By Zheng-Guang Wu

ISBN-10: 3642374964

ISBN-13: 9783642374968

ISBN-10: 3642374972

ISBN-13: 9783642374975

Singular time-delay platforms are very appropriate to explain loads of useful platforms resembling production structures, networked keep watch over platforms, energy structures and electric circuits. therefore, the prior twenty years have witnessed an important growth at the concept of singular time-delay structures, and lots of basic and demanding issues were effectively investigated together with balance research, stabilization, assured price keep an eye on, filtering, observer layout, sliding mode keep watch over etc. the most aim of this publication is to give the most recent advancements and references within the research and synthesis of singular time-delay platforms without or with Markov leaping parameters in a unified framework. The fabrics followed during this e-book are usually in response to study result of the authors. This e-book might be of curiosity to educational researchers operating in singular structures, time-delay structures and Markov bounce structures and to graduate scholars drawn to platforms and keep an eye on theory.

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**Additional resources for Analysis and Synthesis of Singular Systems with Time-Delays**

**Example text**

However, the delay-independent approach has been applied in [76] to obtain the results, which have been proved to be more conservative than the ones with the delay-dependent approach. By the free-weighting matrix method, the problems of robust passivity analysis and passivity-based sliding mode control have been addressed for a kind of uncertain singular time-delay systems in [149], where a delay-dependent suﬃcient condition has been proposed in terms of LMI, which guarantees the sliding mode dynamics to be generalized quadratically stable and robustly passive, and the passiﬁcation solvability condition has been also established.

63), we can obtain −1 T ˆ + Λ1 M α Σ ˇ 2 ΛT Λ2 < 0. 62) holds. 59). This completes the proof. 6, which may lead to some numerical problems when checking such nonstrict LMI condition since equality constraint is often fragile and usually not met perfectly. Applying the similar methods of [175, 178, 200], we introduce the matrix G ∈ Rn×(n−r) satisfying EG = 0 and rank G = n − r, and deﬁne L = L1 E T + GW T , where L1 > 0 and W ∈ Rn×(n−r) . Then, we have the following theorem. 7. 59) hold. 59) are solvable, the desired controller gain is given as K = V (L1 E T + GW T )−1 .

35) εeεt V (xt , t) − λeεt ||x(t)||2 . 35) from 0 to t, we get that t W (xt , t) eεs εV (xs , s) − λ x(s) W (x0 , 0) + 2 ds. 37) that for any t > 0 ke−εt φ(t) x(t)T E T P x(t) ζ1 (t) 2 2 d2 . 38) where α = λmin (P1 )−1 k. 38), a scalar m > 0 can be found such that for any t > 0, e(t) 2 me−εt φ(t) 2 d2 . 40) To study the exponential stability of ζ2 (t), we construct a function as J(t) = ζ2 (t)T Z22 ζ2 (t) − ζ2 (t − d(t))T Z22 ζ2 (t − d(t)). 33) with ζ2 (t)T P4T , we obtain that 0 = 2 ζ2 (t)T P4T ζ2 (t) + ζ2 (t)T P4T Ad4 ζ2 (t − d(t)) + ζ2 (t)T P4T e(t) .

### Analysis and Synthesis of Singular Systems with Time-Delays by Zheng-Guang Wu

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