Copyright 1978, by the author(s). All rights reserved.

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1 Cpyright 1978, by the authr(s). All rights reserved. Permissi t make digital r hard cpies f all r part f this wrk fr persal r classrm use is grated withut fee prvided that cpies are t made r distributed fr prfit r cmmercial advatage ad that cpies bear this tice ad the full citati the first page. T cpy therwise, t republish, t pst servers r t redistribute t lists, requires prir specific permissi.

2 AN INTUITIVE DERIVATION OF A REALIZATION by C. A. Deser ad E. L. L Memradum N. UCB/ERL M78/87 15 December 1978 ELECTRONICS RESEARCH LABORATORY Cllege f Egieerig Uiversity f Califria, Berkeley 94720

3 AN INTUITIVE DERIVATION OF A REALIZATION PROCEDURE BASED ON SINGULAR VALUE DECOMPOSITION C. A. Deser ad E. L. L Departmet f Electrical Egieerig ad Cmputer Scieces ad the Electrics Research Labratry Uiversity f Califria, Berkeley, Califria ABSTRACT This memradum presets a ituitive derivati f the miimal x. realizati f G(s) G ]R(s) based sigular value decmpsiti, The rigial wrk is due t P. Va Dre, et al. Research spsred by the Natial Sciece Fudati Grat ENV

4 I. Itrducti The prblem f cstructig a miimal realizati f a give matrix f ratial fuctis has bee studied i literature, but the umerical aspects f the suggested prcedures have rarely bee csidered. Va Dre has prpsed a umerically stable algrithm fr cstructig a miimal realizati [1]. This paper, based Va Dre1s result, gives a ituitive isight i such a realizati, ad gives a simple prf f miimality. It is well kw that if a strictly prper ratial matrix is decmpsed it the sum f the pricipal parts f its Lauret expasi at each f its ples, say P H(s) = H (s) i=l x the the direct sum f miimal realizatis f each f the H^s is a miimal realizati f H. Thus i secti II, we derive a miimal realizati f a matrix f strictly prper ratial fuctis with a ple f rder 2. secti III, we prve that the prpsed realizati is miimal. secti IV, we geeralize the methd prpsed i secti II; by iducti, we btai a realizati f a matrix f strictly prper ratial fuctis with a sigle ple f arbitrary rder. T illustrate the spirit f the methd, we csider the simple prblem f the miimal realizati f a matrix with a first rder ple.... x, Let G(s) =N^1}/(s-X) where N*1' Gffi. (1) Perfrm a sigular value decmpsiti ^ : N(D. UViV"* where U(1) 61, VU) GC1 I are uitary ad x. (i) ZW =diag(va2,...,ap,0,0,...,0) Gm *. Ut Pj < rak ^. I -2-

5 r(d,<d Let Ux"' ad V(1) dete the first p± clums f U ad V resp, Pi Let Z = diag(a.,a0,...,a ); the (1) = U(1)E(1)V(1) pl pl pl u(1)i(1) pl pl x ~pir s-x S V (1) A D A := XI V (D* = : B C := u(1)z(1) pl pl Sice B ad C have bth rak p^, rak[si-aib] = q±9 Vs e 0 [si-a rak L""cj = Pl, VsG(K Hece the realizati is cmpletely ctrllable ad bservable, hece miimal. II. Miimal realizati f a ple f rder 2. We csider a matrix f ratial fuctis G(s) 3R x., where G(s) is strictly prper. Let G(s) has a sigle ple X f rder 2, hece we write G(s) as 42) 42) G(s) = ~r + (s-x) 2 s-x (1) /ON x. /0x «x4 where N<2) 6 <D \ N<2) 6 I \ -3-

6 (2) N It is clear that t realize the secd rder term j requires (s-x)z (2) at least 2Tak N~ itegratrs. The makig maximum use f these (2) 2*rak N^ N (2) the term s-x * itegratrs with sme additial itegratrs, we will realize (2) xi T determie the rak f N* G, we perfrm a sigular (2) value decmpsiti (abbreviated by SVD) N ad btai (2). <2>E<2)V<2)* (2) /0v x where U(2) e <c (2) imi is uitary; V <C is uitary; M2) x i. Tm.. (2) ZK ' 3R ; Z with a<2) >a<2)... >a(2) > P2 O OJ Hece (2) rak N» = p2' Partitiig bth IT,V^ as fllws, we btai U (2) *-p2+ ^VP2^ U (2) U (2) VP2 V (2) V ^-p.-)- -t-^-p2">- (2) r(2) ±-p2. (3) Let's defie.(2).= f > ~1 (4) (2) p2-4-

7 With the tatis defied i (3) ad (4), Eq. (2) is rewritte as N(2). D(2)z(2)v(2)* (5) P2 P2 P2 Remark: If N(2) = 0, usig (5), a miimal realizati f G(s) is immediate: A := XI VI XI 2J -<-. -*- - x"i,(2)* LP2 J = : B C := <2)E<2) p2 P2 + P2- By ispecti, Vs <E, rak[si-a:b] = 2p ad rak si-a = 2p, hece the realizati is miimal. i Let u S E. dete the iput. Let H2 v -P. := V^ ; u (6) i 2 v ~VP2 L T cstruct a realizati ituitively, csider G(s)u = 'm<2> N2 (s-x) 2 (2)v(2) -= X + (s-x) u(2) +!L_ V(2).v(2)*u s-x (2)v(2) s-x V -p. VP2 (2) Usig the partiti f V frm Eq. (3), we btai -5-

8 N(2)V I Ni(2>V(2> I 2 P2 P2 1 -p2 (2) (2) P2 G(S)U = T^ - TV + r^= V + NrV^ 4- V s-x s-x ~p, s-x ~i"p2 * p2 e~x ~p: (7) Let us use p? itegratrs t realize ~P2 s-x ~p2 the the realizati f the third term f (7) is immediate: (2)v(2) 1 p2,-,. r(2) (2) r-^- v = ; 'V" ' X s-x ~p 1 p "^2 I terms f x ad v, the first tw terms f (7) becme ~p2 ~i"p2 N(2)v(2) N<2V2> 2 p9 _± L z := ^-x + : v s-x ~p( s-x ~i~p2 a [»;" 1 x ~P. V ~i"p2 (8) where N(D :=rnf>vf>infv2>l 1. \ji p2 ; 1 t-p2 j (9) The miimum. f itegratrs required fr realizig (8) is p := rak N-^. T determie px, we perfrm asigular value 1 * decmpsiti!" ad btai (d. D<i)E<i)v(i) (10).. x where IT ' ^ <E *.x x (1) is uitary, V(1) E <E i i is uitary; E^ e m i -6-

9 («.- i i 1 1 (10a) I J with a <«>a<x)... >a(1) > P- Partitiig bth U(1), V(1) as fllws, we btai U (1) _ <!>!U(1) pi! Vpi XD _ r XD,d) rpi. i (11) -HP^'^^-P^ -s-p^ -e-^-p^ We further partiti V; Pl «- Pi -» as fllws: (i) _ (12) \2Lpij We defie (1) CD (1).= (13a) (1) ad p := rak N (1) (13b) -7-

10 With the tatis defied i (11), (12), (13), Eq. (10) is writte as N<X> =U(1)S(1>V(1)*=/u(1>E(1>V(1)*Uv(1)I V<" pl pl Pl Pl Pl Pl pl Pl pl (1>v(1)i v(i) 1 p PX PX (14) ad Eq. (8) becmes 1 P- pl s-x r V(1)*i Lpi! pl _ X ~p2 (15) V _"VP2 _ This shws that z ca be realized by p1 itegratrs. We defie x ~P- s-x A(i)*.; v(d* Lpl i pl. X -p. (16) Remark: Sice N CD.= V L'Vp2. ;(2)v(2)j (2)v(2) 1 frm (9), Jl P2, 1 ±-P2 P2 := rak N<2) =rak N^2)V*2) <rak N*1* =: p±. Based the abve aalysis, G(s) diagram. is realized by the fllwig blck -8-

11 U a) 6 Q. I d a l ih 0 PQ < 4J CO N 60 a 4J O D M 4J CO a J 4J V< O 60 rh CCS 00 a H rh O <4-4 CU 0) N U «4H O «< CO ^ H 1 M is CO 3 T3 + a) /-\ Vi ""s <-< a 1 w CO N»t a) 53 J3 H 1 CO I CT> I 3* PQ if 14-1 u O U-l CO ctf S I Q. I a rh a. >> rh h- a -e. a. TJ O U P. cu 4J CO a. I 0 00 CO» 2 T Q. <> *- Q. * i Q. M r a.» U cd rh <D 1 rh a J L_ a J I a. a. X», rh Q. II ti CD a <

12 Algrithm (2) Step 1 Perfrm the SVD f N^ N(2) = <2> <2) <2)* = ^2)^(2)^2) 2 P2 P2 P2 where p := rak N^ ad V( 'is partitied as ^p2- ^-.-P2-v V (2)_ (2) VP2. Step 2 Defie N(1) : N(2)v(2) 2 P2 (2)v(2) 1.-P2 «- P, VP2 * (1) ad perfrm the SVD f N. N(D. u(i)e(i)vd)*. yd), Pl where p := rak N r(d}, ad Vv (1) vd) Pl Pl ' is partitied as * Pj* * j-pj* V (1) _ v(l) pl V(D Vpi -1 i. X We further partiti V pl as -f-_ -» V (1) ±-P2 $(» pi v(l) LPlJ Step 3 A realizati {A,B,C> f G(s) is -10-

13 A := = : B (17) C := N(1)V(1) 1 Pl III. Prf f Miimality We shw that the realizati {A,B,C} give by Eq. (17) is miimal. x. Therem Csider G(s) m(s) X give by (1). The {A,B,C} give by (17) is a miimal realizati f G(s). Prf Frm the aalysis f secti II, it is clear that {A,B,C} is a realizati f G(s). Hece the remaiig task is t shw miimality, r equivaletly, t shw that {A,B,C} is cmpletely ctrllable ad cmpletely bservable. T shw cmplete ctrllability, we shw that [si-a B] is full rak Vs <D. Nw by (17), Vs i X, [si-a-b] is full rak. Nw fr s = X, we have rak[xi-a:b] = rak +*!+ <-Pf. i iam*,v(i)* m* 'V V pl i"p2 O O V (2) = rak «- p - «- j(d*i ^(d*v(2)* pl! Pl i"p2 (2)* 2-11-

14 rak[xi-a B] = rak A(D* pl V(i)*v(2)* pl i_p2 pi i i. w(2) c "i^i sice V (C OJv<2> is f rak, p2 J v(2)* rak $<D* 'p^io I I p. V(1)* pl pl + p2 because V is full rak ad (12). - C Tsl-A T shw cmplete bservability, we shw that is full rak Vs e? Ij. C J si-ai ' ) is full rak. Nw fr s = X, we have - Agai by (17), Vs ^ X, _pi XI-Al rak = rak L C J (l)v(l) -1 pl A(D* pl (2)v(2) 1 P2J t = rak (l)vd)l ;s; «TAT(D* N(2)v(2) 1 P1 I 1 P2 (1) Nw N^M1^ = U^M1^ is f rak p. because Z^ is square ad f rak 1 Q1 P± P1 1 Pi p, (see (13a)), ad U beig uitary has its first p^ clums, amely U, frmig a Pl A(l)* cz P1XP2 idepedet family. Csider w V ^ C -12-

15 A(D* rak Vv J pl rak jvf^ = rak V1 M2 LOJ rz(iyi)< < pi pi by (12) w2 LOJ (i) pixpi sice Zv ' <C ad is f rak p. =rakfz(1)v(1^ i p2 by (10a), (11) ad (13a).0- =rak)u(1)z(1)v(1)* rak[n<2v2)] 1 P2 L... x sice IT ' G (C ad is f rak. by (9) ad (10) rak[u<2v2>] P2 P2> by (5) = P- (2) because U (2) rak Uv = p0. P«2 w2 is uitary, hece Hece rak I A (D* ; pi (1)V(D (2)v(2).-1 pl! X P2. " Pl + P2 ad the pair (C,A) f (17) is bservable. IV, A iducti step fr the realizati f a ple f rder I > 2. We have cstructed a miimal realizati f a matrix f ratial fuctis with a sigle ple f rder 2. We w csider a matrix f -13-

16 ratial fuctis with a sigle ple f rder > 2: G(s) = N U) N U) (s-x)56 (s-x)* X N (A) s-x (18), v x. where N^; G (E X Vi e {1,2,..., }. The iducti assumpti is that we have a methd fr a miimal realizati f ay matrix f ratial fuctis with a sigle ple f ( -1^ (2,-1} ( -1) rder -1; we dete it by {Av,BV,CV '}. We w cstruct a ( -1) ( -1) (&-1) realizati {A,B,C} f the G(s) f (18) i terms f {A,B,C }. We perfrm a sigular value decmpsiti N ad btai NW. (t)j.(«v(«* =uwia)vu)* p p p (19) (20) where p := rak N. As i (7), we btai G(s)u = + w N N u) _(s-x) (s-x)*"1 4- (s-x) (s-x) -1 NU)VW (sv1,(ovc*> N (A) +... (s-x) NWVU) 1 V<«V( )*u (s-x) _ NU)VW 1-1 -pt s. b-x \ +(s-x)*"1 \~ l (s-x)4"2 S"X ~P* +...+»(«v(«. -If. v 1 P s-x ~p. («VC«1 Vi s-x VP V (21) (22) -14-

17 where we defied -* V -pfl :- VW\ (23) VP Vp Nte that (21) icludes terms ad that the use f the variables v ad --p, (defied i (23)), creates 2 -l terms i (22). (Ideed the first term f (21) leads t ly e term i v.) H We use p itegratrs t realize x ~P s-x ~p ' ** '" " the the realizati f the last term f (22) is immediate. I terms f x ad v, the first ( -1) terms f (22) becme ~PA ~rp z := [N(*-1}] (s-x) -1,(*-!) 'p.i+^«]r»f (s-x) -2 ~rp "rp +... [Nf-1}] """s^x x ~P V VP (24) where N(^1} := [N^VWiNf )VWD ] Vi S{1,2,..., -1}. we i l+l Pj ; 1 i""p By the iducti assumpti, {AU"1),B(^1),ca"1)}is the miimal realizati f (24), we realize G(s) f (18) by the fllwig blck diagram. -15-

18 G(s)u r =it ir a c«-i) fr (s.-a"-11)1 v(m) I 3 g«-l) TfZ II JJr/>( v( V (1) Pi U («., (t) I *>l d <^-/>{ s-x J IT 1 JJL'Jbz-d- IL_ C (si-a)"1 where B^"1^ is partitied as fllws: B (4-1) _ A( -l) ; V( -D ^i B I "P " Hece arealizati {A,B,C} i terms f {A(*"1}.B^"1*,C( -1}} is give as fllws: A := A( -l), A( -l) O i XV V C:= [c^ H<»V<»J v( -l) mv( )* w* rp = : B The realizati f G(s) f Eq. (18) is the btaied iteratively. Fr a prf f miimality, refer t [1]. (25) V. Cclusi Based Va Dre's wrk [1], i secti II, we btai ituitively a realizati f a matrix f ratial fucti with a sigle ple f rder 2; we the prve the miimality. I secti IV, by a iducti -16-

19 step, we btai amiimal realizati f the matrix f ratial fuctis with a sigle ple f rder > 2. Ackwledgemet The authrs wish t thak Dr. Y. T. Wag, ad Mr. Victr H. Cheg fr their stimulatig discussis. -17-

20 REFERENCES [1] P. Va Dre ad P. Dewilde, "State-Space Realizati f a Geeral Ratial Matrix: A Numerically Stable Algrithm," Prceedigs f the Twetieth Midwest Sympsium Circuits ad Systems, Texas Tech Uiversity, August 1977, p