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On invariant polyhedra of continuous-time systems subject to additive disturbances. (English) Zbl 0851.93046
Summary: This paper presents new necessary and sufficient algebraic conditions on the existence of positively ${\cal D}$-invariant polyhedra of continuous-time linear systems subject to additive disturbances. In particular, for a convex unbounded polyhedron containing the origin in its interior, it is also shown that the positive ${\cal D}$-invariance conditions can be split into two lower-dimensional sets of algebraic relations: the first corresponds to disturbance decoupling conditions and the second to positive ${\cal D}$-invariance conditions for bounded polyhedra of a reduced-order system. The stability of the overall system is discussed as well. By exploring the results obtained, a linear programming approach is proposed for solving a state-constrained regulator problem in the presence of additive disturbances.

##### MSC:
 93C73 Perturbations in control systems 93D09 Robust stability of control systems 90C05 Linear programming
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##### References:
 [1] Bitsoris, G.: Existence of polyhedral positively invariant sets for continuous-time systems. Control--theory adv. Technol 7, 407-427 (1991) [2] Blanchini, F.: Feedback control for linear time-invariant systems with state and control bounds in the presence of disturbances. IEEE trans. Autom. control 35, 1231-1234 (1990) · Zbl 0721.93036 [3] Carvalho, A. N.; Milani, B. E. A.: A simple design method for robust linear discrete-time regulators under symmetrical constraints. Proc. American control conf., 778-781 (1994) [4] Castelan, E. B.; Hennet, J. C.: Eigenstructure assignment for state constrained linear continuous-time systems. Automatica 28, 605-611 (1992) · Zbl 0766.93048 [5] Castelan, E. B.; Henet, J. C.: On invariant polyhedra of continuous-time linear systems. IEEE trans. Autom. control 38, 1680-1685 (1993) · Zbl 0790.93099 [6] Dórea, C. E. T.; Milani, B. E. A.: A computational method for optimal L-Q regulation with simultaneous disturbance decoupling. Automatica 31, 155-160 (1995) · Zbl 0825.93248 [7] Fletcher, R.: Constrained optimization. Practical methods of optimization 2 (1981) · Zbl 0474.65043 [8] Linnemann, A.: A condensed form for disturbance decoupling with simultaneous pole placement using state feedback. Proc. 10th IFAC world congress, Munich 9, 92-97 (1987) [9] Porter, B.: Eigenvalue assignment in linear multivariable systems by output feedback. Int. J. Control 25, 483-490 (1977) · Zbl 0347.93018 [10] Schneider, H.; Vidyasagar, M.: Cross-positive matrices. SIAM J. Numer. anal. 7, 508-519 (1970) · Zbl 0245.15008 [11] Tarbouriech, S.; Burgat, C.: Positively invariant sets for constrained continuous-time systems with cone properties. IEEE trans. Autom. control 39, 401-405 (1994) · Zbl 0800.93778 [12] Wonham, W. M.: Linear multivariable control. A geometric approach. (1979) · Zbl 0424.93001