Abstract
This paper considers systems with two-dimensional dynamics (2D systems) described by the continuous-time nonlinear state-space Roesser model. The sufficient conditions of exponential stability in terms of vector Lyapunov functions are established. These conditions are then applied to analysis of the absolute stability of a certain class of systems comprising a linear continuous-time plant in the form of the Roesser model with a nonlinear characteristic in the feedback loop, which satisfies quadratic constraints. The absolute stability conditions are reduced to computable expressions in the form of linear matrix inequalities. The obtained results are extended to the class of continuous-time systems governed by the Roesser model with Markovian switching. The problems of absolute stability and stabilization via state- and output-feedback are solved for linear systems of the above class. The solution procedures for these problems are in the form of algorithms based on linear matrix inequalities.
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References
Rogers, E., Ga-lkowski, K., and Owens, D.H., Control Systems Theory and Applications for Linear Repetitive Processes, Lecture Notes Control Inform. Sci., vol. 349, Berlin: Springer-Verlag, 2007.
Roesser, R.P., A Discrete State-Space Model for Linear Image Processing, IEEE Trans. Automat. Control, 1975, vol. AC-20, pp. 1–10.
Fornasini, E. and Marchesini, G., Doubly Indexed Dynamical Systems: State Models and Structural Properties, Math. Syst. Theory, 1978, vol. 12, pp. 59–72.
Hladowski, L., Galkowski, K., Cai, Z., Rogers, E., Freeman, C.T., and Lewin, P.L., Experimentally Supported 2D Systems Based Iterative Learning Control Law Design for Error Convergence and Performance, Control Eng. Practice, 2010, vol. 18, pp. 339–348.
Yeh, K.-H. and Lu, H.-C., Robust Stability Analysis for Two-dimensional Systems via Eigenvalue Sensitivity, Multidimens. Syst. Signal Proc., 1995, vol. 6, pp. 223–236.
Ooba, T., On Stability Robustness of 2-D Systems Described by the Fornasini-Marchesini Model, Multidimens. Syst. Signal Proc., 2000, vol. 12, pp. 81–88.
Du, C. and Xie, L., Stability Analysis and Stabilization of Uncertain Two-dimensional Discrete Systems: An LMI Approach, IEEE Trans. Circuits Syst. I: Fundament. Theory Appl., 1999, vol. 46, pp. 1371–1374.
Xu, S., Lam, J., Lin, Z., and Galkowski, K., Positive Real Control for Uncertain Two-dimensional Systems, IEEE Trans. Circuits Syst. I: Fundament. Theory Appl., 2002, vol. 49, pp. 1659–1666.
Lam, J., Xu, S., Zou, Y., Lin, Z., and Galkowski, K., Robust Output Feedback Stabilization for Two-dimensional Continuous Systems in Roesser Form, Appl. Math. Lett., 2004, vol. 17, pp. 1331–1341.
Kurek, J.E., Stability of Nonlinear Time-varying Digital 2-D Fornasini-Marchesini System, Multidimens. Syst. Signal Proc., 2012, vol. 23. Available in open access by http://link.springer.com/article/10.1007/s11045-012-0193-4.
Pakshin, P., Galkowski, K., and Rogers, E., Absolute Stability and Stabilization of 2D Roesser Systems with Nonlinear Output Feedback, Proc. 50th IEEE Conf. Decision Control Eur. Control Conf. (CDCECC 2011), Orlando, December 12–15, 2011, pp. 6736–6741.
Yeganefar, Nim., Yeganefar, Nad., Ghamgui, M., and Moulay, E., Lyapunov Theory for 2-D Nonlinear Roesser Models: Application to Asymptotic and Exponential Stability, IEEE Transact. Automat. Control, 2013, vol. 58, pp. 1299–1304.
Lin, Z., Zou, Q., and Ober, R.J., The Fisher Information Matrix for Two-dimensional Data Sets, Proc. IEEE Int. Conf. Acoustics, Speech, Sign. Process. (ICASSP’ 03), 2003, vol. 3. pp. 453-456.
Gelig, A.Kh., Leonov, G.A., and Yakubovich, V.A., Ustoichivost’ nelineinykh sistem s needinstvennym sostoyaniem ravnovesiya (Stability of Nonlinear Systems with Nonunique Equilibria), Moscow: Nauka, 1978.
Mariton, M., Jump Linear Systems in Automatic Control, New York: Marcel Dekker, 1990.
Pakshin, P.V., Diskretnye sistemy so sluchainymi parametrami i strukturoi (Discrete Systems with Random Parameters and Structure), Moscow: Fizmatlit, 1994.
Kats, I.Ya., Metod funktsii Lyapunova v zadachakh ustoichivosti i stabilizatsii sistem sluchainoi struktury (The Method of Lyapunov Functions in Stability and Stabilization Problems for Random Structure Systems), Yekaterinburg: Ural. Gos. Akad. Putei Soobshchen., 1998. Translated under the title Stability and Stabilization of Nonlinear Systems with Random Structures, London: Taylor & Francis, 2002.
Zhang, L. and Boukas, E.-K., Stability and Stabilization of Markovian Jump Linear Systems with Partly Unknown Transition Probabilities, Automatica, 2009, vol. 45, pp. 463–468.
Xiong, J. and Lam, J., Robust H 2 Control of Markovian Jump Systems with Uncertain Switching Probabilities, Int. J. Syst. Sci., 2009, vol. 40, pp. 255–265.
Pakshin, P. and Peaucelle, D., LQR Parametrization of Static Output Feedback Gains for Linear Systems with Markovian Switching and Related Robust Stabilization and Passification Problems, Proc. Joint 48th IEEE Conf. Decision Control and 28th Chinese Control Conf., Shanghai, Dec. 2009, pp. 1157–1162.
Pakshin, P.V., Solov’ev, S.G., and Peaucelle, D., Parametrizing Stabilizing Control in Stochastic Systems, Autom. Remote Control, 2009, vol. 70, no. 9, pp. 1514–1527.
Gao, H., Lam, J., Xu, S., and Wang, C., Stabilization and H ∞ Control of Two-dimensional Markovian Jump Systems, IMA J. Math. Control Inform., 2004, vol. 21, pp. 377–392.
Wu, L., Shi, P., Gao, H., and Wang, C., H∞ Filtering for 2D Markovian Jump Systems, Automatica, 2008, vol. 44, pp. 1849–1858.
Pakshin, P.V., Galkowski, K., and Rogers, E., Linear-Quadratic Parametrization of Stabilizing Controls in Discrete-Time 2D Systems, Autom. Remote Control, 2011, vol. 72, no. 11, pp. 2364–2378.
Krasovskii, N.N., Nekotorye zadachi teorii ustoichivosti dvizheniya (Some Problems in Theory of Motion Stability), Moscow: Fizmatlit, 1959. Translated under the title Problems of the Theory of Stability of Motion, Stanford: Stanford Univ. Press, 1963.
Boyd, S., El-Ghaoui, L., Feron, E., and Balakrishnan, V., Linear Matrix Inequalities in System and Control Theory, Philadelphia: SIAM, 1994.
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.
Matrosov, V.M., Metod vektornykh funktsii Lyapunova: Analiz dinamicheskikh svoistv nelineinykh sistem (Method of Vector Lyapunov Functions: Analysis of Dynamic Properties of Nonlinear Systems), Moscow: Fizmatlit, 2001.
Kojima, C., Rapisarda, P., and Takaba, K., Lyapunov Stability Analysis of Higher-order 2-D Systems, Multidimens. Syst. Signal Proc., 2011, vol. 22, pp. 287–302.
Zhukov, V.P., Polevye metody v issledovanii nelineinykh dinamicheskikh sistem (Field Methods in Analysis of Nonlinear Dynamic Systems), Moscow: Nauka, 1992.
Rantzer, A., A Dual to Lyapunov’s Stability Theorem, Syst. Control Lett., 2001, vol. 42, pp. 161–168.
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Original Russian Text © J.P. Emelianova, P.V. Pakshin, K. Ga-lkowski, E. Rogers, 2014, published in Avtomatika i Telemekhanika, 2014, No. 5, pp. 50–66.
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Emelianova, J.P., Pakshin, P.V., Gałkowski, K. et al. Stability of nonlinear 2D systems described by the continuous-time Roesser model. Autom Remote Control 75, 845–858 (2014). https://doi.org/10.1134/S000511791405004X
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DOI: https://doi.org/10.1134/S000511791405004X