A Convex Programming Approach to the Multiobjective H =H1 Problem 2 1 Seddik M. Djouadi , 1 2 C.D. Charalambous , Systems Eng. Dept., University of Arkansas, Little Rock, AR 72204, 2 SITE, University of Ottawa, Ontario K1N 6N5, Canada, 3 AFRL, Wright-Patterson Air Force Base, Dayton, Ohio 45433 msdjouadi@ualr.edu, chadcha@site.uottawa.ca, daniel.repperger@he.wpafb.af.mil In this paper, Banach space duality theory for the multiobjective H 2 =H 1 problem developed recently by the authors, is used to develop algorithms to solve this problem by approximately reducing the dual and predual reperesentations to nite variable convex optimizations. The Ellipsoid algorithm is then applied to these problems to obtain polynomial-time, nonheuristic programs which nd \nearly" optimal control laws. These algorithms have been implemented numerically to compute an example. a convex function in Q. The problem (1) is innite dimensional in the sense that there is no nite limit to the number of parameters required to specify Q. However, (1) can be approximated by a nite variable convex optimization by restricting Q to lie in the space Pm consisting of analytic trigonometric polynomials of degree at most m of the form a0 + a1 z + + am z m with real coeÆcients, and then discretize the unit circle sufciently nely with respect to m. This yields a convex optimization problem in the variables a0 ; a1 ; ; am . For any xed m, these convex problems generate upper bounds for o and suboptimal control laws, since Q is conned to a proper nite dimensional subspace of H 1 . Such problems are standard applications of convex programming techniques. The technique that will be used here is the Ellipsoid algorithm [6]. This algorithm has the advantage of simplicity, robustness and polynomial time execution. However, it can be slower than other interior point methods. The Ellipsoid algorithm is chose primary for its non-heuristic stopping time [7], that is, for a specied tolerance the algorithm terminates only when the estimate of the optimum is within of the true minimum. Abstract - 1 Introduction In [1] the authors characterized the solution of the multiobjective H 2 =H 1 using Banach space duality theory. Dual and predual spaces were identied, and equivalent maximizations formulated therein. In particular existence of at least one optimal controller was deduced, and shown to satisfy a atness property together with an extremal identity. Subsequently, similar duality results were also proposed in [2]. Predual and dual representations of the multiobjective H 2 =H 1 comprise maximizations or suprema, in contrast to the inmum of the original form, which corresponds to the Pareto optimal solution for the H 2 =H 1 problem [3, 4, 1] o N X = Q2H 1inf(C 3 D.W. Repperger p q) i=1 kHi Ui QVi k + L X 2 N +1 kHi Ui QVi k1 where the matrix transfer functions Hi , Ui , and Vi are stable and of appropriate dimensions, i.e., Hi 2 H 1 (C ni mi ), Ui 2 H 1 (C ni p ), and nally V 2 H 1 (C pmi ). Q is the Youla parameter Q 2 H 1 (C pq ). Each term in (1) represents a particular closed loop ni -input/mi -output transmission from particular exogenous disturbances to particular desired signal outputs [5, 4]. The results of [1] lead naturally to a dual pair of numerical solutions, which approach the optimal o from opposite directions, and have the virtue of producing estimates of o together with tolerance on these estimates. The rst of these solutions, which will be referred to as the \primary", exploits the fact P P that Ni=1 kHi Ui QVi k2 + LN +1 kHi Ui QVi k1 is ! The second or \dual" solution exploits the representation of (1) in the predual in terms of the following maximization (1) 00 11 H1 BB . CC o = sup @@ .. AA 2? S;kk1 H (2) L where ? S is a preorthogonal subspace in the predual, and () is a linear functional to be dened in the sequel. The dual solution as represented by (2) is therefore a convex maximization. The subspace ? S can be restricted to a nite subspace consisting of trigonometric polynomials, which after suÆciently ne discretization of the unit circle, the resulting nite variable constrained yields lower bounds for the optimization represented by o . It can be shown that as m ! 1, the upper and lower bounds obtained by this procedure converge. The multiobjective H 2 =H 1 control problem has received a large attention in the control community, e.g., [8, 5, 9, 4, 10, 11, 12, 13] where several relaxations and approximation methods of bounds have been proposed. Similar ideas to here using nite dimensional approximations based on truncating the Youla parameter Q, system impulse response and/or innite horizons were proposed in [9, 4, 13, 10]. However, the approximations may lead to a controller not feasible for the true closed loop system, and very large optimization problems. An improvement was suggested in [4] by formulating the problem as a semi-denite programming problem (SDP), using the linear matrix inequalities (LMI) formulation of H 2 and H 1 norms [14, 5, 11, 12]. The drawbacks of this approach is that there is no analysis on the rate of convergence to the optimum, or the degree of suboptimality. Moreover the LMI formulation introduces a large number of auxiliary variables [4, 11, 12]. 0 (7) is the distance from B @ 0 . space S := B @ .. L X =1 Expression (1) is equivalent to nding the shortest distance from a vector to a Banach subspace, dened as follows. Let B be the Banach space con Q sisting of the Cartesian product Ni=1 L2(C ni mi ) Q L 1 (C ni mi ) consisting of L matrix-valued L i=N +1 (ni mi ) functions on the unit disk, endowed with the following the norm: B @ kF kB := N X kFi k + L X 2 i=1 i=N +1 kFi k1 ; F =B B . @ .. C C A (3) FL where k k2 and k k1 denote the H 2 and H 1 norms respectively, that is: s kFi k2 = Z 2 0 T r Fi (ei )? Fi (ei ) dm (4) represents the normalized Lebesgue measure dened on the unit circle, and kFi k1 = ess sup jFi (ei )j (5) dm 2[0;2) where j j denotes the largest singular value. Then (1) is equivalent to: o = 0 1 H1 B . C inf @ .. A Q2H 1 (Cpq ) H 0 B @ U1 .. . 1 0 C B AQ@ UL L V1 .. . VL 1 C A (6) B By pre-multiplying the terms in the LHS of (6) by U1?i , , ULi? , the adjoints of the inner factors? of respectively ? U1 , , UL and post-multiplying by V1ci , , VLci , the adjoints of the co-inner factors of respectively V1 , , VL , the H 2 and H 1 norms are preserved, and then (6) is equivalent to: o = Q2H 1inf(C 0 B B @ q) p U1?i H1 V1?ci .. . ? H V? ULi L Lci 1 0 C C A B B @ U1o .. . ULo 1 0 C B CQB A @ V1co .. . VLco 1 C C A B ? Uio Uio i 0 F1 B F2 C .. . ? ? ULi HL VLci 0 V1co .. . C 1 B A H (C pq ) @ ULo S 1 1 ? C A to the sub- 1 C A of B . To VLco ensure closedness of S the following assumption will be made throughout: PL ? (A1) there exists Æ > 0 such that i=1 Uio Uio Æ , PL ? and i=1 Vico Vico Æ for all 2 [0; 2). Under assumption (A1) there exist an outer spectral factor 1 , andP a co-outer spectral factor 2 for PL L ? ? i=1 Uio Uio , and i=1 Vico Vico , respectively, i.e., 2 Banach Space Duality Theory 0 1 U1o ? U1i H1 V1ci = ?1 1; L X =1 ? Vico Vico i = 2 ?2 has then the following equivalent description, let R := 1 R1 .. . C A 2 QL i=1 H 1 (C m p ) i be the outer isometry RL 0 1 U1o . whose range coincides with the range of B @ .. 0 More explicitly that QL R? R i=1 H R has the form = R = I , m a.e. Likewise let 1 (C qn ) be i B @ T C A [15]. ULo 1 1 .. . 1 ULo 1 0 U1o := B @ T1 .. . 1 so C A 1 2 C A TL the co-outer isometry whose range 0 1 coincides with the range of B @ 0 V1co .. . C A. More explicitly VLco 2 1 V1co B .. T has the form T = @ . 2 1 VLco a.e. Then S = RH 1 (C pq )T . 1 C A so that T T ? = I , m Dene the Banach space L1(C ni mi ) as the space of C ni mi -valued functions, under the norm: kF k1 := Z 2 0 ST rF (ei )dm; F 2 L1 (C ni mi ) (8) where ST r denotes the trace-class norm, i.e., X ST r(F (ei ) := T race(F ? F (ei )) = i (F (ei )) (9) 1 2 i where i (F (ei )) are the singular values of F (ei )). Q be the Banach space Ni=1 L2 (C ni mi ) Q L 1 (C n m ) under the norm: L i i i=N +1 (7) kF kB? := max (kF1 k2 ; ; kFN k2 ; kFN +1 k1 ; ; kFL k1 ) (10) Let B? 0 F1 . where F = B @ .. 1 C A 2 B? . Let ? S be the subspace of FL dened by: ? S := (I RR? )B? T ? RH 1 (C pq )T (11) o where denotes the direct sum of subspaces. Ho1 (C pq ) is the subspace of L1 (C pq ) consisting of C pq -valued functions, analytic in the unit disk, and satisfying: B? f 2 Ho1 (C pq ); 1 Z 2 0 f (ei )dm =0 (12) The space H o (C pq ) is simply the subspace obtained by taking the complex conjugate of all functions in Ho1 (C pq ). It turns out that B is the dual space of B? , and ? S is the pre-orthogonal of S . Moreover the following holds [1]: If Q is the restricted to lie in the subspace Pm of analytic trigonometric polynomials of degree m, the resulting optimum is dened to be U ? H1 U1o 1 1 1i m = inf Q 1 ? o (16) U2 i H 2 U2o 1 Q2H 1 B 1 1, then m 1. Therefore Q can If H o H2 B further be restricted to those polynomials in Pm which satisfy the inequality (17) below, without aecting the inmum in (16). kU1?iH1 k2 + kU2?iH2 k1 1 (17) U1o 1 1 Q U2o 1 1 Q If (17) holds then we have that j(U2?i H2 U2o 1 1 Q)(ei )j 1 m a:e: ) kQk1 k1 U2o1 k1 + k1H2 U2o1 k1 (18) Since Q is a degree m polynomial satisfying (18), then by Bernstein's theorem [16] we conclude that kQ0 k1 mk1 U2o1 k1 + k1 H2 U2o1 k1 . Thus, to nd m o it sufo ces to search over the subset of Pm dened by (18). Moreover the derivatives of all such Q are uniformly 0 bounded above by mk1 U2o1 k1 + k1H2 U2o1k1 . This B f @ second point is important because the uniform boundk k edness of this derivative, coupled with the assumption Lco B ? F 2 S that H2 , U2o, U2?i H2 , are uniformly Lipschitz, allows ? ? ? i ; (ULi HL VLci ) )F g (e )dm (13) computation of (16) based on inspection of a suÆciently ne partition of the unit circle. In particular, for each For simplicity the numerical solutions, which are based Q 2 Pm which satises (18), the function of described on convex programming will be summarized here for the by SISO case, two block H 2 =H 1 optimization, that is, Hi , 2 X Ui , Vi , i = 1; 2; and Q are scalar valued functions in j(Uii? Hi Uio 1 1 Q)(ei )j (19) 1 H . i=1 For convenience Hi , i = 1; 2, are normalized such that kH1 k2 1; and kH2 k1 1. is uniformly Lipschitz in , with Lipschitz constant In the SISO case by \absorbing" the inner and co-inner bounded above by factors into common terms, o reduces to: 2 X ? 1 1 Lt := Lip(Uii Hi ) + MQ Lip(Uio 1 ) + kUio 1 k1 mMQ U ? H1 U 1 o 1 i o = min1 Q (14) i=1 U2?i H2 U2o Q2H B Z 2 where Lip() denotes the least upper bounds of all uni ? ? ? ? i = sup (( U H ) ; ( U H ) ) F ( e ) dm formly Lipschitz constants of its argument, and 1i 1 2i 2 kF kB? 1 0 MQ := k1 U2o1 k1 + k1 H2 U2o1 k1 (20) F 2? S 0 1 0 1 ? ? U1i H1 V1ci U1o B C B . C .. = Q2Hmin A @ .. A Q . 1 (Cnn ) @ ? ? ULi HL VLci ULo 1 V1co Z 2 ? ? ? .. C T race ((U1i H1 V1ci ) = sup A . 0 F B? 1 V t 3 Approximate Representation by Convex Optimizations 3.1 The Primary Problem To establish a mixed norm vector problem which approximates the primary optimization in (14) we use the following representation o = U ? H1 1i min U2?i H2 Q2H 1 U1o U2o Q B (15) For a xed > 0, dene Ns := int 2L , where int denotes the smallest integer greater or equal to an argument, and k := Nks for integers k 2 [1; Ns ]. Since the function (19) is uniformly Lipschitz continuous, and Q is uniformly bounded then kU1?i H1 U1o 1 1 Qk2 + kU2?i H1 U2o 1 1 Qk1 0 @ Ns X 1 Ns k=1 j(U ?i H 2 2 j( ? U1i H1 U2o U1o 1 1 )( ik Q e 1 1Q)(eik )j 2[ )j 2 ! 12 + k2max [1;N ] s ] ; (21) The uniform partition of the unit circle may be conservative, in the sense that non uniform partitions with fewer points may be found for which (21) is also satised. This would result in less computations and memory usage. The uniform case is considered here for the sake of clarity. The optimization represented by m o can now be reduced to a nite dimensional convex problem, with an error smaller than in the following manner: Let x 2 Rm+1 be the vector of coeÆcients of Q in increasing powers. Dene Sm the subset of Rm+1 to be the convex set of all coeÆcients x of those Q 2 Pm which satisfy (18). The computation of m o is, within accuracy , approximately equivalent to The constraint that F 2? S must satisfy kF kB? 1, can be expressed as k(I RR?)X + RY kB? 1 (26) and the linear functional associated with H = when evaluated at F can be expressed as ( )= H F Z 2 (U1?i H1 ; U2?i H2 0 ? (I RR ? )X + RY H1 H2 dm (27) For X 2 Gm and Y 2 Hm we will show that the B? norm of every element of ? S be less than or equal unity, 1 m implies that the derivatives of each Xi and Y on the (eik ) U o (eik ) := inf x2Sm Ns unit circle are uniformly bounded. The assumption of Lipschitz continuity of Hi , Uio , Uii? Hi , 8i = 1; 2, enables ? i i ir + k2max j ( U i H (e k ) U o 1( e k) e kj (22) the integral (26) and (27) to be approximated by certain ;Ns r weighted sums. Each integral represents the evaluation of the functional represented by H , and the norm reExpression (22) can be expressed in the following vecstriction in (26) respectively. Similar arguments as in tor form section 3.1 show that expression (26) can be approxi! 1 1 mated by the following nite variable convex optimiza b A m 2 2 p pN x Ns + kb A xklN1s (23) 1 = inf tion x2Sm Ns s l 0v u u @t 1 [1 Ns X ? U1i H1 k=1 2 ] 1 2 2 1 2 m X xr eirk r=0 !! m X 1 1 =0 0 2 where b is the Ns dimensional column vector dened by [bj ]k = Uji? Hj (eik ), Aj is the Ns (m + 1) matrix dened by [Aj ]r;s = Ujo 1 1 (eir )ei(s 1)r , for j = 1; 2, l2Ns is the standard Ns dimensional Euclidean space, Ns and l1 is the Ns dimensional vector space equipped with the uniform norm. max @ j As pointed out previously the dual solution exploits the predual Banach space representation, that is for SISO closed loops = sup kF kB? 1 ? F 2 S Z 2 ? ? ? ? i (( U H ) ; ( U H ) ) F ( e ) dm 1 2 1 i 2 i 0 (24) The preorthogonal ? S in the SISO case takes the following form ? S = f(I RR )X ? + RY : X 1 2 B? ; Y 2 H o g (25) 1 is restricted to lie in the subspace where X = X X2 of B? consisting of trigonometric polynomials of the form a m z m + + a0 + a1 z + + a m z m , where each coeÆcient is a L1 column vector, denoted by Gm . On the other hand, Y is restricted to the subspace of 1 H o consisting m-dimensional anti-analytic trigonometric polynomials of the form a1 z 1 + + a m z m denoted Hm (m is assumed to be even). After suÆciently ne discretization of the unit circle, the resulting nite variable constrained convex optimization yields lower bounds for o . 2 2 Ns where Ns Ns X ? (R2 X k=1 Ns X ? ( R1 X k=1 + R1 Y )(e ik + R2 Y )(e ik 2 ) ! 21 ; ! ) (28) is some positive integer, X = P m x e , and Y = l= 1 yl eilk . The resultr= m r ing problem is then a nite variable convex programming problem in the xr , r = 2m ; ; m2 , and yl , l = m; m + 1; ; 1. Likewise for (27) Pm 2 Ns irk 2 3.2 The Dual Problem o 1 1 2 1 Ns Ns X k=1 (U1i H1?(R2 X + R1 Y ) + U2i H2?( ? R1 X + R2 Y )) (eik ) (29) If x 2 Rm+1 , y 2 Rm are respectively chosen to be the coeÆcients of X and Y in ascending power, then (28) can be written max ! 1 1 x x 2 2 B B Ns ; C C Ns y y l2 l1 (30) where l1Ns is the Ns dimensional vector space equipped with the standard l1 -norm. Likewise (29) can be written T EN s 2 1 D D2 x y (31) where ENT s is an Ns dimensional row vector, whose entries are all 1, and ? ik [B 1 ]k;r = Rp2 (e ) e(ik )(r (Ns ) C1k;r = ik ) R? (i )(r 1 (e e k Ns m 2 m 2 1) 1) [ 2] ; B k;s [ 2] ; C k;s = Rp1 (e ) e (Ns ) ik i = R2 (Nes k ) e ik s ik s U1i H1? R2 U2i H2? R? i (i )(r 1 e k e k Ns U1i H1? R1 +U2i H2? R2 ik ik s e e Ns D1k;r = ( ) ( ) m 2 1) ; [D2 ]k;s = 2 Thus a subgradient of the functional 2 x) 6= 0 is k (~ Re where k is an integer between 1 and Ns , r is an integer between 1 and m +1, and s is an integer between 1 and m. Thus the Euclidean vector convex optimization 2 eT k (b jeTk (b2 A2 x ~) j A2 x ~) A2T e at k x = x~, ! (35) k On the other hand, if k2 (~x) = 0 then x~ is a global minimum for kj and 0 is a subgradient. Thus the subinf Re D1 D2 xy + 2 : xy 2 R2m+1 ; gradient set at x~ for the functional obtained by taking ! the maximum of kj over all integers k between 1 and 1 1 x x 2 2 max B B Ns (32) Ns ; C C Ns contains the element y y l l ! 2 A2 x~) eT is a lower bound for o for the case N = 1, L = 2. ko (b 2 T Re T 2 2 A eko (36) jeko (b A x~)j 1 2 2 4 The Ellipsoid Algorithm Both optimizations of the type (23) and (32) are standard applications of convex programming techniques [7]. We will outline here the Ellipsoid algorithm whose implementation is discussed in details in [7]. The only non standard feature of (23) is that the convex set Sm has no simple explicit description in the coeÆcient space Rm+1 . One way to deal with this diÆculty is to take the starting ellipsoid in the algorithm to contain Sm and then treating (23) as an unconstrained optimization. This is justiable as long as the nal x obtained for (23) lies in the set Sm , which can be tested upon termination of the algorithm. We have found it to hold in the numerical implementation that have been tried. If the nal does not lie in Sm then an alternative to the development here should be used in which the set Sm is replaced by an Euclidean sphere, and new derivative bounds established for Q. This may result in larger upper bounds for Q, and necessitate a ner partition of the unit circle. The implementation of the Ellipsoid algorithm requires the computation of subgradients of the objective convex functionals (23) and (32), and of the constraint functionals for the dual problem (30). Let us now summarize the derivation of the appropriate subgradients. In order to nd the subgradient associated with the functional represented by the unconstrained for (23), that is 1 m 1 2 2 2 = inf k p (b A x)k N + kb A xk N 1 2 x Sm Ns l2 s 1 l s m+1 dened by k be the convex functional on R 2 T 2 2 i T 2 2 A x)j = max Re e ek (b A x) (33) k := jek (b 2[0;2 ) let 2 2 ~ (where 2 (~ k at x k x) 6= 0) con- The subgradient set for tains the subgradient of the convex functional dened by ! 2 A2 x~) eT ( b T 2 2 k x ! Re (34) jeTk (b2 A2 x~)j ek (b A x) where ko2 := arg max1kNs jeTk (b2 A2 x)j. This gives us the subgradient associated to kb2 A2 xklN1s . For the subgradient associated to kb1 Aj xklNs , it suÆces to dierentiate w.r.t. x this yields Re (A1T b1 ) + (A?1 A1 )T x~ (37) The subgradient at corresponding to expression (23) is then 2 ( jeTko (b2 Re ( ! ~) 2T A2 x ~)j A eko2 (38) 2 eT ko b ) + (A?1 A1 )T x~ + Re 1T A b1 A2 x Dene a convex functional on R2m+1 by T 1 D2 ]z (z ) := Re EN s [D The subgradient set for contains the element Re D1T D2T (39) (40) ENs Thus (40) gives a subgradient for the functional dened by (31). In order to obtain a subgradient for the constraint functional (30), consider its rst term f1 (z ) := B 1 B 2 z lNs (41) 2 Then the subgradient of (41) at z , is obtained by differentiating w.r.t. z := Re [B 1 B 2 ]? [B 1 B 2 ] For the second term f2 (z ) = C 1 C 2 z lNs 1 T (42) z (43) 1 dene the convex functional k on R2m+1 , 1 C 2 ]z = max Re ei eT [C 1 k := eT k [C k 2[0;2 ) C 2 ]z (44) The subgradient set for k at z 2 R2m+1 which satises k (z ) 6= 0 contains the vector gk := Re 1 C 2 ]z eT k [C eT [C 1 C 2 ]z k C 1? C 2? ! ek (45) Thus a subgradient of the convex functional (43) is 2 := Ns X k =1 gk (46) Therefore a subgradient for the total convex functional (31) is n := arg max [fj (z )] (47) j =1;2 := n (48) 5 A Numerical Example In this section the method developed in this paper is applied to the following plant 1:9 :1s P = (49) (1 + s)(1:9 + :1s) with weights 3 1 :8 + :2s H1 = :17 ; H2 = 0 1+s 2 3 :1 + 2:1s 1 :8 + :2s ; U2o = :22 U1o = :17 1+s 1+s The mixed objective for this example is given by kH1 U1o P Qk2 + kU2o P Qk1 where the rst term represents performance, while the second term represents robust stability. The lower bound obtained is .41, and is generated by a convex program for the nite variable problem (m = 6) resulting from the inmum in (32) for an accuracy = :01. The upper bound obtained is .44, and is produced by a convex program for the nite variable problem (m = 6) resulting from (23) for an accuracy = :01 . These results are generated by an Ellipsoid algorithm for the nite variable approximations to the primary and dual problems. The optimal 6-th order polynomial Q obtained is Q( z ) = :0162 :0256z :0243z 2 :0192z 3 :0153z 4 :0090z 5 :0051z 6 The Ellipsoid algorithm has the advantage of simplicity and polynomial execution time. However, better results can be achieved by other faster interior point methods such as the method of analytic centers. References [1] S.M. Djouadi, C.D. Charalambous, and D.W. Repperger. On Multiobjective H2 /H1 Optimal Control. Proceedings of the American Control Conference, 4:4091{4096, June, 2001. [2] A. Ghulchak. 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