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Automation and Remote Control

, Volume 80, Issue 2, pp 250–261 | Cite as

On the Optimal Control Problem for a Linear Stochastic System with an Unstable State Matrix Unbounded at Infinity

  • E. S. PalamarchukEmail author
Stochastic Systems
  • 10 Downloads

Abstract

We consider a control problem over an infinite time horizon with a linear stochastic system with an unstable asymptotically unbounded state matrix. We extend the notion of anti-stability of a matrix to the case of non-exponential anti-stability, and introduce an antistability rate function as a characteristic of the rate of growth for the norm of the corresponding fundamental matrix. We show that the linear stable feedback control law is optimal with respect to the criterion of the adjusted extended long-run average. The designed criterion explicitly includes information about the rate of anti-stability and the parameters of the disturbances. We also analyze optimality conditions.

Keywords

stochastic linear-quadratic regulator anti-stability instability superexponential growth Riccati equation 

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References

  1. 1.
    Anderson, B.D.O., Ilchmann, A., and Wirth, F.R., Stabilizability of Linear Time–Varying Systems, Syst. Control Lett., 2013, vol. 62, no. 9, pp. 747–755.MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Bacciotti, A. and Rosierm, L., Liapunov Functions and Stability in Control Theory, New York: Springer, 2006.Google Scholar
  3. 3.
    Dragan, V. and Halanay, A., Stabilization of Linear Systems, Boston: Birkhauser, 1999.CrossRefzbMATHGoogle Scholar
  4. 4.
    Dragan, V., Morozan, T., and Stoica, A.M., Mathematical Methods in Robust Control of Linear Stochastic Systems, New York: Springer, 2006.zbMATHGoogle Scholar
  5. 5.
    Fomichev, V.V., Mal’tseva, A.V., and Shuping, W., Stabilization Algorithm for Linear Time–Varying Systems, Differ. Equat., 2017, vol. 53, no. 11, pp. 1495–1500.MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Phat, V.N., Global Stabilization for Linear Continuous Time–Varying Systems, Appl. Math. Comput., 2006, vol. 175, no. 2, pp. 1730–1743.MathSciNetzbMATHGoogle Scholar
  7. 7.
    Terrell, W.J., Stability and Stabilization: An Introduction, Princeton: Princeton Univ. Press, 2009.CrossRefzbMATHGoogle Scholar
  8. 8.
    Kwakernaak, H. and Sivan, R., Linear Optimal Control Systems, New York: Wiley, 1972. Translated under the title Lineinye optimal’nye sistemy upravleniya, Moscow: Mir, 1977.zbMATHGoogle Scholar
  9. 9.
    Wu, M.–Y. and Sherif, A., On the Commutative Class of Linear Time–Varying Systems, Int. J. Control, 1976, vol. 23, no. 3, pp. 433–444.MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Jetto, L., Orsini, V., and Romagnoli, R., BMI–based Stabilization of Linear Uncertain Plants with Polynomially Time Varying Parameters, IEEE Trans. Automat. Control, 2015, vol. 60, no. 8, pp. 2283–2288.MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Jones, J.J., Modelling and Simulation of Large Scale Multiparameter Dynamical System, Proc. IEEE 1989 National Aerospace and Electronics Conf. (NAECON 1989), New York: IEEE, 1989, pp. 415–425.Google Scholar
  12. 12.
    Levine, J. and Zhu, G., Observers with Asymptotic Gain for a Class of Linear Time–Varying Systems with Singularity, IFAC Proc. Volumes, 1993, vol. 26, no. 2, pp. 145–148.CrossRefGoogle Scholar
  13. 13.
    Karafyllis, I. and Tsinias, J., Non–Uniform in Time Stabilization for Linear Systems and Tracking Control for Non–Holonomic Systems in Chained Form, Int. J. Control, 2003, vol. 76, no. 15, pp. 1536–1546.MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Caraballo, T., On the Decay Rate of Solutions of Non–autonomous Differential Systems, Electron. J. Differ. Equat., 2001, vol. 2001, no. 5, pp. 1–17.MathSciNetzbMATHGoogle Scholar
  15. 15.
    Inoue, M., Wada, T., Asai, T., and Ikeda, M., Non–exponential Stabilization of Linear Time–invariant Systems by Linear Time–varying Controllers, Proc. 50th IEEE Conf. on Decision and Control and European Control Conf., New York, 2011, pp. 4090–4095.Google Scholar
  16. 16.
    Palamarchuk, E.S., On the Generalization of Logarithmic Upper Function for Solution of a Linear Stochastic Differential Equation with a Nonexponentially Stable Matrix, Differ. Equat., 2018, vol. 54, no. 2, pp. 193–200.MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Abou–Kandil, H., Freiling, G., Ionescu, V., and Jank, G., Matrix Riccati Equations in Control and Systems Theory, Basel: Birkhauser, 2012.zbMATHGoogle Scholar
  18. 18.
    Turnovsky, S.J., Macroeconomic Analysis and Stabilization Policy, Cambrigde: Cambridge Univ. Press, 1977.zbMATHGoogle Scholar
  19. 19.
    Palamarchuk, E.S., Optimization of the Superstable Linear Stochastic System Applied to the Model with Extremely Impatient Agents, Autom. Remote Control, 2018, vol. 79, no. 3, pp. 440–451.MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Belkina, T.A. and Palamarchuk, E.S., On Stochastic Optimality for a Linear Controller with Attenuating Disturbances, Autom. Remote Control, 2013, vol. 74, no. 4, pp. 628–641.MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Palamarchuk, E.S., Analysis of the Asymptotic Behavior of the Solution to a Linear Stochastic Differential Equation with Subexponentially Stable Matrix and Its Application to a Control Problem, Theor. Prob. App., 2018, vol. 62, no. 4, pp. 522–533.MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Fischer, J., Optimal Sequence–Based Control of Networked Linear Systems, Karlsruhe: KIT Scientific Publishing, 2015.Google Scholar
  23. 23.
    Aeyels, D., Lamnabhi–Lagarrigue, F., and van der Schaft, A., Eds., Stability and Stabilization of Nonlinear Systems, Berlin: Springer, 2008.Google Scholar
  24. 24.
    Chen, G. and Yang, Y., New Stability Conditions for a Class of Linear Time–Varying Systems, Automatica, 2016, vol. 71, pp. 342–347.MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Palamarchuk, E.S., Stabilization of Linear Stochastic Systems with a Discount: Modeling and Estimation of the Long–Term Effects from the Application of Optimal Control Strategies, Math. Models Comput. Simul., 2015, vol. 7, no. 4, pp. 381–388.MathSciNetCrossRefGoogle Scholar
  26. 26.
    Palamarchuk, E.S., Analysis of Criteria for Long–run Average in the Problem of Stochastic Linear Regulator, Autom. Remote Control, 2016, vol. 77, no. 10, pp. 1756–1767.MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Khasminskii, R., Stochastic Stability of Differential Equations, New York: Springer, 2012, 2nd ed.CrossRefzbMATHGoogle Scholar
  28. 28.
    Mao, X., Stochastic Differential Equations and Applications, Cambridge, UK: Woodhead Publishing, 2007, 2nd ed.zbMATHGoogle Scholar
  29. 29.
    Palamarchuk, E.C., Risk Estimation in Linear Economic Systems under Negative Time Preferences, Ekonom. Mat. Metody, 2013, vol. 49, no. 3, pp. 99–116.Google Scholar

Copyright information

© Pleiades Publishing, Ltd. 2019

Authors and Affiliations

  1. 1.National Research University Higher School of EconomicsMoscowRussia
  2. 2.Central Economics and Mathematics InstituteRussian Academy of SciencesMoscowRussia

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