Download Analysis and Synthesis of Dynamical Systems with Time-Delays by Yuanqing Xia PDF

By Yuanqing Xia

Time-delay happens in lots of dynamical structures comparable to organic platforms, chemical structures, metallurgical processing platforms, nuclear reactor, lengthy transmission strains in pneumatic, hydraulic structures and electric networks. particularly, in recent times, time-delay which exists in networked keep watch over platforms has introduced extra complicated challenge right into a new learn quarter. often, it's a resource of the new release of oscillation, instability and negative functionality. huge attempt has been utilized to assorted facets of linear time-delay platforms in the course of contemporary years. as the advent of the hold up issue renders the method research extra advanced, as well as the problems brought on by the perturbation or uncertainties, within the keep an eye on of time-delay structures, the issues of strong balance and strong stabilization are of significant importance.

This ebook offers a few uncomplicated theories of balance and stabilization of structures with time-delay, that are on the topic of the most ends up in this publication. extra recognition should be paid on synthesis of platforms with time-delay. that's, sliding mode regulate of platforms with time-delay, networked regulate platforms with time-delay, networked facts fusion with random delay.

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Example text

According to the theory of switched systems [172], we have the following result. 5. 1) with time-varying delay τ (k) is stable if there exists a positive definite matrix P ∈ R(¯τ +1)n×(¯τ +1)n such that the following linear inequalities hold for any i ∈ {0, 1, · · · , τ¯}: A¯Ti P A¯i − P < 0. 5, the following example, which is taken from [51], is investigated. 6. 1 where τ (k) is a bounded time-varying delay. 1) is asymptotically stable for all 2 ≤ τ (k) ≤ 7, that is, the maximum delay is τ¯ = 7.

20). 31) t+θ with P , S1 and S2 are symmetric positive-definite matrices to be chosen. It is easy to see that there exist positive scalars β1 and β2 such that β1 ||η(t)||2 ≤ V (η(t), t) ≤ β2 sup ||η(t + θ)||2 . 36) η T (t + θ − τ )S2 η(t + θ − τ )dθ. 37) Since for any vectors u, v and any matrix S > 0 of appropriate dimension, the following inequality holds: 56 5 Robust Delay-Dependent SMC for Uncertain Time-Delay Systems − 2uT v ≤ uT Su + v T S −1 v. 39) μ2 (η(t), t) ≤ τ η (t)P A˜Td1 A˜T1 S2−1 A˜1 A˜d1 P η(t) 0 + −τ η T (t + θ − τ )S2 η(t + θ − τ ).

22) T where V11 = J + ZA11 − Y T A12 + A11 Z − A12 Y + U2T GXGU2T + (ZU2T − Y T U1T ) ×H T X −1 H(U2 Z − U1 Y ) + U2T Gd Xd GTd U2 , V22 = −J + (ZU2T − Y T U1T )HdT Xd−1 Hd (U2 Z − U1 Y ). 16) and Ad11 Z − Ad12 Y V11 T T V22 ZAd11 − Y T Ad12 < 0. Hence, W < 0, which also implies that there exists a sufficiently small ξ1 > 0 such that W+ ξ1 I(n−m)×(n−m) 0(n−m)×(n−m) 0(n−m)×(n−m) 0(n−m)×(n−m) < 0. 19) and above inequality that V˙ (z1 (t), t) ≤ −ξ1 ||P z1 ||2 ≤ −ξ1 λmin (P )||z1 ||2 for all (z1 (t), t) ∈ Rn−m × R.

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