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 STABILIZATION, TRACKING AND DISTURBANCE REJECTION IN MULTIVARIABLE CONVOLUTION SYSTEMS by F. M. Callier ad C. A. Deser Memradum N. UCB/ERL M78/83 7 December 1978 ELECTRONICS RESEARCH LABORATORY Cllege f Egieerig Uiversity f Califria, Berkeley 94720

3 STABILIZATION, TRACKING AND DISTURBANCE REJECTION IN MULTIVARIABLE CONVOLUTION SYSTEMS F. M. Callier, Departmet f Mathematics, Facultes Uiversitaires, Namur, Belgium C. A. Deser, Departmet f Electrical Egieerig ad Cmputer Scieces, ad the Electrics Research Labratry Uiversity f Califria, Berkeley, Califria ABSTRACT.x This paper describes the algebra &(q) x f trasfer fuctis f multivariable distributed systems; f the algebra this is a multivariable extesi &( ) f scalar trasfer fuctis studied i previus papers [1], [2]: a detailed study f s called alright- ad -left-represetatis is de: this is a geeralizati f cprime factrizati thery fr prper ratial trasfer matrices. The paper studies ext feedback system stability f systems with trasfer matrices with elemets i >(a ): a clsed-lp characteristic fucti is defied ad its imprtace discussed. Frthcmig applicatis are precditied by studyig a geeral prblem which is ecutered i cmpesatr desig: this geeralizes t the distributed case a techique used by Yula et al. [3], [4]. Fially the prblem f desigig a feedback cmpesatr fr rbust stabilizati, trackig ad disturbace rejecti f a plat is defied ad slved usig the techiques f the paper. Research spsred by the Natial Sciece Fudati Grat ENG

4 Sme f the results ctaied i this memradum have r will be auced at three cfereces: 1) At the 761st America Mathematical Sciety Meetig i Charlest, S.C., Nv. 3, 1978 uder the title, "Dyamic Output Stabilizati f a Ctrl System". 2) At the 17th IEEE Cferece Decisi ad Ctrl, Sa Dieg, Ca., Ja. 10, 1979, uder the title, "Stabilizati, Trackig ad Distur bace Rejecti i Liear Multivariable Distributed Systems". 3) At the 4th Iteratial Sympsium Mathematical Thery fr Netwrks ad Systems (MTNS 1979) Delft, Hllad (July 3-6, 1979) uder the title, "Matrix Fracti Represetati Thery fr Cvluti Systems" -2-

5 1. itrducti: Mathematical Defiiti ad Facts; Perspective ad Orgaizati f the Paper I previus papers [1], [2] we were ccered with the fllwig mathematical defiitis ad facts ccerig scalar systems. (LTD)+ detes the set f cmplex-valued Laplace trasfrmable distributis with supprt ]R,. Fr a em, ad elemet f e (LTD), is said t belg t CL(gJ O ^ iff, fr t<0,f(t) =0ad, fr t>0,f(t) =f(t) + f± «(t-t.),where i=0» -a t i)f(.)6l (mj:-{f; fim.-hcf f(t) e dt <»}, (ii) tq = a l,a + JQ ad t > 0 fr i = 1,2,..., (iii) fr all i, f <E ad 6(-t ) is the -at. Dirac delta distributi applied at t., (iv) f. e < «>. It is 1 i=l well kw, [7, p. 248] that CL( ) is a cmmutative cvluti Baach algebra with rm defied by iifil,,=c f (t) e"v dt+ f. e"vi (l.d ^ ) J0 a i=0 1 ad with uit elemet 6(0, the Dirac delta distributi; mrever this algebra has divisrs f zer [5, Therem ;38]. Observe als that, fr a = 0, Oj(0) is idetical t the algebra fl. described i [7, p ]; mrever, fr a1 >_ ajj, <X( 0) D<&a") Fr a Gl.a elemet f G (LTD), is said t belg t #_(a ) iff "*" there exists a a, e m, a < a, such that f belgs t #(co. With the 1 ± * <X( )-rm (1.1), <&Ja ) is a rmed cvluti subalgebra f #(c*0) with uit elemet 6 ad with divisrs f zer. Let * dete Laplace trasfrms: i.e., f is the Laplace trasfrm f f. &((J ), < _(<* ) dete cmmutative algebras with pitwise prduct f the ffs where f &(a ),& (a ), respectively: divisrs f zer. "" their uit is 1 ad they have -3-

6 Let <E, := {s ^ <E;Res > }, C, := {s G (C;Res > a } a + a + ad <E := {s (C;Res < a }. a - The fllwig are imprtat prperties f d( ) ad & (a ): (i) f belgs t the cvluti algebra (Xk )» id (<? )resp.), iff f belgs ^^~ t the algebra 4(a ), (tf (a )); (ii) f is a ivertible elemet f (Xk$ ), (tf_(tfq)resp.) iff i bth cases if{ f(s) ;s e (Ea +} > 0; (iiia) if f G 0(1$ ) the is f is buded i <E,, ideed O O T O sup{ f(s) ;s <Ca +} <_ ifh 0fa, ad f is aalytic i «a +; (iiib) if f ^Ql ) the there exists a a. 3R, a., < a, such that f is O X J. O buded i (D, ad aalytic i 3 (D,: as a csequece f has a fiite al+ al + umber f zers i ay cmpact set i <C +; (iv) if f ad g belg t#_(a ) the the pair (f,g) is aq-cprime iff there exist elemets u,v i #_(a ) such that uf + vg = 1 r equivaletly iff 2 if{ (f(s)~, g(s)) ;s e «a +} >0where (-,«) is ay rm i C. A - Let fifca ): = {f; f ^(X (cr ) such that f is buded away frm zer at ifiity «-*--x - A ^ i (C,}: BT( ) is a multiplicative system, [6, p. 46], f d( ) ad each a + - 4» elemet f f a. (a ) has a fiite umber f zers i (D^ +. fi(a ) is the cvluti algebra crrespdig t the pitwise -1 prduct algebra fi(cq) = [(l.c^)] tflzc^)]" i.e. 6(aQ) is the algebra f qutiets f = /d with ^ 6c (a ), d *=CL (. ) ad where, withut lss f - - geerality, the pair (,d) is a -cprime, i.e., ((s),d(s))j ^ 0 fr all s S <c : a pair (,d) which satisfies these cditis is a a -represetati a + ^ f f G )( ): there exists a bijecti betwee the elemets f (O ad the equivalece classes f a -represetatis {(,d)> i which elemets A are equal mdul a multiplicative factr ivertible i < _(a ). ia -4-

7 Imprtat prperties f V(q) are: a A A * i) if f S 6(a ) ad (,d) is a a -represetati f f the: a) there exists. > a such that f is mermrphic i <E, -> <C +, 10 1 is buded at ifiity i (E ad has a fiite umber f ples i (E^ +; b) p (E 0 A,(respectively z (Da +),is a ple, (zer), f f iff d(p) = 0, ((z) = 0); (ii) f is a ivertible elemet f $(a ) iff f is buded away frm zer at ifiity i (E +. Let (E (s) dete the algebra f prper ratial fuctis i s with P A cmplex cefficiets ad let fr a E ]R: &(a ):= <E (s) ^ U ( ) r p = {f; f e (C (s) such that f has ples i G,),IC(a ) := {f;f ^ &(a ) p O T r such that f is zer at ifiity}. H ( ) is a multiplicative system [6,p.46] f the algebra fcq) ad <E (s) =[faj] [(C(aQ) l"1 i.e. Cp(s) is a algebra f qutiets f = /d with Ua0) aru* d tf\. (a ) It fllws that 4_(),4Z(a0), $(a) =[4.^a0^ ^(a^]"1 are extesis f dkq),(j\t(aq),<ep(s) = [#.(aq)] [4C(a)]~ 1-1 fr represetig trasfer fuctis f distributed liear time ivariat systems Nte als that if f 6i( ) the f = f, f«where f. *~ 1 z 1 is a ivertible >* A aoo AOO /\CO elemet f ( (a ) ad f0 belgs t v (a ) : "# (a ) ad (K. (a ) are O 2. OO O essetially the same": i particular tt)(a ) = [# (0 )][# (a )] =[?_<%> ha^)]"1. We shall w be ccered with trasfer matrices f multivariable distributed systems i.e., with matrices with elemets i (LTD)+, d(aq), fi_(a), 0(q). (LTD)^X, A,(a)X, 6(aQ)X are all algebras with a -cmmutative pitwise prduct ad uit I. F =&Xa )x,,*, Nx x.. ^.._.?., Nx,*, xx».,.,.. (fl._(" ) resp.), is ivertible i tt(a ), (# (a ) resp.), iff i x bth cases if{ det F(s) I;s <C,}, i > 0. u. Fa G *= (a ^a )X j is ivertible i ^ vti ^ 1 ' a + r O ( )( ) iff det F is buded away/ frm zer at ifiity i C,. -5-

8 It is the purpse f this paper t establish a prcedure which fr a x. a give plat P 6(a ) 1,-(see Fig. 3.1), fids a utput feedback x cmpesatr C 6(a ) such that the resultig feedback system S, a) is ^-stable while havig a prescribed set f clsed-lp ples i «0 + ' b) tracks asympttically a class f referece sigals ad c) rejects asympttically a class f disturbace sigals. This task is realized as fllws: i secti 2 we establish fr matrices with elemets i &( ) a represetati thery i terms f matrix fractis: A the results 6(a ) f [1], [2] are hereby exteded t multivariable systems; i secti 3 we study feedback system stability f systems with lp matrices with elemets i $(a ): we adpt hereby results f [18], [32] ad defie a clsed lp system characteristic fucti; i secti 4 we study a prelimiary algebraic prblem fr cmpesatr desig which exteds t matrices with elemets i S._{q) a techique used by Yula et al. [3], [4] ivlvig plymials r plymial matrices; i secti 5 we search fr ad fid a cmpesatr desig fr stabilizati, trackig ad disturbace rejecti, usig a set up ispired by [3]-[4] ad [31]-[34]: A. we hadle here a plat with trasfer matrix with elemets i (B(q). The paper is therefre rgaized as fllws: 1. the preset itrducti; 2. matrix fracti represetati thery; 3. feedback system stability; 4. prelimiary algebraic prblem fr cmpesatr desig; 5. cmpesatr desig fr stabilizati, trackig ad disturbace rejecti. Befre startig we shall meti the fllwig cveti i rder t avid the multiple use f the superscript * t idicate Laplace trasfrmed quatities: quatities represeted by script letters are Laplace trasfrmed -6-

9 uless specifically metied. We eed als the fllwig defiitis. Defiiti l.lr. Let fte Aj^ "1 ad # tf_(a)x. We say that the pair (1\Jd) is a -right cprime (a -r.c.) iff there exist elemets He & (a )xm ad Ve<2 (a )X such that 2tfl+2/# = I. Defiiti 1.1&. Let 8^CL (a )x ad H # (a )xm. We say that the - - pair (0,TO is a -left cprime (a -Jt.c.) iff there exist elemets U<=a(c )mx ad Ve 0. (c )x such that^+m^ I Matrix Fracti Represetati Thery. _x± l2 Defiiti 2.1r. Let F (LTD)^ ; the pair (-ft,ft ) is said t be _i_ &r ^r a -right represetati (a -r.r.) f F if 71 e #_(<* ) ad a -x. #r G#_(<>0> such that (i) F=^ /gr1 (ii) the pair (ft,0) is a -right cprime (a -r.c), i.e., there exist a -sx a ^xelemets # ^ (Xj ) 1 ad^g^_() x 1 such that - x (iii)»^c + M? = I.1 det^ G#>Q). a A a -left represetati (a -.r.) f F, F (LTD),is by defiiti a pair (J&,7L) which is similarly defied as (7?,^)') i defiiti 2.1r: chage subscripts r fr A, iterchage the rder f the factrs abve, chse apprpriate dimesis with Jj ad Tf f dimesi X* AJ x : refer t this as Defiiti 2.1&. Remark R2.1 Observe that if =. = 1 the a -represetatis (left ad : i right) reduce t a a -represetati f F = f, [1], [2]. a x. Lemma 2.1 If F G &( ) 1, the F = R + G (2.1) -7-

10 where A x, (i) G<Zd( ) X - (ii) R is a strictly prper elemet f C(s) 1 which is zer if ad A x. Lauret expasis f F at its ples i (C a m-th rder ple at p G <c p e % + x., where i particular F has if ad ly if R has a m-th rder ple at Prf: F = [f..].,--..,=- where fr all i = l,2,...,, fr all ij i,j 3i * * * * j = l,2,...,., f.. = (5(a ), i.e. accrdig t therem 3.3 f [1], f±j =^±j + 8±:j where (i) g±. = &_ (q),(ii) r±.. is astrictly prper ratial fucti which is zer iff f.. ^ CL (a ), (iii) if f..?^l ( ) 13 - ij - the r.. is the sum f the pricipal parts f the Lauret expasis f f at its ples i G,. H Remark R.2.2 The imprtace f the sum decmpsiti f Lemma 1 lies i the fact that it permits t fid a a Jl.r. r a -r.r. fr r 00 A x. F = (c ) """by fidig first such a represetati fr "its ratial pricipal part" R. Nw bserve that, with [2], fi^ ): = {f G C (s); f has ples i <C +} = <D (s) (LJQ) (2.2) p fc( ): ={f G((l(a); f is -zer at ifiity} ^(f{ ) (2.3) 00 <C (s) is a qutiet rig [4((a )][ft (a )]" f R.(a ) with respect t its p 000 multiplicative system $C( ), [2], i.e. if f C (s) the f ca be writte as f = _/d. with. = <jl(a ), d,. = ({ (a ) by usig a scalig plymial r f r e.g. f(s) =(s-l)/(s-2)2 =f/df with f(s) =(s-1)/(s-a+l)2 ad 2 2 d_(s) = (s-2) /(s-a +1) : bserve that i this way e btais a 0 - represetati (f,d ) by makig (f,d ) a -cprime, [1], [2], cacellig ly if FG &_( ) 1, Z 0x- (iii) if F ^ (L_( ) the R is the sum f the pricipal parts f the -8-

11 cmm factrs (s-z)/(s-a) with z (Lq +, a <Cg _: here a pair (f,df) with f S(t(), df S(jC^) ad (f,df) c^-cprime is aqrepresetati. Observe als that (ft(a ) is a Euclidea rig, [9], [10], see als Appedix I. It fllws that every matrix with elemets i x. #Ia ), say ^.^(a ) 1, has a Hermite frm [8, p. 32] btaiable thrugh elemetary peratis [8, p. 34, Th. 22.4]. Hece the same must be true fr triagularizati. Als every cmpatible pair f matrices 71 ad & with elemets i &(a ) has a greatest cmm right divisr (g.c.r.d.), R., [8, p. 35], expressible i the frmt(tl+v& =ft where %L ad Y are matrices with elemets i 6{.( ); furthermre if $_is ivertible i fu ) we say that Ti ad J&are right cprime w.r.t. fisq); te that the matrices *ll,v,9\ca be btaied thrugh elemetary peratis [8, Chapter III, pp ], a variat f this prcedure beig described i [11, pp. 8-9] ad [7, p. 65]; it is als easily see that 71 ad # are right cprime w.r.t (R.( ) iff the matrix 1t has full rak fr all s i C, ad at ifiity; a + mrever if Tlad are right cprime w.r.t. QdJ the they are 0Q-right cprime as i Defiiti l.lr. Similar Facts hld fr a greatest cmm left divisr (g.c.r.d.) ad left cprimeess w.r.t. (ft(aq). The abve suggests that it shuld be relatively easy t fid a ratial aq-r.r. the pricipal ratial part R. f F i (2.1), ce we ca express R as x.x. m R=ftj ~ with ^ e<k(a) i #r ifc(g) x \ det[#r] ^(Q). These suggestis are explited i the prf f the fllwig therem. a x. Therem 2.1 If F 6(a ) 1, the F admits a a -r.r. ad a a -.r.. Mre precisely, 7lr>frr>\>Yr A there exist matrices with elemets i #_( ), amely f -9-

12 such that (i) (7^,%) is a aq-r.r. f F; (ii) (#0,7O is a a -x.r. f F; x x.. i 1 (iii). l r I *r *r :-k -ft! ft U 'x "x-j,«t ; vj J where if we call the matrices the left had side f (2.3),V ad b) respectively, the bviusly Wis a ivertible elemet ("uit") f (.+ )x( + ) /( (a ) ad withut lss f geerality. 1 rl (2.3) det tj= det UT = 1. A 0xi Prf: Withut lss f geerality we assume F %u. (a ) ; therwise chse flt =* ; <% r i ;K ; K =x» i Use w lemma 1 ad recall that each elemet r.. f its ratial pricipal part R admits accrdig t remark R.2.2 a ratial a -admissible repre setati (,d ) with ^.( ) ad d 4L (a ). Recall als r.. r. / r.. r.. 13 ij i3 i3 the structural prperties discussed i Remark R.2,2 ad apply the fllwig prcedure: Algrithm 2.1 Give is F, G ad R as i Lemma 2.1. x. ^.x ^ m Step 1. Fid # S 6L( ) x ad fi A(a ) 1 * with det fl ^ (0 ) c cr r r ad such that Lr r ~ i e.g. by settig & = diag[d.]._1 where the d. are clum least cmm (2.4) demiatrs f R w.r.t. (Pv(a ). -10-

13 Step 2. Csider the ( +.) x ± full rak matrix VI := ui _ A*. J«rJ (_,+ )x. e«(a) i x (2.5) By perfrmig elemetary rw peratis based the Euclidea algrithm perfrmed i the rig $UaQ)» e-s- t8» PP ], [11, p. 8-9], [7, p. 65], upper triagularize #?, i.e. fid a (.+ ) x (.+ ) matrix Ulivertible (.+ )x( + ) i R(a ) X ad a full rak upper triagular matrix.x (ft_g ${( ) 1 1 such that a#i= 1^ (2.6) ad where scalig (multiplyig rws by "uits" i $_(a )) ca be used t get det 20= 1. Step 3. Partiti to ad tp it.. i % % ci ff. ;W'1 = i. i '&. -^ A Y. 8,-J _ A Cmmet: the eight matrices with elemets i ff\() c^ (aq), amely (2.7) 1\T9 8T9tUr9Yr A. A satisfy the cclusis f Therem 1 prvided F has bee replaced by R Step 4. Recallig (2.1), defie t ~ (.+ )x ^ m Ttfe R( ) 1 is full rak because by assumpti, det & e ^v (a ), ~ /> r hece det $ is t the zer-elemet f </ucj ). r t -11-

14 K-K-^6 V%:=\-^% (2-8) Ur %. y.% ' ' \ ad stp. Cmmet: A the eight matrices with elemets i # (a ), amely satisfy the cclusis f Therem 1. We shall w shw that Algrithm 2.1 wrks. Step 1. Sice all elemets r.. f R i (2.1) are elemets f <E (s) ad i3 p have ples ly i <D, they admit a ratial a -admissible represetati O* ><* ) with E (ft(a ) ad d G 0C(a ) with ad d cprime r.. r.. r.. r.. r r 13 i3 13 ij ij rij w.r.t. R.(cr ), ad it is pssible t cstruct a least cmm multiple d G^~(a ) f all demiatrs d.. /fc( ) f clum j, [12, Ch. IV, 3 ij J' * ' 10]. Hece settig r = ii. /d we get that ^ = [ii..] ad A/ H, fy = diag[d.]. _ satisfy the cditis f step 1. r j j=l Step 2. Sice Wlis full rak because by assumpti det & v\- (a ), hece r is t the zer elemet f ft.(a ), step 2 is self explaatry. Step 3. The cmmet f step 3 is true as fllws. Observe that all matrices A *C i (2.5)-(2.7) have elemets i flx ) C^L(a ) with det fif ad _ ^00 1 det 0v. =0v, (a ); mrever frm Jf[= "U) hece A. /V AS -I "1 Frm yj"l)f - I we have -12-

15 Hece (fj,# ) is a a -r.r. f R, with IK a g.c.r.d. f 7?r ad < r, [8, p. 35]. Observe that frm fiffif~ = I, we get als Furthermre sice by cstructi %f is a ivertible elemet f ( + )x( + ) fi^a ),det V(s) teds t a zer cmplex cstat as s ->». Frm the partiti f 2/", (2.7), the [-^ifi^l =-^trj 1^ ] is full rak at ifiity; hece det & G rf?.(a ) teds t a zer cstat at ifiity. Thus (F»^) is a c^-x.r. f R. Step 4. Checkig the cmmet f step 4 fllws easily usig (2.8) ad simple cmputatis, i particular 1iT= vr ;k b«x ;^ i c * i V ; t~~ -G j I S" i J G i I -J i I.. i O 1 0 i ii * i V 0 pr: i -%. 11 i _ ~ i LG! I -J\r\ ru 0 G. I y~x =ti/1 Remark R2.3 Observe that i algrithm 2.1, used i the prf f Therem 2.1, we actually btai that fll G fr ).x. i l det ft e(k"(a ) x ' x. i det #0 e C(c ) X i.e. the "demiatrs" f the aq-r.r. (Jfi^.T^) ad the aq-x.r. Cfe»# ) are ratial! The uiqueess f the represetatis will be treated belw. -13-

16 We have als x. Crllary 2.1. Let F G (LTD) 1, the a x. F fc ) if ad ly if X F admits a a -r.r. 07?,/$) r a a -x.r. Cft'jft). Prf: Oly if: this is a immediate csequece f Therem 2.1. If: Observe that F = [f..].a*.&. Mrever sice F = 1\ ST it fllws by 13 1^,3^ 'r^r J Cramers rule that fr all i ad j fy" [^^[AdJ^l./det^ where frm the clsure prperties f $ (a ), [1], [#]. [Adj<$;]. belgs t fl. (a ) ad by defiiti det fif belgs t^ _(a ). Hece fr all i ad j,- belgs t fi(a0) =^(a^li^c^)]"1, [2]. Remark R2.4 Frm Crllary 2.1 it is bvius that we ca idetify A x. x (a ) 1= {F :Fe (LTD)+ i ad Fadmits a a -r.r. r a a -x.r.} (2.9) This is a suitable geeralizati f [1, Defiiti 3.1] where =. = 1. I the sequel we shall t make ay disticti betwee the tw classes. Ncmmutative fracti rigs, are treated i [13]. a A csequece f Crllary 2.1 is Crllary 2.2. Let (1^9&)9 (resp. (J^,^)) be a pair f matrices such that x. A x. a rtxrt i) i\. s<2j) \ Tr r-ajj ± \ (resp.»t 6d_(), x A H^<2 (a ) 1), ad ii) det # belgs t & (a ), (resp. det ft belgs x r a* t^"(a)). Uder these cditis the pair (/^,JS ) is q - r.c, (resp. the pair -14-

17 j^w (fr Sft ) is a -x.c), if ad ly if rak xx r^w = fr all s <E., (resp. rak a + ^ fifc") f> (.> i = fr all s (E ). O 0 + Prf: We shall restrict urselves t the right-cprime case. : fllws frm [Y \ %L^ & rule. i '<r l. j (s) = I fr all s e <e ad Sylvester's i V A x- <=: Let F = "ft # ad bserve that F G 6(a ) 1. Hece by Therem 2.1 rr - A x. _ a.x Fadmits a a - r. r. (7?r,#r), i.e.,flr #><>) \ #r = a_(q) such that F =7? 8, there exists ft. ^ 6L (a ) l- * ' IT O r r r _"- 1 adrre^ ^0).x a x "= A. 0.-1, with 2L ft +/"ft. =1» ad det ft &CC S0^ Let (H = f ^ ad bserve with r r r r. *-.x. &=& fi +yrj^ that (f^belgs t^(aq) mre fr all s <D, by assumpti. = rak i i * 1with det (H i{x"(aq).further = rak Fffr(s)" &(*) Hece by Sylvester's rule, det (R_(s) ^ 0 fr all s^ <c^ +. Frm the abve A.x it fllws that (ft- is a ivertible elemet f #_( *0) ^ ad there exists x A x % =GL%edA) t ad^ ^ =OT1^ =d\"1^e^(a) 6^.<c) such that Mr+W = x >i^- <^^r> is V^' i Fr future applicatis we have als A x Crllary 2.3. Let FG$(q) iadmit acjq-r.r. (7?r,#.) ad aaq-x.r. (& j^) where a 0. At At O it (p.r.c.f.) f F ad (ft,ft) At The (fl,fr) is a pseud-right-cprime factrizati (p.x.c.f.) f F i the sese f [7, pp ]. At is a pseud-left-cprime factrizati -15-

18 Prf: A A A Apply the defiitis ad the fact that #_(a ) C^Z(0) =#. fr all a < 0. a x We shall w discuss ples f F p(a ) A x Defiiti 2.2. Let p be a ple f F belgig ). The the MacMilla degree f the ple p f F is its maximal rder as a ple f ay mir f ay rder f F. Remark R.2.5 The defiiti f MacMilla degree here is based the fllwig prperties which are true whe F t (s), i.e. is a prper " x. A i ratial trasfer matrix. The characteristic plymial f F G ^ (s) x. is defied t be the least cmm demiatr f all mirs f ay rder f F ad is the characteristic plymial det[si-a] f ay miimal realizati [A,B,C,E] f F, [16], [14]; the MacMilla degree f x. FGC (s) is the degree f its characteristic plymial, [14], [15], [16]: hece the rder f a ple p f F as a zer f its characteristic plymial is its maximal rder as a ple f ay mir f ay rder f F: this ca be called the MacMilla degree f the ple p because this is exactly the MacMilla degree f the term due t p i a partial fracti expasi f F [14], [15]. Mrever let (Nr>Dr), ((D^N^resp.), be a right cprime, (resp. left cprime), plymial matrix factrizati f x. _1 F <E (s) 1, i.e. F = N D~, det D 0, (N,D ) right cprime, p ' rr r rr. (resp. F = D~~TT, det D t 0, (D,N ) left cprime), the det D, X X X X> X * (det D resp.), is equal mdul a zer cstat t the characteristic plymial f F, [11], [17]: hece the MacMilla degree f the ple p f F is the rder f p as a zer f det D, (det D resp.). Smethig. x. A /> i similar ca be de fr ples f F G ^( 0) * a +' -16-

19 x Therem 2.2. Let F >(aq) ad let (#r> r), (Gfy.fl^resp.) be a a -r.r. f F, (resp. a -x.r. f F). Uder these cditis: 0 a) P E <E. is a ple f F, if ad ly if det.fr(p) = 0, (det ^(p) = 0). b) If p j is a ple f F, the the rder f p as a zer f det jy, (det ft resp.), r Prf: Xt is its MacMilla degree, A c) There exists r a ivertible elemet f d ( ) such that det ft = r det ft.. r Fr a) ad b) we shall restrict urselves t a -r.r. ' -1 A ira0 a) Usig F = 1)ft ad the existece f matrices %L e^_(<0.x ad V ed_( ) such that fttt + 7 = I, where all matrices have A elemets i #. (c )» it fllws that g^f+tr-j^1 (2.10) :this expressi ad F are mermrphic i a pe half plae (E,, sme O- < ; furthermre 9/ ad 1/ are aalytic i (D, ad buded i <E,. 1 ^r r a..+ a_ + Let V(p) w be a eighbrhd f p (E, withi C,, the F has a ple 0 ^ -1 at p iff F is ubuded i V(p). Nw if det ft (p) =0 the ft" is ubuded i V(p) ad, because f (2.10), the same must hld fr F: therwise the left had side f (2.10) wuld be buded there. Cversely A, ^ A. _ ^ J_ if F is ubuded i V(p) the det & (p) = 0, therwise F = 7c ft wuld be 1 -* buded there. b) Observe that (#.,$ ) is a -r.c. implies rak ^r(s)~ Lftr(s)J =. Vs (E,, sme ^ <. (2.11) l

20 We fllw w the methd f [18, prf f Fact 2, p. 518]. Let us express -1 ay mir f rder p f F = 7>ft i terms f mirs f rder p f 7) ad ir r 'r mirs f rder.-p f ft. i r By well kw methds ad tatis, A, [19, pp ], we csider the mir f F made f the itersectis f rws i-, i0,...,i ad clums k-,k_,...,k, deted by 1 2 p 1 2 p Jh h \\ \kl k2 " \) =E^(v'''-*p) *"'(S--kp) "lth!-»t<vi> 1< <x <...< p< (2.12) P /i1i2...i\ 2^ VN /klk2'"' k±_p x k i-p l<x.<x0<...<x < 1 2 Q^2 det 0 r where JL<x0<...<x ad ll <!<...<xf, k-<k <...<k ad k'<k*<...<k* 12 p 12.-p 12 p 12.-p frm a cmplete system f idices f {1,2,...,.}. Observe that the umeratr f the abve expressi is prprtial t the Laplace expasi T T T [20, Exercise 7.2.3] f the mir f rder f [&Tflr] by adjigig rws ii...i f ftr t rws 44...k _p f fft- Fr a11 sg^ +' <2,11) implies that at least e such mir rder is zer. Hece fr s = p (E +, 1 at least e umeratr f a expressi (2.12) is zer ad b) fllws usig Defiiti 2.2. c) Csider r = det J3" (det #0)~. Sice det <fr ad det f bth r x r x belg t CL ( ) it fllws that r is a ivertible elemet f B(a) = \{L )] [?C(c )], [1], [2]. Mrever because f b) r has either A /v 1 A H ples r zers i <E,. Hece r ad r belg t (X ( J» [1]» [2]. c + 18-

21 Remark R2.6 By a similar reasig as i the prf f Therem 2.2c), i.e. by usig Therem 2.2b), it is easily shw that if (?,/%) ad a x. y, (ittff) are tw -r.r.'s f FS fj the there exists r a ivertible elemet f&jaj such that det ^ =rdet fr., ad similarly a Qx^ if Gfy^) ad (Aj^) are tw c^-x.r.'s f Fe^(aQ) the there exists ra ivertible elemet f ^_(Q) such that det ft% =rdetfl^. Mrever the latter elemets r, (icludig the e metied i Therem 2.2c)), will ivertible elemets f tfk^) if the demiatr determiats actually belg t tfc(a). This is the case i algrithm 2.1. We are w ready t lk at the uiqueess f aq-admissible represetatis A x. f F G {5( ). This is a geeralizati f Therem 3.4 f [1], a x. Therem 2.3. Let F G #(a ) 1 ad let (#.,&) ad <fi\fi\) be tw 1 r j. t jl a -r.r.'s f F, (respectively let C#»# ) ad (ftj^j) be tw c^-x.r.'s f F). Uder these cditis there exists a.x. a x aea()1 i, (resp.^ed (a ) ) (2.13) such that ad A.x. jj ^ is ivertible i #_(aq) 1,(resp.. is ivertible i a x a.(0) ) <2-14> %-%&.%- * «. (reap. ^ =/^, ^ =tftj - Mrever if ^, #f, jft, #1 elemets i 6^0 ). Prf. have elemets i i.( ) the &ad have r r x x We shall restrict urselves t -r.r.'s with elemets i CL_iaQ)' * f a.x; Defie ft = (ft') ft. Observe that, sice ft ad ft belg t & (a ) 1 x r r r r <-> with det fr ad det ft1 i#~(a ), it fllws by Cramer's Rule that/lad (2'15> -19-

22 a 1 A x (fl are elemets f (E)(q ). Mrever frm # = Fjfr' ad 7>r = FJ9f "r t r r it fllws that (2.15) hlds. Observe fially that /# + Yft. = I ad icjy + 1[U? = * where all matrices have elemets i CL (q ): hece by 1 1 (2.15), Uft + Vft a#c ad $ fl + ff } =^ where all matrices the A left had sides have elemets i & (a ): s sice CL (a ) is a algebra, - A.x. /Rad (IC belg t < _(q ) 1 x, i.e. (2.13)-(2.14) hld. «We give w a defiiti ad a crllary eeded fr further develpmets. Defiiti 2.3. We say that the pair (f) 9Jfr ), ((>&,# )resp.), is a ~ "*-^-^~ r r x x x a -right represetati ( x. a -r.r.), (resp. is a x. q -left represetati ( x a -x.r.)), iff A x A x A x (i) ^ e &_(qq) i ad ^ ^d_(q) ± \ (resp. fr^&jj ad a x. A a i (ii) the pair (/?,# ) is q -r.c, (resp. the pair Oft,,ft,) is a -x.c); r r x x (iii) det^- ^Ci(a ), (resp. det ft0 ^Ci(a )). r - x - Remark R.2.7 It fllws frm Cramerfs rule that if (# 9ft) is a _, a x. x, q -r.r. the F = # vjt G $(q ) ; mrever if we defie tw i r r x q -r.r.'s (1} 9ft) ad ( #',ft) t be equivalet if there exists (f^a ivertible elemet f CLJQ)± ±such that (1^,%.) = (f±fi>ftjd the accrdig t Therem 2.3, there exists a bjecti betwee the set f equivalece classes f x. a -r.r. fs {(/J,ft)} ad the elemets F * i r r x f fe(q ). As a csequece, mdul a equivalece class, e x A ai q -r.r. represets e elemet F G fc( ) ad vice-versa. Smethig similar is als true fr a x. q -x.r. ($ 9i)0). i xx rtx Crllary 2.4x. Let F b( ) x. The fr ay_ qq-x.r. (ft^jly) f F A there exist matrices with elemets i CL (a )» amely t^. ^;r?r,^r,ar,rr -20-

23 such that (i) <#,Ji ) is a aq-r.r. f F. i i Vri V*' X\-%? CA\ *t. 1%: rj L i, ' i (2.16) where if we call the matrices the left had side f (2.16), 'Z/respectively (i+)x(±+) %r, the bviusly &/is a ivertible elemet f &_(c ) ad withut lss f geerality det V= det U1 e 1. (2.16a) Prf: Apply Therem 2.1 ad use Therem 2.3 fr idetificati purpses. Remark R2.8 It is bvius that a similar Therem is valid whe we start frm ay q -r.r. (fl >A>) f F: call this Crllary 2.4r. 3. Feedback System Stability Csider the multi-iput multi-utput feedback system S shw i Fig. 3.1, where all relevat expressis are described i the frequecy A, A, dmai: i) usually P ad C are the plat ad ctrller trasfer fuctis with respective iputs u, u ad utputs y, y ; ii) u is the system iput ad w. the plat iput disturbace; iii) y = y is the system utput ad e = u - y = u the system errr. s s s c * Nte that if we had additive disturbaces applied at the plat utput, say w, the their effect is equivalet t a additial system iput -w. Frm Fig. 3.1 the system equatis are -21-

24 u. u w ṖJ. 1. i i! p- ~at t 1 u. l. A> "C ' 1 i 0~ 0 ' p J u (3.1) Let G =. l ~0. l -C 0 P~", J =. l r! I -. -I 0 i L. J (3.2) ad bserve that -1- = J G. Hece the system's iput-errr trasfer fucti H : (u,w ) \» (u,u ) J * e s p v c p ad iput-utput trasfer fucti H : (u,w ) * (y,y ) satisfy r yspcp A. A, 1 He = (I+G) X, (3.3) JH = I - H. y c (3.4) We have als the fllwig: System Assumptis Al) Fr sme a < 0 A x. a.x P <E( >(q ) x ad 6e6(q ) X (3.5) where P has a qq-x.r. (^,7^^), (3.6) C has a q -r.r. (/), ). lcr cr (3.7) A2) det [I + 6] = det [I +CP] is buded away frm zer at ifiity i i (3.8) Cq

25 Csequetly by the prperties f the algebras AAA &.= $.(0),#,_(<J0) cd p fr q < 0, &(q ), [1], [2] ad by (3.1)-(3.8): (.+ )x( + ) GG g(a) X x (3.9) A ( + )x( + ) A( + )x( + ) Jad J"1 belg ta_(a) * X C^L i i, (3.10) det[i+g]_1 =det[i +PC]"1 =det[i +CP]"1 ^&(q)9 (3.11) i H ad H belg t < >(q (.+ )x(.+ ) ) x X, (3.12) e y A(.+ )x(.+ ) ( + )x( + ) fi ea i ± h ea X X, (3.13) e y ( + )x( + ). A ( + )x( + ) He^(a)i 10~He&(a)i0 (3.14) e ^^ y - It makes therefre sese t have the fllwig: Defiiti 3.1 [18]. The feedback system S described by (3.1)-(3.8) is said t be #.-stable iff bth its iput-errr trasfer fucti He ad its iput-utput trasfer fucti H belg t (X Remark R3.1 Frm (3.13) ce system S is ^-stable the its iput-utput map (u,w ) \ + (u = e,u ;y,y = y ) will (i), fr ay p G [1,«], sp c spcp s take L -iputs it L -utputs with fiite gai ad (ii) will take ctiuus P P ad buded iputs, (peridic iputs, almst peridic iputs, resp.) it utputs belgig t the same classes, [7], [21], By (3.5)-(3.7) the fucti x defied i <E (fr sme q. < a ) by: X==detVcr+W <3-15) A is a elemet f (k ( ) ad is called characteristic fucti f S (i <Ca +). The imprtace f x Is discussed ext. -23-

26 Therem 3.1. Csider a feedback system S specified by (3.1)-(3.8). Csequetly (3.9)-(3.14) hld. Uder these cditis: (i) the system S is ^-stable if ad. ly if X(s) * 0 fr all s e (c+ ; (3.16) (ii) p C, is a zer f x(') (3.17) q t if ad ly if pe <c ^ is a ple f 5 (3.18) r q + r e if ad ly if p e <D ^ is a ple f H ; (3.19) v a + r y (iii) the MacMilla degrees f p (E^ A A, + as a ple f H ad H are the same ad equal t the multiplicity f p as a zer f x(*)» Prf f (i): First frm the defiiti (3.15) ad (3.5)-(3.7) X= det[i +PC] detj&;r det# (3.20). * 00 a. Hece by (3.8) ad sice bth det ft ^ ad det & ~ belg t&_(q), x is buded away frm zer at ifiity i (E,. Thus (3.16) is equivalet t if{ x(s) : s c } > 0. Nw the cclusi fllws by cditi (35) f A A. * Therem 1 f [18]: ideed G ad G2 f [18] crrespd t the preset C ad P; by Crllary 2.3 (ft ^tj 9 (Mcr>A%r),resp.), is a pseud leftcprime factrizati f P, (resp. pseud right-cprime factrizati f C); fially, as idicated i the cclusis f [18], Therem 1 f [18] applies t rectagular systems (i.e. ^.). a x. Prf f (ii) ad (iii): Sice P (&(qq),by Therem 2.1 it fllws that Phas ac-r.r. <^r./3 r). (3'21) mrever by (3.21), (3.6) ad Therem 2.2c): -24-

27 A there exists r a ivertible elemet f #_(%) ysuch that det -r det^pr. (3.22) Recall w relatis (3.2)-(3.4), (3.21), (3.7) ad csider the fllwig A matrices with elemets i^_(q), amely:?h. i '0 Upr -ii i L cr i, f~ fc cr 0. l ft pr (3.23) The it fllws easily usig Crllary 2.2: (.+ )x(4+^) l, Til. (tljv) is a q -r.r. f G 6 $( ) i ad similarly, usig H = (I+G) = ft(r+tl) ad H = J^Gd+G)"1 = j"1/?^/))"1: A (i+)x(i+q) (ft,jd+7)) is a a -r.r. f H e (qq) (i+)x(.+) (J_1^,6+7» is a 0Q-r.r. f Hy ft(qq) (3.24) (3.25) Nw, sice by (3.23), (3.21) ad (3.7) &+=. jbu: ^ 'cr i i.-a -> pr "^.! A i L "cr pr. «A -c, I il r J I i % "ft cr det [8+fl] = det [I +PC] det^cr det #pr. Hece cmparig (3.20), (3.26) ad (3.22):. i ' ft» pr_j A there exists r a ivertible elemet f tf_() such that (3.26) X = r det[j&+tf]. Recallig that r is buded ad buded away frm zer i Qq +9 sme ± < q9 it fllws by Therem 2.2a) ad (3.24), (3.25), (3.27) that the equivaleces (3.17) <=* (3.18) <=* (3.19) hld; similarly cclusi iii) is a csequece f Therem 2.3b) ad (3.24), (3.25), (3.27) (3.27)

28 Remarks. R3.2 Equati (3.15) defiig x is t the ly pssible expressi fr a characteristic fucti f S i (E.: bserve that ay q + J A A A elemet rx f (X ( ), where r is a ivertible elemet f CL Ak used istead f x fr havig the defiig (a ) ca be prperties f a characteristic fucti required i Therem 3.1. We call therefre characteristic fucti f S i (E, ay elemet f the equivalece class f elemets f d (q ) beig equal t x> defied by (3.15), mdul a ivertible elemet f A Q_(gQ). Observe that such a characteristic fucti is btaied if i (3.15) i) the <JQ-x.r. (fi^^ti^) f P is replaced by ather a -*.r. GB1^/?^) r if the tfq-r.r. (flcr»je r) a f C is replaced by ather a -r.r. (fl',#* ) (use Therem 2.3), ad ii) if we use left r right a -represetatis A, A, fr P ad/r C, (use (3.20) ad Therem 2.2c), see als Therem 1 f [18]). The characteristic fucti x give by (3.15) was chse because it suits best ur preset purpses. R3.3 Cditi (3.16) ca be checked by the graphical methds, [22], [23]. R3.4 Nte that accrdig t [32, Therem 3], the ^-stability f clsed lp system S is rbust. 4. Prelimiary "Algebraic" Prblem fr Cmpesatr Desig a x. We are give F &( ) where * x W ±S ay a~*"tm f * e^>(a0) * (4-i) We wat t slve prblem (COMP) (COMP) : Uder assumpti (4.1) defied by a x fr ay $ Q-._( ) slve the equati a x A x. Recall frm Defiiti 2.1 that ft e CL ( ), fl e CL (a ) 1, X - O X - O ' the pair (ft^h^) is qq-x.c. ad det & Ed^a0)- -26-

29 *.*+$#-# (4.2) fr A x A x 9Ge^_(a) i ad/p< _(q) (4.3) Prelimiary ifrmati: because f (4.1), accrdig t Crllary 2.4x, A there exist six matrices with elemets i^_(q ) amely V^^r'Ar'^r'K such that (i) (Hr,ftr) is a aq-r.r. f F (4.4) (ii) i,. i i ~ra UA p-rl-^1 u. i = LAi AJL*r j*lj w 1 ; i I -J where if we call the matrices the left had side f (4.5) respectively 2i/~~ Ci ( ), the bviusly hfis a ivertible elemet f ad withut lss f geerality \f9 (4.5) det'i^= det If1 = 1. Recall further by remark R2.7) that,mdul a equivalece class, e ± x ( a x q -r.r. (il9ft) represets e elemet f $)(a ) We are the lead t the fllwig: Therem 4.1 Csider the prblem (COMP). Uder the assumptis ad tatis specified abve: (i) All the slutis U il] f (COMP) are give by i r-aq TL LtJ 7L/ = ^-i L#J (4.5a) i i.e.-x=%tl-^;^ =^+r l9' (4.6) -27-

30 where "71 is a arbitrary elemet f /7 (q ).x i Mrever, by (4.5), (4.6) is equivalet t, "VlI [_erj r-xi.. -V IhI. l i.e. = -rrx+^ ft =\x+ftzif (4.7) ad (0C>tf) is q -r.c. if ad ly if (11$) is q -r.c. (4.8) (ii) If i additi F(s) +0 as s -» i C, x. ' q + i (4.9) the (OQW is a x a -r.r. if ad ly if (72,&) is a. x q -r.r. i (4.10) Hece accrdig t Remark R2.7) all slutis (XiU) f (COMP) resultig i elemets f ( >(q ) A "" w are geerated by (4.6) by the class a x m^cl (cr ) x ; (7?,fr) is a, x -r.r.} (4.11) O 1 Prf f(i): Nte that if (X,ip is give by (4.6), the usig (4.5), OCM) is a sluti f (4.2)-(4.3), i.e. f (COMP). Nw let (X*ft) be a sluti f (COMP), i.e., f (4.2)-(4.3). OCU) = (a^,]^ft): f The a particular sluti is ideed by (4.5) #^ +ft^ = * Xt remais t ^ add t this particular sluti the geeral sluti f the hmgeeus equati crrespdig t (4.2), amely #X+#0& = 0 (4.12) We claim that ay sluti (X,^) f (4.12) ca be put it the frm x (X,^) = (-J^,7?r7\) fr sme 71E cl_(q) i. T prve this, let (JC,U) be ay sluti f (4.12) ad defie 1\E >(a ) i by: (4.13) -28-

31 Hece by (4.12), (4.1) ad (4.4) tf-ftyaa-wl- (4,14) (4.13) ad (4.14) shw that ay sluti has the required frm but it remais.x t be shw that T\ CL_(0) 1. Use (4.5), (4.13) ad (4.14) t btai ~YX + Yfy =yi> where all matrices the left had side have elemets i a* r «.x <$.(a). Therefre ft^d (a ). KA~- - The equivalece f (4.6) ad (4.7) is a csequece f (4.5). Equivalece (4.8) is als a csequece f (4.5) ad Crllary 2.2. Prf f (ii): Observe that by (4.1) ad (4.9) det &z ed!!(a0) (4'15) ad 7? (s) -» 0 as IsI -> - i C. (4.16) <xv ' x. ' ' q + i where (4.16) fllws by # =j^f, (4.9) ad because all elemets f ft% are i CJL ( ), therefre are buded i <E.. Nw by (4.7) ad (4.16), - fr ay sequece (s.)^ c «a +with \s±\ -» <», i»- «, we have lim if det &{b±) =lim if det ^(Sj) det ^(s±) : sice by (4.15) det O is buded ad buded away frm zer at ifiity i VQ * fllws that +9 it tet%^cf_(aq) if ad ly if det Jfr^tf"^). (4.17) Hece equivalece (4.10) is established usig equivaleces (4.8) ad (4.17). * Remark R4.2. Prblem (COMP) discussed abve is a geeralizati f a methd fr cmpesatr desig i the lumped case fud i [3], [4]. I the sequel the sluti f this prblem will be used t shw cstructively A x. A that ay plat P G &( ) fr sme < 0, with P(s) ^ 0 as s + «> -29-

32 i C,»ca be stabilized by dyamic utput feedback i the sese f Fig. 3.1: mre precisely a cmpesatr C, (see Fig. 3.1), shuld be fud such that the clsed lp system S is CL-stable ad bth the A, A. iput-errr- ad iput-utput trasfer fuctis H resp. H have a give set f ples i the vertical strip [q,0) with specified MacMilla degrees. Mrever we wuld like that C wuld be such that the clsed lp system S is a rbust servmechaism. Kw stabilizati techiques i the lumped case iclude the desig f a state estimatr ad the use f state feedback r the desig f a ctrller, [16], [24], [25], [26], Multivariable servmechaisms are discussed i [27], [28], [29], [30], [31], [32], [33], [34]. 5. Cmpesatr Desig fr Stabilizati, Trackig ad Disturbace Rej ecti A We are give a plat P such that where A x P <B(q ) fr sme q < 0 Phas aq -*.r.c0 t^pt) with J3p 6KJ x (5.1) (5.2) the elemets f P = " [p] are real-valued Laplace trasfrmable distributis with supprt l+; (5.3) P(s) -* 0 as Is -* «> i <E x. ' ' q + i (5.4) Referece sigals (t be tracked) are geerated as fllws: x (0) is a arbitrary vectr i ]R ad s i(t) = a x (t), u (t) = c x (t), vt e m, s s s s s s + ^ where x x x (t) e m s, a ems s, c em0 s s ' s ' s (C,A ) is a cmpletely bservable pair; C5.5) thus u(s) = C(sl-A)""1x (0). -30-

33 Disturbace sigals (t be rejected) are geerated as fllws: w x (0) is a arbitrary vectr i ]R ad w * (t) - v«(t)» wp(t) =Vw(t) vt e m+ -\ where (C,A ) is a cmpletely bservable pair; WW (5.6) thus w (s) =C (si-a )-1x (0) p W WW J Furthermre, with q(...)detig the spectrum f the square matrix betwee the paretheses, we assume that q(aw) U q(ag) C (D+. (5.7> Let w i k ad \J>. dete the miimal plymials f A respectively Aw '\, ad let <j> := mic least cmm multiple f i >. ad \\> (5.8) q = degree f (j> =: 3<j> (5.9) Let Z.M dete the list f zers f <j>, i.e., let z. be a zer f < >, m dete its multiplicity ad let <f> admit k distict zers, the *-[$] (z.»...z.,; Z,...,z«;... ; z,,...,z,) ^i 3 ^_ j ^ (5.10) "l m2 *k {z1,...,zk> = q(aw) U(Ag), (5.11) q = 2 i=l m, z 6 [<}>]«=> z G2[«x (5.12) -31-

34 ad the maximal rder f z as a ple f ay elemet -1 l f Cr(sl-As) xr(0) ad C^sl-A^) xw<0) ay xg(0) GS S \ (5.13) ay xw(0) G]R w is m± (see Appedix 2). Fr trackig ad disturbace rejecti purpses, x pe6(0) i: [33], we assume fr i> (5.14) rak[7jp (s)] =Q V s e CA^ U0(A8). (5.15) Let fially A be a give fiite list f pits f the vertical strip [,0) with the prperty that JlGA^IeA. (5.16) We shall w discuss the Stabilizati, Trackig ad Disturbace Rejecti Prblem (STDP):.x the give data (5.1)-(5.16) fid a ctrller C G 6(q ) Fr, crrespdig t real valued distributis, such that the feedback system S, (3.1)-(3.8), (Fig. 3.1): (i) is ^.-stable; (ii) j [X;tt ] i.e., the list f zers f x (the characteristic fucti + f S defied by (3.15)) i (E is exactly A; q + ^ (iii) V x (0) G3R S, V xw(0) ]R the referece sigals u ( ) defied by (5.5) will be tracked asympttically ad the disturbaces w (*) defied by (5.6) will be rejected asympttically; mre precisely, with Fig. 3.1 i mid: the system errr e ( ) geerated by (u (*)»w ( )) defied by (5.5) ad (5.6) satisfies, fr sme < 0, e (t) = (e ) as t >» s i.e. (5.17) lim e (t)/ea = 0; t-x» s ' -32-

35 x. * i (iv) prperty (iii) is maitaied fr ay perturbed plat P fr which the feedback system S, (3.1)-(3.8), remais ^-stable. A x Remarks. R5.1 C is required t be i 8(0 ) 9 hece 6 is buded at ifiity i <E. This crrespds t C beig a prper ratial matrix i the lumped case. R5.2 Assumpti (5.4) is satisfied by all realistic mdels f physical plats: it reflects the iertia-like prperties f physical plats: it x implies als that, fr CGft), det[i +PC] = det[i +CP] -j- 1 as r. 1 A ix0 as IsI -» 00 i <E :hece, fr ay C sd( ),cditi (3.8) will be + satisfied ad the iput-errr- ad iput-utput trasfer fuctis He ad H f system S (see secti 3) will belg t 6(0 ) a (i+)x(i+) R5.3 Accrdig t the Therem 3.1,cditi (ii) f the (STDP) guara tees that simultaeusly the iput-errr- ad iput-utput trasfer fuc tis H ad H f feedback system S (Fig. 3.1) will have a prescribed set f ples i (E with specified MacMilla degrees amely the distict + pits f A with their give multiplicities. Observe that i the lumped case a similar ple specificati is de fr all f (E. The ituitive idea here is t place the "dmiat ples." Fially it shuld be stressed A, A. tis: we cat say which elemet f H ad H will btai a ple. e y R5.4 Cditi (iii) f the (STDP) will t ly guaratee that feedback system S is a servmechaism: it, i fact, guaratees that the system errr e ( ) s due t the referece ad disturbace sigals cvergest zer faster t tha e as t >«fr sme a < 0. R5.5 Cditi (iv) is a rbustess prperty guarateeig that as lg as the feedback System S remais ^-stable the referece sigals will be tracked ad disturbaces will be rejected asympttically: see als [32, Therem 3]. A. that we place here ples f H A, ad H csidered as matrix-valued fuc- -33-

36 I rder t slve the (STDP) we start by givig a prelimiary defii ti ad result. Fr q G]R csider the fucti space {00 0 e"at f(t) dt< c} (5.18) Lemma 5.1. Let q < 0. Let g G tf(a). Let u G L ad u G <2(q). The the l, q cvluti y = g * u satisfies y(t) = (e ) as t + «. (5.19) -qt Prf. Fr ay f G d(a), let f bee aeiiea defied by Dy f x (t) \zj := e f(t). Frm y = g * u ad y = g * u, we btai ya=ga*ua (5.20) ya = 8a * ua (5.21) J ly (t^ldt1 -* 0 q as t -*- c ad y_(t) -* cstat, say, b as t + ". Frm (5.20), y G L. q j.,q sice u G L-. Csequetly the cstat b = 0; equivaletly, y (t) -> 0 as t -)». Sice y(t) = e0ty (t), (5.19) fllws. «We are w ready fr the sluti f the (STDP). We shall dete by 2[f ;Q] the list f zers f the fucti f i the set Q, ad by [f] the list f zers f f. Algrithm 5.1. Data: We are give the descripti f a plat P, f referece- ad distur A bace-sigals (u («),w ( )), f the plymial <J> ad the lists 2I<}>] ad A: s p see (5.1)-(5.16). Step 1. Pick d ay mic plymial i ]R[s] such that 8d = 3d) = q ad d(s) ^ 0 fr all s G <E qq+ (5.23) -34-

37 Cmmet 1: Observe that ^ G ft (q ) CQ ( ) with real cefficiets, (5.24) zt ] =zw with xez[fi "f *G a. (5.25) Step 2. Pick a x <0 G u (q ), crrespdig t real valued distributis, - ft such that det < >G#(q ) ad Z[det*0;<E ] = A. - q + (5.26) Cmmet 2: The cditis fr JO ca be met by chsig D G R( ) crrespdig t real-valued distributis. x Step 3. Observe that F=P?6*«0>, A x with a -t.r.c^.iy:- C0pJl±,7?p,> (5.27) crrespdig t real-valued distributis, ad fid, usig the techique f Crllary 2.4l9 six matrices with elemets i d ( ) crrespdig t "N real valued distributis, amely %> ifa T> >T, %> vt (5.28) such that: i) i "V. -\ V. i r x.*. I i 0 "* 0 I _ (5.29) where if we call the matrices the left had side f (5.23) "^respectively (.+ )x(.+ ) rl i i %), the bviusly ij is a ivertible elemet f$ (q ) ad withut lss f geerality we ca scale it s that t1. det 2/= det[&j ] = 1 ; (5.29a) 35-

38 ii) Wr»«Q.) is a q -.r. f F. (5.30) Cmmets 3: I (5.28) elemets i CL ( ) crrespdig t real-valued "" distributis are btaied by grupig cmplex cjugate ples ad crres pdig residues. Step 4. Observe F i (5.27) satisfies F(s) -> 0 xj i as Is I > * i (E a+ (5.31) ad,usig (5.26)-(5.31), slve (COMP), defied by (4.2)-(4.3), as fllws:.x i) Pick 7?G CL ( ) i - the class crrespdig t real-valued distributis i A x {TiedjJ 1 : 0ft9P) is a.x q -r.r.} i ii) Set -X := ±71 -\*>i Ui=yjl+ 1^0. (5.32) (5.33) Cmmet 4: (i) the chice (5.32) is equivalet (by Crllary 2.2), t ix pickig 7? e#_(q ) that crrespdig t real valued distributis such. r- -* rak J0(s) = fr all s G A. (5.34) (ii) (X,^) as give by (5.33) is a.x a -r.r crrespdig t real valued distributis (5.35) (iii) Usig (5.29) e has als by (5.33): 7? =- 1/X+ 2^ ; <0 =7? #+ JD Tf.. (5.36) -36-

39 Step 5. Set 7iCI'-= as.-dcr'-h (5'37) C:=7?.O"1, (5.38) cr cr ad STOP. Cmmet 5: (i) A x CGB()1 with q - r.r. (7?.< > ) cr cr crrespdig t real valued distributis (5.39) (ii) C slves the (STDP). (5.40) Therem 5.1. Csider Algrithm 5.1. The C, as give by (5.38), belgs a 4X t ( ) with q - r.r. (7? 90 ) ad slves the (STDP). cr cr Prf: We shall shw that algrithm 5.1 wrks. Step 1 ad Step 2: These steps are self explaatry. Step 3. Because f Crllary 2.4 we ly eed t shw (5.27). Sice by * A XI^ A xl (5.24), 4 e &(a ) ad sice P G 8( ) it fllws that F = P^-G <B(q ) 1, (J) O O (J) Mrever (C Ajl J is a q -.r. f P^-. Ideed P^= C0 A)"1?? ad by v pad>/lpr <J) (j) pfcd p& (5.2), (5.8)-(5.15) det^^e^^a^ ad rak[eopa(s) ^- 7?p (s)] =Q fr fr all s G (E, i.e. by Crllary 2.2 (P ^97l ) is a - Jl.c.. qq+* ' ' p d p^ Step 4. Because f Therem 4.1 ad Crllary 2.2, we eed ly t shw (5.31). Nw bserve that this fllws frm (5.24), (5.27) ad (5.4). Step 5. a) (5.39) is true by the fact that (3C^4) is a ±x q -r.r.. Ideed bserve that the equati describig (COMP) is give by -37-

40 < =7^+4?=v*+i^ P* where we used (5.27) ad where by (5.25) ad (5.10)-(5.11) J3(s) = "ft 0(s) 3T(s) px, s G a(ag) Uq(Aw). Therefre by (5.7), (5.26), (5.16), (5.14), (5.15), rakbocs)] =, V s G c(a ) U a(a ). Hece this ad (5.35) imply that 0C(s) rak v>ss = V s G (D V Nw (5.35) ad (5.24) imply that det [ft -] G$~(a ), ad hece by Crllary 2.2 the pair (X,^J) A. is q - r.c b) We shw w that (5.40) is true. i) First fr the feedback system S (3.1)-(3.8) (Fig. 3.1), (where the plat trasfer fucti P is is give by (5.1)-(5.4) ad (5.14)-(5.15), ad where the ctrller trasfer fucti satisfies (5.37)-(5.39)), the A, A. /V trasfer fuctis H. ad H are well defied ad have elemets i C(a ) e y v (see Remark R5.2). ii) The equati (4.1) f (COMP) reads here: <D = 7? X+ >Ur= JO A, (5.41) I l\j p cr pit cr' x ' where we used (5.27) ad (5.37). Hece the characteristic fucti (3.15) f S satisfies here: X = det 3, (5.42) such that by usig (5.26) -38-

41 Hx;«J =?[detio;«j = A- (5.43) _+ Hece als by (5.16): q_+ X(s) * 0 V s G <E+. (5.44) It fllws that by (5.43) prperty (ii) f the (STDP) is verified, while prperty (i) fllws frm (5.44) ad Therem 3.1. iii) We shall w shw that the trackig prperty (iii) f the (STDP) hlds. If H resp. H dete trasfer fuctis f System S (Fig. 3.1) es,us " e^»w«defied by: H : u H" e with w = 0, (5.45) e,u s s p s' s r H : w "" e with u = 0, (5.46) e,w p s s s* p the usig (5.2), (5.39), (5.37), (5.41), K. "t1^"1 -^cr^par-vcrl^p* "Ij^pt (5'47) s s H =- [I+PC]'1^ =-8 L0 J) +7? 07) rhf 0=-4^""^ <5'48> e,w L J cr^pflrcr 'p& cr p d p s p Observe w that by (5.26) ad [7, Appedix D], det<# belgs t x /» CL() fr sme q G (q,0). Nw, sice 6 ( ) C^?(q), it fllws there a x fre, by (5.2) ad sice Ij Gd_(Q), that t a x x -F Veaw * (5-50) -39-

42 Furthermre by (5.5)-(5.13) ad (5.23): Vxs(0),^1[ cs(si-as)-1 xs(0)] GL^ ^ ( (5.51) ad its derivative belgs t C((), Vxw(0),^-1[^w(sI-V-1 ^(0)] Gl"^ 1 ad its derivative belgs t Ct(a). (5.52) Csider w the system errr e due t (u,w ) give by (5.5)-(5.7), the s s V xs(0)' V *W(0) A A, A, A, ^ e = H u + H w s e,u s e,w p s s s p ^ =^V [d Cs(sI"As)"1 x(0> 1+If\l t Cw[sl-Aw)-Ixw(0) ] whe we used (5.45)-(5.46), (5.47)-(5.48), (5.5)-(5.6). Therefre by (5.49)-(5.53) ad Lemma 5.1 (5.53) V xg(0), V xw(0), eg(t) =(e0t) ast+«. Q.E.D. iv) Prperty (iv) f the (STDP) is shw t be true as fllws. 3 Xi Let P G 8(a) be ay perturbed plat fr which the feedback system S (3.1)-(3.8) remais ^-stable. By Therem 2.1 P admits - A.r.CO ftl) ad the characteristic fucti x (3.15) becmes where X = det JO, - x -V>cr+Vke«-( ) Mrever (3.8) ad (3.20) read w respectively -40-

43 det [I +PC] is buded away frm zer at ifiity i <E, j^a. _ a+ X = det[i +PC]det jo det JO.. cr p r A It fllws that x is buded away frm zer at ifiity i <E ad by U-stability ad Therem 3.1 x(s) f 0 fr all s i d. I fact mre is A *^ true: sice x ca ly have a fiite umber f zers i the strip [,0), it fllws that ]a tq,0) such that x(s) i 0V sg <c_. Hece 7, i.e. det ^ is buded away frm zer i <E q<0: this implies, [7, Appedix D], q+ -_-1 A _ X A A that ft belgs t QX), (bserve als that CL (c ) c #(q) such that A _ x - #G< (q) ). Observe w that the trasfer fuctis i (5.47) ad (5.48) read \,us =W%. -* \, p= -i?r\t --- A_x A x where jtf"1^ <2() ad^xrty Gtf(G) i; ad qg[aq,0). The reasig f (ii) ca w be repeated t shw that V x (0), V x (0), eg(t) = (e ) as t -> «with ^ [a,0). Hece prperty (iii) f the (STDP) is maitaied. -41-

44 Appedix 1: R(a ) is a Euclidea Rig Recall that a pricipal ideal rig R, [12], is called a Euclidea rig. [8], if the fllwig prperties hld: 1. Assciated with every zer elemet f R is egative umber y(a) called the gauge f a; 2. Fr every pair a, b f R, b 0 there exist tw elemets r ad q f R such that a = bq + r ad either r = 0 r else y(r) < y(b). Recall that (ft(q ) is a pricipal ideal rig, [9], ad that the fllw ig fact hlds. Fact Al. Let a G <fl(q ), let tt(s) = s - a +1, the a = e,/try(a) a a*t* (Al.l) where e is a ivertible elemet f fl(q ), a ' i is a plymial which is zer at all zers f a ad where else, y(a) = umber f zers f a i (E ad at ifiity. (A1.2) v Cmmet: If a is ivertible i fl( ), the y(a) = 0. w Prf: a = /d where ad d are cprime plymials. Factrizig 1 CL ci cl cl = where (respectively ) takes it accut the zers f a a a+ a at a i <E (respectively (E ), ad bservig that d = d ad y(a) = 3(,) ", + 9(d ) - 3( ) «3(d ) - 3( ), we get a = e /iry^a; with a a 3. a~ a a*i e = TTY(a) /d. (A1.3) a a- a +We use Siglerfs term "gauge," [36], istead f MacDuffee's "stathm," [8]: "degree" culd als have bee used but wuld be misleadig sice we hadle plymials at the same time. -42-

45 Observe that e is ivertible i fl(q ). CL O We are w able t defie a Euclid Algrithm fr (L(q) with the gauge defied by (A1.2). Euclid Algrithm fr (R,(q): Give aad bi 6{(q),b^0, fid rg6{(q) ad q efi( ) such that a = bq + r where r = 0 r y(r) < y(b). Step 1. If y(b) _< y(a) g t step 2, else a = bo + a i.e. r = a,q = 0 ad y(r) = y(a) < y(b). Stp. Step 2. Apply Fact Al.l t a ad b, i.e. Step 3. Develp a+/iry(a) ad b+/iry(b) as plymials i w:= it = (s- +1)~, i.e., a+,,. _ v k Y(a)-k w* k (^at)(w)" a+w where, G C, a+ 3(,.) (Jfe<-)- E ^(b)'1 where ^ad bserve that the degree i w f these plymials is the gauge f a ad b resp.. + Step 4. Divide the plymial ( frr) «by the plymial ( fcj) (*) : T V<*/ TT "i- -43-

46 the there exist plymials x(w) ad y(w) such that (~^at)(w) =<-^y>^)x(w) +y(w) (A1.5) h+ with either y=0r 3(y(w)) <W-^yMw)) = Y(b). (A1.6) Step 5. Reitrduce the ivertible elemets e ad e, f (A1.4) t btai a = bq + r where q(s) = ^ ea(s) x(^-^j-) 1, (A1.7) r(s) =ea(s) y(7^i), (A1.8) ad bserve that either r = 0 r Y(r) < Y(b). (A1.9) Stp. a Justificati f the Euclid Algrithm We check the result f Step 5. Observe that q ad r as give by (A1.7)-(A1.8) are ft( ). Hece we must shw that if r ^ 0 the Y(r) < y(b). Observe w that by (A1.2) V a, b G (S((a ), a 4 0, b i 0, Y(ab) = Y(a) + y(b). Hece by (A1.8), with y(e ) = 0 we get Y(r) = a Y(y( +7"^* Hece, i view f (Al.6), if we ca shw that Y(y) = Y(y(s,Q +1)) 3(y(w)) the we are de. Nw y G ${( ), s we get by Fact Al.1-44-