Proc. 4th IEEE Mediterranean Symposium on New Directions in Control and Automation,1996, pp.286-291
COMPUTATION OF MAXIMAL ADMISSIBLE SETS OF CONSTRAINED LINEAR SYSTEMS Carlos E.T. D´orea∗ and
Jean-Claude Hennet
LAAS-CNRS, 7, Avenue du Colonel Roche, 31077 Toulouse, FRANCE e-mail :
[email protected] [email protected]
ABSTRACT In this paper, efficient algorithms are provided to determine the maximal sets of admissible initial states for linear discrete-time systems subject to linear constraints. The generic algorithm is presented for autonomous systems. It is shown that only non-redundant constraints are generated at each iteration. Application of the algorithm to the controlled case is made possible through an explicit characterization of the maximal one-step admissible domain. The property of (A,B)-invariance of polyhedral domains is then introduced and analytically characterized.
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INTRODUCTION
A basic requirement for many dynamical systems is to maintain a given output function in a pre-specified bounded region. For a stable autonomous system, this requirement may be satisfied by restricting the set of admissible initial states to a subset, which is both positively invariant and included in the state region associated with the output constraints. In this paper, an efficient algorithm is provided to compute the maximal set of admissible initial states in the case of autonomous linear discrete-time systems. This algorithm has the same theoretical convergence properties as the one proposed by Gilbert and Tan [5], but it has the advantage of only generating constraints which are not redundant at each iteration. Similar output constraints can also be imposed to controlled systems, which may be open-loop stable or unstable. Some of the control objectives are to stabilize the system and to maintain its output trajectory within the domain of constraints. An explicit state-space description of the maximal one-step admissible domain is provided. The (A,B)-invariance of a polyhedral domain is then characterized by algebraic conditions. Fi∗ This author has a research studentship from CNPq Brazil.
nally, we propose an efficient technique to construct the maximal domain of initial states for which the state trajectory can be maintained in the domain of constraints by a suitable control. The system models considered in this paper are deterministic discrete-time linear models. However, the presented results can be extended to systems with uncertainties on parameters and subject to bounded additive disturbances. Notations. The components of a matrix M are noted Mjk , and its row vectors Mj . By convention, inequalities between vectors and inequalities between matrices are componentwise. Matrix In denotes the identity matrix in
