Stochastic Stability Analysis and Synthesis of Continuous-Time Linear Networked Systems

In this paper, we study the problem of stability analysis and controller synthesis of continuous-time linear networked systems in the presence of stochastic uncertainty. Stochastic uncertainty is assumed to enter multiplicatively in system dynamics through input and output channels of the plant. We used the mean square notion for stochastic stability to address the analysis and controller synthesis problems of linear networked systems. These results generalize existing results on stability analysis and controller synthesis from discrete-time linear systems to continuous-time linear systems with multiplicative uncertainty. Necessary, sufficient conditions for mean square exponential stability are expressed in terms of the input-output property of deterministic or nominal system dynamics captured by the mean square system norm and variance of channel uncertainty. The stability results can also be interpreted as small gain theorem for continuous-time stochastic systems. Linear Matrix Inequalities (LMI)-based optimization formulation is provided for the computation of mean square system norm for stability analysis and controller synthesis. For a special case of single input channel uncertainty, we also prove a fundamental limitation result that arise in the mean square exponential stabilization of continuous-time linear systems. Simulation results are presented for WSCC 9 bus power system to demonstrate the application of developed framework.

[1]  Xu Ma,et al.  Mean Square Limitations of Spatially Invariant Networked Systems , 2013, CPSW@CISS.

[2]  Sambarta Dasgupta,et al.  Control of systems in Lure form over erasure channel , 2014, ArXiv.

[3]  Bruno Sinopoli,et al.  Foundations of Control and Estimation Over Lossy Networks , 2007, Proceedings of the IEEE.

[4]  M. Mackey,et al.  Chaos, Fractals, and Noise: Stochastic Aspects of Dynamics , 1998 .

[5]  Amit Diwadkar,et al.  Limitations for Nonlinear Observation Over Erasure Channel , 2013, IEEE Transactions on Automatic Control.

[6]  P. McLane.  Optimal stochastic control of linear systems with state- and control-dependent disturbances , 1971 .

[7]  J. Willems,et al.  Frequency domain stability criteria for stochastic systems , 1971 .

[8]  P. Khargonekar,et al.  Robust Stabilization of Uncertain Systems , 1988, 1988 American Control Conference.

[9]  Robert E. Skelton,et al.  Mean-square small gain theorem for stochastic control: discrete-time case , 2002, IEEE Trans. Autom. Control..

[10]  Fernando Paganini,et al.  A Course in Robust Control Theory , 2000 .

[11]  Jan C. Willems,et al.  Feedback stabilizability for stochastic systems with state and control dependent noise , 1976, Autom..

[12]  Nicola Elia,et al.  Limitations of Linear Control Over Packet Drop Networks , 2011, IEEE Transactions on Automatic Control.

[13]  Miroslav Krstic,et al.  Stabilization of stochastic nonlinear systems driven by noise of unknown covariance , 2001, IEEE Trans. Autom. Control..

[14]  Umesh Vaidya,et al.  Limitations of nonlinear stabilization over erasure channels , 2010, 49th IEEE Conference on Decision and Control (CDC).

[15]  P. Florchinger.  Lyapunov-like techniques for stochastic stability , 1994, Proceedings of 1994 33rd IEEE Conference on Decision and Control.

[16]  J. Willems,et al.  Robust Stabilization of Uncertain Systems , 1983 .

[17]  Sai Pushpak,et al.  Stability analysis and controller synthesis for continuous-time linear stochastic systems , 2015, 2015 54th IEEE Conference on Decision and Control (CDC).

[18]  A. Pritchard,et al.  Stability radii of linear systems with respect to stochastic perturbations , 1992 .

[19]  Ruth J. Williams,et al.  Stabilization of stochastic nonlinear systems driven by noise of unknown covariance , 1998, Proceedings of the 1998 American Control Conference. ACC (IEEE Cat. No.98CH36207).

[20]  A. Pritchard,et al.  A Riccati equation approach to maximizing the stability radius of a linear system by state feedback under structured stochastic Lipschitzian perturbations , 1993 .

[21]  L. Ghaoui.  State-feedback control of systems with multiplicative noise via linear matrix inequalities , 1995 .

[22]  V. Dragan,et al.  A small gain theorem for linear stochastic systems , 1997 .

[23]  Ian Postlethwaite,et al.  Multivariable Feedback Control: Analysis and Design , 1996 .

[24]  D. Bernstein.  Robust static and dynamic output-feedback stabilization: Deterministic and stochastic perspectives , 1987 .

[25]  Nicola Elia,et al.  Remote stabilization over fading channels , 2005, Syst. Control. Lett..

[26]  Charles R. Johnson,et al.  Matrix analysis , 1985, Statistical Inference for Engineers and Data Scientists.

[27]  Munther A. Dahleh,et al.  Feedback stabilization of uncertain systems in the presence of a direct link , 2006, IEEE Transactions on Automatic Control.

[28]  Mario A. Rotea,et al.  The generalized H2 control problem , 1993, Autom..

[29]  Bassam Bamieh,et al.  Structured stochastic uncertainty , 2012, 2012 50th Annual Allerton Conference on Communication, Control, and Computing (Allerton).

[30]  W. Wonham.  Optimal Stationary Control of a Linear System with State-Dependent Noise , 1967 .

[31]  Sekhar Tatikonda,et al.  Control over noisy channels , 2004, IEEE Transactions on Automatic Control.

[32]  Amit Diwadkar,et al.  Limitations and tradeoffs in synchronization of large-scale networks with uncertain links , 2014, Scientific Reports.

[33]  Richard M. Murray,et al.  Optimal LQG control across packet-dropping links , 2007, Syst. Control. Lett..

[34]  Amit Diwadkar,et al.  Robust synchronization in nonlinear network with link failure uncertainty , 2011, IEEE Conference on Decision and Control and European Control Conference.

[35]  Amit Diwadkar,et al.  Stabilization of linear time varying systems over uncertain channels , 2014, 1408.6913.

[36]  C. Scherer,et al.  Multiobjective output-feedback control via LMI optimization , 1997, IEEE Trans. Autom. Control..

[37]  M. Rotea.  The Generalized H_2 Control Problem , 1991 .

[38]  R. P. Marques,et al.  Discrete-Time Markov Jump Linear Systems , 2004, IEEE Transactions on Automatic Control.

[39]  Amer AL-Hinai,et al.  WSCC 9-Bus System , 2000 .