On the polyhedral method of solving problems of control strategy synthesis in discrete-time systems with uncertainties and state constraints

We consider control synthesis problems for linear and bilinear discrete-time systems under uncertainties and state constraints. Two types of problems are studied: when controls are additive and when they appear in the system matrix. For both problems we consider cases without uncertainty and cases with uncertainty, including additive parallelotope-bounded uncertainties and interval uncertainties in the matrix of the system. We continue to develop the methods of “polyhedral” control synthesis with the use of polyhedral (parallelotope-valued) solvability tubes. Namely, the technique proposed by the author earlier for solving the first problem is expanded for the case of matrix uncertainties. Further, for both problems, a uniform solution scheme is developed, which makes it possible to construct control strategies by explicit formulas. Descriptions of the polyhedral solvability tubes in the form of systems of nonlinear recurrence relations are given. Control strategies, which can be constructed using these polyhedral solvability tubes, are described. For the first problem, both techniques produce the same polyhedral solvability tubes, but the control strategies turn out to be different; an interrelation between the controls of both types is indicated. Results of computer simulations are presented.