Automation and Remote Control

, Volume 74, Issue 3, pp 426–441 | Cite as

Stochastic H 2 /H -control for a dynamical system with internal noises multiplicative with respect to state, control, and external disturbance

  • M. E. Shaikin
Topical Issue


We consider an optimal control problem for a dynamical system under the influence of disturbances of both deterministic and stochastic nature. The system is defined on a finite time interval, and its diffusion coefficient depends on the control signal. The controller in the feedback circuit is assumed to be static, nonstationary, linear in the state vector, and satisfying the condition ‖L < γ that bounds the norm of operator L: v ↦z for the transition of external disturbance to the controllable output signal. Solving the optimization H 2/H -control problem, we get three matrix functions satisfying a system of two differential equations of Riccati type and one matrix algebraic equation. In the special case of a stochastic system whose diffusion coefficient does not depend on the control signal, the system is reduced to two related Riccati equations.


Remote Control Linear Matrix Inequality Style Equation External Disturbance Riccati Equation 
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  1. 1.
    Athans, M., Editorial on the LQG Problem, IEEE Trans. Automat. Control, 1971, vol. 16, no. 6, p. 528.CrossRefGoogle Scholar
  2. 2.
    Jacobson, D.H., Optimal Stochastic Linear Systems with Exponential Performance Criteria and Their Relation to Deterministic Differential Games, IEEE Trans. Automat. Control, 1973, vol. 18, no. 2, pp. 124–131.zbMATHCrossRefGoogle Scholar
  3. 3.
    Doyle, J.C. and Stein, G., Robustness with Observers, IEEE Trans. Automat. Control, 1979, vol. 24, no. 4, pp. 607–611.MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Zames, G., Feedback and Optimal Sensitivity: Model Reference Transformations, Multiplicative Seminorms, and Approximate Inverses, IEEE Trans. Automat. Control, 1981, vol. 26, no. 2, pp. 301–320.MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Doyle, J.C., Glover, K., Khargonekar, P.P., and Francis, B.A., State Space Solutions to Standard H 2 and H Control Problems, IEEE Trans. Automat. Control, 1989, vol. 34, no. 8, pp. 831–847.MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Zhou, K., Glover, K., Bodenheimer, B., and Doyle, J., Mixed H 2 and H Performance Objectives. I. Robust Performance Analysis, IEEE Trans. Automat. Control, 1994, vol. 39, no. 8, pp. 1564–1574.MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Doyle, J., Zhou, K., Glover, K., and Bodenheimer, B., Mixed H 2 and H performance objectives. II. Optimal control, IEEE Trans. Automat. Control, 1994, vol. 39, no. 8, pp. 1575–1587.MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Boltyanski, V.G. and Poznyak, A.S., Robust Maximum Principle: Theory and Applications, Boston: Birkhauser, 2012.zbMATHCrossRefGoogle Scholar
  9. 9.
    Mustafa, D. and Glover, K., Minimum Entropy H -Control, Lecture Notes in Control and Information Sciences, Berlin: Springer-Verlag, 1991.Google Scholar
  10. 10.
    Vladimirov, I.G., Kurdjukov, A.P., and Semyonov, A.V., State-Space Solution to Anisotropy-Based Stochastic H -Optimization Problem, Proc. 13th IFAC World Congress, San-Francisco, 1996, pp. 427–432.Google Scholar
  11. 11.
    Letov, A.M., Analytical Controller Design. I–IV, Autom. Remote Control, 1960, vol. 21, no. 4, pp. 303–306; no. 5, pp. 389–393; no. 6, pp. 458–461; 1961, vol. 22, no. 4, pp. 363–372.zbMATHGoogle Scholar
  12. 12.
    Poznyak, A.S., Sebryakov, G.G., Semenov, A.V., and Fedosov, E.A., H -teoriya upravleniya: fenomen, dostizheniya, perspektivy, otkrytye problemy (H -Control Theory: Phenomenon, Achievements, Prospects, and Open Problems), Moscow: GosNIIAS, 1990.Google Scholar
  13. 13.
    Arov, D.Z. and Krein, M.G., On Computing Entropy Integrals and Their Minima in Generalized Continuation Problems, Acta Scienta Matematica, 1983, vol. 45, pp. 33–50.MathSciNetzbMATHGoogle Scholar
  14. 14.
    Vladimirov, I.G., Diamond, P., and Kloeden, P., Anisotropy-Based Robust Performance Analysis of Finite Horizon Linear Discrete Time Varying Systems, Autom. Remote Control, 2006, vol. 67, no. 8, pp. 1265–1282.MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    Balandin, D.V. and Kogan, M.M., Sintez zakonov upravleniya na osnove lineinykh matrichnykh neravenstv (Design of Control Laws based on Linear Matrix Inequalities), Moscow: Fizmatlit, 2007.Google Scholar
  16. 16.
    Hinrichsen, D. and Pritchard, A.J., Stochastic H , SIAM J. Control Optim., 1998, vol. 36, no. 5, pp. 1504–1538.MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    Chen, B.S. and Zhang, W., Stochastic H 2 /H Control with State-Dependent Noise, IEEE Trans. Automat. Control, 2004, vol. 49, no. 1, pp. 45–56.MathSciNetCrossRefGoogle Scholar
  18. 18.
    Zhang, W., Zhang, H., and Chen, B., Generalised Lyapunov Equation Approach to State-Dependent Stochastic Stabilisation/Detectability Criterion, IEEE Trans. Automat. Control, 2008, vol. 53, no. 7, pp. 1630–1642.MathSciNetCrossRefGoogle Scholar
  19. 19.
    Wu Zhen, Forward-Backward Stochastic Differential Equations, Linear Quadratic Stochastic Optimal Control and Nonzero Sum Differential Games, J. Syst. Sci. Complexity, 2005, vol. 18, no. 2, pp. 179–192.zbMATHGoogle Scholar
  20. 20.
    Ma, J., Protter, P., and Yong, J., Solving Forward-Backward Stochastic Differential Equations Explicitly. A Four Step Scheme, Probab. Theory Related Fields, 1994, vol. 98, pp. 339–359.MathSciNetzbMATHCrossRefGoogle Scholar
  21. 21.
    Bulinskii, A.V. and Shiryaev, A.N., Teoriya sluchainykh protsessov (Theory of Random Processes), Moscow: Fizmatlit, 2003.Google Scholar
  22. 22.
    Yong, J. and Zhou, X.Y., Stochastic Controls: Hamiltonian Systems and HJB Equations, New York: Springer, 1999.zbMATHGoogle Scholar
  23. 23.
    Peng, S. and Wu, Z., Fully Coupled Forward-Backward Stochastic Differential Equations and Applications to Optimal Control, SIAM J. Control Optim., 1999, vol. 37, pp. 825–843.MathSciNetzbMATHCrossRefGoogle Scholar

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© Pleiades Publishing, Ltd. 2013

Authors and Affiliations

  • M. E. Shaikin
    • 1
  1. 1.Trapeznikov Institute of Control SciencesRussian Academy of SciencesMoscowRussia

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