Infinite horizon H2/H∞ control for discrete-time time-varying Markov jump systems with multiplicative noise

In this paper we consider the infinite horizon H"2/H"~ control problem for discrete-time time-varying linear systems subject to Markov jump parameters and state-multiplicative noise. A stochastic version of a bounded real lemma is firstly developed for a general class of discrete-time time-varying Markov jump systems with state- and disturbance-multiplicative noise. By which we present a necessary and sufficient condition for the solvability of the H"2/H"~ control problem in terms of four coupled discrete-time Riccati equations. Moreover, the obtained design is applied to a macroeconomic problem to verify its effectiveness.

[1]  V. Dragan,et al.  The linear quadratic optimization problems for a class of linear stochastic systems with multiplicative white noise and Markovian jumping , 2004, IEEE Transactions on Automatic Control.

[2]  U. Shaked,et al.  H-infinity Control and Estimation of State-multiplicative Linear Systems , 2005 .

[3]  Oswaldo Luiz V. Costa,et al.  Indefinite quadratic with linear costs optimal control of Markov jump with multiplicative noise systems , 2007, Autom..

[4]  James Lam,et al.  Robust filtering for discrete-time Markovian jump delay systems , 2004, IEEE Signal Processing Letters.

[5]  Uri Shaked,et al.  H∞ output-feedback control of discrete-time systems with state-multiplicative noise , 2008, Autom..

[6]  Uri Shaked,et al.  H∞ OUTPUT-FEEDBACK OF DISCRETE-TIME SYSTEMS WITH STATE-MULTIPLICATIVE NOISE , 2005 .

[7]  U. Shaked,et al.  $H_\infty$ Control and Estimation of Retarded State-Multiplicative Stochastic Systems , 2007, IEEE Transactions on Automatic Control.

[8]  G. Zames.  Feedback and optimal sensitivity: Model reference transformations, multiplicative seminorms, and approximate inverses , 1981 .

[9]  Oswaldo L. V. Costa.  MIXED H 2 / H ~ CONTROL OF DISCRETE-TIME MARKOVIAN JUMP LINEAR SYSTEMS * , 1996 .

[10]  Weihai Zhang,et al.  Finite Horizon $H_{2}/H_{\infty}$ Control for Discrete-Time Stochastic Systems With Markovian Jumps and Multiplicative Noise , 2010, IEEE Transactions on Automatic Control.

[11]  B. Anderson,et al.  A Nash game approach to mixed H2/H∞ control , 1994, IEEE Transactions on Automatic Control.

[12]  D. Hinrichsen,et al.  Stochastic $H^\infty$ , 1998 .

[13]  Shengyuan Xu,et al.  Robust H∞ control for uncertain discrete‐time stochastic bilinear systems with Markovian switching , 2005 .

[14]  Shengyuan Xu,et al.  Robust Control and Filtering of Singular Systems , 2006 .

[15]  Vasile Dragan,et al.  Mathematical Methods in Robust Control of Discrete-Time Linear Stochastic Systems , 2009 .

[16]  Yong Wang,et al.  On eigenvalue sets and convergence rate of Itô stochastic systems with Markovian switching , 2010, Proceedings of the 29th Chinese Control Conference.

[17]  Weihai Zhang,et al.  Infinite horizon stochastic H2/Hºº control for discrete-time systems with state and disturbance dependent noise. , 2008 .

[18]  P. Khargonekar,et al.  State-space solutions to standard H/sub 2/ and H/sub infinity / control problems , 1989 .

[19]  V. Ugrinovskii.  Robust H∞ infinity control in the presence of stochastic uncertainty , 1998 .

[20]  I. Petersen,et al.  Robust Control Design Using H-? Methods , 2012 .

[21]  D. Hinrichsen,et al.  Stochastic H∞ , 1998 .

[22]  James D. Hamilton.  A New Approach to the Economic Analysis of Nonstationary Time Series and the Business Cycle , 1989 .

[23]  Yuguang Fang,et al.  Stochastic stability of jump linear systems , 2002, IEEE Trans. Autom. Control..

[24]  Bor-Sen Chen,et al.  Generalized Lyapunov Equation Approach to State-Dependent Stochastic Stabilization/Detectability Criterion , 2008, IEEE Transactions on Automatic Control.

[25]  Lihua Xie,et al.  An improved mixed H2/H∞ control design for hard disk drives , 2005, IEEE Trans. Control. Syst. Technol..

[26]  B. Anderson,et al.  A Nash game approach to mixed H/sub 2//H/sub /spl infin// control , 1994 .

[27]  Mi-Ching Tsai,et al.  Robust and Optimal Control , 2014 .

[28]  W. L. Koning,et al.  Discrete-time Markovian jump linear systems , 1993 .

[29]  Isaac Yaesh,et al.  H-Control and Estimation of State-multiplicative Linear Systems , 2005 .

[30]  Gang George Yin,et al.  Markowitz's mean-variance portfolio selection with regime switching: from discrete-time models to their continuous-time limits , 2004, IEEE Transactions on Automatic Control.

[31]  Vasile Dragan,et al.  Asymptotic Properties of Input-Output Operators Norm Associated with Singularly Perturbed Systems with Multiplicative White Noise , 2002, SIAM J. Control. Optim..

[32]  V. Dragan,et al.  Stability and robust stabilization to linear stochastic systems described by differential equations with markovian jumping and multiplicative white noise , 2002 .

[33]  J. Doyle,et al.  Robust and optimal control , 1995, Proceedings of 35th IEEE Conference on Decision and Control.

[34]  D. Hinrichsen,et al.  H∞-type control for discrete-time stochastic systems , 1999 .

[35]  Yulin Huang,et al.  Infinite horizon stochastic H2/Hinfinity control for discrete-time systems with state and disturbance dependent noise , 2008, Autom..

[36]  Riccardo Muradore,et al.  Mixed H2/Hinfinity control: the discrete-time case , 2005, Syst. Control. Lett..

[37]  Marcelo D. Fragoso,et al.  Stationary filter for continuous-time Markovian jump linear systems , 2005, 2004 43rd IEEE Conference on Decision and Control (CDC) (IEEE Cat. No.04CH37601).

[38]  Ian R. Petersen,et al.  Robust Control Design Using H-infinity Methods , 2000 .

[39]  Tamer Başar,et al.  H1-Optimal Control and Related Minimax Design Problems , 1995 .

[40]  T. Basar,et al.  H∞-0ptimal Control and Related Minimax Design Problems: A Dynamic Game Approach , 1996, IEEE Trans. Autom. Control..

[41]  Bor-Sen Chen,et al.  Stochastic H2/H∞ control with state-dependent noise , 2004, IEEE Trans. Autom. Control..