Analysis and Design of Markov Jump Systems with Complex by Lixian Zhang, Ting Yang, Peng Shi, Yanzheng Zhu

By Lixian Zhang, Ting Yang, Peng Shi, Yanzheng Zhu

The e-book addresses the keep watch over concerns comparable to balance research, keep an eye on synthesis and filter out layout of Markov bounce structures with the above 3 sorts of TPs, and therefore is principally divided into 3 components. half I experiences the Markov bounce platforms with partly unknown TPs. assorted methodologies with diverse conservatism for the elemental balance and stabilization difficulties are constructed and in comparison. Then the issues of country estimation, the regulate of platforms with time-varying delays, the case concerned with either in part unknown TPs and unsure TPs in a composite method also are tackled. half II bargains with the Markov bounce platforms with piecewise homogeneous TPs. Methodologies which could successfully deal with keep an eye on difficulties within the situation are built, together with the only dealing with the asynchronous switching phenomenon among the at the moment activated method mode and the controller/filter to be designed. half III specializes in the Markov bounce structures with reminiscence TPs. the idea that of σ-mean sq. balance is proposed such that the soundness challenge may be solved through a finite variety of stipulations. The platforms concerned with nonlinear dynamics (described through the Takagi-Sugeno fuzzy version) also are investigated. Numerical and functional examples are given to ensure the effectiveness of the bought theoretical effects. eventually, a few views and destiny works are offered to finish the book.

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3 Stabilization 37 and ∀s ∈ {1, 2, . . 5) such that the closed-loop system is stochastically stable. 26). 19) holds. 19) is equivalent to ⎡ −Pi ∗ ∗ ⎢ √πiK1 PK1 Ai −PK1 ∗ ⎢ √ ⎢ πiK2 PK2 Ai 0 −PK2 ⎢ ⎢ .. .. ⎢ . . ⎢√ ⎢ πiKm PKm Ai 0 0 i i ⎣ (i) 1 − πK Pj Ai 0 0 ∗ ∗ ∗ .. 40). 37). 14, that is, the underlying system is stochastically stable. 26). 22 In contrast with the continuous-time case, the discrete-time case is relatively simpler since all the elements in the TPM are nonnegative and we need not distinguish the cases of diagonal elements known or unknown.

10), one can also readily obtain Θi < 0. 3), which completes the proof. 8 Note that if IU(i)K = ∅, ∀i ∈ I, the underlying system is the one with completely known TRs, which becomes the MJLS in the usual sense. 6). , the TRs are completely unknown, then the system can be if IK viewed as a switching linear system under arbitrary switching. 10) becomes Ai Pi + Pi Ai ≤ −Pj , ∀i = j ∈ I. , Pi = P. 10) are reduced to Ai P + PAi = −P < 0, namely, a latent quadratic common Lyapunov function will be shared among all the modes.

0, ⎥ . √ (i) (i) ⎥ πiKm PKm Ai ⎦ (i) −πK Pi 1 1 −Pj Pj Ai ∗ −Pi < 0. 36), respectively. 31). 32). e. the underlying system is stochastically stable. 26). This completes the proof. From the development in the above theorems, one can clearly see that our obtained stability and stabilization conditions actually cover the results for the usual MJLS and the switching linear systems under arbitrary switching (all the TPs are unknown). Therefore, the systems considered and corresponding criteria explored in the section are more general in hybrid systems field.

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