Exact linearization of nonlinear systems with outputs - Springer Link

Math. Systems Theory 21, 63-83 (1988)

Mathematical Systems Theory ©1988 Sonnger-Verlag New York Inc.

Exact Linearization of Nonlinear Systems with Outputs* D. Cheng, ~ A. Isidori, 2 W. Respondek, 3 and T. J. Tarn 4 =Institute of Systems Science, Academyof Sciences, Beijing, People's Republic of China 2 Department of Information and Systems Science, Universita di Roma "'La Sapienza', Rome, Italy 3 Institute of Mathematics, Polish Academyof Sciences, Warsaw, Poland 4 Department of Systems Sciences and Mathematics, Washington University, St. Louis, Missouri, USA

Abstract. This paper discusses the problem of using feedback and coordinates transformation in order to transform a given nonlinear system with outputs into a controllable and observable linear one. We discuss separately the effect of change of coordinates and, successively, the effect of both change of coordinates and feedback transformation. One of the main results of the paper is to show what extra conditions are needed, in addition to those required for input-output-wise linearization, in order to achieve full linearity of both state-space equations and output map.

I.

Introduction

In the last years there has been an increasing interest in the problem of compensating the nonlinearities of a nonlinear control system. Among all the problems of this kind, the most natural question is that of when there exists a local (or global) change of coordinates, i.e., a diffeomorphism, in the state space that carries the given nonlinear system into a linear one. Krener [1] showed the importance of the Lie algebra of vector fields associated with the system in studying such a question and gave an answer to this problem. Brockett [2] enlarged the class of

*This research was supported in part by the National Science Foundation under Grants ECS-8515899, DMC-8309527, and INT-8519654.

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D. Cheng, A. Isidori, W. Respondek, and T. J. Tam

transformations by also allowing a certain form of feedback. Following Brockett's paper, Jakubczyk and Respondek [3], Su [4], and Hunt and Su [5] considered the full feedback group and gave necessary and sufficient conditions for linearization of the state-space equations. Studying the input-output behavior, Claude et al. [12] gave sufficient conditions for the immersion of a system into a linear one and Isidori and Ruberti [13] found necessary and sufficient conditions, together with a constructive procedure, for linearization of the input-output response. The growing literature concerning linearization now includes, besides the above problem, the consideration of different classes of transformations [6], global questions [7]-[11], [28], [29], partial linearization [14]-[16], [34], and many others [22], [23]. A survey of linearization problems can be found in [25] and [36]. In this p a p e r we consider a system of the form ~ g,(x)ui,

g=f(x)+

i=1

Yi = hi(x),

(1) i= 1,...,p,

where f, g~ . . . . ,gm are C ~ vector fields in R n, h = (hl • • • hp) r is a C ~ vectorvalued function. We assume we are given an initial state Xo~ R n around which we consider our system and we also assume, throughout the paper, that f(xo) = 0 and h(xo) = O. We study various kinds of equivalence of the above system to a linear system of the form

= A z + ~ biui, i=l

yi=Ciz,

i=l,...,p.

(2)

Without the assumption f(xo) = O, h(xo) = 0 all results still remain valid but it is necessary to add constant terms to A z and C~z, respectively. All the results shown below are local. It means that the conditions we state need to hold locally around the initial state Xo and that the linearizing transformations exist locally around this point. Section 2 deals with linearization without feedback. We state a series of conditions which describe those nonlinear systems which can be transformed, via local diffeomorphisms, to controllable and observable linear ones. We call this exact linearization. In Section 3 we consider a single-input system and we deal with the search of a feedback and coordinates change in the state space which transforms both the dynamics and the output maps into a linear form. We call this exact linearization via feedback. In Section 4 we provide an improved version of a previous test [13] for linearization o f the input-output response. In Section 5 we use this improved version in order to solve the problem of exact linearization via feedback for a multi-input system. In particular, we identify the extra conditions needed, in addition to those required for linearization of the input-output response, in order to achieve full linearity at a state-space level.

Exact Linearization of Nonlinear Systems with Outputs 2.

65

Linearization Without Feedback

In this section we illustrate solutions to the p r o b l e m o f turning (1) into a linear system by means o f change of coordinates only. We assume the reader is familiar with the notions o f a Lie bracket o f vector fields, a Lie derivative of a function along a vector field, one-forms, and distributions and codistributions. Introduction to these subjects may be found, e.g., in [27], where special emphasis on the use o f these concepts in control theory is given. Since we are only interested in controllable and observable linear systems, we introduce a "controllability" condition (C) and an "observability" condition (O) in the following way. Let D j denote the distribution s p a n n e d by the set o f vector fields {ad~ -1 gi: l < - i < - m , l < - q < - j } • Let E j denote the codistribution s p a n n e d by the set o f one-forms {dL~--~hi: l j).

(3) m

Observe that K1 ->. • • >- Km > 0 and ~i=1 Ki = n. N o t e that, for a linear system, (C) represents the controllability condition exactly and (K~ . . . . , Kin) the set o f controllability (or Brunovsky) indices [21], [32]. Observe that if (C) holds then we can reorder the g~'s in such a way that D j = Sp{ad~ -1 gi: 1 --rK(Mk). Using this notation it is possible to state the following result, a synthesis of the main theorem of [13] and Lemma 6. Theorem 7. System (1) is input-output linearizable around Xo by the feedback (11) if and only if, for all k