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

The e-book addresses the keep an eye on concerns comparable to balance research, keep watch over synthesis and clear out layout of Markov bounce structures with the above 3 sorts of TPs, and hence is principally divided into 3 components. half I reports the Markov bounce structures with in part unknown TPs. diversified methodologies with assorted conservatism for the fundamental balance and stabilization difficulties are constructed and in comparison. Then the issues of kingdom estimation, the keep an eye on of structures with time-varying delays, the case concerned with either in part unknown TPs and unsure TPs in a composite manner also are tackled. half II bargains with the Markov leap platforms with piecewise homogeneous TPs. Methodologies that may successfully deal with regulate difficulties within the state of affairs are built, together with the only dealing with the asynchronous switching phenomenon among the at the moment activated procedure mode and the controller/filter to be designed. half III makes a speciality of the Markov leap structures with reminiscence TPs. the concept that of σ-mean sq. balance is proposed such that the soundness challenge could 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 useful examples are given to ensure the effectiveness of the received theoretical effects. eventually, a few views and destiny works are awarded to finish the book.

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Additional resources for Analysis and Design of Markov Jump Systems with Complex Transition Probabilities

Example text

4) is stochastically j∈IK i stable (or stable for any switching sequence if IK = ∅, for all i ∈ I) with an H∞ performance index γ. 6). 11) i ˜ i i where β1 = inf{λmin (−( A˜ i PK Pi )), i ∈ I } and β2 = inf{(1 − πK ) min[λmin Ai − πK j (−( A˜ i P j A˜ i − Pi ))], i ∈ I }. 1. Note that β will reduce to only β1 (respectively, β2 ) if all the TPs are known (respectively, unknown). Now, to establish the H∞ performance for the system, consider the following performance index: J ∞ E e (k)e(k) − γ 2 w (k)w(k) k=0 under zero initial condition, V (x(k), ˜ rk ) |k=0 = 0, and we have ∞ J≤E ∞ = where ζ(k) k=0 k=0 x˜ (k) w (k) i e (k)e(k) − γ 2 w (k)w(k) + ζ (k) V i ζ(k) and A˜ i P¯ i B˜ i + C˜ i D˜ i A˜ i P¯ i A˜ i − Pi + C˜ i C˜ i 2 ∗ −γ I + B˜ i P¯ i B˜ i + D˜ i D˜ i P¯ i πi j P j + i j∈IK By Schur complement, i πi j P j = PK + i j∈IU K i < 0 is equivalent to: ⎡ i πi j P j i j∈IU K −P¯ i ⎢ ∗ ⎢ ⎣ ∗ ∗ ⎤ 0 P¯ i A˜ i P¯ i B˜ i −I C˜ i D˜ i ⎥ ⎥ < 0.

4 x1 Fig. 2 0 x 1 Fig. 3 ? 2 2 ? 1 ? 3 ? 4 ? 3 ? 3 ? ? 2 ? 2 ? 3 ? 4 ? 3 ? 8 , B2 = , B3 = , B4 = . 2. 4] under two different modes evolution. It is seen from the curves in Figs. 4 that, despite the partially unknown TPs, the designed controllers are feasible and effective ensuring the resulting closed-loop systems are stable, in the continuous-time or in discrete-time cases, respectively. The validity and the reduction of conservatism of the results obtained above are verified by the following numerical examples.

Now, the following example gives the verification on the results for the discretetime counterpart. 3 ? 2 ? ? 3 ? 5 ? 4 ? 3 ? 1 2 3 4 1 ? 3 ? 2 ? 5 ? 3 ? ? 4 x1 Fig. 2 0 x 1 Fig. 3 ? 2 2 ? 1 ? 3 ? 4 ? 3 ? 3 ? ? 2 ? 2 ? 3 ? 4 ? 3 ? 8 , B2 = , B3 = , B4 = . 2. 4] under two different modes evolution. It is seen from the curves in Figs. 4 that, despite the partially unknown TPs, the designed controllers are feasible and effective ensuring the resulting closed-loop systems are stable, in the continuous-time or in discrete-time cases, respectively.

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