On stabilizabilityholdability problem for linear discrete time systems
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Consider the following problem. Given a linear discretetime system, find if possible a linear statefeedback control law such that under this law all system trajectories originating in the nonnegative orthant remain nonnegative while asymptotically converging to the origin. This problem is called feedback stabilizabilityholdabiltiy problem (FSH). If, in addition, the requirement of nonnegativity is imposed on the controls, the problem is a positive feedback stabilizabilityholdabiltiy problem (PFSH). It is shown that the set of all linear state feedback controllers that make the openloop system holdable and asymptotically stable is a polyhedron and the external representation of this polyhedron is obtained. Necessary and sufficient conditions for identifying when the openloop system is not positive feedback R+ninvariant (and therefore there is no solution to the PFSH problem) are obtained in terms of the system parameters. A constructive linear programming based approach to the solution of FSH and PFSH problems is developed in the paper. This approach provides not only a simple computational procedure to find out whether the FSF, respectively the PFSH problem, has a solution or not but also to determine a linear state feedback controller (respectively, a nonnegative linear state feedback controller) that endows the closedloop (positive) system with a maximum stability margin and guarantees the fastest possible convergence to the origin.
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