Model reduction of discrete Markovian jump systems with time‐weighted H2 performance

Summary This paper is concerned with the optimal time-weighted H2 model reduction problem for discrete Markovian jump linear systems (MJLSs). The purpose is to find a mean square stable MJLS of lower order such that the time-weighted H2 norm of the corresponding error system is minimized for a given mean square stable discrete MJLSs. The notation of time-weighted H2 norm of discrete MJLS is defined for the first time, and then a computational formula of this norm is given, which requires the solution of two sets of recursive discrete Markovian jump Lyapunov-type linear matrix equations. Based on the time-weighted H2 norm formula, we propose a gradient flow method to solve the optimal time-weighted H2 model reduction problem. A necessary condition for minimality is derived, which generalizes the standard result for systems when Markov jumps and the time-weighting term do not appear. Finally, numerical examples are used to illustrate the effectiveness of the proposed approach. Copyright © 2015 John Wiley & Sons, Ltd.

[1]  Jianbin Qiu,et al.  Model Approximation for Discrete-Time State-Delay Systems in the T–S Fuzzy Framework , 2011, IEEE Transactions on Fuzzy Systems.

[2]  Patrizio Colaneri,et al.  Almost Sure Stability of Markov Jump Linear Systems With Deterministic Switching , 2013, IEEE Transactions on Automatic Control.

[3]  Wuneng Zhou,et al.  Time-delay dependent H∞ model simplification for singular systems with Markovian jumping parameters , 2011 .

[4]  Jun Wang,et al.  A gradient flow approach to on-line robust pole assignment for synthesizing output feedback control systems , 2002, Autom..

[5]  Angelika Bunse-Gerstner,et al.  h2-norm optimal model reduction for large scale discrete dynamical MIMO systems , 2010, J. Comput. Appl. Math..

[6]  Biao Huang,et al.  H2 approximation of multiple input/output delay systems , 2004 .

[7]  Daizhan Cheng,et al.  Adaptive control of linear Markov jump systems , 2006, Int. J. Syst. Sci..

[8]  Yongduan Song,et al.  ${\cal H}_{\infty}$ Model Reduction of Takagi–Sugeno Fuzzy Stochastic Systems , 2012, IEEE Transactions on Systems, Man, and Cybernetics, Part B (Cybernetics).

[9]  J. Geromel,et al.  A Convex Programming Approach to the H2-Control of Discrete-Time Markovian Jump Linear Systems , 1995, Proceedings of 1995 34th IEEE Conference on Decision and Control.

[10]  Wei Xing Zheng,et al.  Weighted H∞ model reduction for linear switched systems with time-varying delay , 2009, Autom..

[11]  John B. Moore,et al.  A gradient flow approach to decentralized output feedback optimal control , 1996 .

[12]  Yun Zou,et al.  Robust exponential stabilization for Markovian jump systems with mode-dependent input delay , 2007, Autom..

[13]  Nan Xiao,et al.  Stabilization of Markov jump linear systems using quantized state feedback , 2010, Autom..

[14]  Guang-Hong Yang,et al.  H∞ model reduction of linear continuous-time systems over finite frequency interval-LMI based approach , 2009, 2009 American Control Conference.

[15]  P. Fortin,et al.  Comments on "The design of linear regulators optimal for time multiplied performance indices" , 1972 .

[16]  Shengyuan Xu,et al.  Delay‐dependent passivity and passification for uncertain Markovian jump systems with time‐varying delays , 2012 .

[17]  Gilles Ferreres,et al.  Reduction of dynamic LFT systems with LTI model uncertainties , 2004 .

[18]  J.C. Geromel,et al.  ${\cal H}_{\infty}$ Filtering of Discrete-Time Markov Jump Linear Systems Through Linear Matrix Inequalities , 2009, IEEE Transactions on Automatic Control.

[19]  Lixian Zhang,et al.  Model reduction for a class of nonstationary Markov jump linear systems , 2012, J. Frankl. Inst..

[20]  Shengyuan Xu,et al.  Optimal time-weighted H 2 model reduction for Markovian jump systems , 2012, Int. J. Control.

[21]  James Lam,et al.  An approximate approach to H2 optimal model reduction , 1999, IEEE Trans. Autom. Control..

[22]  E. Boukas,et al.  Stability and stabilization of discrete‐time singular Markov jump systems with time‐varying delay , 2010 .

[23]  James Lam,et al.  On H2 model reduction of bilinear systems , 2002, Autom..

[24]  Biao Huang,et al.  H ∞ structured model reduction algorithms for linear discrete systems via LMI-based optimisation , 2009, Int. J. Syst. Sci..

[25]  James Lam,et al.  H ∞  model reduction for discrete‐time positive systems with inhomogeneous initial conditions , 2015 .

[26]  Michael Athans,et al.  Command and control (C2) theory: A challenge to control science , 1986 .

[27]  Zikuan Liu,et al.  Jump linear quadratic regulator with controlled jump rates , 2001, IEEE Trans. Autom. Control..

[28]  Victor Sreeram,et al.  Optimal model reduction of non-causal systems via least squares frequency fitting , 2007 .

[29]  James Lam,et al.  Mixed Additive/Multiplicative H∞ Model Reduction , 2013 .

[30]  James Lam,et al.  A gradient flow approach to the robust pole-placement problem , 1995 .

[31]  Z. K. Liu,et al.  Guaranteed Cost Control of a Markov Jump Linear Uncertain System Using a Time-Multiplied Cost Function , 2002 .

[32]  Frank L. Lewis,et al.  Optimal output tracker using a time‐weighted performance index , 1998 .

[33]  R. Prasad,et al.  Time domain model order reduction using Hankel matrix approach , 2014, Journal of the Franklin Institute.

[34]  Ian R. Petersen,et al.  Guaranteed cost control of uncertain systems with a time-multiplied quadratic cost function: An approach based on linear matrix inequalities , 1997, 1997 European Control Conference (ECC).

[35]  James Lam,et al.  H∞ model reduction of Markovian jump linear systems , 2003, Syst. Control. Lett..

[36]  Anders Rantzer,et al.  Balanced Truncation for Discrete Time Markov Jump Linear Systems , 2010, IEEE Transactions on Automatic Control.

[37]  Shafishuhaza Sahlan,et al.  Improved results on frequency‐weighted balanced truncation and error bounds , 2012 .

[38]  D. Wilson.  Optimum solution of model-reduction problem , 1970 .

[39]  Fei Liu,et al.  Fuzzy model‐based fault detection for Markov jump systems , 2009 .

[40]  Cishen Zhang,et al.  A Gradient Flow Approach to Optimal Model Reduction of Discrete-Time Periodic Systems , 2002, J. Glob. Optim..

[41]  Shengyuan Xu,et al.  Time-Multiplied Guaranteed Cost Control of Linear Delay Systems , 2008 .

[42]  Satakshi,et al.  Order reduction of linear discrete systems using a genetic algorithm , 2005 .

[43]  Marcelo D. Fragoso,et al.  A New Approach to Detectability of Discrete-Time Infinite Markov Jump Linear Systems , 2005, Proceedings of the 44th IEEE Conference on Decision and Control.

[44]  Yao-Lin Jiang,et al.  Model order reduction based on general orthogonal polynomials in the time domain for coupled systems , 2014, J. Frankl. Inst..

[45]  Victor Sreeram,et al.  A new approach for frequency weighted L2 model reduction of discrete-time systems , 1998 .

[46]  V. Ugrinovskii *,et al.  Decentralized control of power systems via robust control of uncertain Markov jump parameter systems , 2005 .

[47]  El Kebir Boukas.  On reference model tracking for Markov jump systems , 2009, Int. J. Syst. Sci..