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# Article

 Title: Infinite-dimensional LMI approach to analysis and synthesis for linear time-delay systems (English) Author: Ikeda, Kojiro Author: Azuma, Takehito Author: Uchida, Kenko Language: English Journal: Kybernetika ISSN: 0023-5954 Volume: 37 Issue: 4 Year: 2001 Pages: [505]-520 Summary lang: English . Category: math . Summary: This paper considers an analysis and synthesis problem of controllers for linear time-delay systems in the form of delay-dependent memory state feedback, and develops an Linear Matrix Inequality (LMI) approach. First, we present an existence condition and an explicit formula of controllers, which guarantee a prescribed level of $L^2$ gain of closed loop systems, in terms of infinite-dimensional LMIs. This result is rather general in the sense that it covers, as special cases, some known results for the cases of delay- independent/dependent and memoryless/memory controllers, while the infinity dimensionality of the LMIs makes the result difficult to apply. Second, we introduce a technique to reduce the infinite-dimensional LMIs to a finite number of LMIs, and present a feasible algorithm for synthesis of controllers based on the finite-dimensional LMIs. (English) Keyword: time-delay system Keyword: linear system Keyword: LMI MSC: 93B50 MSC: 93B51 MSC: 93C23 MSC: 93D05 idZBL: Zbl 1265.93146 idMR: MR1859098 . Date available: 2009-09-24T19:41:20Z Last updated: 2015-03-26 Stable URL: http://hdl.handle.net/10338.dmlcz/135423 . Reference: [1] Azuma T., Ikeda, K., Uchida K.: Infinite-dimensional LMI approach to $H^\infty$ control synthesis for linear systems with time-delay.In: Proc. of ECC99, 1999 Reference: [2] Azuma T., Kondo, T., Uchida K.: Memory state feedback control synthesis for linear systems with time delay via a finite number of linear matrix inequalities.In: Proc. of IFAC Workshop on Linear Time Delay Systems 1998, pp. 183–187 Reference: [3] Azuma T., Watanabe, R., Uchida K.: An approach to solving parameter-dependent LMI conditions based on finite number of LMI conditions.In: Proc. of American Control Conference 1997, pp. 510–514 Reference: [4] Boyd S., Ghaoui L. El, Feron, E., Balakrishnan V.: Linear Matrix Inequalities in System and Control Theory.SIAM Stud. Appl. Math. 15 (1994) Zbl 0816.93004, MR 1284712 Reference: [5] Souza C. E. de: Stability and stabilizability of linear state-delayed systems with multiplicative noise.In: Proc. of IFAC Workshop on Linear Time Delay Systems 2000, pp.21–26 Reference: [6] Dugard L., (eds.) E. I. Verriest: Stability and Control of Time-Delay Systems.(Lecture Notes in Control and Information Sciences 228.) Springer–Verlag, Berlin 1997 Zbl 0901.00019, MR 1482570 Reference: [7] Fattouh A., Senme, O., Dion J.-M.: $H_\infty$ controller and observer design for linear systems with point and distributed time-delay.In: Proc. of IFAC Workshop on Linear Time Delay Systems 2000, pp. 225–230 Reference: [8] Gu K.: Constrained LMI set in the stability problem of linear uncertain time-delay systems.In: Proc. of American Control Conference 1997, pp. 3657–3661 Reference: [9] Gu K.: Discretization of Lyapunov functional for uncertain time-delay systems.In: Proc. of American Control Conference 1997, pp. 505–509 Reference: [10] Hale J., Lunel S. M. V.: Introduction to Functional Differential Equations.Springer–Verlag 1993 Zbl 0787.34002, MR 1243878 Reference: [11] He J., Wang, Q., Lee T.: $H^\infty$ disturbance attenuation for state delayed systems.Systems Control Lett. 33 (1998), 105–114 MR 1607812, 10.1016/S0167-6911(97)00114-X Reference: [12] Ikeda K., Azuma, T., Uchida K.: A construction method of convex polyhedron in infinite number LMI approach for linear time-delay systems.In: Proc. of Annual Meeting of IEEJ 2000, pp. 1006–1007 (in Japanese) Reference: [13] Ikeda K., Uchida K.: Analysis of state reachable sets for linear time-delay systems.In: Proc. of SICE2000, #105A-1 (in Japanese) Reference: [14] Lee J. H., Kim S. W., Kwon W. H.: Memoryless $H_\infty$ controllers for state delayed systems.IEEE Trans. Automat. Control 39 (1994), 159–162 MR 1258692, 10.1109/9.273356 Reference: [15] Li X., Souza C. E. de: LMI approach to delay-dependent robust stability and stabilization of uncertain linear delay systems.In: Proc. of Conference on Decision $\&$ Control 1995, pp. 3614–3619 Reference: [16] Loiseau J. J., Brethe D.: An effective algorithm for finite spectrum assignment of single input systems with delay.In: Proc. of Symposium Modeling, Analysis and Simulation, IEEE–IMACS Conference Computational Engineering in Systems Applications 1996 Reference: [17] Louisell J.: A stability analysis for a class of differential-delay equations having time-varying delay.(Lecture Notes in Mathematics 1745.) Springer–Verlag, Berlin 1991, pp. 225–242 Zbl 0735.34063, MR 1132034 .

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