On polyhedral control synthesis for dynamical discrete-time systems under uncertainties and state constraints

Elena K. Kostousova

doi:10.3934/dcds.2018153

Article Contents

2018, Volume 38, Issue 12: 6149-6162. Doi: 10.3934/dcds.2018153

This issue Previous Article Combinatorial approach to detection of fixed points, periodic orbits, and symbolic dynamics Next Article Principal Floquet subspaces and exponential separations of type Ⅱ with applications to random delay differential equations

On polyhedral control synthesis for dynamical discrete-time systems under uncertainties and state constraints

Elena K. Kostousova

Krasovskii Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciences, 16, S.Kovalevskaja street, Ekaterinburg, 620990, Russia

Received: August 31, 2017

Revised: November 30, 2017

Published: November 2018

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Abstract

We deal with a problem of target control synthesis for dynamical bilinear discrete-time systems under uncertainties (which describe disturbances, perturbations or unmodelled dynamics) and state constraints. Namely we consider systems with controls that appear not only additively in the right hand sides of the system equations but also in the coefficients of the system. We assume that there are uncertainties of a set-membership kind when we know only the bounding sets of the unknown terms. We presume that we have uncertain terms of two kinds, namely, a parallelotope-bounded additive uncertain term and interval-bounded uncertainties in the coefficients. Moreover the systems are considered under constraints on the state ("under viability constraints"). We continue to develop the method of control synthesis using polyhedral (parallelotope-valued) solvability tubes. The technique for calculation of the mentioned polyhedral tubes by the recurrent relations is presented. Control strategies, which can be constructed on the base of the polyhedral solvability tubes, are proposed. Illustrative examples are considered.

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Mathematics Subject Classification: Primary: 93C41, 93C55, 93B52, 93C05, 93C10; Secondary: 93B40, 52B12.

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Figure 1. Results of polyhedral synthesis for case (A) using the following controls: only control $u $, only $U $, both $u$ and $U$

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Figure 2. Results of polyhedral synthesis with both controls $u$ and $U$ for cases (A), (B, ⅱ), and (B, ⅱ; SC)

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Figure 3. Suitable polyhedral tubs $\mathcal{P}^{-}{[\cdot]}$ and corresponding controlled trajectories with both $u$ and $U$ for cases (A) and (B, ⅱ; SC)

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Figure 4. Corresponding controls $u$ and $U$ for cases (A) and (B, ⅱ; SC)

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