By Yuanqing Xia

Time-delay happens in lots of dynamical structures similar to organic structures, chemical structures, metallurgical processing platforms, nuclear reactor, lengthy transmission strains in pneumatic, hydraulic structures and electric networks. particularly, lately, time-delay which exists in networked regulate structures has introduced extra advanced challenge right into a new study sector. often, it's a resource of the iteration of oscillation, instability and terrible functionality. enormous attempt has been utilized to varied features of linear time-delay structures in the course of contemporary years. as the creation of the hold up issue renders the method research extra advanced, as well as the problems attributable to the perturbation or uncertainties, within the regulate of time-delay platforms, the issues of strong balance and powerful stabilization are of serious importance.

This ebook provides a few uncomplicated theories of balance and stabilization of structures with time-delay, that are on the topic of the most leads to this booklet. extra recognition can be paid on synthesis of structures with time-delay. that's, sliding mode keep watch over of platforms with time-delay, networked regulate structures with time-delay, networked information fusion with random delay.

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**Example text**

9. [30] The following statements are equivalent i There exists a positive deﬁnite matrix P such that AT P A − P < 0. 16) There exist a positive deﬁnite matrix P and a matrix V such that −Q V T AT AV Q − V − V T < 0. 7 with less conservatism. 10. 11) is stable if there exist a positive deﬁnite V11 0 matrix Q ∈ R(τ +1)n×(τ +1)n and matrices V = with V11 ∈ V21 V22 Rn×n , V21 ∈ Rτ n×n , V22 ∈ Rτ n×τ n and Y ∈ Rm×n such that the following inequality holds ¯T −Q V T A¯T + I1T Y T B T ¯ ¯ AV + BY I1 Q−V −V < 0.

11) is stable if there exist a positive deﬁnite maQ11 0 trices Q = with Q11 ∈ Rn×n , Q22 ∈ Rτ n×τ n and a matrix 0 Q22 Y ∈ Rm×n such that the following inequalities holds ¯T −Q QA¯T + I1T Y T B ¯ ¯ AQ + BY I1 −Q < 0. 14) Then, the state feedback can be chosen as K = Y Q−1 11 with which the resulting closed-loop system is stable. Proof. 11) is stable if and only if there exists positive deﬁnite matrix P such that A˜T P A˜ − P < 0. 15) By Schur complement, the above inequality is equivalent to the following inequality, −P A˜T < 0.

In order to reduce the conservativeness, an extra variable can be introduced, and Q is still in a general form. 9. [30] The following statements are equivalent i There exists a positive deﬁnite matrix P such that AT P A − P < 0. 16) There exist a positive deﬁnite matrix P and a matrix V such that −Q V T AT AV Q − V − V T < 0. 7 with less conservatism. 10. 11) is stable if there exist a positive deﬁnite V11 0 matrix Q ∈ R(τ +1)n×(τ +1)n and matrices V = with V11 ∈ V21 V22 Rn×n , V21 ∈ Rτ n×n , V22 ∈ Rτ n×τ n and Y ∈ Rm×n such that the following inequality holds ¯T −Q V T A¯T + I1T Y T B T ¯ ¯ AV + BY I1 Q−V −V < 0.