On the polyhedral method of control synthesis for an enhanced evasion problem for discrete-time systems with bilinearity and state constraints
E. K. Kostousova N.N. Krasovskii Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciences, Ekaterinburg
Abstract:
The evasion problem under uncertainty is considered for discrete-time systems with an initially linear structure and state constraints, where controls
$u$,
$U$, and
$v$ act;
$u$ and
$v$ enter additively, and
$U$ enters into the system matrix. In the considered control synthesis problem, which we call the enhanced evasion problem, the aim of
$v$ is either to avoid the trajectory to hit a given terminal set at a given final moment, as well as a sequence of sets specified at previous moments, or to violate at least one of the state constraints, whatever the admissible realizations of
$u$ and
$U$. The presence of
$U$ introduces nonlinearity into the systems and leads to bilinear type systems. It is assumed that the terminal and intermediate sets are parallelepipeds, the controls
$u$ and
$v$ are constrained by parallelotope-valued constraints,
$U$ by interval constraints, and the state constraints are specified in the form of zones. A polyhedral method for synthesizing controls
$v$ is developed using polyhedral (parallelepiped-valued) tubes, which can be found from recurrence relations using explicit formulas. To obtain a solution to the problem under consideration, a solution to an auxiliary one-step polyhedral evasion problem with bilinearity is found. Its connections with the problems of interval analysis concerning the so-called sets of quantifier solutions to interval equations are noted. Examples illustrating the efficiency of the method are given.
Keywords:
uncertain systems, evasion problem, control synthesis, bilinear systems, state constraints, polyhedral methods, parallelepipeds, interval analysis.
UDC:
517.977
MSC: 93C41,
93C55,
93C10,
93B52,
52B12 Received: 04.02.2025
Revised: 14.03.2025
Accepted: 17.03.2025
DOI:
10.21538/0134-4889-2025-31-2-125-140
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