Journal of Optimization Theory and Applications

, Volume 152, Issue 2, pp 394–412 | Cite as

Optimal Guaranteed Cost Control of Linear Systems with Mixed Interval Time-Varying Delayed State and Control

  • M. V. Thuan
  • V. N. PhatEmail author


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.


Guaranteed cost control Stability Stabilization Interval delay Lyapunov method Linear matrix inequality 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Hale, J.K., Verduyn Lunel, S.M.: Introduction to Functional Differential Equations. Springer, New York (1993) zbMATHGoogle Scholar
  2. 2.
    Kolmanovskii, V., Myshkis, A.: Applied Theory of Functional Differential Equations. Kluwer Academic, Dordrecht (1992) Google Scholar
  3. 3.
    Udwadia, F.: Noncollocated control of distributed-parameter nondispersive systems with tip Inertias using time Delays. Appl. Math. Comput. 47(1), 47–75 (1992) MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Udwadia, F.E., Kumar, R.: Time-delayed control of classically damped structural systems. Int. J. Control 60(5), 687–713 (1994) MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    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) Google Scholar
  6. 6.
    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) CrossRefGoogle Scholar
  7. 7.
    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) MathSciNetCrossRefGoogle Scholar
  8. 8.
    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) CrossRefGoogle Scholar
  9. 9.
    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) Google Scholar
  10. 10.
    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) MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Petersen, I.R., Macfarlane, D.C.: Optimal guaranteed cost control and filtering uncertain linear systems. IEEE Trans. Autom. Control 39(10), 1971–1977 (1994) zbMATHCrossRefGoogle Scholar
  12. 12.
    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) Google Scholar
  13. 13.
    Yu, L., Chu, J.: An LMI approach to guaranteed cost control of linear uncertain time-delay systems. Automatica 35(6), 1155–1159 (1999) MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    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) MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    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) MathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    Park, J.H.: Delay-dependent criterion for guaranteed cost control of neutral delay systems. J. Optim. Theory Appl. 124(3), 491–502 (2005) MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    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) MathSciNetzbMATHCrossRefGoogle Scholar
  18. 18.
    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) MathSciNetzbMATHCrossRefGoogle Scholar
  19. 19.
    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) MathSciNetzbMATHCrossRefGoogle Scholar
  20. 20.
    Yang, D., Cai, K.Y.: Reliable guaranteed cost sampling control for nonlinear time-delay systems. Math. Comput. Simul. 80(10), 2005–2018 (2010) MathSciNetzbMATHCrossRefGoogle Scholar
  21. 21.
    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) MathSciNetzbMATHCrossRefGoogle Scholar
  22. 22.
    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) MathSciNetzbMATHCrossRefGoogle Scholar
  23. 23.
    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) MathSciNetzbMATHCrossRefGoogle Scholar
  24. 24.
    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) MathSciNetzbMATHCrossRefGoogle Scholar
  25. 25.
    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) MathSciNetzbMATHCrossRefGoogle Scholar
  26. 26.
    Shao, H.: New delay-dependent stability criteria for systems with interval delay. Automatica 45(3), 744–749 (2009) MathSciNetzbMATHCrossRefGoogle Scholar
  27. 27.
    Yoneyama, J.: Robust guaranteed cost control of uncertain fuzzy systems under time-varying sampling. Appl. Soft Comput. 11(2), 249–255 (2011) CrossRefGoogle Scholar
  28. 28.
    Gu, K., Kharitonov, V.L., Chen, J.: Stability of Time-Delay Systems. Birkhauser, Boston (2003) zbMATHCrossRefGoogle Scholar
  29. 29.
    Boyd, S., Ghaoui, El., Feron, E., Balakrishnan, V.: Linear Matrix Inequalities and Control Theory. SIAM Studies in Applied Mathematic, vol. 15, SIAM, Philadelphia (1994) CrossRefGoogle Scholar
  30. 30.
    Gahinet, P., Nemirovskii, A., Laub, A.J., Chilali, M.: LMI Control Toolbox for Use with MATLAB. The MathWorks, Natick (1995) Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  1. 1.Department of MathematicsThai Nguyen UniversityThai NguyenVietnam
  2. 2.Institute of MathematicsVASTHanoiVietnam

Personalised recommendations