Abstract
This paper deals with the problem of optimal guaranteed cost control for linear systems with interval time-varying delayed state and control. The time delay is assumed to be a continuous function belonging to a given interval, but not necessary to be differentiable. A linear–quadratic cost function is considered as a performance measure for the closed-loop system. By constructing a set of augmented Lyapunov–Krasovskii functional combined with Newton–Leibniz formula, a guaranteed cost controller design is presented and sufficient conditions for the existence of a guaranteed cost state-feedback for the system are given in terms of linear matrix inequalities (LMIs). Numerical examples are given to illustrate the effectiveness of the obtained result.
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Hale, J.K., Verduyn Lunel, S.M.: Introduction to Functional Differential Equations. Springer, New York (1993)
Kolmanovskii, V., Myshkis, A.: Applied Theory of Functional Differential Equations. Kluwer Academic, Dordrecht (1992)
Udwadia, F.: Noncollocated control of distributed-parameter nondispersive systems with tip Inertias using time Delays. Appl. Math. Comput. 47(1), 47–75 (1992)
Udwadia, F.E., Kumar, R.: Time-delayed control of classically damped structural systems. Int. J. Control 60(5), 687–713 (1994)
Udwadia, F.E., Hosseini, M., Chen, Y.: Robust control of uncertain systems with time-varying delays in control input. In: Proc. of the American Control Conference, USA, pp. 3840–3845 (1997)
Udwadia, F.E., von Bremen, H., Kumar, R., Hosseini, M.: Time delayed control of structural systems. Earthquake Eng. Struct. Dyn. 32(2), 495–535 (2003)
Udwadia, F.E., Phohomsiri, P.: Active control of structures Using time delayed positive feedback proportional control designs. Struct. Control Health Monit. 13(1), 536–552 (2006)
Udwadia, F.E., Hubertus von, B, Phohomsiri, P.: Time-delayed control design for active control of structures: principles and applications. Struct. Control Health Monit. 14(1), 27–61 (2007)
Fu, D., Bai, Y., Sun, M.: Delay-dependent H ∞ dynamic output feedback control for systems with time-varying delay. In: IEEE International Conference on Control and Automation Christchurch, New Zealand, IEEE, New York (2009)
Chang, S.S.L., Peng, T.K.C.: Adaptive guaranteed cost control of systems with uncertain parameters. IEEE Trans. Autom. Control 17(3), 474–483 (1972)
Petersen, I.R., Macfarlane, D.C.: Optimal guaranteed cost control and filtering uncertain linear systems. IEEE Trans. Autom. Control 39(10), 1971–1977 (1994)
Fischman, A., Dion, J.M., Dugard, L., Neto, A.T.: A linear matrix inequality approach for guaranteed cost control. In: Proc. 13th IFAC World Congress, San Fransisco, USA, vol. 4, pp. 197–202 (1996)
Yu, L., Chu, J.: An LMI approach to guaranteed cost control of linear uncertain time-delay systems. Automatica 35(6), 1155–1159 (1999)
Li, H., Niculescu, S.L., Dugard, L., Dion, J.M.: Robust guaranteed cost control of uncertain linear time-delay systems using dynamic output feedback. Math. Comput. Simul. 45, 349–358 (1998)
Costa, E.F., Oliveira, V.A.: On the design of guaranteed cost controllers for a class of uncertain linear systems. Syst. Control Lett. 46(1), 17–29 (2002)
Park, J.H.: Delay-dependent criterion for guaranteed cost control of neutral delay systems. J. Optim. Theory Appl. 124(3), 491–502 (2005)
Shi, P., Boukas, E.K., Shi, Y., Kagarwal, R: Optimal guaranteed cost control of uncertain discrete time-delay systems. J. Comput. Appl. Math. 157(3), 435–451 (2003)
Chen, W.H., Guan, Z.H., Lu, X.: Delay-dependent guaranteed cost control for uncertain discrete-time systems with both state and input delay. J. Franklin Inst. 341, 419–430 (2004)
Zuo, Z.Q., Wang, Y.J.: Novel optimal guaranteed cost control of uncertain discrete systems with both state and input delays. J. Optim. Theory Appl. 139(1), 159–170 (2008)
Yang, D., Cai, K.Y.: Reliable guaranteed cost sampling control for nonlinear time-delay systems. Math. Comput. Simul. 80(10), 2005–2018 (2010)
Yang, J., Luo, W., Li, G., Zhong, S.: Reliable guaranteed cost control for uncertain fuzzy neutral systems. Nonlinear Anal. Hybrid Syst 4(2), 644–658 (2010)
Kwon, O.M., Park, J.H., Lee, S.M.: Exponential stability for uncertain dynamic systems with time-varying delays: LMI optimization approach. J. Optim. Theory Appl. 137(3), 521–532 (2008)
Hien, L.V., Phat, V.N.: Exponential stability and stabilization of a class of uncertain linear time-delay systems. J. Franklin Inst. 346, 611–625 (2009)
Nam, P.T., Phat, V.N.: Robust stabilization of linear systems with delayed state and control. J. Optim. Theory Appl. 140(2), 287–299 (2009)
Phat, V.N, Ha, Q.P., Trinh, H.: Parameter-dependent H ∞ control for time-varying delay polytopic systems. J. Optim. Theory Appl. 147(1), 58–70 (2010)
Shao, H.: New delay-dependent stability criteria for systems with interval delay. Automatica 45(3), 744–749 (2009)
Yoneyama, J.: Robust guaranteed cost control of uncertain fuzzy systems under time-varying sampling. Appl. Soft Comput. 11(2), 249–255 (2011)
Gu, K., Kharitonov, V.L., Chen, J.: Stability of Time-Delay Systems. Birkhauser, Boston (2003)
Boyd, S., Ghaoui, El., Feron, E., Balakrishnan, V.: Linear Matrix Inequalities and Control Theory. SIAM Studies in Applied Mathematic, vol. 15, SIAM, Philadelphia (1994)
Gahinet, P., Nemirovskii, A., Laub, A.J., Chilali, M.: LMI Control Toolbox for Use with MATLAB. The MathWorks, Natick (1995)
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Thuan, M.V., Phat, V.N. Optimal Guaranteed Cost Control of Linear Systems with Mixed Interval Time-Varying Delayed State and Control. J Optim Theory Appl 152, 394–412 (2012). https://doi.org/10.1007/s10957-011-9920-5
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DOI: https://doi.org/10.1007/s10957-011-9920-5