Application of Generalized Polynomials to the Decoupling of Linear Multivariable Systems

Transcription

1 Applcaton of Generalzed Polynomals to the Decouplng of Lnear Multvarable Systems,,3 J. Ruz-León and D. Henron. CINVESTAV-IPN, Undad Guadalajara P.O. Box 3-438, Plaza la Luna, 4455 Guadalajara, Jalsco, Mexco. E-mal: LAAS-CNRS 7 Avenue du Colonel Roche 377 Toulouse, France E-mal: henron@laas.fr 3. Insttute of Informaton Theory and Automaton Academy of Scences of the Czech Republc Prague, Czech Republc. Abstract. In ths paper, propertes of generalzed polynomals and generalzed polynomal matrces are appled to the problem of decouplng wth -stablty of lnear square multvarable systems,.e. decouplng of lnear systems wth closedloop poles n a regon of the complex plane. We present frst some extensons of well-known results about generalzed polynomals, whch are bascally defned as ratonal functons wth poles n a symmetrc regon of the extended complex plane. Then, an applcaton of the concepts to the problem of decouplng wth -stablty of lnear square multvarable systems s presented. The condtons for ths problem to have a soluton are stated n terms of the row and global zero structureofthesystemoutoftheregon. Keywords : Generalzed Polynomals, Canoncal Forms, Lnear Systems, Decouplng, Stablty. Introducton Algebrac concepts are fundamental n practcally any branch of engneerng, and control theory s by no means the excepton. The analyss and desgn of dynamc control systems reles to a deep extent on the notons and propertes from algebrac objects. For nstance, polynomals, proper ratonal functons, proper and stable ratonal functons, matrces wth elements from these sets, and canoncal forms related to these matrces are a key tool n system descrpton, analyss and desgn. These ratonal functons can be regarded as partcular cases of what s known as generalzed polynomals. There exst many contrbutons n the lterature about generalzed polynomals and ts applcatons to control theory. The concept of ratonal functons wth poles n a prescrbed regon of the complex plane was orgnally studed n [], where t was shown that ths set s a prncpal deal doman. In [] t s shown that the set IR S of all transfer functons wth denomnators n a prescrbed set S s a prncpal deal doman, and system nvarants under feedback and cascade control are characterzed by the free IR S-module generated by the columns of the system transfer functon. Generalzed polynomals are ntroduced n [3] as ratonal functons wthout poles n a certan regon of the fnte complex plane, and they are used todescrbethesetofalltransfer functons for partcular sgnals usng a set of admssble controllers. In [4] t s shown that the rng of proper and stable ratonal functons s a Eucldean doman, and the Eucldean dvson Correspondng author.

2 process n ths rng s characterzed. The algebra of proper and Ω-stable ratonal functons and matrces s examned n [5], where Ω s a symmetrc regon of the fnte complex plane C, contanng + C := {s C, Re(s) }, and excludng at least one negatve real number. Generalzed polynomals are usually lmted n the prevous references to proper ratonal functons wth poles (or wth no poles) n a regon of the fnte complex plane. We present n ths paper an extenson of ths concept, by ncludng the pont at nfnty, and consderng a regon of the extended complex plane. Algebrac propertes of generalzed polynomals are studed, and extensons of well-known results are presented. Wth a sutable defnton of a degree functon, t s shown that the set of -generalzed polynomals s a Eucldean rng, and canoncal forms for -generalzed polynomal matrces are derved. We consder here only the propertes of generalzed polynomals that are needed to study the problem of decouplng wth -stablty of lnear square multvarable systems. Even though most of the propertes and results presented here about generalzed polynomals and generalzed polynomal matrces can be consdered as extensons of standard results, we beleve our approach s more comprehensve and provdes a nce nterpretaton of concepts and results n terms of poles and zeros of the functons belongng or not to the regon. For nstance, concepts lke the degree of a generalzed polynomal, or the characterzaton of a unt n the rng of generalzed polynomal, have a nce nterpretaton n terms of the zeros of the functon out of the regon. Also, a wder set of ratonal functons lke polynomals and nonproper ratonal functons can be consdered as partcular cases of -generalzed polynomals. Afterwards, we present an applcaton of the notons and propertes of -generalzed polynomals to the problem of decouplng wth -stablty of lnear square multvarable systems,.e. decouplng of lnear systems wth closed-loop poles n a regon of the complex plane. Roughly speakng, decouplng of dynamc systems mples ndependence between the nputs and the outputs of the system, and from the practcal pont of vew t s of nterest to acheve decouplng because t s often desrable to control the outputs of the system ndependently. We are nterested n the rowby-row decouplng of lnear multvarable systems by statc state feedback. Ths problem has been extensvely studed snce the 96 s, and t has been solved for the case of regular state feedback (see for nstance [6,7]), whch s the knd of state feedback that has to be appled to decouple systems wth the same number of nputs and outputs (square systems). The regular decouplng problem wth stablty by statc state feedback has been solved n [8,9]. In ths paper, t s proved that a lnear system s decouplable wth -stablty f and only f the so-defned -stable nteractor of the system s a dagonal matrx, or equvalently, f and only f the row and global zero structure of the system out of the regon concde. The -stable nteractor and the row and global zero structure of the system out of are defned va the column HermteformandtheSmthformofthesystemtransferfunctonovertherngof-generalzed polynomals. If the system s decouplable wth -stablty, a procedure to compute a decouplng state feedback s presented. It s also shown that the condtons for the row-by-row decouplng problem and decouplng wth stablty can be easly obtaned as partcular cases of our soluton. Ths work s organzed as follows. In secton, -generalzed polynomals are ntroduced and some of ther algebrac propertes are consdered. Secton 3 deals wth -generalzed polynomal matrces, and the generalzed Hermte form and the Smth form of a matrx are presented. These canoncal forms can be consdered as the natural extensons of smlar concepts for polynomal matrces. No proofs of the results presented n secton and secton 3 are provded, snce they can be consdered as extensons of well-known results. An applcaton of the propertes of -generalzed polynomals and matrces s presented n secton 4, where we study the problem of decouplng wth -stablty of lnear multvarable systems. Fnally, we end wth some conclusons. -Generalzed polynomals We wll frst ntroduce some notaton. Through ths work, IR wll denote the feld of real numbers, C wll stand for the complex plane and C for the extended complex plane,.e. C = C { }. Also, C wll denote the open left half complex plane, and C ts extended verson,.e. C = C { }. The set of polynomals n the varable s wth coeffcents n IR wll be denoted by IR[s], and the set of ratonal functons over IR wll be denoted by IR(s). The pont s = w C s sad to be a

3 s w s w zero of f(s) f lm f(s) =, and s = w C s sad to be a pole of f(s) f lm f(s) =. Accordngtothsdefnton, f zeros and poles are counted wth ther own multplctes, then any ratonal functon has the same number of poles and zeros snce the pont at nfnty s taken nto account. The ratonal functon f(s) = a(s)/b(s), where a(s) andb(s) are coprme polynomals, s sad to be proper f deg b(s) deg a(s), strctly proper f deg b(s) > deg a(s), and proper and stable f t s proper, and ts poles le n the open left half complex plane C. A symmetrc subset of C s a set C, contanng at least one pont of the real axs, and wth the property that f s = α + jω, thens = α jω. These restrctons on are bascally due to the fact that we are dealng wth ratonal functons wth real coeffcents. As a specal case, we can also defne = { }. The man concept of ths secton s the followng: Gven a symmetrc set C,thesetof -generalzed polynomals, denoted IR (s), s defned as the set of ratonal functons whose poles are n. Example. In ths example we specfy some regons for and fnd ther correspondng set IR (s): If s taken as the extended complex plane C, then IR (s) s the set of ratonal functons IR(s). If s defned as the complex plane C, then IR (s) s the set of proper ratonal functons. In ths case f(s) IR (s) has to have all ts poles n fnte postons wthn the complex plane as the proper ratonal functons do. The set of proper ratonal functons wll be denoted by IR p(s). If = { } then IR (s) s the set of polynomals IR[s]. Ths can also be seen from the fact that any polynomal can be consdered as a ratonal functon f(s) = a(s)/b(s), where b(s) =,.e. a ratonal functon wth no fnte poles. If = C, then IR (s) s the set of stable ratonal functons. If = C, then IR (s) s the set of proper and stable ratonal functons, hereafter denoted IR (s). ps Wth the usual defntons of addton and multplcaton of ratonal functons, t can be seen that IR (s) s a commutatve rng (wth multplcatve unt and addtve unt ), and t s also an ntegral doman. The next step s to show that IR (s) s a Eucldean rng. To ths end, for any nonzero element f(s) IR (s), we defne the functon γ(f), called the degree of f(s) and denoted by deg f(s), as the number of zeros of f(s) lyngoutsdeof. Remark. Observe that the prevous defnton agrees completely wth the defnton of the degree of an element n the rngs IR[s], IR p(s) and IR ps(s), where: If a(s) IR[s] s a polynomal ( = ), then γ(a) =dega(s), whch corresponds to the number of fnte zeros of a(s). If f(s) =a(s)/b(s) IR p(s) s a proper ratonal functon ( = C), then γ(f) =degb(s) deg a(s), whch corresponds to the number of nfnte zeros of f(s). If g(s) IR ps(s) s a proper and stable ratonal functon ( = C ), then γ(g) =numberof unstable + nfnte zeros of g(s),.e. number of zeros of g(s) outsdeofc. Through ths work, the degree of the proper ratonal functon f(s) IR p(s) wll be denoted by degp f(s), and the degree of the proper and stable ratonal functon g(s) IR ps(s) wll be denoted by deg g(s). ps Wth ths defnton of deg f(s), t s evdent that the unts of the rng IR (s) are the nonzero elements f(s) IR (s)suchthatdegf(s) =. From the algebrac pont of vew, the followng s a fundamental property of IR (s). Theorem. Gven a symmetrc set, then IR (s) s a Eucldean rng, wth the degree functon as defned prevously,.e. as the number of zeros of f(s) lyngoutsdeof. 3

4 Some more concepts are further ntroduced next. Lemma. Let h(s), f(s) IR (s). Then h(s) dvdes f(s) (.e. there exsts x(s) IR (s) such that f(s) =h(s)x(s)), denoted h(s) f(s), f and only f each zero of h(s) outsdeof s also a zero of f(s), at least wth the same multplcty. The functon h(s) IR (s) s sad to be a greatest common dvsor (gcd) of f(s) andg(s) IR (s) f h(s) sacommondvsoroff(s) andg(s),.e., h(s) f(s) andh(s) g(s), and any other common dvsor of f(s) andg(s) sadvsorofh(s). A gcd of a set of more than functons can be defned n an analogous manner. Let h(s) IR (s) be a greatest common dvsor of f(s) andg(s) IR (s). Then h (s) :=h(s)u(s) s also a gcd of f(s) andg(s), where u(s) s a unt of IR (s). Thus, a greatest common dvsor of two generalzed polynomals s unque up to multplcaton by unts of the rng IR (s). Remark. The relevant nformaton n determnng the gcd of some functons f (s) sthe set of zeros outsde of whch are common to all these functons. Indeed, let h(s) beagcdofa fnte set of -generalzed polynomals f (s), f (s),..., f k(s). From Lemma. t can be seen that deg h(s) = n, wheren sthenumberofzerosoutsdeof whch are common to these functons f (s), f (s),..., f k(s). Accordng to the prevous remark, a gcd of a set of functons can be obtaned analytcally by nspecton of the zeros of these functons. A gcd can also be obtaned n an algorthmc way. The algorthm to obtan a gcd for polynomals s well known n the lterature (see, for nstance []) and t s based on the prncple of polynomal dvson. The extenson of ths procedure to IR (s), whch s not consdered n ths paper, can be readly obtaned snce Eucldean dvson also apples to -generalzed polynomals. Let h(s) beagcdoff(s) andg(s) IR (s). Then f(s) andg(s) are called coprme f h(s) s a unt of IR (s). The followng result s mmedate from Remark.. Lemma. The -generalzed polynomals f(s) andg(s) IR (s) are coprme f they do not have any common zeros outsde of. 3 -Generalzed polynomal matrces Matrces whose entres are -generalzed polynomals wll be consdered n ths secton. The set of m n matrces of dmensons m n whose entres belong to IR (s) wll be denoted by IR (s). Wth the usual defntons of addton and multplcaton of matrces, the set of square matrces whch m m belong to IR (s) s readly seen to be a non commutatve rng wth respect to these operatons. m m m m A square nonsngular matrx U(s) IR (s) s sad to be a unt of the rng IR (s) f there m m m m exsts V (s) IR (s) such that U(s)V (s) =V (s)u(s) =I m. Unts of IR (s) wll be called m m IR (s)-unmodular matrces. Then, the unts of IR (s) for polynomal, proper, and proper and stable ratonal matrces wll be called, respectvely, IR[s]-unmodular, IR p(s)-unmodular and IR (s)-unmodular matrces. ps m m Lemma 3. A square nonsngular matrx U(s) IR (s) s IR (s)-unmodular f and only f det U(s) s a unt of IR (s),.e. a nonzero ratonal functon whose degree s zero. Next, we wll ntroduce the generalzed Hermte form and the Smth form of a matrx, whch are canoncal forms defned va factorzatons usng IR (s)-unmodular matrces. m n Theorem 3. Let A(s) IR (s), and suppose for smplcty that rank A(s) = m. Then n n m n there exsts a IR (s)-unmodular matrx U(s) IR (s) and a matrx H(s) IR (s), called the 4

5 generalzed (column) Hermte form of A(s) and unque up to unts of IR (s), such that h (s) ()... A(s)U(s) =H(s) =... () (3.) h (s)... h (s). where, for >j, m mm h (s) =, or deg h (s) < deg h (s). (3.) j j Proof. The proof s constructve. The matrx H(s) s obtaned by elementary column operatons on A(s) over IR (s) and usng the dvson algorthm for -generalzed polynomals. The procedure s smlar to the case of the Hermte form of polynomal matrces (see []). It seems msleadng to call H(s) a canoncal form for A(s), when t happens that t s not completely unque but that t s unque up to unts of IR (s). We can overcome ths dffcultybyntroducng theconceptofamonc-generalzed polynomal. Ths noton, however, has to be establshed for every partcular case of -generalzed polynomals. We wll defne t for the rngs we are manly nterested n ths work. Consder the rngs IR[s], IR p(s), and IR ps(s), then: A polynomal wll be taken to be monc as usually understood,.e. a(s) IR[s] s monc f the coeffcent correspondng to ts hghest power s equal to. The proper ratonal functon f(s) IR p(s) wll be called monc f t s of the form f(s) = s n, where n =degp f(s). The proper and stable ratonal functon g(s) IR ps(s)wllbecalledmoncftsofthe ²(s) form g(s) = π where²(s) IR[s] s a monc polynomal wth roots out of the open left half n complex plane C (antstable polynomal), π = s + α wth α C,andn =degps g(s). The basc dea behnd the prevous defnton s that from any -generalzed polynomal, we can obtan the smplest element (a monc one) by extractng unts of IR (s) from t. Regardng Theorem3., fwefurtherspecfythath (s) n(3.) are monc -generalzed polynomals, then H(s) s a unque matrx. We wll consder that ths s the case when speakng about the Hermte form of some matrx. m n m m Theorem 3. Let A(s) IR (s) wth rank A(s) = r mn(m, n). Then, there exst IR (s)- unmodular matrces V (s) IR (s) and V (s) IR n n (s) such that " # r dag{ψ (s)} = V (s)a(s)v (s) =Ψ(s) = (3.3) where ψ (s) dvdes ψ + (s),.e. ψ (s) ψ (s), =,...,r. (3.4) + The -generalzed polynomals ψ (s) are unque up to multplcaton by unts of IR (s). We wll consder them to be monc, and they wll be called the nvarant factors of A(s), where as we stated prevously, the defnton of a monc -generalzed polynomal depends on the rng IR (s). Under ths assumpton the matrx Ψ(s), hereafter called the -generalzed Smth form, or smply the Smth form of A(s), s a canoncal form for A(s). If we defne the determnantal dvsors of A(s) IR m n (s) as (s) := monc gcd of all mnors of A(s), =,...,r, r =ranka(s), (3.5) then t can be seen that where (s) :=. (s) ψ (s) =, =,...,r, (3.6) (s) 5

6 4 Applcaton to decouplng wth -stablty We tackle n ths part of the paper the problem of decouplng wth -stablty of lnear multvarable square systems, whch wll be defned as the problem of decouplng plus the modes of the closedloop system beng located at a specfc symmetrc regon of the complex plane. The problem statement s presented n secton 4.. In secton 4. we defne the -stable nteractor va the column Hermte form of the system transfer functon over the rng IR (s), and show that ths matrx s the part of the system that s nvarant under the acton of a state feedback whch preserves -stablty. The man results and some llustratve examples are presented n secton Problem statement We consder n ths work square lnear multvarable and controllable systems descrbed by ½ ẋ(t) =Ax(t)+Bu(t) (A, B, C) y(t) =Cx(t) n p p where x IR, u IR and y IR are, respectvely, the state, nput and output vectors of the system. The system (A, B, C) s sad to be row-by-row decouplable by statc state feedback f there exsts astatefeedback (F, G) : u(t) =Fx(t)+Gv(t) p n p p where F IR and G IR are constant matrces, wth G nonsngular, and v(t) sanew nput vector, such that the nput v (t) controls the output y (t), =,...,p,wthoutaffectng the other outputs. Let be a symmetrc subset of the complex plane C. The system (A, B, C) ssadtoberow-by-row decouplable wth -stablty by statc state feedback, f t s decouplable and the egenvalues of the matrx (A + BF) arelocatedn. Observethat s taken here as a subset of the complex plane C, and not as a subset of the extended complex plane C as n secton, snce the egenvalues of the matrx (A + BF) areallfnte. From the nput-output pont of vew, the prevous formulaton s equvalent to the exstence of a state feedback (F, G) such that the transfer functon T F,G(s) of the closed loop system (A + BF,BG,C) softheform T (s) =C(sI A BF) BG = dag {w (s),...,w (s)} F,G p and the egenvalues of the matrx (A+BF) arelocatedn, whch mples also that w (s) IR (s), =,...,p. The egenvalues of the matrx (A + BF) are the modes of the closed-loop system (A + BF,BG,C). Thepolesofthesystemareasubsetofthesystemmodes,andbothsetsconcdefthesystem s controllable and observable. The dynamcs of the closed loop system (A + BF,BG,C) depends on the poston of the egenvalues of the matrx (A + BF), thus the mportance of assgnng these egenvalues. Decouplng assures that every nput s controllng every output ndependently; the modes locaton assures requred system dynamcs, for nstance stablty or faster response. We can suppose wthout loss of generalty that the modes of the system (the egenvalues of matrx A), are n,.e. that (A, B, C) s-stable; f not, there always exsts a prelmnary state feedback whch places the modes n, snce we are consderng the system (A, B, C) to be controllable. If the modes of the system are n, then the entres of the system transfer functon T (s) =C(sI A) B are -generalzed polynomals, because they are ratonal functons whose poles are n. Then, T (s) can be consdered as a -generalzed polynomal matrx. Ths consderaton wll be helpful when obtanng the column Hermte form and the Smth form of T (s) over the rng IR (s), and so that the condtons for decouplng wth -stablty can be drectly tested from the transfer functon of the system. 6

7 4. State feedback wth -stablty Let (F, G) be a statc state feedback appled on the -stable system (A, B, C), and such that the closed-loop system (A + BF,BG,C) s also -stable. The closed-loop transfer functon s gven by After some manpulatons we obtan T (s) =C(sI A BF) BG. F,G T F,G(s) =C(sI A) {z B [I F (si A) } {z B] G } T (s) R(s) where T (s) s the transfer functon of the system (A, B, C), and R(s) :=[I F (si A) B] G. Snce the closed-loop system s supposed to be -stable, then R(s) must be clearly a -generalzed polynomal matrx. Further, from R (s) =G [I F (si A) B] = G [det(si A)I F Adj(sI A)B] det(si A) t can be seen that R (s) s also a -generalzed polynomal matrx, snce the roots of det(si A) are n. Then, we have the followng result. Lemma 4. Let T (s) be the transfer functon of a -stablesystem(a, B, C). Then, the effect of a statc state feedback (F, G), G nonsngular, whch preserves -stablty can be represented n transfer functon terms as a IR (s)-unmodular matrx postmultplyng T (s). Ths fact establshes a natural restrcton on the type of feedback we can use whle tryng to acheve decouplng wth -stablty: For our purposes, the state feedback (F, G) wll be sad to be an admssble state feedback f ts effect on the system (A, B, C) can be represented as a IR (s)- unmodular precompensator actng on the system transfer functon T (s). Ths can be consdered as the matrx nterpretaton of the fact that we are nether allowed to ntroduce -unstable poles (poles out of ) nor to cancel out -unstable zeros n order to keep the nternal -stablty of the closed-loop system. At ths stage, t s mportant to consder the structural nformaton of the system (A, B, C) whch remans nvarant under the acton of IR (s)-unmodular compensaton, and consequently, nvarant under the acton of an admssble state feedback. 4.. The -stable nteractor Snce the acton of an admssble state feedback on (A, B, C) can be represented as multplcaton of T (s) on the rght by a IR (s)-unmodular matrx, the nformaton of the system that s nvarant under such a feedback s contaned n the column Hermte form of T (s) overtherngir (s)(see Theorem 3.), presented next. Let T (s) be the transfer functon of (A, B, C). Then, there exsts a IR (s)-unmodular matrx p p p p U(s) IR (s) and a nonsngular lower trangular matrx Φ (s) IR (s), unque up to unts of the rng IR (s), such that ϕ (s) ().. T (s)u(s) =Φ (s) =.., (4.) ϕ p(s)... ϕ pp(s) where the ratonal functons ϕ j(s) IR (s) satsfy, for >j, ϕ (s) =, or deg ϕ (s) < deg ϕ (s), (4.) j j and they are of the form ² (s) ϕ (s) = (4.3) πk 7

8 and for >j ² j(s) ϕ j(s) =, (4.4) πk j where ² (s) s an antstable polynomal, or polynomal wth only -unstable roots (roots out of ), π = s + α s a stable term ( α ), ² j(s) IR[s] s a polynomal, and k, kj are postve ntegers. The matrx U(s) represents elementary column operatons on T (s) overir (s). The ratonal matrx Φ (s), whch s the nverse of Φ (s), wll be called the -stable nteractor of the system, snce ths matrx can be consdered as the extenson of the classc nteractor [] for the set. NotcethatΦ (s) s not unque but t s unque up to unts of the rng IR (s). For a fxed π = s + α, thematrxφ (s) sunque,andhavngthsnmndswhywecalledt the stable nteractor of the system. Actually, the algebrac propertes of Φ (s) do not depend on the choce of π, nor the results stated n the next sectons based on these propertes. The matrx Φ (s) s n general a ratonal matrx havng only -unstable poles. Ths can be seen from the fact that the numerator of the determnant of Φ (s) s the product of the antstable polynomals ² (s), =,...,p.observethatf(a, B, C) hasnofnte zeros out of, thenφ (s) s a polynomal matrx. The -stable nteractor s a state feedback nvarant whch contans the -unstable structural nformaton of the system,.e. zeros out of of the system, ncludng nfnte zeros. 4.. Feedback realzaton of precompensators It was shown before that the effect of an admssble state feedback s equvalent n transfer functon termsasactngair (s)-unmodularprecompensator on the system transfer functon T (s). The converse problem for a proper compensator s the followng: A gven proper compensator P (s) s sad to be feedback realzable on the system (A, B, C) f there exsts a state feedback (F, G) such that P (s) =[I F (si A) B] G. The followng result, used n the proof of Theorem 4., states the condtons for a proper compensator to be realzable. Lemma 4. [3] Let the matrces N (s) andd(s) be a rght coprme matrx fracton descrpton (MFD) of the system (A, B, I n)wthd(s) column reduced, and let P (s) beanonsngular proper compensator. Then P (s) s state feedback realzable on (A, B, I n) f and only f P (s) s bproper, and P (s)d(s) s a polynomal matrx. 4.3 Man results The followng result presents the condtons for decouplng wth -stablty. Theorem 4. The controllable and -stable square system (A, B, C) s decouplable wth - stablty f and only f ts assocated -stable nteractor Φ (s) s a dagonal matrx. Proof. Necessty. Suppose that (A, B, C) s decouplable wth -stablty. Then there exsts a state feedback (F, G) such that T F,G(s) =C(sI A BF) BG = T (s)[i F (si A) B] G = W (s). Snce [I F (si A) B] G s a IR (s)-unmodular matrx, and W (s) =dag{w (s),...,w p(s)}, then t follows that Φ (s) s dagonal. Suffcency. Supposng that Φ (s) s dagonal, we wll prove ths part by showng that the IR (s)- unmodular matrx U(s) appearng n (4.) s feedback realzable. Let N (s) and D(s) be a rght coprmemfdofthesystem(a, B, I n)wthd(s) column reduced. Accordng to Lemma 4., U(s) wll be proved to be feedback realzable f U (s)d(s) s polynomal. From (4.) we have T (s)u(s) =CN (s)d (s)u(s) =Φ (s), 8

9 and from ths, we get Φ (s)cn (s) =U (s)d(s). (4.5) Snce the left hand sde of the last equaton has only poles out of, and the rght hand sde has only poles n, then clearly U (s)d(s) s a polynomal matrx, whch proves the result. Gven that U (s)d(s) s polynomal, then there exsts a constant soluton X, Y wth X no sngular to the polynomal matrx equaton [4] XD(s)+YN (s) =U (s)d(s). (4.6) Then, a state feedback (F, G) suchthatu(s) =[I F (si A) B] G s gven by F = X Y, G = X. (4.7) Ths state feedback decouples the system (A, B, C), preserves nternal -stablty and produces Φ (s) as the transfer functon of the closed-loop system. Remark 4. From the fact that the IR (s)-unmodular matrx U(s) n(4.) s feedback real- zable, then Φ (s) (dagonal or not) can be consdered as the transfer functon of the closed-loop system (A + BF,BG,C), whose -stable nteractor s Φ (s). Thus, the matrx U(s) canbere- garded as the nput-output result of a statc state feedback whch assgns poles of the system at the postons of -stable zeros producng cancellaton, and the remanng poles beng located at s = α. The necessary and suffcent condtons for decouplng wth -stablty, stated n Theorem 4. n terms of the -stable nteractor of the system, can also be expressed n terms of the row and global zero structure of the system out of, whch wll be defned as follows. Let n be the number of zeros out of (-unstable zeros), fnte or nfnte, multplctes n- cluded, of the th row of T (s). The ntegers {n, n,..., n p} wll be called the row zero structure of the system out of. Remark 4. Let Φ (s) be the column Hermte form of T (s) asgvenby(4.). Because of property (4.), and snce U(s) n(4.) s IR (s)-unmodular, t can be seen that f ϕ (s) sthe only entry dfferent from zero of the th row of Φ (s), then n =deg ϕ (s) =k, otherwse n < deg ϕ (s) =k. The global zero structure of the system out of wll be defned va the Smth form of the system transfer functon (see Theorem 3.). Let Ψ(s) be the Smth form of the system transfer functon T (s) overtherngir (s)foragven,.e. there exst IR (s)-unmodular matrces V (s) and V (s) such that p V (s)t (s)v (s) =Ψ(s) =dag{ψ (s)} (4.8) = where the ratonal functons ψ (s) IR (s) satsfy the dvsblty condtons over IR (s), and they are of the form ψ (s) ψ (s), =,...,p, + δ (s) ψ (s) =, =,...,p, π n where δ (s) s a polynomal wth only -unstable roots, π = s + α s a stable term, and n s a postve nteger. The global zero structure of the system out of wll be defned as the set of ntegers {n,n,...,n }, p 9

10 whch correspond to the degree of the ratonal functons ψ (s) n the Smth form of the system transfer functon T (s),.e. n =deg ψ (s), =,...,p. Lemma 4.3 Let Φ of T (s) over IR (s) as gven by (4.) and (4.8). Then (s) and Ψ(s) be, respectvely, the column Hermte form and the Smth form k = n (4.9) = = Proof. From (4.) and (4.8), and due to the fact that the degree of the determnant of a IR (s)- unmodular matrx s zero (see Lemma 3.), t can be seen that Then (4.9) results from the prevous equaton. deg [det Φ (s)] = deg [det Ψ(s)]. Theorem 4. The -stable nteractor of the system (A, B, C) s dagonal (equvalently: the system s decouplable wth -stablty) f and only f the row and global zero structure of the system out of s the same,.e. f and only f n = n. (4.) = = Proof. Necessty. Suppose that Φ (s) s dagonal. Then, from Remark 4. we have that Thus, from Lemma 4.3 we have (4.). n =deg ϕ (s) =k, =,...,p. Suffcency. Suppose that (4.) holds, but that Φ (s) s not dagonal. From Remark 4. we have that n < k for some =,...,p, mplyng also that Followng Lemma 4.3, ths means that n < k. = = contradctng our assumpton. n < n = = The next two examples llustrate the man results. Example 4. Let the system (A, B, C) begvenby A =, B =, 4 3 C =.5.5 whose transfer functon s (s+) T (s) =C(sI A) B =. s (s+) (s+) 4 3

11 Let us suppose that we want to decouple ths system by statc state feedback, such that the closedloop system s nternally stable,.e. such that the egenvalues of the matrx (A + BF) arenthe open left half complex plane C. It can be seen that the -stable nteractor of ths system for = C (IR (s)=ir ps(s)) s not a dagonal matrx, because the entry (, ) of T (s) cannotbe cancelled out by an elementary column operaton over IR (s), thus the system s not decouplable wth stablty. In terms of the row and global zero structure of the system out of = C Theorem 4.), we have that n =, n =, n =, n =3, and (4.) does not hold. The reason for ths s that the row nfnte zeros and the global nfnte zeros concde, but not the row and global unstable zeros, snce s = s an unstable global zero but t s not a row zero of T (s). Snce the row nfnte zeros {, } and the global nfnte zeros {, } concde, ths system s decouplable, but not decouplable wth stablty. Indeed, the decouplng compensator " # P (s) = s+ (s+)(s ) s s feedback realzable, and the state feedback realzng P (s) can be obtaned from a constant solu- ton X, Y wth X nonsngular to the polynomal matrx equaton XD(s)+YN (s) = P (s)d(s), where N (s), D(s) s a rght coprme MFD of (A, B, I n)wthd(s) column reduced. Then the state feedback F =, G = produces the decoupled system (A + BF,BG,C) whose transfer functon s ps (s+) T (s) =C(sI A BF) BG =, F,G (s+) but the closed-loop system s not nternally stable, snce one of the egenvalues of the matrx (A + BF) s equal to. Example 4. Let the system (A, B, C) begvenby 5 6 A = 5 9 6, B =, C = whose transfer functon s (s+) T (s) =C(sI A) B =. s (s+) (s )(s+) (s+) 4 3 Agan, let us suppose that we want to decouple ths system by statc state feedback, such that the closed-loop system s nternally stable. The -stablenteractorofthssystemfor = C (IR (s)=ir ps(s)) s a dagonal matrx " # π π Φ (s) = =, s π π or equvalently, the row and global nfnte and unstable zeros of the system concde, meanng that the system s decouplable wth stablty. s (see

12 From (4.), and wth π = s +,wehavethat " # U(s) = s+. (s+)(s+) s+ Then, from a constant soluton X, Y, wthx nonsngular, to the polynomal matrx equaton XD(s) +YN (s) =U (s)d(s), where N (s), D(s) s a rght coprme MFD of (A, B, I n)wth D(s) column reduced, we obtan the state feedback F =, G = whch produces the decoupled and stable system (A + BF,BG,C) whose transfer functon s (s+) T (s) =C(sI A BF) BG =. F,G s (s+) Actually, ths system can be decoupled wth stablty wth closed-loop transfer functon where α,,3,4 (s+α )(s+α ) T (s) = F,G are arbtrary postve real numbers. s (s+α 3)(s+α 4) Let us suppose now that we want to decouple ths system wth -stablty, where s the re- gon shown n Fg. 4., startng at s = 3, and formng an angle of ±45 wth the negatve real axs. Therowzerostructureofthesystemoutof s and the global zero structure out of s Fg. 4.. Regon n =, n =, n =, n =3. Snce these structures do not concde due to the fact that the global zero at s = snotarow zero of the system, then decouplng wth -stablty by statc state feedback can not be acheved. Ths can also be seen from the fact that the -stable nteractor of ths system s not a dagonal matrx. Notce that the angle of the regon wth respect to the real axs plays a role only f the system has complex fnte zeros wth magnary part dfferent from zero, whch s not the case n ths example. Snce the symmetrc regon of the complex plane can be arbtrarly chosen, t s to be expected that prevous results on decouplng can be deduced from our results. Indeed, the condtons for decouplng of square lnear multvarable systems, and decouplng wth stablty can be easly deduced from our soluton. For these two cases, the condtons of Theorem 4. can be nterpreted as follows:

13 Consder the decouplng problem wth closed-loop modes at any fnte poston. In ths case = C and the regon out of s =. Then the system (A, B, C) s decouplable f and only f the row and global nfntezerostructureofthesystemsthesame[6,7]. In the decouplng problem wth stablty, t s requred that all modes of the closed-loop system are n the left hand complex plane,.e. = C and = C+.Thenthesystem s decouplable n ths case f and only f the row and global zero structure of the system out of s the same,.e. f and only f the row and global nfnte zeros, plus the row and global zeros n C+ are the same [8,9]. 5 Conclusons In ths paper we have examned algebrac propertes of -generalzed polynomals, and appled these propertes to the problem of decouplng wth -stablty of lnear square multvarable systems. The structural necessary and suffcent condtons for ths problem to have a soluton are stated n terms of the -stable nteractor of the system or n terms of the row and global zero structure of the system out of. Summarzng: decouplng as such,.e. each system output beng controlled ndependently by one and only one nput regardless of the poston of the closed-loop modes, can be acheved f and only f the row nfnte zero structure of the system s the same as the global nfnte zero structure. If the system s decouplable, decouplng wth closed-loop modes n a partcular regon of the complex plane can be acheved f and only f the global fnte zeros out of of the system, multplctes ncluded, are also row fnte zeros of the system. Observe that f a system(a, B, C) wth no fnte zeros s decouplable, then t can be decoupled wth closed-loop modes located at any fnte poston of the complex plane. If a decouplable system has fnte zeros, then for the system to be decouplable wth -stablty, the regon must contan every global zero that t s not a row zero of the system. For nstance, n Example 4., the only global fnte zero that t s not a row fnte zero s s = ; then the system can be decoupled wth -stablty f and only f the regon contans ths pont. Theorems 4. and 4. state the necessary and suffcent condtons for decouplng wth -stablty, Theorem 4. n terms of the -stable nteractor of the system, and Theorem 4. n terms of the global and row zero structure of the system out of. From the practcal pont of vew, snce elementary operatons n a partcular rng could be very nvolved, n general s smpler to test the condtons of Theorem 4. than to obtan the -stable nteractor of the system. The row zero structure of the system out of can be obtaned for nstance from a gcd of the -th row of T (s), and the global zero structure of the system out of can be obtaned from the Smth form of T (s) usng the determnantal dvsors of T (s), avodng thus the use of elementary operatons n IR (s). Ths procedure, however, can also be very complcated. If the system (A, B, C) s decouplable wth -stablty, a decouplng state feedback can be computed as ndcated n the proof of Theorem 4.. Observe that the man feature of our extenson to generalzed polynomals as presented n ths paper, namely the ncluson of the pont at nfnty, s not fully exploted n the study of the decouplng problem wth -stablty. Indeed, though generally defned as a subset of the extended complex plane, for the problem of decouplng the regon s taken as a subset of the complex plane, snce the poles of a lnear system (A, B, C) areallfnte. Infnte poles may appear n the study of the so-called sngular systems,.e. systems of the form Eẋ(t) =Ax(t)+Bu(t) y(t) =Cx(t) where E s not square, or s sngular. But even n ths case, snce nfnte pole behavor s undesrable from the practcal pont of vew, control strateges are usually desgned to deal only wth fnte poles. It s then hoped that the concepts of generalzed polynomals wll be appled n a near future to solve n a unfed way other multvarable control problems besdes decouplng. As shown n Theorems 4. and 4., obtanng the Hermte and Smth canoncal forms s a cornerstone of the desgn algorthm to acheve multvarable decouplng. For the same reason as Gaussan elmnaton wthout pvotng can be numercally unstable n the feld of real or complexvalued matrces, t s now recognzed that elementary operatons on polynomal or ratonal objects 3

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