Yves Genin, Yurii Nesterov, Paul Van Dooren. CESAME, Universite Catholique de Louvain. B^atiment Euler, Avenue G. Lema^tre 4-6

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1 Submtted to ECC 99 as a regular paper n Lnear Systems Postve transfer functons and convex optmzaton 1 Yves Genn, Yur Nesterov, Paul Van Dooren CESAME, Unverste Catholque de Louvan B^atment Euler, Avenue G Lema^tre - B-18 Louvan-La-Neuve, Belgum Fax : , Tel : E-mal : vdooren@anmauclacbe Abstract Recently, a compact characterzaton of scalar postve polynomals on the real lne and on the unt crcle was derved by Nesterov [] In ths paper we show how to extend ths result to pseudo-polynomal matrces, and also present a new proof based on the postve real lemma The characterzaton s very smlar to the scalar case and also allows the use of fast algorthms for computng the central pont of the correspondng convex set 1 Introducton Postve transfer functons play a fundamental role n systems and control theory: they represent eg spectral densty functons of stochastc processes, show up n spectral factorzatons, and are also related to the Rccat equatons When such transfer functons are ratonal, t s known snce the work of Youla [] that they possess ratonal spectral factorzatons Later on t was shown that usng state-space models of postve transfer functons one could express the condton of postvty n terms of lnear matrx nequaltes (see eg []) Postve transfer functons obvously form a convex set, and recently they were also beng studed by people n convex optmzaton [1], [] In order to optmze over the set of postve functons, t s mportant to have a compact (say \mnmal") parameterzaton of these functons and recently Nesterov presented such a parameterzaton for scalar postve polynomals In ths paper we look at the same problem va the state-space theory and hence va the more general class of ratonal matrx functons Frst of all, we recall the basc results of para-hermtan transfer functons, a concept we need when lookng at the matrx case of postve transfer functons, snce postve matrces nherently requre some knd of symmetry Then we recall the postve real lemma whch essentally gves the lnear matrx nequalty descrbng the postvty condtons for postve transfer functons From ths general theory, we then derve Nesterov's parameterzaton of postve polynomals and also extend t to the matrx case and to pseudo-polynomal transfer matrces We end by showng a partcular applcaton to systems and control, where the above results are of drect use 1 Ths paper presents research supported by NSF contract CCR-9-91 and by the Belgan Programme on Inter-unversty Poles of Attracton, ntated by the Belgan State, Prme Mnster's Oce for Scence, Technology and Culture The scentc responsblty rests wth ts author 1

2 Postve para-hermtan transfer functons Let (s) be a m m ratonal para-hermtan transfer functon, e (s) : = (?s) (1) = (s): It s a well known result of state-space theory [] that any proper transfer functon of that type admts mnmal realzatons of the form (s) = h B (?s I n? A )?1 ; I m H (s In? A)?1 B ; () I m where H s some approprate hermtan matrx Note that the assumpton (s) proper (e (s) bounded at s = 1) s made for the sake of smplcty and could be lfted wth the help of generalzed state-space representatons or of an approprate transformaton of the varable s Clearly, H s not unquely dened from (s) Indeed, let us replace H by H(X), dened as follows : X A + A X X B H(X) = H + () B X 0 and where X s any n n hermtan matrx (s) s easly vered by drect nspecton not to be aected by ths substtuton, whch clearly preserves the hermtan property of the realzaton Let us now consder the subset of para-hermtan transfer functons that are nonnegatve dente on the magnary axs (<s = 0) and partton H conformably wth () and () (!) = (!) 0 for? 1! 1 () H = H11 H 1 : () H 1 H It appears from () and () that (1) = H so that H s nonnegatve dente Let us now further assume that there exsts a hermtan matrx X such that H(X) s nonnegatve dente Then, H(X) can be factorzed nto " L H(X) = () W # h L; W for approprate r n and r m matrces L and W, respectvely, and wth r the rank of H(X) Therefore, f G(s) s dened as G(s) = L(s I n? A)?1 B + W; () one has (s) = G (s)g(s): (8)

3 Hence, the exstence of a hermtan matrx X such that H(X) s nonnegatve dente appears to be a sucent condton for (s) to be a para-hermtan transfer functon nonnegatve on the whole of the magnary axs (<s = 0) Moreover, t can be shown under mld assumptons that ths exhausts the class of all possble hermtan realzatons of (s) wth gven (A; B) par, and hence, ths condton s also necessary for H(X) [] Clearly, nether the matrx X nor the factorzatons () (8) are unquely dened It can be shown however that, f H s postve dente, there exst solutons X such that H(X) s nonnegatve dente wth rank H(X) = m These specal solutons X can be obtaned from the algebrac Rccat equaton H 11 + XA + A X? (H 1 + XB)H?1 (H 1 + B X) = 0 (9) and yeld a spectral factorzaton of (s) as (s) = G (s) G(s), where G(s) s dened from () and (), and s now squared and nvertble Extenson to the real lne and unt crcle Para-hermtan transfer functons exhbt a symmetry property wth respect to the magnary axs It s a well known fact that equvalent classes of transfer functons can be dened by substtutng to the magnary axs other specal contours of the complex plane, namely the real axs and the unt crcle Let us brey recall the correspondng substtutons As the varable transformaton s = x maps the magnary axs <s = 0 onto the real axs =x = 0, one can transform any para-hermtan transfer functon (s) nto a hermtan transfer matrx (x) = (x) = ( x) and conversely Therefore, by settng A :=? A and B := B, one can transform any para-hermtan transfer functon (s) nto a hermtan transfer functon (x) admttng hermtan realzatons of the form (x) = h B (x I n? A )?1 ; I m H (xin? A)?1 B ; (10) I m where H s the same hermtan matrx as before Furthermore, f (x) s nonnegatve dente for real x, one derves from () and () that there must exst skew-hermtan matrces X =?X such that the hermtan matrx?x A + A X?X B H(X) = H + (11) B X 0 s nonnegatve dente Smlarly, the varable transformaton s = (z? 1)=(z + 1) maps the magnary axs <s = 0 onto the unt crcle jzj = 1 Therefore, one can transform a para-hermtan transfer functon (s) nto a para-recprocal transfer functon (z) = [(z? 1)=(z + 1)] and conversely n the sense that t veres the equalty (z) = (1=z) Furthermore, t can be shown by

4 elementary algebrac manpulatons that, f one makes the substtutons " # I I (I? A)?1 B H := H ; B (I? A )?1 I I B := (I? A)? B; A := (I? A)?1 (I + A) (z) admts the hermtan realzaton (z) = h z B (I n? z A )?1 ; I m H (z In? A)?1 B : (1) I m Fnally, f (z) s nonnegatve dente on the unt crcle (jzj = 1), the equvalent form of () s found to be A X A? X A H(X) = H + X B (1) B X A B X B so that, f (z) s nonnegatve dente on the unt crcle (jzj = 1), there must exst a hermtan matrx X = X such that H(X) s nonnegatve dente Postve pseudo-polynomal matrces Pseudo-polynomal matrces are matrces wth a nte expanson n postve and negatve powers of the ndependent varable (e s, x or z) For a transfer functon (s) that s postve on the magnary axs (<s = 0) t suces to consder an m m matrx polynomal of the varable s?1 tx (s) = Q s? (1) =0 snce one can elmnate postve powers of s by multplyng by (?s )?` (whch s postve on the magnary axs) So let us nvestgate the condtons under whch ths matrx polynomal s nonnegatve dente on the magnary axs (<s = 0) A rst observaton s that ts degree s even (t = k) and one has Q = Q and Q =?Q for even and odd, respectvely Settng s = x, one transforms (s) nto a polynomal (x) = ( x) of the varable x?1 (x) = kx =0 P x? (1) that s nonnegatve dente wth (s) (on the real axs =x = 0) and where P = Q = P Therefore, these two problem formulatons are dentcal so that t s sucent to consder ts (x) formulaton In case A has 1 as an egenvalue, one may use equvalently the varable transformaton s = (1?z)=(? z) wth any unt modulus complex number

5 Let us then ntroduce the followng hermtan block trdagonal matrx H = P k P k?1 = P k?1 = P k? P k? = P k? = P1 = P 1 = P 0 (1) and observe that (x) can be rewrtten as (10) wth A the kmkm matrx and B the kmm matrx, respectvely, dened by A = 0 I m 0 I m 0 Im 0 ; B = I m : (1) From (11), t then appears that (x) wll be nonnegatve dente f and only f there exsts a skew-hermtan matrx X =?X such that the matrx Y = H : : : 0 0 X 11 X 1 : : : X 1k 0 X 1 X k 0 X k1 X k : : : X kk 0? 0 X 11 X 1 : : : X 1k 0 X 1 X k 0 X k1 X k : : : X kk : : : 0 s nonnegatve It turns out that ths characterzaton of matrx polynomals nonnegatve on the real axs extends a result earler obtaned by Nesterov [] for scalar polynomals Ths s made explct n the followng theorem Theorem 1 A pseudo-polynomal matrx of form (1) s nonnegatve dente on the real axs f and only s there exsts a nonnegatve dente matrx such that (assumng Y ;j = 0 for > k and j > k) : P = k? X j=0 (18) Y = [Y s;t ; s; t = 0; : : :; k] (19) Y k??j;j for = 0; : : :; k: (0) Proof : The proof s a drect consequence of the lnear matrx nequalty Y 0 snce the pattern of the X j blocks mples exactly condton (0)

6 Let us now brey consder the same characterzaton problem when the unt crcle (jzj = 1) s substtuted for the real axs (=x = 0) The problem reads as follows : nd a necessary and sucent condton such that the pseudo-polynomal matrx (z) = X+k =?k P z (1) s nonnegatve on the unt crcle : (e ) 0 for 0 A form (1) s easly obtaned for ths transfer matrx by usng the same A and B matrces as n the real axs stuaton and by denng the hermtan matrx H = 0 0 : : : 0 P k 0 0 P k?1 0 0 : : : 0 P 1 P?k P?k+1 : : : P?1 P 0 ; () where P k = P?k Therefore, t appears from (1) that (z) wll be nonnegatve on the unt crcle f and only f there exsts a km km hermtan matrx X such that the matrx Y = H : : : 0 0 X 11 X 1 : : : X 1k 0 X 1 X k 0 0 X k1 X k : : : X kk? X 11 X 1 : : : X 1k 0 X 1 X k 0 X k1 X k : : : X kk : : : 0 0 s nonnegatve dente Ths then leads to the followng theorem whch extends Nesterov's characterzaton of scalar postve functons to the matrx case Theorem A pseudo-polynomal matrx of form (1) s nonnegatve dente on the unt crcle f and only s there exsts a nonnegatve dente matrx such that : P = () Y = [Y s;t ; s; t = 0; : : :; k] () Xk? j=0 Y j;+j for = 0; : : :; k; () Proof : The proof s a drect consequence of the lnear matrx nequalty Y 0 snce the pattern of the X j blocks mples exactly condton () Let us remark that the characterzatons (18) and () nvolve n practce the soluton of a lnear matrx nequalty, e to nd a hermtan or skew-hermtan matrx X such that Y 0, and that ecent numercal methods exst nowadays to solve such knd of problems

7 Applcaton to checkng controllablty An mportant robustness problem n systems and control s to check whether or not an (A; B) par remans controllable under perturbatons of norm d or less The perturbed system (A + A; B + B) s then sad to be robustly controllable A smple test for controllablty states that (A + A; B + B) s controllable f and only f h n A + A? I n ; B + B > 0; 8 C: () Usng standard perturbaton theory for sngular values, one nds that f d < ^d, where h ^d := mn n A? I n ; B C () then () holds for all perturbatons of -norm less or equal to d : h k A; B k d: (8) Moreover, t s shown n [] that ^d = sup d for whch ths holds Fnally, ()-(8) mply that d = h A? I n ; B A? I n? d I B n 0; 8 C: (9) Wth r = jj; z = =jj, one has = r:z so that d can be rewrtten as d (r; z) = AA + BB? d I n? r(zb + z?1 B) + r I n 0; 8r <; z e < (0) It s not easy to test f ths two varable pseudo-polynomal matrx s postve, snce ts elements or sngular values are not convex functons of the varable = r:z But f one \freezes" r = or z =, then the problem s reduced to checkng the postvty of d (r; ) over the real lne for the varable r, or of d (; z) over the unt crcle for the varable z So for a gven and wth d (r; ) = Z 0 + rz 1 + r Z, where Z 0 = AA + BB? d I n ; Z 1 =?(B +?1 B); Z = I n ; (1) the results of the precedng sectons can be appled to establsh that one wll have d (r; ) > 0; 8r < f and only f there exsts a matrx X =?X such that " # I n?(b +?1 B)=? X 0: ()?(B +?1 B)= + X AA + BB? d I n or equvalently, by puttng Y = X? (B??1 B)=, f and only f there exsts a matrx Y =?Y such that " # I n?b? Y 0: ()??1 B + Y AA + BB? d I n Smlarly, f for a gven one expresses d (; z) as (; z) = z?1 R?1 + R 0 + zr 1 where R 1 = R?1 =?B ; R 0 = AA + BB? d I n + I n ; ()

8 one can use the same argument as above to prove that d (; z) > 0; 8z e < f and only f there exsts a matrx X = X such that (AA + BB? d I n + I n )=? X?B 0 ()?B (AA + BB? d I n + I n )= + X or equvalently, by puttng Y = X? (AA + BB? d I n? I n )=, f and only f there exsts a matrx Y = Y such that I n? Y?B 0: ()?B AA + BB? d I n + Y Both these problems are tractable by themselves snce they can be solved va sem-dente programmng technques A procedure to nd ^d would then be to maxmze d such that d (r:z) 0 and for checkng the above condton one can use ether () for all values of or () for all values of Concluson In ths paper we extended the condtons that a polynomal s nonnegatve on the real axs or the unt crcle, to the case of pseudo-polynomal matrces We beleve that ths extenson wll have mportant applcatons n systems and control, and we ndcated a smple but basc problem where such matrx problems do ndeed occur The characterzaton s elegant n the sense that t has an nherent Hankel or Toepltz structure Ths structure can be exploted wth fast FFT-based algorthms for solvng the basc Newton teratons needed n the semdente programmng technques References [1] S Boyd, L El-Ghaou, E Feron, V Balakrshnan, Lnear Matrx Inequaltes n Systems and Control Theory, Studes n Appled Mathematcs, 1, SIAM, Phladelpha, PA, 199 [] R Esng, \The dstance between a system and the set of uncontrollable systems", n Symp on the Mathematcal Theory of Networks & Systems, pp 0{1, 1988 [] Yu Nesterov, \Squared Functonal Systems and optmzaton Problems", submtted for publcaton, March 1998 [] V Popov, Hyperstablty of Control Systems, Sprnger Verlag, Berln, 19 (Roumanan verson n 19) [] J C Wllems, \Least squares statonary Optmal Control and the algebrac Rccat equaton", IEEE Trans Aut Contr, AC-1, pp1-, 191 [] D Youla, \On the factorzaton of ratonal matrces", IRE Trans Inform Theory, IT, pp 1-189, 191 [] K Zhou wth JC Doyle and K Glover, Robust and optmal control, Upper Saddle Rver, Prentce Hall, 199 8