Informatics and Applications

2025, Volume 19, Issue 3, pp 10-18

STABILIZATION OF THE TRAJECTORY OF A LINEAR SYSTEM WITH JUMPING DRIFT UNDER INTEGER CONTROL CONSTRAINTS

  • A. V. Bosov

Abstract

An integer-valued version of the stabilization problem for a linear stochastic differential system is considered where the drift evolves in a jumping manner determined by a Markov chain. The control objective is formalized via a quadratic cost functional. Depending on the conditions, both the full information case (the state of the chain is known) and the indirect observation case (the system state serves as an indirect observation of the unknown chain state) are possible. A distinguishing feature of the formulation lies in the integer constraints on the admissible control values. Unlike the previously solved unconstrained problem, the existence conditions for a solution are not satisfied in the "integer" formulation; therefore, an e-optimal solution is investigated. An e-optimal control can be obtained by discretizing the optimal solution of the unconstrained problem and applying mixed-integer nonlinear programming. However, the stochastic nature of the problem and the large number of switching scenarios prevent the guaranteed computational feasibility of solving it using dynamic programming. For practical implementation, a relaxation method is used: a heuristic approximation is computed as the result of an integer transformation of the e-optimal control in the unconstrained problem. Three variants of such transformations are proposed. A numerical experiment was conducted using the same applied model as in previous works on unconstrained control (position dynamics of a simple mechanical actuator). The results primarily confirm the applicability of the proposed solutions in terms of the stabilization objective and also allow for a comparison of the nature of the relaxation strategies.

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