UNIVERSITA POLITECNICA DELLE MARCHE - FACOLTA DI INGEGNERIA Dpartmento d Ingegnerua Informatca, Gestonale e dell Automazone LabMACS Laboratory of Modelng, Analyss and Control of Dynamcal System Zeros and Zero Dynamcs for Lnear, Tme-delay System
Tme delay systems are nterestng n connecton wth: ndustral applcatons (where delays are unavodable effects of the transportaton of materals) tele-operated systems, networked systems, large Integrated Communcaton Control Systems or ICCS (where delays orgnates from dspatchng nformaton through slow or very long communcaton lnes). In the last years, a great research effort has been devoted to the development of analyss and synthess technques for tme delay systems (Proc. IFAC Workshop on LTDS 2, 21, 23, 25).
The noton of ZERO and of ZERO DYNAMICS play an mportant role n several control problems, especally when solutons requre some sort of nverson. For classcal lnear systems, ZERO and ZERO DYNAMICS can be characterzed n abstract algebrac terms by the noton of ZERO MODULE (Wyman,San-1981). The noton of ZERO MODULE can be generalzed to other classes of dynamcal systems, notably to that of systems wth coeffcents n a rng. By explotng the relatons between systems wth coeffcents n a rng and tme-delay systems, sutable notons of ZERO and ZERO DYNAMICS can be defned these latter.
WS Zero Module for Lnear Systems Zero Module and Zero Dynamcs for Systems over Rngs Tme-delay Lnear Systems Zero Module and Zero Dynamcs for Tme-delay Lnear Systems Applcaton to nverson and trackng problems
Tme delay system wth uncommensurable delays: Σ d x& (t) y(t) k a 1 j k c 1 j A C j j x(t x(t jh jh ) + ) k b 1 j B j u(t jh ) + t R, contnuous tme axs x X R n, state value space {statesfunctons x: [T-ah,T) R n } ( -dm. R-vector space) u U R m, nput value space (m-dm. R-vector space) y Y R p, output value space (p-dm. R-vector space) A j, B j, C j real matrces of sutable dmensons h for 1,...,k are fxed tme delays
Gven a rng R, a system Σ wth coeffcents n R s a quadruple (A,B,C,X) where A, B, C are matrces of dmensons n n, n m, p n wth entres n R and X R n. Dynamcal nterpretaton: Σ x(t + 1) Ax(t) y(t) Cx(t) + Bu(t) t Z, ordered set of nteger numbers (dscrete tme) x X R n, state module (n-dm. free R-module) u U R m, nput module (m-dm. free R-module) y Y R p, output module (p-dm. free R-module)
Tme delay system Σ d x&(t) y(t) δ δ Σ d x& (t) y(t) a c A x(t h) + C x(t h) a c A C δ δ x(t) + x(t) A B C b B u(t a A b B c C b B δ δ δ δ u(t) h) Substtute δ wth the ndetermnate Δ System w. c.. the rng R R[Δ] Σ x(t + 1) Ax(t) y(t) Cx(t) + Bu(t)
Tme delay system Σ d x&(t) y(t) a c A x(t C x(t h) h) + b B u(t h) t, x, u, and y have dfferent meanngs n the two frameworks: e.g. x(t) R n n the tme delay fremework, x(t) R n n the rng framework; notatons are kept equal by abuse. System w. c. n the rng R R[Δ] Σ x(t + 1) Ax(t) y(t) Cx(t) + Bu(t)
x(t + 1) Ax(t) + Bu(t) Σ y(t) Cx(t) R[z] rng of polynomals n the ndetermnate z wth coeffcents n R R(z) S -1 R[z] localzaton at the multplcatve set S of all monc polynomals Transfer Functon Matrx G Σ C(zI-A) -1 B (entres n R(z)) R(z)-morphsm C(zI-A) -1 B : U R(z) Y R(z) t u(z) U R(z), u(z) u t z, u t U (nput sequence) y(z) Y R(z), y(z) y t z, y t Y (output sequence) t t t
Σ x(t + 1) Ax(t) + Bu(t) y(t) Cx(t) G Σ C(zI-A) -1 B : U R(z) Y R(z) R[z]-modules ΩU U R[z] U R(z) ΩY Y R[z] Y R(z) Defnton (CP-1983, after WS-1981) Gven the system Σ (A,B,C,X) wth coeffcents n the rng R and transfer functon matrx G Σ, the Zero Module of Σ s the R[z]-module Z Σ defned by Z Σ (G -1 Σ (ΩY) + ΩU)/(KerG + ΩU)
Σ x(t + 1) Ax(t) + Bu(t) G Σ C(zI-A) -1 B : U R(z) Y R(z) y(t) Cx(t) (G -1 (ΩY) + ΩU)/(KerG + ΩU) Z Σ Proposton (CP-1983) Let G Σ D(z) -1 N(Z) be a coprme factorzaton. The canoncal projecton p N : ΩY ΩY/NΩU nduces an njectve R[z]-homomorphsm α: Z Σ Tor(ΩY/NΩU). As a consequence of the above Proposton we have the followng foundamental result Z Σ s a fntely generated, torson R[z]-module
Tme delay system Σ d x&(t) y(t) δ δ Σ d x& (t) y(t) a c A x(t h) + C x(t h) a c A C δ δ x(t) + x(t) A B C b B u(t a A b B c C b B δ δ δ δ u(t) h) Substtute δ wth the ndetermnate Δ System w. c. n the rng R R[Δ] Σ x(t + 1) Ax(t) y(t) Cx(t) + Bu(t)
Tme delay system Σ d x&(t) y(t) a c A x(t C x(t h) h) + b B u(t h) Z Σ (G -1 (ΩY) + ΩU)/(KerG + ΩU) Z Σd ZERO MODULE System w. c. n the rng R R[Δ] Σ x(t + 1) Ax(t) y(t) Cx(t) + Bu(t)
Tme delay system Σ d x&(t) y(t) a c A x(t C x(t h) h) + b B u(t h) Defnton Gven a tme-delay system Σ d, let Σ be the assocated system wth coeffcents the rng R. The Zero Module Z Σd of Σ d s the Zero Module Z Σ of Σ. System w. c. n the rng R R[Δ] Σ x(t + 1) Ax(t) y(t) Cx(t) + Bu(t)
If the zero module Z Σ of Σ s free, over the rng R, then t can be represented as Z Σ (R m, D), where D: R m R m s an R- homomorphsm. Then, we can consder the followng noton. Defnton Gven a system Σ wth coeffcents n the rng R, whose zero module Z Σ can be represented as the par (R m,d), the Zero Dynamcs of Σ s the dynamcs nduced on R m by D, that s by the dynamc equaton z(t+1) Dz(t), for z R m. Remark that n case Z Σ cannot be represented as a par (R m,d), the Zero Dynamcs s not defned.
Σ (A,B,C,X) wth coeffcents n R: a controlled nvarant submodule (c..s.) of X s a submodule V X such that A(V) V + ImB feedback property: there exsts an R-morphsm F: X U such that (A+BF)V V (F s called a frend of V). V* maxmum c..s. contaned n KerC R* mnmum c..s. contanng ImB V* Proposton Gven Σ (A,B,C,X), w.c.n R and C(zI-A) -1 B D -1 N coprme, let N(U R(z)) be a drect summand of Y R(z) and let V* have the feedback property wth a frend F. Then, V*/R* endowed wth the R[z] structure nduced by (A+BF) s somorphc to Z Σ.
If V* has the feedback property wth a frend F and V*/R* s a free R-module, say V*/R* R m, lettng D be a matrx that represents (A + BF) V*/R* wth respect to the canoncal bass of R m, t s be possble to represent the Zero Dynamcs of Σ as the dynamcs nduced on R m by D: z(t+1) Dz(t), for z R m. The above characterzaton of Zero Dynamcs allows us to analyse t n a smple, practcal way, avodng the necessty of workng wth R[z]-modules and of nvolved computatons. Unfortunately, t holds only f V* has the feedback property (strong requrement).
Proposton Gven Σ (A,B,C,X), w.c.n R, t s possble to construct, n a canonc way, a dynamcal extenson Σ e of Σ such that V e * has the feedback property wth a frend F e. Then, f V e */ R e * s free, t s somorphc to the largest free submodule of V*/ R*. Proposton In the above context and wth the above notatons, assumng that the Zero Dynamcs of Σ s defned, let V e */ R e * be a free R-module of dmenson m and let D be a matrx representng the R-morphsm (A e + B e F e ) Ve*/ Re* wth respect to the canoncal bass of R m. Then, the ZeroDynamcs of Σ s the dynamcs nduced on R m by D z(t+1) Dz(t), for z R m.
Defnton Gven a tme-delay system Σ d, let Σ be the assocated system wth coeffcents the rng R. The Zero Dynamcs of Σ d s that of Σ, f the latter s defned. Proposton Gven the tme-delay system Σ d, wth commensurable delays, let Σ be the assocated system wth coeffcents the rng R. Then, f Σ s left nvertble and V* s free, the Zero Dynamcs of Σ s defned and so s that of Σ d.
Hurwtz set: a set H of monc polynomals n R[z] H contans at least one lnear monomal z + a wth a R; H s multplcatvely closed; any factor of an element n H belongs to H. Defnton A system Σ (A,B,C,X) w.c.. R s sad to be H-stable f det(zi-a) belongs to H H-mnmum phase f ts zero dynamcs s defned and H- stable. For systems assocated to tme-delay ones, R R[Δ] H {p(z,δ) R[z, Δ], such that p(s,e -hs ) for all s C wth Re(s) } H-stablty (phase mn.) n the rng framework Asymptotc stablty (phase mn.) n the tme-delay framework
Proposton Gven a left (respectvely, rght) nvertble system Σ (A,B,C,X) wth coeffcents n the rng R and transfer functon G, let G nv denote a left (respectvely, rght) nverse of G and let Σ nv (A nv, B nv, C nv, X nv ) be ts canoncal realzaton. Then, the relaton G nv G Identty nduces an njectve R[z]-homomorphsm ψ : Z Σ X nv (respectvely, the relaton G G nv Identty nduces a surjectve R[z]-homomorphsm. ϕ: X nv Z Σ ) between the Zero Module of Σ and the state module X nv of the canoncalrealzaton of G nv. In case the Zero Dynamcs of Σ s defned, but not mnmum phase, the above Proposton allows us to say that Σ has no H- stable nverses.
Problem Gven a SISO tme-delay system Σ d and the correspondng system Σ w.c.n R R[Δ], consder the problem of desgnng a compensator whch forces Σ d to track a reference sgnal r(t). Consder the extended system Σ E. x(t + 1) Ax(t) + e(t) cx(t) r(t) bu(t) whose output s the trackng error and apply the Slverman Inverson Algorthm.
Ths gves the relaton k. k 1 e(t + k ) ca x(t) + ca bu(t) r(t + k ) Then, choosng a real polynomal p(z) z + az n such a way that t s n the Hurwtz set H, we can construct the compensator Σ C z(t + 1) Az(t) + bu(t). u(t) (ca (ca k whose acton on Σ causes the error to evolve accordng to the equaton k e(t k ) 1 k + + a e(t + ) ca (x(t) z(t)) k k 1 1 b) b) 1 1 (ca k k 1 k 1 z(t) r(t a e(t + ) + k )) +
The compensator Σ C z(t + 1) u(t) (ca solves the trackng problem. Az(t) + bu(t) (ca k k 1 1 b) b) 1 1 (ca k k 1 z(t) r(t a e(t + ) + k )) + H-stablty of the compensator s a key ssue and, snce ts. constructon s based on nverson, t can be dealt wth by usng phase mnmalty. If Σ d and Σ, and hence Σ E, are H-mnmum phase (that s: ther zero dynamcs s H-stabe), an H-stabe compensator s obtaned.
Example. Consder the tme-delay system and the assocated system Σ (A,B,C,X) wth coeffcents n R R[Δ] and matrces V* span (Δ ) T R 3, V* s not of feedback type (because t s not closed), but t s free. R* {}. The Zero Dynamcs s defned and t can be represented as a sutable par (R m,z) n order to check, for nstance, phase mnmalty.
Example (contnued). To analyze the Zero Dynamcs, we consder the extenson Σ e (A e,b e,c e,x e ), wth V e * span (Δ 1) T R 4 s of feedback type, F e The dynamc matrx A c (A e + B e F) of the compensated system s A c
Example (contnued). The Zero Dynamcs turns out to be gven by (R, [ 1]) or, n other terms, by the dynamc equaton. ξ(t + 1) ξ(t) ξ(t) ξ(t) n the tme-delay framework. We can conclude that the system s mnmum phase.
The notons of Zero Module and of Zero Dynamcs have been ntroduced n the tme-delay framework, by explotng the correspondence between system wth coeffcents n a rng and tme-delay systems and the algebrac characterzaton of Zeros. Stablty of the Zero Dynamcs and Phase Mnmalty can then be defned and used for the constructon of stable nverses and of stable solutons to trackng problems n the tme delay framework.