , IEEE 1'ra.ns. , Vol. AC-21, No. 2, 1976. [Pol.2) Polak, E., and Wardi, Y. Y., "A nondiffe r e ntiable optimization algorithm for the d e s ign of con t rol syste ms subj e ct to singular value ine qua lities ove r a fr e que n cy range", Auto matic a., Vol. 18, NO . 3, pp. 267-28 3, 1982. [Pol.3) Polak, E. , " Semi-infinite optimization in engine ering design", in Lecture Notes in Economics and Mathematical Systems, Vol. 215: Semi-Infinite Progra.mming a.nd /!pplica.lions, Edited by A. V. Fiacco and K. O. Kortanek , Spring erVerlag, Berlin, New York, Tokyo, 1983. [Pol.4-) Polak, E., "A Modified Nyquist stability criterion for use in computer-aided design", IEEE 1'ra.ns. on Automa.tic Control , Vol. AC-29, No. 1, pp 91-93, 1984-. [Pol.5) Polak. E., and Mayne, D. Q., "Algorithm models for nondifferentiable optimization", University of California, Electronics Research Laboratory Memo UCB/ ERL No. M82/ 34-, 5/ 10/ 82. [Pol.6) Polak, E., and Stimler, D. M. ,"Optimization-Based Design of Control Systems with Unce rtain Plant:Pr oblem Formulation", Memo No UCB/ ERL 83/ 16,University of California,Berkeley, 1983. [Sti.l) Stimler, D. M., "Optimization-Based De sign of Control Systems with Uncertain Plant, " Ph.D. The sis, Univers ity of California, Berkeley, 1984-.
Fi g. 1
CO :'\TROL API'LJCATIO:'\S OF :'\O:'\LJ:,\L-\R PROGR .·\\I\II:,\( ~ II
ON THE DESIGN OF LINEAR CONTROL SYSTEMS WITH PLANT UNCERTAINTY VIA NONDIFFERENTIABLE OPTIMIZA TION* E. Polak and D. M. Stimler Depa rt ml'1lt of t:lectncal E nf,{l1u'l'r1nK and Compll tl'r SCIeNce" aud Ih t' E Il'Ctro llln [" 1II1'er,,1.' 0/ CtlIi/onl/tI. B erkfil~. CA 'J./ 720. ["SA
ABSI'RAcr We present a three phase semi-infinite optimization algorithm which is motivated by worst case, single-input single-output control system design problems. When suitably interpreted, the algorithm can also be used for multivariable control system design with performance requirements that lead to semi-infinite inequality constraints.
R l'~e(l r(h
Labomlun,
meetinl!> this goal is to carry out a worst ca.se design which ensures that specifications will be met for all mathematical plant models in a prescribed class. It was shown in [Sti.1] that worst case design of single-input single-output (SI SO) control systems, in the frequency domain, can be expressed as a somewhat unusual semi-infinite optimization problem. In this section we shall summarize the findings presented in [Sti.1] so as to justify the structure of the algorithm to be presented in the next section. We shall use the system configuration of Figure 1 for illustration. We assume that the plant transfer function is of the form,
1. Introduction Over the last decade, the control system designer's ability to make intell~ent decisions has been tremendously enhanced by the introduction of interactive computing packages which enable the designer to define problems by means of graphical editors, simulate specified system responses and display graphically the results. As the power of these definition. simulation and display systems grows, designers are becoming more and more aware of the fact that a major obstacle to obtaining significantly better p erformance than is currently possible is traceable to the limited ability humans have for searching through alternatives in a multidimensional design parameter space. In response to this phenomenon in control engineering as well as in other engineering areas, there has arisen a new area of research in optimization: semi-1nfinite optimization, i.e., optimization over a finite dimensional desi~n vector which must satisfy a continuum of inequalities (see, e.~., [Bec.!. Gon.!. Gon.2, May. 1, PoLl, Pol.2, Pol.3, Pol.4, Pol.5 J). These inequalities are referred to as semi-infinite. Such inequalities arise when system responses are constrained to lie in envelopes over frequency or time . They also arise when the plant depends on uncertain parameters which leads to a worst case design situation. An important issue in optimization-bas ed computeraided design is that of problem formulation . In particuler, care must be taken to formulate design requirements as constraints which involve functions that are compatible with the requirements of the optimization algorithm and are computationally tractable . 1n order that we may use nondifferentiable optimization techniques [Cla.1], the constraint functions must be locally Upschitz continuous. In Section 2, we show that the worst case design of single-input single-output (SISO) control systems can be reduced to a tractable, tbough ratber unusual semi-infinite optimization problem. The unusual properties of this problem are due to the fact that a c losed loop system whose component subsystems have rational transfer functions may have transfer functions which are dis continuous as a function of frequency. Since there are no algorithms in the literature that solve this type of semi-infinite optimization problem, we have constructed a special algorithm for its solution. The algorithm is presented in Section 3. (When suitably modified. the algorithm can also be used for multiinput mUlti-output control system design).
P(s,a,l(s)) ~ P.(s,a)l(s)
(2.1)
Po(s,a) ~ KTI(s + z');TI(s + pi)
(2.2)
where ,
i
with a a vector whose components are the gain K, the zeros z' and the poles pi and l (-) EL is a perturbation function which is used to express unstructured uncertainty attributable to unmodelled dynamics, high frequency measurement errors, etc. We introduce the possibility of structured v:ncertainty in our model by assuming that the gain K, the zeros z' and the poles pi lie in confidence int er vals of the formt
The product of all the uncertainty intervals in (2.3) will be denoted by A; it is a compact subset of !Rn.. The set of unstructured perturbation functions L is assumed to consist of all rational transfer functions, l (s), which have equal numerator and denominator degree and which are bounded in magnitude and phase as follows:
l,v(r.»";; Il(jr.»
1 ,.;;
lAl(r.»
V r.>E!R+
(2.4a) (2.4b)
where l,v, rAl,1.t,~ :R+"'R are bound functions which satisfy
o
continuously differentiable
,.;; 1,.;; [Al(r.»
V r.> E R.-
(2.4c) (2.4d)
It is convenient to use the faclored form, below, for the compensator transfer functions :
C(%,s)
=
Kc{s + de) I](s2+2ats +(bt-)2) (s +dg) TI(s2+2ats +(bC)2) i
(25) .
A similar form will be used for F(% ,s). The design vector EIR", to be determined by optimization, is composed of the free coefficients in these representations of C and F. It was shown in [Pol.6,Sti.1] that many design requirements (including disturbance rejection, saturation avoidance, closed loop stability, etc.) may be state d as inequality constraints on the design vector. These in equalities are of the form
%
2. Worst Case Design as a Semi-Infinite Optimization Problem A major goal in control system design is to ensure satisfactory system performance in the pres ence of uncertainty in the mathematical model of the plant. One approach to
ct%,r.>,a,l(jr.»)";; 0
V r.>EO, V aEAand V l EL ,(2.6)
• Research spmwored by the National Science Foundation under grant t For complex zeros and poles which must occur in complex cO::1jugate EC8-S121l49, the otfice of Naval Research \IDder contract NOOOI4-83-K-0602 the Jdr Farce otfice of Scienililc Research grant AFOSR-B3-0361, the Sem: pairs we assume that the confidence "intervals" are in fac t rect angles in C lconductor Re""arch Comortium grant SRC-82-1 HlOB, and a Universi t y of !(u,v) ER2 IRe,(w')";; u,.;; Re(wi).Im,(wi),.;; v ,.;; Sydney TraveJlin& Scholandrip.
~l~w'] ~
16 15
Ir~'{(wi)l
E. Polak and D. M. Stimler
1616
where 0 c JR is compact. Verification that (2.6) is satisfied is equivalent to evaluating max ~(x.(.).a.l(j(.)) . ~~~ 'EL
(2.7)
description to ( in (2.9a) . Finally. taking all these considerations into account. when a cost function f :Rn .. lR is introduced (which we will assume to be continuously differentiable). the worst case design problem assumes the form of the semi-infinite optimization problem below: minI! (x) Ihi(x) S O.i
S'mce this is computationally extremely difficult. we have developed in [Pol.6.Sti.1] a set of techniques for replacing inequalities of the form (2.6). by slightly more conservallve. but computation ally much more tractable inequalities of the form
where the differentiable.
(2.13)
= 1.2•. ..1;gi(x) S O.j = 1.2 . . ..m;x EX! f .h' :Rn"lR,i = 1.2•... .1 are continuously gl (x) ~ max(i (x .(.).v)
(2.14)
"EO, VEN
where N is a compact subset of R4. By (2.8) being more conservative than (2.6) we mean that any x ElRn which satisfies (2.8) also satisfies (2.6). The function { in (2.8). which arises from tl1e process descri~d in [StL1J. is upper semi-continuous on Xx lR, x R4. where X is the set of design parameters which stabilize all the closed loop systems within the uncertainty range. The fun,!,ltion ( may become unbounded outside of this domain. On Xx OxJR'. (has the form (.) E [(.)0.(.)1) (x .(.).v)
=
where for i
with O · a compact interval for each j EI1.2 ... .. ml and X define~ as in (2.12a). The functions (I are assumed to have the same structure as the function (in (2.9a).
3. Development of the Optimal Design Algorithm We shall present a Phase (}-Phase I-Phase 11 algorithm for solving design problems of the form (2.13). Under a mild hypothesis. in the phase 0 stage. starting from xoEJRn . an arbitrary initial point. this algorithm constructs a finite I(X'(.)'V) sequence. Ixd:~1 such that XlooEX Next. in phase Ithe algo(10 (x.(.).v) (.) E ( (.)10-1.(.)10 ) .k E I 2.3 ..... s l 10 ( rithm computes a sequence Ix.; ldlo o' such that Xi EX for all (.H(X . (.). v) (.)~ (.),.(.).. 1] i;;" k o. In general there is a finite index. k l • such that XIo is I axl(1o (x .(.). v). (10+ I(X .(.).v)j (.) - (.)~.k E 11.2 ..... s l a feasible point for (2.13). In phase II. the algorithm constructs a sequence Ix.dt= 10,'1 such that x.; is feasible for the (2.9a) problem (2.13) for all i Elk l +1.k l +2.k l +3 .... l and the cost
~
1.2 •.. ..s. (.)i
DO
E 0 ~ [c.>o.(.).. I]
(2.9b)
and for k Ell. 2 •...• s + lJ. (10 is continuously differentiable in x at (z.(.).v) for all XEX for all (.)EO. for all vER'. We shall use the notation
sequence.
I! (x.;)li =k 1+ 1 decreases monotonically.
To simplify notation we shall restrict our discussion to the simplest version of problem (2.13). Le .• minI! (x) 19 (x)
~ TE~ (Cx .(.). v)
S
o.X EX!
(3.1a)
vEM
(2.9c)
where
X~ Ix ElRn Ig(x) ~ max (x.(.).v) S Ol . "EO
It follows (see [Cia. 1]) that
vEM
9 (z) ~ max ~(x .(.).v) "EO
(2. 10)
vEM
is a regular function and that its generalized gradient is given by
co I Vz (I; (X. (.). v) 1 «(.).v) Ero(x)l if (.) E «(.)1;-10(.)1; ).k E 11.2.... .s + II /lg (z)
= coIVzMx.(.).v) lj Elk.k+1l. (i(z.(.).v)
=(Cx,(.).v).
«(.).v) Ero(x)jif (.) = (.)1;. k EI1.2 •.... s+1l.
(2.11a)
R
where ro:R" .. 2 ,xJl< is defined by ro(x) ~ 1«(.). v) EOxN I (.) is a global maximizer of (Cx ... ·) over 0 x Nl . (2.11 b) The set se does not have a particularly convenient description for use ~ an optimization algorithm. Hence we define a subset X of X which is determined by a set of inequalities which constitute a sufficient condition for closed loop stability (see [Pol.4.Sti.1]). Hence we obtain that
X~ Ix ElR" lil'(x) SO. i
= 1.2... .,[; gi(x) S
O. j
= 1.2.... .m:l (2.12a)
where the il' :R" .. lR, i E 11.2 ..... fj are continuously differentiable. and for each j EI1.2 •. ..•m:l the gl are of the form. (2. 12b) It was shown in [Sti.1] that for each j E 11.2 ..... m:l. the functions (1:R"XR.XR~"lR are regular on R.:'xlR.-OwlxR4. where OdU is as in (2.9c) and that the have a similar
e
(3.1b)
The extension of the algorithm to the more general case of problem (2. 13) is straightforward. In the presentation to follow. we shall use many concepts and results in nondifferentiable optimization which may be found in [Cla1]. We shall omit proofs and indicate where in the literature these results may be found. 3.1 An Algorithm Model The algorithm we are going to present is based on a general algorithm theory. presented in [Pol.5]' A key aspect of this theory is the characterization of search direction finding rules. Our development of algorithms has been greatly assisted by the use of certain abstract models of the direction-finding process. We begin by defining such a model for computing descent directions for a function -t:R""lR which is locally Lipschitz continuous (hereafter abbreviated to l.L.c.) [Pol.5]' Definition 3.1: [ ? ] Let -y,:lRn .. R be l.L.c. A family of maps Rft I G,¥'Ol. ~ 0 such that G.-y,:R" .. 2 is said to be a family of convergent directianfinding (c.d.f.) maps for -y, if: (i) For all x ElR". Ga-y,(x) 8-y,(x). the Clarke generalized radient [CIa 1] of -y, at x. ii) For all x E lR" . if £ < t·. then G.-y,(x) c G.<1/I(x). ill) (a) For any t ;;" 0 and for all x E lRn. G.-y,(x) is cOnYel[. (b) G..1/I(x) is u~per semicontinuous (u.s.c.) in the sense of Berge [Ber. 1J in (t.x) at (O.x) for all x ERn. (iv) Given any E Rn . t > 0 and '3 > O. there exists a p > 0 such that for any x E B(x .p) and any fj E B-y,(x). there exists an '7 E ~-y,(x) such that 1'7-fj ~ So. • It is easy to show that if xERn is such that O~ ~i) and £ ;;" 0 is such that 0 It G.-y,(x). then
=
~
x
h.(x) ~ argmin!}~ 11 h 11 2+ max (v . h) l vEG.t(f)
(3.2)
is a descent direction for -y, at X. Setting t = 0 for all iterations results in a steepest descent like algorithm for solving min-y,(x) which may fail to zER"
converge (When -y, is not differentiable) because of the
Design of Linear Control Syst ems possibly discontinuous behavior of the search direction defined by (3.2) . The function of the parameter t is to "smooth" out this discontinuous behavior to the extent that convergence can be established. Now consider the problem min!t{x) 11/I(x),; O!
(3.3)
where /:R" -o1R is continuously differentiable and 1/I:lRn -oR is l.L.c. We show how the c.d.f. map for 1/1 may be used to construct a convergent algorithm for (3.3). Let t .., 0 and let 1/1+ {x ) ~ max!0.1/I{x)!
(3.4)
d/{x;h) =
(3.5)
d ./,{ 'h) tl
.yx.
-
maJi;
r E
(v.h)
G,'I'{z) -00
if 1/I{x) .., -t if1/l{x)
For "I > 0.11 ~ (ho. hi) e:lRn +' with h' e:Rn. we define the phase I - phase IT t-search direction for problem (3.3) by ri.(x)
\6\ 7
Remark 3 .1: The hypothesis that for all x e: Rn such that '!/I(x) > O. d 01/l(x;h),t0. in Theorem 3.1. ensures that the alg~ rithm does not jam up at infeasible point. _ 3.2 Convergent Direction FIDding M:aps for Problem (2. 13) We shall now develop an algorithm for the problem (2.13) by developing a pair of c.d.f. maps for the l.L.c. functions ff and '!/I where 1/I(x) ~ max!ff(x).g(zH .
(3.12)
The convergence properties of the algorithm that we shall state will follow directly from Theorem 3.1. First we develop_a c.d.f. map for ff under the simplifying assumption that ~(-. ... ) is continuously ditferentiable on Rn X ~ XR4. We shall need the following hypothesis. Assumption S. l : For each x e: IR". the function f(x .. .. ) has a finite number of local exlrema in OxN . -
Let r:~xR"-o;!l+xR' be defined by
f.(x) ~ !('->.II) e: OX N I ('->.11)
=(h~ (x ).hi (x»
is a local maximizer of f(x. · .. ) over OxN and
~ argminB~ II11 112 + max
g(x) - f(x .'->.11) ,; t!
Ii'ER"+l
(3 .13)
We now let !d/(x;h') -hoV11f+(x). d.1/I(x;h')!! .
(3.7)
It can be shown using the Von Neumann minimax theorem [Ber.1] that
11.(x)
=
~f~p!*11V11 2Iv
e:co![Vv1r;»].
[~'ll
).
(3.8)
lEG,-;(Z)
Let a.(x) denote the minimum value of the quadratic program on the right hand side of (3.8) . Note that 9a(x) = 0 is a standard first order optimality condition for (3.3) (see. e.g .. [Cia. 1. Pol.5]). Next. we must introduce a mechanism for reducing the "smoothing" parameter. t. as we approach a solution point of problem (3.3) (see [Pol.5]). since otherwise the resulting algorithm would converge to points that satisfy an unnecessarily weak optimality condition. For 6 e: (0.1). let
E~ !0!v!6"" Ik e:7l+!
(3.9)
(3. 14) where co denotes the convex hull. The next result was established in [Sti.1]. Proposition 3.2: Suppose Assumption 3.1 holds. Then the sets !G'.gO!.;o.o are a family ofc .d.f. maps for ff . When the function f(x .. •11) is of the form (2.9a) we must modify (3. 14) to the following form:
G.§(x) ~
(3.10) Combining the above search direction rule with the Armijo stepsize rule [Arm. 1] and an update rule. we obtain the following Phase I - Phase D Algorithm M:odel 3.1 Parameters: a .{J.6 e: (0.1). "I > O. Data: Xo e: Rn . . Step 0 : Set i O. Step 1: Compute t(",) and the direction. f4 ~ hi,,"!) (",). Step 2: If 1/1(",) > O. compute the largest stepsize Si e:! 1.(3.(32 •... ! such that
=
1/I(:r;
+ Sif4) -1/I(Xi) ,; -Siae(:r;) .
(3.lla)
If 1/1(",) ,; O. compute the largest stepsize Si e: !1.(3.~ • ... ! such that
(3.11b) and 1/I(Xi + sif4) ,; O.
(3.11c)
(3.15)
Theorem 3.3: Suppose Assumption 3.1 holds. Then the sets !G.oO!.;o.o are a family of c.d.f. maps for ~ 9(x) - mrf
-
"E ~E
G.,,(:z:)
~
O. we let
~ CO{lllz (X.'->.II) I('->.11) e: ri(x)! v!llz ~(x .'->.11) I (3.16)
where
rl(x) ~ !('->.II) e:OxN I ('->.11) is a local maximizer off(x ... ·) aver OXN.and1/l(x) - f(x.'->.II)'; e!
(3 . 17)
and r:(:z:) is obtained by replacing f by t in the above definition. It has been shown in [Sti.1] that. if t(x .... ) has a finite number of local extrema in OXN. for each x e: IR". then the family I G.",' H. ~ is a family of c.d.f. maps for 1/1.
°
3.3 APhaae I>-Pbase I-Pbase D Algorithm
We shall now construct a phase 0 - phase I - phase 11 algorithm. based on Algorithm Model 3.1. for solving problem (2.13) . We define the phase 0 e-search direction at :z:e:R" by (3. 18)
Step 3: Set :r;+, i
= :r; + Sif4
=i
+
1
and go to Step 1. _ The following convergence result was established in [Pol.5]' Theorem 3 .1: Suppose that for all x e: Rn such that '!/I(x) > O. d 01/l(x;h),t0. If £ is any accumulation point of a sequence !:r; !i-=O. constructed by Algorithm Model 3.1. then '!/I(£) ,; 0 and 9 0 (£)
Ilzt(x.'->.v) .
In [Sti.l] we find the following result.
Sunilarly. for t
and let
co
''' .~)E r"Z)
= O.
-
and the phase I - phase 11 e-search direction at x e:R" by 11.{x)
={h~ (x ).hi(x» (3. 19)
We denote by G.{x) and 9.{x) the minimum value of the quadratic program in (3. 18) and (3. 19). resp ec tively. Next. we
E. Polak and D. M. Stiml er
1618
introduce a pair of smoothing functions,
&(x) ~ max!I:E:E I8.(x),;; -I:l
(3.20)
l:(x)~max!I:E:E I 6.(x),;; -I:l .
(3.21)
(3.22)
[Gon.2) Gonzaga. C., and Polak, E., "On constraint dropping schemes and optirnality functions for a class of outer approximations algorithms", J. SIAM Control and Optimiza.tion Vol. 17, 1979.
(3 .23)
[May.1) Mayne, D. Q. , and Polak, E., "A qua dra tic ally c onvergent algorithm for solving infinite-dim e nsional ine qualities" , Jour . 0/ /!ppl . Ma.th . a.nd. Optimiza.tion, VoL 9, pp. 25-4-0, 1982.
F'mally we define the phase 0 search direction at x by h(x) ~ hc(z)(x)
and the phase I - phase II search direction at x by
h(x)~h.l(z)(X) .
Remark 3.2: We note that the computation of h(x) and h (x) involves a loop. • We now state the algorithm. Optimal Design Algorithm 3.2: Parameters: Cl, p, 6 E: (0,1), '"1 > O. Data: Xo . Step 0 : Set i = O. _ Step 1: If o9(X,»OA com{>ute &(x;) and ht ~ h(xi)' Else, computel:(x,) and ht = h(x,). Step 2: If o9(x,»O, compute the largest Si E:!l,P,p2, ... l such that
(3. 23) If 09 (x;) ,;; 0 and ..p(x,) such that
> 0, compute the largest Si E: !l,P,p2,,,.l o9(x, + s,ht) ,;; 0
(3. 24-a)
..p(x, + s,ht) - ..p(Xi) ,;; -SiClI:(x;) .
(3. 24-b)
If ..p(x,) ,;; 0, compute the largest Si E: !1,p,{l2,
[Gon.1) Gonzaga. C., Polak, E., and Trahan, R., "An improved algorithm for optimization problems with function a l inequality constraints", IEEE 1'rans., Vol. AC-25, No . 1, 1980.
...l
such
that
/ (x, + s,ht) - / (Xi)';; -SiClI:(X,)
(3.25a)
,,(X, + s,ht),;; 0 .
(3.25b)
and
Step 3: Set (3.26) replace i by i + 1 and go to Step 1. • The following convergence result was established in [Sti.1]. Theorem 3 .4: (a) Suppose that for all x E:lRn such that o9(x);;" 0, 90 (x) < O. Then for each s e quenc e !x,l c onstructe d by Algorithm 3.2 the re exists a finite positive integer, ko, such that g(x/e~ ,;; O. (b) Suppose that for all x E:IRn such that ..p(x);;" 0, 8 0 (x)<0 . If fi is any accumulation point of a s e quence !xi It=o constructed by Algorithm 3.2, then ..p(fi) ,;; 0 and 6 0 (fi) O. •
=
4. Conclusion We have shown that the worst case design of SISO control systems leads to rather unusual semi-infinite optimization problems which are outside of the cla ss solvable b y existing optimization algorithms . We have also shown that the theory of nonditIerentiable algorithms pre sent e d in [Pol. 5) provides a map which can be used to construct an optimal design algorithm for these problems. 5. References
[Arm.1) Armijo, 1.., "Minimization of functio ns having continuous partial derivatives", Pacific J. Math ., vol. 16, pp. 1-3, 1966. [Bec .1) Becker, R. G., Heunis, A. J., and Mayne, D. Q., "Computer-aided design of control systems via optimization", Proe . lEE, vol. 126, no. 6, 1979. [Ber.1) Berge, C., Topological Spaces , Macmillan, New York, N.Y., 1963. [Cla. 1) Clarke, F. H., Optimizaton and Nons'l7LlJoth Analysis, Wiley-Interscience, New York, N.Y., 1963.
[PoLl) Polak, E. and Mayne, D. Q. , "An Algorithm for optimization problems with fun c tional ine quality constraints" , IEEE 1'ra.ns. , Vol. AC-21, No. 2, 1976. [Pol.2) Polak, E., and Wardi, Y. Y., "A nondiffe r e ntiable optimization algorithm for the d e s ign of con t rol syste ms subj e ct to singular value ine qua lities ove r a fr e que n cy range", Auto matic a., Vol. 18, NO . 3, pp. 267-28 3, 1982. [Pol.3) Polak, E. , " Semi-infinite optimization in engine ering design", in Lecture Notes in Economics and Mathematical Systems, Vol. 215: Semi-Infinite Progra.mming a.nd /!pplica.lions, Edited by A. V. Fiacco and K. O. Kortanek , Spring erVerlag, Berlin, New York, Tokyo, 1983. [Pol.4-) Polak, E., "A Modified Nyquist stability criterion for use in computer-aided design", IEEE 1'ra.ns. on Automa.tic Control , Vol. AC-29, No. 1, pp 91-93, 1984-. [Pol.5) Polak. E., and Mayne, D. Q., "Algorithm models for nondifferentiable optimization", University of California, Electronics Research Laboratory Memo UCB/ ERL No. M82/ 34-, 5/ 10/ 82. [Pol.6) Polak, E., and Stimler, D. M. ,"Optimization-Based Design of Control Systems with Unce rtain Plant:Pr oblem Formulation", Memo No UCB/ ERL 83/ 16,University of California,Berkeley, 1983. [Sti.l) Stimler, D. M., "Optimization-Based De sign of Control Systems with Uncertain Plant, " Ph.D. The sis, Univers ity of California, Berkeley, 1984-.
Fi g. 1