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Parametrizing stabilizing controls in stochastic systems

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Abstract

We consider discrete linear systems with noises that depend on both state and control. We give necessary and sufficient stabilizing conditions that yield a parametric description (parametrization) of all linear stabilizing controls with static output feedback that provide exponential stability in average of a quadratic closed system. We also give constructive sufficient conditions that allow computing the amplification matrix of a stabilizing control with an auxiliary optimization problem with constraints in the form of linear matrix inequalities without iterative procedures. The results extend to random structure systems and simultaneous stabilization problems. We show two possible applications of our results to robust control problems. The first application is a special case of the simultaneous stabilization problem, and the second is based on considering a stochastic model as a comparison system for a system with affine indeterminacies. We give an example of solving the simultaneous stabilization problem for a hypothetical aircraft.

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References

  1. Trofino-Neto, A. and Kucera, V., Stabilization via Static Output Feedback, IEEE Trans. Automat. Control, 1993, vol. 38, pp. 764–765.

    Article  MATH  MathSciNet  Google Scholar 

  2. Kucera, V. and de Souza, C.E., A Necessary and Sufficient Condition for Output Feedback Stabilizability, Automatica, 1995, vol. 31, pp. 1357–1359.

    Article  MATH  Google Scholar 

  3. Iwasaki, T. and Skelton, R.E., Parametrization of All Stabilizing Controllers via Quadratic Lyapunov Functions, J. Optim. Theory Appl., 1995, vol. 85, pp. 291–307.

    Article  MATH  MathSciNet  Google Scholar 

  4. Syrmos, V.L., Abdallah, C.T., Dorato, P., and Grigoriadis, K., Static Output Feedback: A Survey, Automatica, 1997, vol. 33, pp. 125–137.

    Article  MATH  MathSciNet  Google Scholar 

  5. Gadewadikar, J., Lewis, F.L., Xie, L., Kucera, V., and Abu-Khalaf, M., Parameterization of All Stabilizing H ∞ Static State-feedback Gains: Application to Output-feedback Design, Automatica, 2007, vol. 43, pp. 1597–1604.

    Article  MATH  MathSciNet  Google Scholar 

  6. Pakshin, P.V. and Mitrofanov, I.N., Parametrization of Stabilizing Controllers and Robust Stabilization via Static Output Feedback for Jump Linear Systems, Proc. 6 IFAC Sympos. Nonlinear Control Syst., Stuttgart, 2004, vol. 3, pp. 1151–1156.

    Google Scholar 

  7. Pakshin, P.V. and Retinskii, D.M., Robust Stabilization of Random-structure Systems with Feedback Switched According to the Plant Output, Autom. Remote Control, 2005, vol. 66, no. 7, pp. 1153–1161.

    Article  MATH  MathSciNet  Google Scholar 

  8. Pakshin, P.V. and Solov’ev, S.G., Synthesis of Robust Discrete-time Systems Based on Comparison with Stochastic Model, Izv. Ross. Akad. Nauk, Teor. Sist. Upravlen., 2007, no. 6, pp. 5–15.

  9. Pakshin, P.V., Exponential Dissipativeness of the Random-structure Diffusion Processes and Problems of Robust Stabilization, Autom. Remote Control, 2007, vol. 68, no. 10, pp. 1852–1870.

    Article  MATH  MathSciNet  Google Scholar 

  10. Nurdin, H.I., James, M.R., and Petersen, I.R., Quantum LQG Control with Quantum Mechanical Controllers, Proc. 17 IFAC World Congress, Seoul, 2008, pp. 9922–9927.

  11. 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.

    Article  MATH  MathSciNet  Google Scholar 

  12. 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.

    Article  MATH  MathSciNet  Google Scholar 

  13. Pakshin, P.V., Diskretnye sistemy so sluchainymi parametrami i strukturoi (Discrete Systems with Random Parameters and Structure), Moscow: Fizmatlit, 1994.

    Google Scholar 

  14. Rosinova, D., Vesely, V., and Kucera, V., A Necessary and Sufficient Condition for Static Output Feedback Stabilizability of Linear Discrete-time Systems, Kybernetika, 2003, vol. 39, pp. 447–459.

    MathSciNet  Google Scholar 

  15. Yu, J-T., A Convergent Algorithm for Computing Stabilizing Static Output Feedback Gains, IEEE Trans. Automat. Control, 2004, vol. 49, pp. 2271–2275.

    Article  MathSciNet  Google Scholar 

  16. Crusius, C.A.R. and Trofino, A., Sufficient LMI Conditions for Output Feedback Control Problems, IEEE Trans. Automat. Control., 1999, vol. 44, pp. 1053–1057.

    Article  MATH  MathSciNet  Google Scholar 

  17. Krasovskii, A.A., Sistemy avtomaticheskogo upravleniya poletom i ikh analiticheskoe konstruirovanie (Systems of Automatic Control of Flying and Their Analytic Constructing), Moscow: Nauka, 1973.

    Google Scholar 

  18. Johnson, C.D., The “Unreachable Poles” Defect in LQR Theory: Analysis and Remedy, Int. J. Control, 1988, vol. 47, pp. 697–709.

    Article  MATH  Google Scholar 

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Authors and Affiliations

  1. Arzamas Polytechnic Institute of Alekseev Nizhni Novgorod State Technical University, Arzamas, Russia

    P. V. Pakshin

  2. Lobachevskii State University, Nizhni Novgorod, Russia

    S. G. Solov’ev

  3. LAAS-CNRS, Université de Toulouse, Toulouse, France

    D. Peaucelle

Authors
  1. P. V. Pakshin
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  2. S. G. Solov’ev
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  3. D. Peaucelle
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Additional information

Original Russian Text © P.V. Pakshin, S.G. Solov’ev, D. Peaucelle, 2009, published in Avtomatika i Telemekhanika, 2009, No. 9, pp. 85–99.

This work was supported in part by the Russian Foundation for Basic Research, projects nos.07-01-92166-NTsNI_a, 08-01-97036-r_Povolzh’e_a, and Centre National de la Recherche Scientifique, PICS project no.4281.

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Pakshin, P.V., Solov’ev, S.G. & Peaucelle, D. Parametrizing stabilizing controls in stochastic systems. Autom Remote Control 70, 1514–1527 (2009). https://doi.org/10.1134/S0005117909090057

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