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Uniform Stabilizability of Parameter-Dependent Systems with State and Control Delays by Smooth-Gain Controls

  • Valery Y. GlizerEmail author
Technical Note
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Abstract

A linear time-invariant system with multiple point-wise and distributed delays in state and control is considered. The feature of the system is that its coefficients depend on a parameter, varying in some finite closed interval. An exponential stabilizability of this system by a memory-less state-feedback control with a parameter-dependent gain is studied. Using the linear matrix inequality approach, sufficient conditions for such a stabilizability with a smooth gain in the control are derived. An illustrative example is presented.

Keywords

Linear parameter-dependent system State and control delays Memory-less parameter-dependent smooth-gain control Stabilizability Lyapunov–Krasovskii functional Linear matrix inequality 

Mathematics Subject Classification

34K06 93C23 93D15 

Notes

References

  1. 1.
    Dontchev, A.L.: Perturbations. Approximations and Sensitivity Analysis of Optimal Control Systems. Springer, New York (1983)CrossRefzbMATHGoogle Scholar
  2. 2.
    Kokotovic, P.V., Khalil, H.K., O’Reilly, J.: Singular Perturbation Methods in Control: Analysis and Design. Academic Press, London (1986)zbMATHGoogle Scholar
  3. 3.
    Vasil’eva, A.B., Butuzov, V.F., Kalachev, L.V.: The Boundary Function Method for Singular Perturbation Problems. SIAM, Philadelphia (1995)CrossRefzbMATHGoogle Scholar
  4. 4.
    Gershon, E., Shaked, U., Yaesh, I.: \(H_{\infty }\) Control and Estimation of State-Multiplicative Linear Systems, Lecture Notes in Control and Information Sciences, vol. 318. Springer, London (2005)zbMATHGoogle Scholar
  5. 5.
    Boltyanski, V.G., Poznyak, A.S.: The Robust Maximum Principle: Theory and Applications. Birkhauser, Boston (2010)zbMATHGoogle Scholar
  6. 6.
    Fridman, E.: Introduction to Time-Delay Systems. Birkhauser, New York (2014)CrossRefzbMATHGoogle Scholar
  7. 7.
    Briat, C.: Linear Parameter-Varying and Time-Delay Systems: Analysis, Observation, Filtering & Control. Springer, New York (2015)CrossRefzbMATHGoogle Scholar
  8. 8.
    Glizer, V.Y.: Stabilizability and detectability of singularly perturbed linear time-invariant systems with delays in state and control. J. Dyn. Control Syst. 5, 153–172 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Apkarian, P., Tuan, H.D.: Parameterized LMIs in control theory. SIAM J. Control Optim. 38, 1241–1264 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    De Souza, C.E., Trofini, A., De Oliveira, J.: Parametric Lyapunov function approach to \(H_{2}\) analysis and control of linear parameter-dependent systems. IEEE Proc. Control Theory Appl. 150, 501–508 (2003)CrossRefGoogle Scholar
  11. 11.
    Bliman, P.-A.: From Lyapunov–Krasovskii functionals for delay-independent stability to LMI conditions for \(\mu \)-analysis. In: Niculescu, S.I., Gu, K. (eds.) Advances in Time-Delay Systems, Lecture Notes in Computational Science and Engineering, vol. 38, pp. 75–85. Springer, New York (2004)CrossRefGoogle Scholar
  12. 12.
    Glizer, V.Y.: Novel controllability conditions for a class of singularly perturbed systems with small state delays. J. Optim. Theory Appl. 137, 135–156 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Glizer, V.Y.: \(L^2\)-stabilizability conditions for a class of nonstandard singularly perturbed functional-differential systems. Dyn. Contin. Discrete Impuls. Syst. Ser. B Appl. Algorithms 16, 181–213 (2009)Google Scholar
  14. 14.
    Han, X., Liu, Z., Li, H., Liu, X.: Output feedback controller design for polynomial linear parameter varying system via parameter-dependent Lyapunov functions. Adv. Mech. Eng. 9, 1–9 (2017)Google Scholar
  15. 15.
    Glizer, V.Y.: Euclidean space controllability conditions and minimum energy problem for time delay system with a high gain control. J. Nonlinear Var. Anal. 2, 63–90 (2018)CrossRefzbMATHGoogle Scholar
  16. 16.
    Malanowski, K., Maurer, H.: Sensitivity analysis for state constrained optimal control problems. Discrete Contin. Dyn. Syst. 4, 241–272 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Balas, G.J.: Linear parameter-varying control and its application to aerospace systems. In: Grant I. (ed.) Proceedings of of 23rd International Congress of Aeronautical Sciences, 2002, Toronto, Canada, Paper ICAS 2002-5.4.1, Optimage Publishers Ltd., London (2002)Google Scholar
  18. 18.
    Glizer, V.Y.: Suboptimal solution of a cheap control problem for linear systems with multiple state delays. J. Dyn. Control Syst. 11, 527–574 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Glizer, V.Y.: Correctness of a constrained control Mayer’s problem for a class of singularly perturbed functional-differential systems. Control Cybern. 37, 329–351 (2008)MathSciNetzbMATHGoogle Scholar
  20. 20.
    Bonnans, J.F., Silva, F.J.: Asymptotic expansion for the solutions of control constrained semilinear elliptic problems with interior penalties. SIAM J. Control Optim. 49, 2494–2517 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Gaitsgory, V., Rossomakhine, S.: Averaging and linear programming in some singularly perturbed problems of optimal control. Appl. Math. Optim. 71, 195–276 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Glizer, V.Y.: Dependence on parameter of the solution to an infinite horizon linear-quadratic optimal control problem for systems with state delays. Pure Appl. Funct. Anal. 2, 259–283 (2017)MathSciNetzbMATHGoogle Scholar
  23. 23.
    Delfour, M.C., McCalla, C., Mitter, S.K.: Stability and the infinite-time quadratic cost problem for linear hereditary differential systems. SIAM J. Control 13, 48–88 (1975)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Vinter, R.B., Kwong, R.H.: The infinite time quadratic control problem for linear systems with state and control delays: an evolution equation approach. SIAM J. Control Optim. 19, 139–153 (1981)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Phat, V.N., Nam, P.T.: Exponential stability and stabilization of uncertain linear time-varying systems using parameter dependent Lyapunov function. Int. J. Control 80, 1333–1341 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Botmart, T., Niamsup, P.: Robust exponential stability and stabilizability of linear parameter dependent systems with delays. Appl. Math. Comput. 217, 2551–2566 (2010)MathSciNetzbMATHGoogle Scholar
  27. 27.
    Glizer, V.Y., Fridman, E.: Stability of singularly perturbed functional-differential systems: spectrum analysis and LMI approaches. IMA J. Math. Control Inf. 29, 79–111 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Glizer, V.Y., Fridman, E., Feigin, Y.: A novel approach to exact slow-fast decomposition of linear singularly perturbed systems with small delays. SIAM J. Control Optim. 55, 236–274 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Glizer, V.Y.: Controllability of nonstandard singularly perturbed systems with small state delay. IEEE Trans. Autom. Control 48, 1280–1285 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Glizer, V.Y.: Observability of singularly perturbed linear time-dependent differential systems with small delay. J. Dyn. Control Syst. 10, 329–363 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Glizer, V.Y.: Euclidean space output controllability of singularly perturbed systems with small state delays. J. Appl. Math. Comput. 57, 1–38 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Glizer, V.Y.: Controllability conditions of linear singularly perturbed systems with small state and input delays. Math. Control Signals Syst. 28(1), 1–29 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Stoorvogel, A.A., Saberi, A.: Continuity properties of solutions to \(H_2\) and \(H^{\infty }\) Riccati equations. In: Proceedings of the 34th IEEE Conference on Decision and Control, New Orleans, LA, IEEE Publishing, vol. 4, pp. 4335–4340 (1995)Google Scholar
  34. 34.
    Zhang, X., Tsiotras, P., Iwasaki, T.: Lyapunov-based exact stability analysis and synthesis for linear single-parameter dependent systems. Int. J. Control 83, 1823–1838 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Bliman, P.-A.: An existence result for polynomial solutions of parameter-dependent LMIs. Syst. Control Lett. 51, 165–169 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Gusev, S.V.: Parameter-dependent \(S\)-procedure and Yakubovich lemma. arXiv:math/0612794v1 [math.OC], p. 11 (2006)
  37. 37.
    Bachelier, O., Henrion, D., Yeganefar, N., Mehdi, D.: On the solutions to complex parameter-dependent LMIs involved in the stability analysis of \(2D\) discrete models. arXiv:1409.4321v2 [math.OC], p. 13 (2015)
  38. 38.
    Gu, K.: An integral inequality in the stability problem of time-delay systems. In: Proceedings of the 39th IEEE Conference on Decision and Control, Sydney, Australia, IEEE Publishing, vol. 3, pp. 2805–2810 (2000)Google Scholar
  39. 39.
    Boyd, S., Ghaoui, L.E., Feron, E., Balakrishnan, V.: Linear Matrix Inequalities in Systems and Control Theory. SIAM, Philadenphia (1994)CrossRefzbMATHGoogle Scholar
  40. 40.
    Bliman, P.-A., Prieur, C.: On existence of smooth solutions of parameter-dependent convex programming problems. In: De Moor, B., Motmans, B. (eds.): Proceedings of the 16th International Symposium on Mathematical Theory of Networks and Systems, Leuven, Belgium, Katholieke Universiteit Leuven Publishing, p. 7 (2004)Google Scholar
  41. 41.
    Oliveira, R.C.L.F., Peres, P.L.D.: LMI relaxations for homogeneous polynomial solutions of parameter-dependent LMIs. IFAC Proceedings Volumes, Vol. 39, Issue 9, 5th IFAC Symposium on Robust Control Design, Toulouse, France, pp. 543–548 (2006)Google Scholar

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Authors and Affiliations

  1. 1.Department of MathematicsORT Braude College of EngineeringKarmielIsrael

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