Automation and Remote Control

, Volume 68, Issue 10, pp 1852–1870 | Cite as

Exponential dissipativeness of the random-structure diffusion processes and problems of robust stabilization

  • P. V. Pakshin
Stability of Systems


Consideration was given to the class of systems described by a finite set of the controllable control-affine diffusion Ito processes with stepwise transitions defined by the evolution of the uniform Markov chain (Markov switchings). For these systems, the notion of exponential dissipativity was introduced, and its theory was developed and used to estimate the possible variations of the output feedback law under which the system retains its robust stability. For the set of linear systems with uncertain parameters, proposed was a two-step procedure for determination of the output feedback control based on comparison with the stochastic model and providing their simultaneous robust stabilization. At the first step, the robust stabilizing control is established by means of an iterative algorithm. Then, the possible variations of the feedback law for which the robust stability is retained are estimated by solving a system of matrix linear inequalities. An example was presented.

PACS number



Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Boyd, S., El Ghaoui, L., Feron, E., and Balakrishnan, V., Linear Matrix Inequalities in Control and System Theory, Philadelphia: SIAM, 1994.zbMATHGoogle Scholar
  2. 2.
    Polyak, B.T. and Shcherbakov, P.S., Robastnaya ustoichivost’ i upravlenie (Robust Stability and Control), Moscow: Nauka, 2002.Google Scholar
  3. 3.
    Balandin, D.V. and Kogan, M.M., Sintez zakonov upravleniya na osnove lineinykh matrichnykh neravenstv (Design of the Control Laws on the Basis of Linear Matrix Inequalities), Moscow: Nauka, 2007.Google Scholar
  4. 4.
    Bernstein, D.S., Robust Static and Dynamic Output-Feedback Stabilization: Deterministic and Stochastic Perspectives, IEEE Trans. Automat. Control, 1987, vol. AC-32, pp. 1076–1084.CrossRefGoogle Scholar
  5. 5.
    Ait Rami, M. and El Ghaoui, L., LMI Optimization for Nonstandard Riccati Equation Arising in Stochastic Control, IEEE Trans. Automat. Control, 1996, vol. 41, pp. 1666–1671.zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Ait Rami, M. and Zhou, X.Y., Linear Matrix Inequalities, Riccati Equations, and Indefinite Stochastic Linear Quadratic Controls, IEEE Trans. Automat. Control, 2000, vol. 45, pp. 1131–1143.zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Willems, J.C., Dissipative Dynamic Systems, I. General Theory, Arch. Rat. Mech. Anal., 1972, vol. 45, pp. 321–351.zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Willems, J.C., Dissipative Dynamic Systems, II. Linear Systems with Quadratic Supply Rates, Arch. Rat. Mech. Anal., 1972, vol. 45, pp. 352–393.zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Hill, D.J. and Moylan, P.J., The Stability of Nonlinear Dissipative Systems, IEEE Trans. Automat. Control, 1976, vol. 21, pp. 708–711.zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Hill, D.J. and Moylan, P.J., Connection between Finite-Gain and Asymptotic Stability, IEEE Trans. Automat. Control, 1980, vol. 25, pp. 931–936.zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Byrnes, C.I., Isidori, A., and Willems, J.C., Passivity, Feedback Equivalence, and the Global Stabilization of Minimum Phase Nonlinear Systems, IEEE Trans. Automat. Control, 1991, vol. 36, pp. 1228–1240.zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Polushin, I.G., Fradkov, A.L., and Hill, D.J., Passivity and Passification of Nonlinear Systems, Avtom. Telemekh., 2000, no. 11, pp. 3–37.Google Scholar
  13. 13.
    Florchinger, P., A Passive System Approach to Feedback Stabilization of Nonlinear Control Stochastic Systems, SIAM J. Control Optimiz., 1999, vol. 37, pp. 1848–1864.zbMATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Thygesen, U.H., On Dissipation in Stochastic Systems, in Proc. Am. Control Conf., San Diego: IEEE, 1999, pp. 1430–1434.Google Scholar
  15. 15.
    Borkar, V. and Mitter, S., A Note on Stochastic Dissipativeness, in Directions in Mathematical Systems Theory and Optimization, Lecture Notes in Control Inform., Berlin: Springer, 2003, vol. 286, pp. 41–49.CrossRefGoogle Scholar
  16. 16.
    Aliyu, M. D.S., Dissipative Analysis and Stability of Nonlinear Stochastic State-Delayed Systems, Nonlinear Dynam. Syst. Theory, 2004, vol. 4, pp. 243–256.zbMATHMathSciNetGoogle Scholar
  17. 17.
    Shaked, U. and Berman, N., H Control for Nonlinear Stochastic Systems: The Output-Feedback Case, in Preprints 16th IFAC World Congr., Prague, 2005, CD-ROM, pp. 1–6.Google Scholar
  18. 18.
    Zhang, W. and Chen, B.S., State Feedback H Control for a Class of Nonlinear Stochastic Systems, SIAM J. Control Optimiz., 2006, vol. 44, pp. 1973–1991.zbMATHCrossRefGoogle Scholar
  19. 19.
    Kats, I.Ya., Method funktsii Lyapunova v zadachakh ustoichivosti i stabilizatsii sistem sluchainoi struktury (Method of the Lyapunov Functions in the Problems of Stability and Stabilization of the Randomstructure Systems), Ekaterinburg: Ural. Gos. Akad. Putei Soobshch., 1998.Google Scholar
  20. 20.
    Mao, X., Stability of Stochastic Differential Equations with Markovian Switching, Stoch. Process. Appl., 1999, vol. 79, pp. 45–67.zbMATHCrossRefGoogle Scholar
  21. 21.
    Pakshin, P.V. and Ugrinovskii, V.A., Stochastic Problems of Absolute Stability, Avtom. Telemekh., 2006, no. 11, pp. 122–158.Google Scholar
  22. 22.
    Blondel, V. and Tsitsiklis, J.N., NP-hardness of Some Linear Control Design Problems, SIAM J. Control Optimiz., 1997, vol. 35, pp. 2118–2127.zbMATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    Yu, J-T., A Convergent Algorithm for Computing Stabilizing Static Output Feedback Gains, IEEE Trans. Automat. Control, 2004, vol. 49, pp. 2271–2275.CrossRefMathSciNetGoogle Scholar
  24. 24.
    Pakshin, P.V., Robust Stability and Stabilization of the Family of Jumping Stochastic Systems, Nonlinear Anal. Theory, Methods Appl., 1997, vol. 30, pp. 2855–2866.zbMATHCrossRefMathSciNetGoogle Scholar
  25. 25.
    Krasovskii, A.A., Sistemy avtomaticheskogo upravleniya poletom i ikh analiticheskoe konstruirovanie (Systems of Automatic Flight Control and their Analytical Design), Moscow: Nauka, 1973.Google Scholar

Copyright information

© Pleiades Publishing, Ltd. 2007

Authors and Affiliations

  • P. V. Pakshin
    • 1
  1. 1.Arzamas Polytechnical InstituteNizhni-Novgorod State Technical UniversityArzamasRussia

Personalised recommendations