feedback control of linear systems under input

Congresso Brasileiro de Automática, Congresso Latino Americano de Controle Automático, Rio de Janeiro, Brésil, 1994, pp.49-54

FEEDBACK CONTROL OF LINEAR SYSTEMS UNDER INPUT CONSTRAINTS Jean-Claude HENNET, Eugênio B. CASTELAN

∗

Laboratoire d’Automatique et d’Analyse des Systèmes du C.N.R.S., 7, avenue du Colonel Roche, 31077 Toulouse FRANCE

Abstract This paper shows that the problem of control under linear symmetrical input constraints of linear stabilizable systems can always be solved by restricting the set of admissible initial states to a polyhedral domain of the state space. This result can be obtained by an appropriate closed-loop eigenstructure assignment. Two subsequent problems are then addressed: the maximal size of the domain of admissible initial states and the robustness of the control scheme. Uncertainties on the system parameters usually reduce the actual admissible domain to a bounded set. Nevertheless, it is important to construct for the unperturbed system an admissible domain which correctly approximates the maximal controllable domain. In particular, the proposed design technique is based on the construction of an admissible positively invariant domain having the same infinite directions as the maximal set of initial states.

Keywords : Constraint theory, Eigenvalue assignment, Eigenvector assignment, Invariance, Multivariable control systems,

1

Introduction

Pole assignment has been recognized for a long time as a very efficient and convenient technique for controlling linear systems. But, except in the monovariable case, it does not completely characterize the behaviour of the closed-loop system. As shown by [Moore,1976], the choice of a set of closed-loop eigenvalues generally lets the designer with some degrees of freedom in the choice of the associated eigenvectors. The eigenstructure assignment can be performed in a simple and general way by selecting one eigenvector in the transmission subspace associated with each selected eigenvalue. Several different rules in the choice of the eigenvectors can be used to provide the closed-loop system with some desired properties beyond stability. In particular, some disturbance decoupling problems can be solved by confining the image of the disturbance in an (A,B)-invariant subspace of the kernel of the output matrix, and by making ∗

The research support of the second author is provided by CAPES, Brazil.

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this subspace invariant by an appropriate eigenstructure assignment [Wonham, 1985]. Also, the robustness of the control scheme can be improved by selecting the matrix of the eigenvectors so as to minimize its condition number [Kautsky et al., 1985]. If the closed-loop system is asymptotically stable, its eigenvectors and eigenvalues define the directions and the rates of contraction along them, relatively to a polyhedral norm which can be used as a Lyapunov function. The unit ball for this norm is a positively invariant set of the system in the sense of [Kalman and Bertram, 1960]. Any homothetic of the unit ball is also positively invariant. Thus, in the absence of constraints on the input vector, the whole state space has the same directions and rates of contraction as the unit ball. The picture becomes very different when taking into account control limitations, which almost always exist in real systems. The closed-loop eigenstructure obtained by state feedback only applies as long as the input vector remains in the admissible domain. If a constraint is violated by the linear feedback law, an admissible control vector can still be obtained by saturating the feedback control. But the closed-loop system then becomes non-linear, and its stability region has to be investigated [Tarbouriech and Burgat, 1990]. In the generic case of symmetrical linear constraints on the input vector, the design problem can thus be decomposed into two stages: • Construct a positively invariant domain where the linear state feedback control law satisfies the constraints, • Extend the set of controllable initial states by allowing the use of the saturated feedback law as long as the closed-loop behaviour remains stable. The positively invariant domain to be constructed in the first stage should be as large as possible. An algorithm was recently proposed by [Lasserre, 1991] to compute the maximal controllable set of a linear system subject to symmetrical input constraints. The ideal case would be to obtain the positive invariance of this set by state feedback. The second stage would then become useless. But this set is usually defined by a great number of constraints, and the existence of a constant feedback law solving this invariance problem is generally not guaranteed. However, it is shown in this paper that it is always possible to construct a simple symmetrical positively invariant domain having the infinite directions of the maximal controllable set. The use of the saturated feedback will then enlarge the set of admissible initial states to an extent which is related to the robustness of the control scheme, and in particular to the condition number of the matrix of closed-loop eigenvectors. Some basic results on positively invariant domains and on the maximal controllable domain under input constraints are presented in section 2. Then, in section 3, some design techniques are proposed in the two basic cases: when the number of inputs is greater (3.1) and smaller (section 3.2) than the number of unstable open-loop poles. Section 4 describes how to construct an admissible domain of initial states for which the proposed control laws are robust enough in the case of structured perturbation, and how to extend this domain by allowing saturated controls. The two examples given in section 5 illustrate the proposed control schemes.

2

Preliminary results

This study is presented in the continuous-time framework. But it can also be directly used for discrete-time linear models. The only changes are related to spectral conditions. The correspondence between the spectral regions of stability is well known, and the correspondence between the 2


spectral regions of invariance will be specified in the sequel. Consider the following linear dynamical model: x(t) ˙ = Ax(t) + Bu(t) for t ≥ 0

(1)

with x(t) ∈ m, the kernel of SF cannot be both invariant and stable. Therefore, condition r ≤ m is necessary for the existence of a gain matrix F for which the closedloop system is stable and admits S(SF, 1s ) as a positively invariant symmetrical polyhedron. It will now be shown that in the case s = m =rank S, this condition is also sufficient. 6


3.1

Positive invariance of S(SF, 1m ) when r ≤ m

In this chapter, it is first assumed, for simplicity, that matrix S is full rank, with s = m. The dimension of the open-loop unstable subspace, r, is supposed to be less than or equal to the number of inputs, m, with rank B = m. Furthermore, if the Jordan blocks associated with unstable eigenvalues are controllable, the system is stabilizable [Chen, 1984], and the following result can be obtained by an appropriate eigenvalue-eigenvector assignment. Lemma 3.1 : In the case r ≤ m, it is possible to construct a feedback gain matrix F for which the closed loop system admits S(SF, 1m ) as a positively invariant domain . Proof: The proof of this result will be constructive. The positive invariance property of S(SF, 1m ) can be obtained by an appropriate eigenstructure assignment. Actually, to obtain the largest positively invariant domain , matrix S is decomposed into two submatrices: # " S1 S= ; S1 ∈ 0). Once the appropriate closed-loop eigenvalues and eigenvectors have been selected, a satisfactory gain matrix can be directly computed.

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