Read e-book online Stochastic Switching Systems: Analysis and Design (Control PDF

By El-Kabir Boukas

ISBN-10: 0817637826

ISBN-13: 9780817637828

An introductory bankruptcy highlights fundamentals suggestions and sensible types, that are then used to resolve extra complex difficulties in the course of the e-book. incorporated are many numerical examples and LMI synthesis equipment and layout ways.

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Mean exponentially stable (MES) if there exist positive constants α and β such that the following holds for any initial condition (x0 , r0 ): E x(t) 2 |x0 , r0 ≤ α x0 e−βt . 9) Remark 4. From the previous definitions, we can see that MES implies MSS and SS. 1) as follows. Definition 2. 1) is said to be 1. 7) holds for any initial condition (x0 , r0 ) and for all admissible uncertainties; 2. 9) holds for any initial condition (x0 , r0 ) and for all admissible uncertainties. Remark 5. From these definitions, we can see that RMES implies RSS.

18) By Dynkin’s formula, we obtain t E[V (x(t), i)] − V (x0 , r0 ) = E L V (x(s), r(s))ds 0 t x⊤ (s)x(s)ds|(x0 , r0 ) , ≤ − min{λmin (Q(i))}E i∈S 0 implying, in turn, t min{λmin (Q(i))}E i∈S x⊤ (s)x(s)ds|(x0 , r0 ) 0 ≤ E [V (x(0), r0 )] − E [V (x(t), i)] ≤ E [V (x(0), r0 )] . This yields that t E 0 x⊤ (s)x(s)ds|(x0 , r0 ) ≤ E [V (x(0), r0 )] mini∈S {λmin (Q(i))} holds for any t > 0. 1) is exponentially mean square stable and therefore stochastically stable. This completes the proof of sufficiency.

2 . 0). 0689 . 1378 As can be seen, the two matrices are symmetric and positive-definite, therefore based on the result of the previous theorem, the studied system is stochastically stable. Example 8. In this example let us consider a system with two modes. 0 . 2 The transition probability rate matrix between these modes is given by Λ= −1 1 . 2). 0 π1 + π2 = 1, which gives π1 = 56 and π2 = 61 . This means that the system spends more time in the stable mode, which therefore affects the stability of the global system.

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Stochastic Switching Systems: Analysis and Design (Control Engineering) by El-Kabir Boukas


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