Strong dynamic input-output decoupling: from linearity to nonlinearity

We study the strong dynamic input-output decoupling problent (SDIODP) for nonlinear systetls' It is shown that, given a generically satisfied assumption, the solvability of tlie SDIODP around an equilibriurn point is equivalent to the solvability ofthc satne problem for the linearization of the system arourrd this equilibrium point. We introduce the Singh compensator, a dynamic state feedback of miuinral order that solves the SDIODP. It is showu that, given the assumption mentioned above' the linearization of the Singh compensator around an equilibriutn point is a Singh compensator for the linearization of the original nonlinear system around this equilibrium poiut' L Introduction Since for output regulation tasks input-output decoupled systerns are relatively easy to hanclle, the so called input-output decoupling problern has received a lot of attention in the literature. Today, this probIem is quite well understood for nonliuear corttrol systems, see e.g. [16], [1]'[13] where a complete solution is described in terms of a dynanric contpeusator' Au important feature of the decoupling compensator we consider in this paper' the so called Singh colnpensator, is that it is a decoupling competlsator of minimal dimension, see [9], and therefore it is intuitively of minimal " complexitY" . Control engineers are often led to handle a specific control problern in a concrete nonlinear systeru by linearizing the given model around an equiliblium point, and afterwards solve, if possible, t\e given coutrol problem for the liuearization. We show in the present paper that at least in a local sense such au approach may be successfully used, provided ihe syst,em fulfills a generically satisfied regularity assumption. In particular it follows that the decoupling Singh compensator for the nonlinear system beccmes, wheu linearized around the equilibrium point, a decoupling compensator for the linearization. I\rrtherrnore' we have, again under the same regularity assumption, a converse of the above result: for a decoupling Singh compensator for the linearization of a nonlinear system there exists a decoupling Singh compensator for STRONG DYNAMIC INPUT-OUTPUT DECOUPLING: FROM LINEARITY TO NONLINEARITY HJ.C. Huijberts* and H. Nijmeiier** *Depa,mentor"''!"tr;::;#.?:tr#:;í:i^::";i#;:;;"i;x,vorrectnotogv' **Department of Applied Mathematics,IJniversity ofTwente, P.O.8ox217,7500 AE Enschefu,The Netlwrlards the nonlinear sytem having the formentioned compensator as its linearization. The practical conse' quence of this is that, at least in a sufficiently small neighborhood, the linear solution ofthe input-output decoupling problem for the linearization of a nonlinear system will act as a first order approximate solution of the decoupling problem for the original nonlinear system. In this sense, we consider this paper as a justification of engineering practice. Of course it rernains to be studied for each specific system what an acceptable size of the region of applicability of this result wil l be. With the hereforementioned relation between nonlinear and linearized input-output decouplability in mind, we conclude this paper with some remarks relating algebraic and geometric structure at infinity of the nonlinear system. These comments should be considered as a complement to the work in [11]. 2 Preliminaries Consider a square nonlinear system P, given by equations of the forrn p { i = Í @ ) + s ( t ) u [ e = h ( r ) ( 1 ) with c col(r1, " ',tn) € ft ' Iocal coordinates for the state space manifold X, u e IR denoting the controls, and y € ftdenoting the outputs. Furthermore, rve assume all data to be analytic. Recall that a meromorphic function q is a function of the form 4 rf 0, where r and 0 are analytic functions. Assume that the control functions u(Í) are n times coutinuously differentiable. Then define s(0) ;= u, u ( i+ l ) . @ ld lu$ ) . V iew c , u , . . . , , r ( n -1 ) as va r i ables and let f denote the field consisting ofthe set of ratioual functions of (u,' .. ,u("-r); with coefficients that are merornorphic in c. For the system (1) we define in a natural way (with y(0) :y) y (L+ l ) O (k+ t ) ( c , u , . . . , 2 ( t ) ) = 0Y&) ' " ' ' Ë-1 n"'í[) filr\xt + g(r)ul+ ,f, ffi"{'*'r Note that U, i , . . . ,y( ' ) so def ined have components in the field K.