Conte, G.

Moog, C. H.

Perdon, A. M.

Zheng, Y. F.

English

Institute of Mathematics AS CR, v. v. i.

KybernetikaAnna Maria Perdon; Yu Fan Zheng; Claude H. Moog; Giuseppe ConteDisturbance decoupling for nonlinear systems: A unified approachKybernetika, Vol. 29 (1993), No. 5, 479--484Persistent URL: http://dml.cz/dmlcz/124534Terms of use:© Institute of Information Theory and Automation AS CR, 1993Institute of Mathematics of the Academy of Sciences of the Czech Republic provides access to digitizeddocuments strictly for personal use. Each copy of any part of this document must contain theseTerms of use.This paper has been digitized, optimized for electronic delivery and stamped withdigital signature within the project DML-CZ: The Czech Digital Mathematics Libraryhttp://project.dml.czKYBERNETIKA — VOLUME 29 (1993), NUMBER 5, PAGES 479-484 DISTURBANCE DECOUPLING FOR NONLINEAR SYSTEMS: A UNIFIED APPROACH1 A . M . PERDON, Y . F . ZHENG, C.H. MOOG AND G. CONTE This note presents an exposition which unifies various (static or dynamic feedback) solutions given in the literature to the disturbance decoupling problem since the early development of modern nonlinear theory up to the end of the 70's. This is possible thanks to some recently introduced generalized transformations depending on a finite number of time derivatives of the input. In this way, some classical controlled invariant distributions can be replaced by a related elementary linear subspace by which a NSC for disturbance decoupling can be derived. 1. INTRODUCTION AND PRELIMINARIES In this paper we give a general and unified description for the solvability of the Dis-turbance Decoupling Problem which is solved either by static state feedback (DDP), dynamic state feedback (DDDP) or generalized state feedback (GDDP). Differently from other results ([3,5,6]), our description does not rely on any (structure) algo-rithm. For this purpose a linear algebraic setting is used. Consider the nonlinear system E of the form E=(x = f(x)+g(x)u \ y = h(x) where x G IRn, u £ Mm and y e IRP. The entries of f(x), g(x) and h(x) are meromorphic functions. As done in [2], let K := K(x,«,«,..., u(k\...), that is the differential field of meromorphic functions of x, u, it,..., u^k\..., for k > 0. Over this field we can define a differential vector space, £ := span{<fs; x € K}. £ can be decomposed into the sum of two subspaces, £ = X (&U, where X :=spanJC{rfx} U := span*; {d«<*>, k > fj}. 'This work was performed with the financial support of NATO. The work of Y.F. Zheng was done during his visit at Laboraloire d'Automatique de Nantes with the support of C.N.R.S. and E.C.N. 480 A.M. PERDON, Y.F.ZHENG, C.H. MOOG AND G. CONTE Define the output differential space y := spanK {dt/fc), k > o\ . Thus ycs = x®u. (2) Recalling the results from the geometric linear control theory ([10]) the key concept to solve the DDP is the so-called supremal (A,B)-invariant subspace contained in the kernel of C, denoted V*. This notion has been generalized to nonlinear system theory by A*, the largest controlled-invariant distribution contained in the kernel of the output map ([7]). In a dual form, it was pointed out that the subspace X (iy represents the subspace of X whose observability is not affected by the input and is invariant under regular feedback [11]. For linear systems, remarking that X contains in a natural way an isomorphic copy of the finite dimensional state space X, we have that V* is isomorphic to X n (X n ^fc)1, where 34 := span^jdj/, dy,..., dy(k^}, for k large enough. For nonlinear control systems the subspace X C\y, in general, is not closed, in the sense that there does not exist a basis for Xf\y which consists of closed (or locally exact) one-forms only. If we consider generalized transformations, i.e. the state space coordinates transformations, feedbacks and output injections which are parametrized in the input and its derivatives [4], we show that the subspace X C\ y plays a similar role as it does for linear control systems. More precisely X n y is shown to describe the maximal observable subsystem with respect to all possible generalized (or quasi static [1]) feedbacks. Let us recall the definitions (that could be given in different ways) of generalized transformations. Definition 1.1. Given an isomorphism $ from 8 to 8, $ defines a generalized-state transformation if $(X) is closed and $(X)®U = X®U. (3) Let $(X) = spanK{d6, i = 1,2, . . . , n}. From (3), J ^ i i g l l l l l M Js nonsingular 0(X\, #2i • • • > Xn) and (£1,^2, • • • ,6 .) defines a generalized-state coordinate system. Definition 1.2. Given an isomorphism $ from 8 to 8, $ defines a regular gener-alized state feedback if - $(U) = span^dv^, k > 0} where v := (v\,..., vm) and «,- € !C, i = 5 1 , . . . , m, - and X®$(U) = X®U. (4) Let V = $(W) = span^d t /* ) , it > 0}. The equality (4) implies that (*-»•• •>""») d(ui,...,um) is invertible. Thus, there exist relations as «,• = Ui(x,v,v,...) and in the rest of the paper v is considered to be the new input of the closed loop system, i. e., after generalized state feedback vi,... ,vm and their time derivatives are independent variables. Disturbance Decoupling for Nonlinear Systems: a Unified Approach 481 From the above definition, one may also write $ : E — t. For the closed-loop system E there is a background field K, which is the differential field of meromorphic functions of £, v, v,..., v^k\ ..., for k > 0, and a differential vector space £ := span^{dn, 77 £ K}. The two background fields K and K can be identified. As £ = £(z, u , u , . . . ,«( r)), t> = ^;(x, u, it,..., u(s>), then every function i] £ K can be considered as a function in K, i.e. i] = f/(x,u,u,.. .,u^q>) £ K, where k, s, r, q are some properly defined integers. Therefore, the transformation $ sets up a linear map from £ to £ over the same field. Thus, for any dn £ £ we can define di) := <b(di]) £ £. Especially, let j/> ' be the fcth derivative of the ith output j/,- and let dy\ be its differential, then dy\ := <&(dy\ '), where y\ is the A:th derivative of the z'th output j),- of the closed-loop system E. The Disturbance Decoupling Problem can now be dealt with in a general and uni-fied way. The definition and solvability conditions of DDDP are given in [3,5, 6, 8, 9]. For the GDDP it is required to find a generalized-state space change of coordinates and a regular generalized-state feedback to fulfil the disturbance decoupling. More precisely, consider a nonlinear system E of the form { x = f(x) + g(x) u + p(x) w (5) y = h(x) which satisfies the general assumptions as system (1). The background field is defined by K := K(x, u,ii,..., u^k\ ... ,w,w,..., w^r>,...). Generalized Dis turbance Decoupling Prob lem Formulation Find, if possible, an isomorphism $ from £ to £, which defines a generalized-state transformation and a regular generalized-state feedback such that $|w is the identity and the differential output space y* of the closed loop system satisfies y* c *(#) © *(u) = *(x) e v. From a practical point of view, the GDDP reduces to find a generalized coordinates system, £,• = £i(x, u,ii,..., u^'^) i = 1, 2 , . . . , n, and new inputs Vj = Vj(x, u,ii,... .. . , i / 5 ' ) ) j = 1,2,.. . , m such that ——'•— m . is invertible. 9 (wi ,u 2 , . . . , t t m ) Under the new coordinates £ = (£i>£2> • • • ,£n)T and the new input v = (vltV2,- •• • • •, vm)T the system (5) becomes f i = f(Z,v,v,...,v(r\w) E, = < (6) \y = h(0 where r is a positive integer and the output y is independent of the disturbance w, i. e. the output differential space 3^* of (6) satisfies y* c *(x) © $(u) = $(x) © v. 482 A.M. PERDON, Y.F. ZHENG, C.H. MOOG AND G. CONTE Remark: (6) has not the most general form after a generalized-state space change of coordinates and a regular generalized-state feedback. More generally, (6) could contain time derivatives of w. In the next section, one verifies that the solvability conditions for DDDP and GDDP are equivalent. More precisely, we show that DDDP, or GDDP, is solvable if and only if ,Vn3^Cspan ; e {p(x)}x . (7) Condition (7) can be viewed as a natural generalization of the solvability condition of the DDP given in the linear control system theory. In fact, in a dual form, we have X n (X n 34) ~ V*, thus, (7) is exactly the same condition appearing in [10]. 2. MAIN RESULT First one shows that condition (7) is a necessary and sufficient condition for the GDDP. After that it will be shown that the solvability conditions for DDDP and GDDP are equivalent. In order to solve the GDDP it is necessary to find a proper generalized-state space change of coordinates and a regular generalized-state feedback. Let pi, 1 < i < p, denote the orders of the zeros at infinity; then one has Lemma 2 .1 . (X ny)elt = spanjc {t*y{fc); k < Pi, l<i<pj®U. (8) T h e o r e m 2.2. For system (5) the GDDP is solvable if and only if Xr\ycspa,njt{p(x)}x (9) where K, := K(x, « , « , . . . , u^h\ ...,w,w,..., w^r\ ...). P r o o f . Sufficiency. Rewrite system (5) as * = f(x) + g(x) u + p(x) w = f(x) + g(x) u V = h(x) (10) where g(x) = (g(x)p(x)) and u = Qj). u, is considered to be the input of system (10). Without loss of generality, assume px < p2 < • • • < pq < oo, q < p, then a ( p i - i ) since y[pl~l) = y\pl~1)(x), we have that rfy^1_1) = ^ dx belongs, by (9), to t ^ dv{pl~l) span^JXa:)}-1-. Hence y[pl) = ^ - (f + gu +pw) = <j>01(x) + </>n(x)u. Start defining the generalized state feedback by Vi = <t>01(x) + <f>n(x) u (11) Disturbance Decoupling for Nonlinear Systems: a Unified Approach 483 fhere v\ is a new independent input. From the definition of p2 (p\ < p2) it follows that f (x, y[>, A >pЛ îor a,ny k < p2, ,/*) .- „(-) 2/2 — 2/2 *•> = ^ < / + J „ + P . , + £ M Г > > > . Since #->-E^# = ̂ Г^,n,, from (9), j / 2 = ^02 f^i2/i ) + ^12 (£,2/i ) u- The second step for the definition of the generalized state feedback is then v2 = cj>02 (x, v\,..., v[k)) + <t>l2 (x, vi,..., v[k)) u. (12) Repeating the foregoing process, one defines v\,...,vq which can be arbitrarily com-pleted to define a regular generalized feedback solution to the GDDP. Necessity. Assume that (9) is not true, then there exists C S X n y such that (C,p) ^ 0. By Lemma 2.1, there exists dy\ ' with k < pi, 1 < i < p, such that dy\ —C,+ai\dva + ai2 dvi2 -\ \-ais dvis where dv,j G span{dui ,dv\,... ,dvq, dvq, ...}. Thus, applying any generalized transformation <3> the output <b(dy\ )) of the closed loop system equals <$(£) + <L> (a t l dv{\ + ai2 dvi2 + .. . + ais dvis). Since $ ( a n dva + ai2 dvi2 + ... + ais dvis) £ <*>(X) © V, and $(C) £ $(X) © V, then $ (dy\k)) <£ $(;f) © V, which stands in contradiction. • From the proof of Theorem 2.2 one gets also a structural condition for the solvabil-ity of the GDDP, i.e. the GDDPis solvable if and only if span{dt;1,... ,d t ; J C X@U. By this condition and by the algorithm used in [3] one derives a dynamic state feed-back solution to the DDP of (5), consequently one has: Theorem 2 .3 . DDDP is solvable if and only if GDDP is solvable. Condition (9) is an alternative condition to Theorem 2.3 in [5]. Example [6]. X\=X2U\, X2 = 5, X3~X2 + X4 + X4U\, x4 = u2, i 5 = x\U\ + w, 2/1 = X\ J/2 = x3. One computes y2 = x2 + X4 + x4 —, thus X(~\y = span I dxlt dx3, I 1 L̂_L ] dx2 x2 { \ xl2 ) + I 1 + — J dx4 > and condition (9) is satisfied although A* is zero. A.M. PERDON, Y.F.ZHENG, C.H. MOOG AND G. CONTE 3. C O N C L U S I O N A necessary and sufficient condit ion for t he solvability of the G D D P and D D D P has been given. T h e condit ion is intr insic to t he sys tem and does no t depend on any a rb i t ra ry s t a t e space extension. Only the d a t a of the original sys tem are in-volved in (9) . T h e equivalence between G D D P and D D D P should be viewed as t h e general izat ion of t he equivalence of D D P and D D D P known for l inear sys tem. (Received March 3, 1993.) R E F E R E N C E S [1] E. Delaleati: Sur les derivees de l'entree en representation et commande des systemes non lineaires. P h . D . Thesis dissertation, Universite Paris-Sud 1993. [2] M. D.Di Benedetto, J .W. Crizzle and C. H. Moog: Rank invariants of nonlinear sys-tems. SIAM J. Control Optim. 27(1989), 658-672. [3] L. Cao and Y. F. Zheng: Disturbance decoupling via dynamic feedbacks. Internat. J. Systems Sci. 23 (1992), 683-694. [4] M. Fliess: Generalized controller canonical forms for linear and nonlinear dynamics. IEEE Trans. Automat. Control 35 (1990), 994-1001. [5] H . J . C . Huijberts, H. Nijmeijer and L. L.M. Van Der Wegen: Dynamic disturbance decoupling for nonlinear systems: the nonsquare and noninvertible case. In: Con-trolled Dynamical Systems (B. Bonnard, B. Bride, J. P. Gauthier and I. Kupka, eds.), Birkhauser, Boston 1991, pp. 243-252. [6] H . J . C . Huijberts, H. Nijmeijer and L. L. M. Van Der Wegen: Dynamic disturbance decoupling for nonlinear systems. SIAM J. Control Optim. 30 (1992), 336-349. [7] A. Isidori: Nonlinear Control Theory. Second edition. Springer-Verlag, New York 1989. [8] W. Respondek: Disturbance decoupling via dynamic feedback. In: Controlled Dynam-ical Systems (B. Bonnard, B. Bride, J. P. Gauthier and I. Kupka, eds.), Birkhauser, Boston 1991, pp. 347-357. [9] L. L.M. van der Wegen: Local disturbance decoupling with stability for nonlinear systems. (Lecture Notes in Control and Information Sciences 166.) Springer-Verlag, Berlin 1991. [10] W.M. Wonham: Linear Mul t ivar ia te Control: a Geometric Approach. Third edition. Springer-Verlag, New York 1985. [11] Y. F. Zheng and L. Cao: Reduced inverses for controlled systems. Math. Control Signals Systems, to appear. Professor A. M. Perdon, Dipartimento di Matematica "V. Volterra", Universita di An-cona, via Brecce Bianche, 60131 Ancona. Italy. Professor Y. F. Zheng, Applied Mathematics Research Laboratory, East China Normal University, Shanghai 200062. China. Dr. C. H. Moog, Laboratoire d'Automatique de Nantes Unite Associee au C.N.R.S, E.C.N./ Universite de Nantes, 1 Rue de la Noe, 44072 Nantes Cedex 03. France. Professor G. Conte, Dipartimento di Automatica ed Elettronica, Universita di Ancona, via Brecce Bianche, 60131 Ancona. Italy. 

Disturbance decoupling for nonlinear systems: A unified approach

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Disturbance decoupling for nonlinear systems: A unified approach

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Institute of Mathematics AS CR, v. v. i.