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An improved characterization of bounded realness for singular delay systems and its applications. (English) Zbl 1284.93117
Summary: This paper is concerned with establishing a delay-dependent bounded real lemma (BRL) for singular systems with a time delay. Without resorting to any bounding techniques for some cross terms and model transformation, a new version of BRL for such systems is proposed, which guarantees a singular system to be regular, impulse free and stable while satisfying a prescribed $$H_{\infty }$$ performance level for any delays smaller than a given upper bound. Based on this, an $$H_{\infty }$$ state feedback controller is designed via a linear matrix inequality approach. The BRL, stability as well as $$H_{\infty }$$ results developed in this paper are less conservative than existing ones in the literature, which is demonstrated by providing some numerical examples.

##### MSC:
 93C15 Control/observation systems governed by ordinary differential equations
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##### References:
 [1] Theory of Functional Differential Equations. Springer: New York, 1977. · doi:10.1007/978-1-4612-9892-2 [2] Jeung, Automatica 32 pp 1229– (1996) [3] . Introduction to the Theory and Applications of Functional Differential Equations. Kluwer Academic Publishers: Dordrecht, 1999. · doi:10.1007/978-94-017-1965-0 [4] Delay Effects on Stability: A Robust Control Approach. Springer: Berlin, 2001. [5] Verriest, Kybernetika 37 pp 229– (2001) [6] He, Systems and Control Letters 51 pp 57– (2004) [7] Jeung, IEEE Transactions on Automatic Control 43 pp 971– (1998) [8] Li, Automatica 33 pp 1657– (1997) [9] Xu, IEEE Transactions on Automatic Control 46 pp 1321– (2001) [10] Xu, International Journal of Systems Science 33 pp 1195– (2002) [11] Singular Control Systems. Springer: Berlin, 1989. · doi:10.1007/BFb0002475 [12] Lewis, Circuits, Systems and Signal Processing 5 pp 3– (1986) [13] Lin, Linear Algebra and its Applications 297 pp 133– (1999) [14] Verghese, IEEE Transactions on Automatic Control 26 pp 811– (1981) [15] Lin, International Journal of Control 73 pp 407– (2000) [16] Lin, IEEE Transactions on Automatic Control 44 pp 1768– (1999) [17] Liu, Linear Algebra and its Applications 263 pp 377– (1997) [18] Liu, International Journal of Systems Science 32 pp 1205– (2001) [19] Zhang, International Journal of Control 72 pp 39– (1999) [20] Xu, IEEE Transactions on Automatic Control 47 pp 1122– (2002) [21] Xu, IEEE Transactions on Circuits and Systems I 49 pp 551– (2002) [22] Xu, International Journal of Robust and Nonlinear Control 13 pp 1213– (2003) [23] Fridman, Linear Algebra and its Applications 351352 pp 271– (2002) [24] Fridman, IEEE Transactions on Automatic Control 47 pp 253– (2002) [25] Fridman, IEEE Transactions on Automatic Control 47 pp 1931– (2002) [26] . Robust Control and Filtering of Singular Systems. Springer: Berlin, 2006. · Zbl 1114.93005
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