J.M. Schumacher* ABSTRACT

Transcription

1 Jue, 1982 LIDS-P ALMOST STABILIZABILITY SUBSPACES AND HIGH GAIN FEEDBACK by J.M. Schumcher* ABSTRACT The clss of "lmost stbilizbility subspces" is itroduced s the stte-spce log of the clss of stble but ot ecessrily proper trsfer fuctios. Almost stbilizbility subspces c be cosidered s cdidte closed-loop eigespces ssocited with ifiitely fst d stble modes. We derive the bsic properties of these subspces, d show tht they c be pproximted by regulr stbilizbility subspces. The reltio with high gi feedbck is elborted upo i umber of pplictios. Lbortory for Iformtio d Decisio Systems, M.I.T., Cmbridge, MA 02139, U.S.A. This reserch ws supported by the Netherlds Orgiztio for the Advcemet of Pure Scietific Reserch (ZWO).

2 1. Itroductio The cocept of "lmost-ivrice" ws itroduced by J.C. Willems i series of recet ppers [1-3] s geometric mes of studyig high gi feedbck d more geerlly, symptotic pheome i lier systems. I sese, lmost ivrit subspces provide stte-spce prllel to the frequecy-domi use of oproper trsfer fuctios. Aother such prllel ws mde quite explicit by Hutus [4], who liked the clss of "stbilizbility subspces" (which hd lredy ppered, without beig med s such, i the work of Wohm d his collegues i the seveties [5]) to the set of stble proper trsfer fuctios, which plys promiet role i recet reserch like [6-8]. Of course, importt role is lso plyed by the clss of stble but ot ecessrily proper trsfer fuctios. I this pper, we shll idetify the correspodig stte-spce cocept, which we shll term "lmost stbilizbility subspce". These subspces c be thought of s cdidte closed-loop eigespces ssocited with ifiitely fst d stble modes. The forml defiitio will be give i Sectio 2, log with umber of bsic properties. I Sectio 3, the key result is prove tht every lmost stbilizbility subspce c be obtied s the limit of sequece of stbilizbility subspces. Applictios re give i Sectio 4. A few exmples will be discussed of kow results tht c be re-iterpreted i terms of lmost stbilizbility subspces. Most of the sectio, however, is devoted to ew results o the problem of stbiliztio by high gi feedbck. Throughout this pper, we shll work with fixed fiite-dimesiol time-ivrit lier system, give by

3 -2- x' (t) = Ax(t) + Bu(t) (x(t) e X, u(t) e U) (1.1) (ugmeted by observtio equtio i subsectio 4D). The stte spce X, the iput spce U, the system mppig A:X -+ X, d the iput mppig B:U + X re ll tke over the rel field 1R, but the obvious complexifictios will be used where eeded without chge of ottio. The complex umber field is deoted by. The ullspce d the rge of lier mppig M will be writte s ker M d im M, respectively.

4 -3-2. Defiitio d Bsic Properties Recll the followig defiitio from [4]: Defiitio 2.1: A subspce S of X is stbilizbility subspce if there exists F: X + U such tht (A+BF)S C S d the restrictio of A+BF to S is stble. We hve the followig chrcteriztios of this cocept. Propositio 2.2 [4]. S is stbilizbility subspce if d oly if for every x e S there exist stble strictly proper rtiol fuctios i(s) d (s) such tht i(s) e S (for ll s) d x = (si-a) (s) + B (s). (2.1) Propositio 2.3 [9]. S is stbilizbility subspce if d oly if (si-a)s + im B = S + im B (2.2) for ll s i the right hlf-ple. Almost (A,B)-ivrit subspces d lmost cotrollbility subspces were itroduced i [1] (see lso [10] for pure lgebric tretmet, bsed o discrete-time iterprettio). Recll the bsic result [1] tht subspce V is lmost (A,B)-ivrit if d oly if it c be writte s the sum of (A,B)-ivrit subspce [5] d lmost cotrollbility subspce. This is oe motivtio for the followig defiitio. Defiitio 2.4. A subspce S of X is lmost stbilizbility subspce if it c be writte i the form S = S + R, where S is stbilizbility subspce, d R is lmost cotrollbility subspce.

5 -4- Further motivtio is provided by the ext result. Propositio 2.5. The followig re equivlet: (i) S is lmost stbilizbility subspce. (ii) For every x e S there exist stble rtiol fuctios i(s) d (s) such tht i(s) e S (for ll s) d x = (si-a)t(s) + BW(s). (2.3) (iii) The iclusio S C (si-a)s + im B (2.4) holds t ll poits s i the right hlf-ple. The proof follows closely the lies of [10], d will therefore be omitted. Additiol support for Def. 2.4 comes from the followig propositio. Propositio 2.6. A subspce S is stbilizbility subspce if d oly if it is both (A,B)-ivrit subspce d lmost stbilizbility subspce. Proof. Combie Prop. 2.3 d Prop. 2.5 (iii) with the observtio tht subspce S is (A,B)-ivrit if d oly if (si-a)sc S + im B (2.5) for some s e CI (cf. [5], p. 88), which is esily see to be equivlet to the sttemet tht (2.5) holds for ll s e ce. The followig direct-sum decompositio is immedite cosequece of the geerl decompositio give i [2].

6 -5- Propositio 2.7. Every lmost stbilizbility subspce c be writte i the form S = R + R + 5, where R is cotrollbility subspce, R is slidig subspce (i.e., lmost cotrollbility subspce tht does ot coti y ozero cotrollbility subspce, see [2]), d S is costig subspce (i.e., (A,B)-ivrit subspce tht does ot coti y ozero cotrollbility subspce, see gi [2]) such tht the restrictio of A+BF to S is stble for y F such tht (A+BF)S C S. It is esily checked tht the sum of two lmost stbilizbility subspces is gi lmost stbilizbility subspce. Hece, there is uique lrgest lmost stbilizbility subspce i y give subspce K, d we shll deote it by S* (K). The followig chrcteriztio closely prllels the oe give by Hutus [4] for the lrgest stbilizbility subspce i K, which we shll deote by S*(K). The proof follows the lies of [4] d [10] d will be omitted. Propositio 2.8. S*(K) equls the set of ll x e K for which there exist stble rtiol fuctios i(s) d w(s) such tht i(s) e K d x = (si-a) (s) + BL(s) (2.6) Removig the restrictio "xek", we obti the followig subspce, which will tur out to be more useful. Defiitio 2.9. S*(K) equls the set of ll x e X for which there exist stble rtiol fuctios i(s) such tht i(s) e K d (2.6) holds. This subspce c be iterpreted s the set of ll vectors tht c serve s iitil vlues for stble d possibly impulsive trjectories tht sty i K for ll time ([2], [10]). Geometriclly, S*(K) c be

7 -6- chrcterized s follows. We write R* (K) for the lrgest lmost cotroll bility subspce i K, d defie R*(K) = AR*(K) + im B s i [2]. 'b Propositio S*(K) = S*(K) + R (K). The proof c be give without difficulty, usig the methods of [2] d/or those of [10]. Actully, [2] uses the bove formul s the defiitio of the subspce S*(K) (see Thm. 18; ote the pritig error). A coveiet wy of fidig out whether give subspce is lmost stbilizbility subspce is give by the followig rk test (cf. [10]). Propositio Let S be give subspce. Write dim S = k, dim (S + AS + im. B) = r. Choose bsis for X such tht the first k bsis vectors sp S d the first r bsis vectors sp S + AS + im B. Let the mtrices of A d B with respect to this bsis d give bsis i U be All A12 13 A A21 A22 23, B B2 (2.7 A32 A33 0 The S is lmost stbilizbility subspce if d oly if si-a 1 1 rk = r (2.8) -A21 B for ll s i the right hlf-ple. A ltertive route leds vi the defiitio d the lgorithms give i [2] d [5].

8 -7-3. Asymptotic Properties We shll use the followig (quite commo) otio of covergece for sequeces of subspces. Defiitio 3.1. A sequece {V} of subspces of fixed dimesio k is sid to coverge to k-dimesiol subspce V if the followig is true. Let {x 1,...I x k} be bsis for V. For every >0, oe c fid N such tht for ll > N there exists bsis {l,...,x k } of V which stisfies x-xill xi < for ll i = 1,...,k. The mi result of this sectio is: Theorem 3.2. For every lmost stbilizbility subspce S, there exists sequece of stbilizbility subspces {Sj which coverges to it. The proof will be give through series of lemms. The first oe of these c be proved by stdrd mes. Lemm 3.3. Suppose tht V = V V r, d suppose lso tht we hve sequeces {V } covergig to V.j, for ech j = l,..,r. The the subspces V, V re idepedet for ll sufficietly lrge. Moreover, if we ''''' r defie V = V V for these, the the sequece {V} coverges ~1 r to V. Lemm 3.4. Every lmost cotrollbility subspce R c be writte s direct sum R = R, where for ech j = l1,...,r there exist mppig F: X + U, vector b e im B, d iteger k > 0, such tht RJ = sp {b, (A+BF)b,..., (A+BF) kb}. (3.1) Proof. This is immedite from the geometric chrcteriztio of lmost cotrollbility subspces give i [1].

9 -8- Lemm 3.5. Every subspce R of the form (3.1) c be obtied s the limit of sequece defied by S = S + sp {b, (A+BF)b,..., (A+BF) k-} (3.2) where S is (oe-dimesiol) stbilizbility subspce. Proof. For ll sufficietly lrge, the mppig I + -(A+BF) will be ivertible, d so we c defie S by S = sp{(i+ (A+BF))-lb}. (3.3) With 1 give by (3.1) d S by (3.2) d (3.3), it is esily verified tht S = (I + -(A+BF)) R (3.4) which shows tht the sequece {S } coverges to R. It remis to show tht is stbilizbility subspce. To see this, ote tht (si-a) (I + -(A+BF))-lb = = (si-(a+bf))(i + -1 (A+BF)) -b + BF(I + 1 (A+BF))- b = (3.5) = (s+)(i + (A+BF))-lb + BF(I + 1(A+BF))-lb - b As cosequece, we hve (si-a)s + im B = S + im B (3.6) for ll s i the right hlf-ple (i fct, for ll s # -). By Prop. 2.3, this gives the desired result. Proof (of Thm. 3.1). It follows from Prop. 2.7 d Lemm 3.4 tht S c be writte s the direct sum of stbilizbility subspce d umber of subspces of the form (3.1). Usig Lemm 3.3, we c ow complete the proof

10 -9- by repeted pplictio of Lemm 3.5. If we thik of lmost cotrollbility subspces s ivrit subspces for the ifiite modes of closed-loop system (clled ito existece by ifiite-gi feedbck), the the theorem c be iterpreted s syig tht, i this cotext, "ifiity" c lwys be red s "mius ifiity". It is essetil for the proof tht the stble prt of the complex ple cotis poits of rbitrrily lrge modulus. I the discrete-time situtio, oe tkes the iterior of the uit circle s the stble prt of the complex ple, d it would be possible to modify the defiitios of "stbilizbility subspce" d "lmost stbilizbility subspce" ccordigly. But it is the o loger true, s c be see from simple exmples, tht every lmost stbilizbility subspce c be obtied s the limit of sequece of stbilizbility subspces.

11 Applictios A. "Almost" Disturbce Decouplig The followig set-up is cosidered i [2]. Let x' (t) = Ax(t) + Bu(t) + Gq(t) (z(t) = Hx(t) (4.1) costitute lier system, i which x(t) d u(t) hve the usul meigs, q(t) is cosidered s disturbce iput, d z(t) represets "cost" fuctio which we wt to keep close to zero. After stte feedbck u(t) = Fx(t), the system (4.1) becomes x' (t) = (A+BF)x(t) + Gq(t) [z(t) = Hx(t) (4.2) d the ifluece of q(t) o the cost fuctio z(t) is give by t (t) = He(A+BF) (t-s) Gq(s)ds (4.3) 0 As mesure of the degree to which the feedbck F succeeds i tteutig the disturbce, we my use the rtio of the L -orm of z(-) to the L - P P orm of q(-), where p stisfies 1 < p < m: p (F) = ZIII Lj L0,)(0(4.4) We c ow stte the followig result, which idetifies the situtios i which it is possible to elimite the ifluece of the disturbce 'lmost' completely, while the stbility of the closed-loop system is ssured.

12 -11- Theorem 4.1. For the system (4.1), the followig two sttemets re equivlet: (i) For every C > 0, there exists stte feedbck mppig F such tht A+BF is stble d (F c ) < E. (ii) Thepir (A,B) is stbilizble (i.e., there re o ustble ucotrollble eigevlues), d im G C S*(ker H). (4.5) The proof of this sttemet c be give log the lies of [2], d it will be omitted here. The correspodig result i [2] does ot require the stbility of the closed-loop system, so tht there is o stbilizbility requiremet o the pir (A,B) d (4.5) is replced by the coditio tht im G should be cotied i V (ker H). b B. Sigulr Optiml Cotrol Cosider the lier system (x'(t) = Ax(t) + Bu(t), x(0) = x 0 (4.6) z (t) = Hx(t) with ssocited cost fuctiol OO J(x 0 ) = mi J (I Iz(t) 2 + 2I ju(t) I 2 )dt.(4.7) u O It is ssumed tht the pir (A,B) is stbilizble, the pir (H,A) is detectble, B is full colum rk, d H is full row rk. Uder these coditios, the followig result hs bee give by Frcis [11].

13 -12- Theorem 4.2. J (x ) teds to 0 s c + 0 if d oly if xo e S*(ker H) (4.8) where the stbility set from which the subspce S*(ker H) is defied hs to be tke equl to the closed left hlf-ple. A similr result is give i [2]. Of course, Frcis did ot formulte his theorem i terms of lmost ivrit subspces, but it c be see from the lgorithms he uses tht the subspce costructed i [11] is exctly S*(ker H), i the bove iterprettio. The fct tht the closed left hlf-ple is importt here (rther th the ope LHP) c be mde plusible by cotiuity rgumet. C. Solvbility of Rtiol Mtrix Equtio Let Rl(s) d R 2 (s) be strictly proper rtiol trsfer mtrices, of sizes pxm d pxr, respectively. Uder vrious circumstces, it is importt to kow whether the equtio R l (s)x(s) = R2(s) (4.9) hs solutio i the set of stble rtiol trsfer mtrices. Suppose tht we hve reliztio for the trsfer mtrix [Rl(s) R 2 (s)]: -1 [R 1 (s) R 2 (s)] = H(sI-A) [B G] (4.10) The we should be ble to stte the solvbility coditios for (4.9) i terms of the mtrices H,A,B, d G. Ideed, the followig result ws essetilly proved by Beqtsso [12]: Theorem 4.3. Suppose tht the reliztio give by (4.10) is observble. The the equtio (4.9) hs stble rtiol solutio if d oly if

14 -13- im G C S*(ker H). (4.11) This coditio is quite closely relted to the coditio of Thm. 4.2, d, i fct, the proof of Thm. 4.2 i [11] is bsed o Begtsso's result. Obviously, there is lso close coectio betwee Thm. 4.3 d Thm Coectios of this sort re discussed i [3] d [4]. D. High Gi Feedbck We shll ow cocetrte o pplictio of differet type, ivolvig the otio of "lmost stbilizbility subspce" itself rther th S*-spce. Our cocer will be with dymic output feedbck rther th stte feedbck, so we cosider the cotrolled d observed lier system x' (t) = Ax(t) + Bu(t) (4.12) (y(t) = Cx(t) I dditio to our erlier ottiol covetios, we deote the output spce by Y, d we set dim Y = p. We shll eed the followig cocept. A subspce T of X will be clled miimum-phse iput subspce if there exists mppig T: Y+X such tht: T = im T (4.13) det CT # 0 (4.14) -1 det(si-a)det C(sI-A) T # 0 for s e +. (4.15) Of course, (4.15) sys tht the trsfer fuctio C(sI-A)- 1 T should hve o ustble zeros ([13], p.41), d this motivtes the termiology.

15 -14- Note, however, tht we lso require CT to be ivertible. A chrcteriztio i terms of the subspce T itself c be give s follows. Lemm 4.4. A subspce T is miimum-phse iput subspce if d oly if T D ker C = X d the restrictio of PA to ker C is stble, where P is the projectio oto ker C log T. Proof. Suppose tht T is miimum-phse iput subspce, d let T be mppig stisfyig ( ). It follows from (4.14) tht T e ker C = X. We see tht T(CT) lso stisfies ( ), so we my s well ssume tht CT = I. The the projectio P oto ker C log T is simply give by P = I - TC. (4.16) The subspce ker C is, of course, ivrit for PA, d the fctor mppig iduced by PA o the quotiet spce X/(ker C) is clerly 0. Therefore, we hve det(si-pa) = s P det(si-pa ker C(4.17) Now, cosider the followig mipultios, i which we use the determit equlity det(i+mn) = det(i+nm), the rule A (si-a) ssi - d the fct tht CT = I. det(si-pa) = det(si-(i-tc)a) = = det (si-a) det (I+TCA (si-a) ) = = det (si-a) det (I+CA (si-a) T) = (4.18) = det (si-a) det(i-ct + sc(si-a) T) = = s P det (si-a) det C(sI-A)-T.

16 Comprig this with (4.17), we see tht (4.15) implies tht PAker C is stble. For the coverse, suppose tht we hve subspce T tht stisfies the coditio of the lemm. The it is esily verified tht there exists (uique) mppig T: Y+-X such tht TC = I-P, d tht this mppig stisfies (4.13) d (4.14). (I fct, CT=I.) Moreover, it is immedite from (4.17) d (4.18) tht (4.15) holds. The followig result ws proved i [14] (Lemm 2.12), see lso [15] (Thmi. 4.4 d Lemm 5.1). Theorem 4.5. Suppose tht, for the system (4.12), we hve stbilizbility subspce V tht cotis miimum-phse iput subspce. Let the dimesio of V be k. The the system (4.12) c lwys be stbilized by dymic output feedbck of the form w'(t) = Acw(t) + Gcy(t), w(t) e W u(t) = F w(t) + Ky(t) (4.19) where the order of the feedbck dymics (i.e., dim W) is equl to k-p. I prticulr, if k-p, the the system (4.12) c be stbilized by direct output feedbck loe. We ow wt to prove the sme theorem, but with "stbilizbility subspce" replced by "lmost stbilizbility subspce". The ide is tht there re stbilizbility subspces rbitrrily close to give lmost stbilizbility subspce (Thm. 3.2) d, o the other hd, y subspce tht is close eough to miimum-phse iput subspce will itself be. miimum-phse iput subspce. Let us first formlly estblish the ltter fct.

17 -16- Lemm 4.6 Let T be miimum-phse iput subspce, d let {T } be sequece of subspces covergig to T. The T is miimum-phse iput subspce for ll sufficietly lrge. Proof. Let T be mppig stisfyig ( ); s i the proof of Lemm 4.4, we my ssume tht CT = I. By the defiitio of covergece, there exists bsis {xl,...,x } for T d correspodig bsis {x 1... for ech T, such tht {x } coverges to x. for ech i=l,...,p. Defie 1 1 yi = Cxi (i=l,...,p). The {Yl',..,Yp is bsis for Y, d x i = Ty i (i=l,...,p). Defie T :Y + X for ech by T i x = (i = 1,...,p). Ob 1 viously, we hve T + T. Hece, we lso hve CT + CT = I which implies tht CT will be ivertible for ll sufficietly lrge. I other words, T $ ker C = X for these vlues of. The projectio log T oto ker C is give by P = I - T ) C. (4.20) By the cotiuity of mtrix iversio d multiplictio, it follows tht P + I-T(CT) C = I-TC = P, the projectio oto ker C log T. This implies, i prticulr, tht the sequece of mppigs {P Aker C } coverges to PAIker C' By the cotiuity property of the eigevlues ([16], p. 191), it follows tht PAlker C is stble for ll sufficietly lrge. A ppel to Lemm 4.4 ow completes the proof. We re ow i positio to prove the mi result of this sectio. Theorem 4.7. Suppose tht, for the system (4.12), we hve lmost stbilizbility subspce V tht cotis miimum-phse iput subspce

18 -17- T. Let the dimesio of V be k. The the system (4.12) c lwys be stbilized by dymic output feedbck of the form (4.19), where the order of the feedbck dymics is equl to k-p. Proof. Let {V } be sequece of stbilizbility subspces covergig to V. The there exists sequece tt }, with T C V for ech, such tht {T } coverges to T. Accordig to Lemm 4.6, T will be miimum-phse iput subspce for ll sufficietly lrge. Tke such, d pply Thm. 4.5 to the correspodig V d T. We immeditely hve the followig corollry. Corollry 4.8. Suppose tht the system (4.12) is squre d miimum-phse, d lso suppose tht the mtrix CB is ivertible. The the system (4.12) c be stbilized by direct output feedbck loe, i.e., there exists K such tht A+BKC is stble. Proof. It suffices to ote thtim B is lmost cotrollbility subspce d tht the ssumptios of the corollry imply tht im B is lso miimum-phse iput subspce. The result the follows from pplictio of Thm I order to obti gi mtrix K s i the corollry, wht oe hs to do is to form sequece of stbilizbility subspces tht will pproximte the lmost stbilizbility subspce im B. I other termiology, this comes dow to selectig ptter log which p closed-loop poles will go off to ifiity. For the purposes of illustrtio, let us suppose tht we wt to crete closed-loop eigevlue with multiplicity -1 p t the poit -, for E>0. The correspodig stbilizbility subspce is

19 -18- V = (I+EA) (im B). (4.21) The mppig T6 which stisfies CT = I d im T = V is, of course, give by T s = (I+EA) B [C (I+SA) -B] d it is esy to verify tht F e F(V ) if d oly if F (I+SA) B = -s I. (4.23) As idicted i [15] (see the proof of Lemm 5.1), the correspodig gi mtrix K is the give by K = FT = [C(I+SA) B]. (4.24) From the theory developed bove, it follows tht the closed-loop mppig A+BK C will hve eigevlue of multiplicity p t the poit -s, while the other -p eigevlues will pproch the zeros of det(si-a)det C(sI-A) B s goes to zero. Of course, this c lso be verified by direct computtio. The hypotheses of Cor. 4.8 re well-kow to provide excellet circumstces for study of high-gi feedbck. I root-locus terms, they me tht there re oly first-order symptotic root loci, d tht the fiite termitio poits re ll stble. Recetly, these hypotheses hve tured up i study of robust cotroller desig vi LQG techiques [17] d ivestigtio ito cotroller desig for lrgely ukow systems [18]. For sigle-iput-sigle-output systems, it is quite esy to prove Cor. 4.8 by root-locus rgumet. Ideed, i this cse there is oly oe

20 -19- pole tht goes off to ifiity s the gi is icresed, d so stbility c be gurteed by selectig the right sig of the gi. The moder multivrible root-locus techiques (see, e.g., [19]) llow this rgumet to be exteded to the multi-iput-multi-output cse. However, it seems tht the followig re dvtges of the 'geometric' pproch tht hs bee developed here. First, we hve essetilly broght dow the mout of symptotic lysis ivolved i obtiig the result to mere pplictio of some stdrd cotiuity rgumets from mtrix theory. Secod, our result is geerl i the sese tht it is ot tied up with y specific symptotic pole ptter. Third, the result ppers here s corollry of more geerl theorem (Thm. 4.5). The lst poit is perhps the most importt: it suggests tht the methods preseted here might lso be helpful i studyig more difficult situtios (higher-order symptotics, o-miimum-phse systems). Cotiued reserch will hve to show if this is ideed true.

21 Coclusios The purpose of this pper hs bee to itroduce the cocept of "lmost stbilizbility subspces". We gve umber of equivlet chrcteriztios of this clss of subspces, d liked it to the clss of stble but ot ecessrily proper trsfer fuctios. We estblished the importt fct tht lmost stbilizbility subspces c be viewed s limits of regulr stbilizbility subspces. Severl pplictios were discussed, d specil emphsis hs bee plced o the role tht lmost stbilizbility subspces c ply i the study of high gi feedbck. The results tht we obti suggest tht we might hve wy here to develop geerl theory, which escpes the oe-prmeter frmework tht is so ofte chrcteristic both for root-locus d for LQG techiques. However, our results re oly prelimiry, d much work i this directio remis to be doe.

22 -21- REFERENCES [1] J.C. Willems, "Almost A (Mod B)-Ivrit Subspces", Asterisque, vol , pp , [2] J.C. Willems, "Almost Ivrit Subspces: A Approch to High Gi Feedbck Desig - Prt I: Almost Cotrolled Ivrit Subspces", IEEE Trs. Automt. Cotr., Vol. AC-26, pp , [3] J.C. Willems, "Almost Ivrit Subspces: A Approch to High Gi Feedbck Desig - Prt II: Almost Coditiolly Ivrit Subspces", IEEE Trs. Automt. Cotr., to pper (1982). [4] M.L.J. Hutus, "(A,B)-Ivrit d Stbilizbility Subspces, A Frequecy Domi Descriptio", Automtic, Vol. 16, pp , [5] W.M. Wohm, Lier Multivrible Cotrol: A Geometric Approch (2d ed.), Spriger-Verlg, New York, [6] L. Perebo, Algebric Cotrol Theory for Lier Multivrible Systems, Ph.D. Thesis, Lud Istitute of Techology, [7] C.A. Desoer, R.W. Liu, J. Murry, R. Seks, "Feedbck System Desig: The Frctiol Represettio Approch to Alysis d Sythesis", IEEE Trs. Automt. Cotr., Vol. AC-25, pp , [8] B.A. Frcis, M. Vidysgr, "Algebric d Topologicl Aspects of the Servo Problem for Lumped Lier Systems", S&IS Report No. 8003, Uiv. of Wterloo, [9] J.M. Schumcher, "Regultor Sythesis Usig (C,A,B)-Pirs", IEEE Trs. Automt. Cotr., to pper (1982). [10] J.M. Schumcher, "Algebric Chrcteriztios of Almost Ivrice", Report LIDS-P-1197, Lb. for Iformtio d Decisio Systems, MIT, April [11] B.A. Frcis, "The Optiml Lier-Qudrtic Time-Ivrit Regultor with Chep Cotrol", IEEE Trs. Automt. Cotrol, Vol. AC-24, pp , [12] G. Begtsso, "Feedforwrd Cotrol of Lier Multivrible Systems - The Nocusl Cse", Cotrol System Report 7506, Uiversity of Toroto, [13] H. Kwkerk, R. Siv, Lier Optiml Cotrol Systems, Wiley, New York, 1972.

23 -22- [14] J.M. Schumcher, Dymic Feedbck i Fiite- d Ifiite-Dimesiol Lier Systems, MC Trct 143, Mthemticl Cetre, Amsterdm, [15] J.M. Schumcher, "Compestor Sythesis Usig (C,A,B)-Pirs", IEEE Trs. Automt. Cotr. Vol. AC-25, pp , [16) J.N. Frkli, Mtrix Theory, Pretice Hll, Eglewood Cliffs, N.J., [17] J.C. Doyle, G. Stei, "Multivrible Feedbck Desig: Cocepts for Clssicl/Moder Sythesis", IEEE Trs. Automt. Cotr., Vol. AC-26, pp. 4-16, [18] D.H. Owes, A. Choti, "High Performce Cotrollers for Ukow Multivrible Systems", Automtic, to pper (1982). [19] D.H. Owes, Feedbck d Multivrible Systems, Peter Peregrius, Stevege, 1978.