External polyhedral estimates of reachable sets of discrete-time systems with integral bounds on additive terms

Elena K. Kostousova

doi:10.3934/mcrf.2021015

Article Contents

2021, Volume 11, Issue 3: 625-641. Doi: 10.3934/mcrf.2021015

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External polyhedral estimates of reachable sets of discrete-time systems with integral bounds on additive terms

Elena K. Kostousova

Krasovskii Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciences, 16, S.Kovalevskaja street, Ekaterinburg, 620108, Russia

Received: January 31, 2019

Revised: September 30, 2019

Early access: March 2021

Published: September 2021

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Abstract

We deal with the reachability problem for linear and bilinear discrete-time uncertain systems under integral non-quadratic constraints on additive input terms and set-valued constraints on initial states. The bilinearity is caused by an interval type uncertainty in coefficients of the system. Algorithms for constructing external parallelepiped-valued (shorter, polyhedral) estimates of reachable sets are presented. For linear time-invariant systems, two techniques for constructing touching external estimates with constant orientation matrices are described and compared. For time-dependant bilinear systems, parallelepiped-valued estimates are constructed. For bilinear systems with constant coefficients, nonconvex estimates are proposed in the form of unions of parallelepipeds. Evolution of all estimates is determined by systems of recurrence relations.

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Mathematics Subject Classification: Primary: 93B03, 93C05, 93C10, 93C55, 93C41; Secondary: 52B12.

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Figure 1. Touching external estimates for $ \mathcal{X}[\cdot] $ and $ \mathcal{X}[N] $ from $ \mathfrak{P}^{1+} $ (figures (a), (b)) and $ {\hat {\mathfrak{P}}}^{2+} $ (figures (c), (d)) in Example 1

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Figure 2. External estimates for $ \mathcal{X}[\cdot] $ and $ \mathcal{X}[N] $ in Example 2

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Figure 3. External estimates for $ \mathcal{Y}[N] $ for three cases in Example 3: $ \mathcal{P}^{II+}{[N]} $ from Theorem 5.2 (dashed lines), $ \mathcal{P}^{+, {\rm mod}}{[N]} $ from (49) (solid thick lines), and estimates in the form of unions (48) of several parallelepipeds, not intersections as in previous figures

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