On invariant polyhedra of continuous-time linear systems. IEEE Transactions on Automatic Control, Vol. 38, No.11, 1993, pp.1680-85.
ON INVARIANT POLYHEDRA OF CONTINUOUS-TIME LINEAR SYSTEMS E.B. CASTELAN∗, J.C. HENNET† Laboratoire d’Automatique et d’Analyse des Syst`emes du CNRS, 7, avenue du Colonel Roche, 31077 Toulouse FRANCE Phone: (33) 61 33 62 00 - Telefax: (33) 61 55 35 77 Telex LAASTSE 520930F
Abstract : This paper presents some conditions of existence of positively invariant polyhedra for linear continuous-time systems. These conditions are first described algebraically, then interpreted on the basis of the system eigenstructure. Then, a simple state-feedback placement method is described for solving some linear regulation problems under constraints. Jean-Claude Hennet (Author to whom correspondence should be addressed) Telephone (33) 61 33 63 13 e-mail
[email protected] Eugˆ enio B. Castelan Telephone (33) 61 33 64 16 e-mail
[email protected] Keywords : Continuous time linear dynamical system, Positive invariance, Stability, Convex polyhedron, M -matrix.
∗ †
On leave from LCMI/EEL/UFSC, Florian´ opolis, Brazil, with research support from CAPES, Brazil. Also member of G.R. Automatique CNRS Pole SARTA.
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On invariant polyhedra of continuous-time linear systems. IEEE Transactions on Automatic Control, Vol. 38, No.11, 1993, pp.1680-85.
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Introduction
Any locally stable dynamical system admits some domains in its state-space from which any state-vector trajectory cannot escape. These domains are called positively invariant sets of the system. If a system is subject to constraints on its state vector and can be controlled, the purpose of a regulation law can be to stabilize it while maintaining its state-vector in a positively invariant set included in the admissible domain. Under a state feedback regulation law, this design technique can also be used to satisfy constraints on the control vector, possibly by transferring these constraints onto the state-space. The existence and characterization of positively invariant sets of dynamical systems is therefore a basic issue for many constrained regulation problems. If a norm or a semi-norm of the state vector, n(x), decrases along any admissible trajectory, then the system admits positively invariant sets defined by n(x) ≤ µ for some real positive numbers µ. In particular, many cases of constrained o linear systems can be solved by using polyhedral semin i| with G a linear operator of
