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RECENT ADVANCES IN ROBUST CONTROL – NOVEL APPROACHES AND DESIGN METHODSE Part 15

Tham khảo tài liệu 'recent advances in robust control – novel approaches and design methodse part 15', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | Model Based p-Synthesis Controller Design for Time-Varying Delay System 409 8 x t Ax t Adx t 5 t 6 where 0 5 t 5 Q1 XAdA XAdAd 61 1X 0 62 1X 0 7 8 Q1 5 1 A Ad TX X A Ad 61 1 62 1 X. 9 Theorem 2 Park 1999 The system given in 6 with time-varying delay is asymptotically stable for any delay 5 t satisfying condition 7 if there exist matrix P 0 Q 0 V 0 and W such that Q2 WTAd Q ATaTv ATATB -V 0 0 0 0 10 V where Q2 A Ad TP P A B WtB BtW Q 11 0 Ỗ WT P . 12 Theorem 3 Tang Liu 2008 The system given in 6 with time-varying delay which satisfies 7 is asymptotically stable for any delay 5 t which satisfies 13 if there exist matrices P 0 Q 0 Z 0 Y and W such that the following linear matrix inequality LMI holds 0 5 t 5 5 t V 1 13 where Q3 Y PAd WT YdATZ W WT 1 v Q WdATZ Z 0 Z 0 Q3 PA AtP Y Yt Q. 14 15 In packet based networked system the condition 5 t 1 implies that the preceding packet is not caught up by the successive packet. 2.4 Delay dependent stability analysis in frequency domain In frequency domain based Nyquist criterion gives necessary and sufficient condition and the eigen value based analysis described below is another option of the analysis. Lemma 2 The system 2 is asymptotically stable for all 5 6 0 5 if and only if ý jw 5 0 xw 0 where ý s 5 sIn A Ade s5 . Corollary 1 The system 2 is asymptotically stable for all 5 6 0 5 if and only if det In G j w jỗ w 0 w 0 16 410 Recent Advances in Robust Control - Novel Approaches and Design Methods Fig. 2. Robust control based method where G s F sIn A 1 H Ad HF A A Ad and Ỗs ỗs Iq ỗs e Ỗs 1. Lemma 2 and Corollary 1 requires solving a transcendental equation. Thus another set A jw which covers Ỗs is chosen. This selection of set A jw seriously effects on conservativeness. Zhang proposed very less conservative method using modified Pade approximation. It gives very less conservative Ỗ which is very close to the Nyquist criterion. The eigen value analysis including Pade based method can be only applicable for time invariant .