Finite Horizon $H_{2}/H_{\infty}$ Control for Discrete-Time Stochastic Systems With Markovian Jumps and Multiplicative Noise

In this note, we consider the finite horizon mixed <i>H</i> <sub>2</sub>/<i>H</i> <sub>∞</sub> control problem for discrete-time stochastic linear systems subject to Markov jump parameters and multiplicative noise. Firstly, we derive a stochastic bounded real lemma (SBRL), which is used to establish a necessary and sufficient condition for the existence of the mixed <i>H</i> <sub>2</sub>/<i>H</i> <sub>∞</sub> control via the solvability of four coupled difference matrix-valued recursions (CDMRs). Moreover, a state feedback <i>H</i> <sub>2</sub>/<i>H</i> <sub>∞</sub> controller is designed by means of the solutions of CDMRs.

[1]  Gao Wen-hua.  State feedback control for a class of stochastic nonlinear systems with time-delay disturbs , 2009 .

[2]  Eduardo D. Sontag,et al.  An example of a GAS system which can be destabilized by an integrable perturbation , 2003, IEEE Trans. Autom. Control..

[3]  V. Dragan,et al.  Mathematical Methods in Robust Control of Linear Stochastic Systems , 2006 .

[4]  Eduardo Sontag.  Smooth stabilization implies coprime factorization , 1989, IEEE Transactions on Automatic Control.

[5]  I. Petersen,et al.  Absolute Stabilization and Minimax Optimal Control of Uncertain Systems with Stochastic Uncertainty , 1997 .

[6]  Ricardo Paulino Marques,et al.  Mixed / -Control of Discrete-Time Markovian Jump Linear Systems , 1998 .

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

[8]  Peng Shi,et al.  Control of Markovian jump discrete-time systems with norm bounded uncertainty and unknown delay , 1999, IEEE Trans. Autom. Control..

[9]  R. P. Marques,et al.  Mixed H2/H∞-control of discrete-time Markovian jump linear systems , 1998, IEEE Trans. Autom. Control..

[10]  A. Astolfi,et al.  Nonlinear observer design using invariant manifolds and applications , 2005, Proceedings of the 44th IEEE Conference on Decision and Control.

[11]  P.V. Kokotovic,et al.  The joy of feedback: nonlinear and adaptive , 1992, IEEE Control Systems.

[12]  Y. Huang,et al.  Stochastic H2/H8 control for discrete-time systems with state and disturbance dependent noise. , 2007 .

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

[14]  M. Mariton,et al.  Jump Linear Systems in Automatic Control , 1992 .

[15]  Uri Shaked,et al.  Static H2 and H∞ output-feedback of discrete-time LTI systems with state multiplicative noise , 2006, Syst. Control. Lett..

[16]  Alessandro Astolfi,et al.  Invariant Manifold Based Reduced-Order Observer Design for Nonlinear Systems , 2008, IEEE Transactions on Automatic Control.

[17]  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.

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

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

[20]  X. Zhou,et al.  Stochastic Controls: Hamiltonian Systems and HJB Equations , 1999 .

[21]  Min Chen,et al.  Perturbation Analysis of a Dynamic Priority Call Center , 2010, IEEE Transactions on Automatic Control.

[22]  Murat Arcak,et al.  A relaxed condition for stability of nonlinear observer-based controllers , 2004, Syst. Control. Lett..

[23]  Yulin Huang,et al.  Stochastic H2/Hinfinity control for discrete-time systems with state and disturbance dependent noise , 2007, Autom..

[24]  David Angeli,et al.  A Unifying Integral ISS Framework for Stability of Nonlinear Cascades , 2001, SIAM J. Control. Optim..

[25]  Anton A. Stoorvogel,et al.  The H ∞ control problem: a state space approach , 2000 .

[26]  El-Kébir Boukas,et al.  Robust production and maintenance planning in stochastic manufacturing systems , 1995, IEEE Trans. Autom. Control..

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

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

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

[30]  Ricardo Paulino Marques,et al.  Robust Hz -Control for Discrete-time Markovian Jump Linear Systems , 1998 .

[31]  Alessandro Astolfi,et al.  Observer design for a class of nonlinear systems using dynamic scaling with application to adaptive control , 2008, 2008 47th IEEE Conference on Decision and Control.

[32]  Petar V. Kokotovic,et al.  Observer-based control of systems with slope-restricted nonlinearities , 2001, IEEE Trans. Autom. Control..

[33]  V. V. Dombrovskii,et al.  A Linear Quadratic Control for Discrete Systems with Random Parameters and Multiplicative Noise and Its Application to Investment Portfolio Optimization , 2003 .

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

[35]  David J. N. Limebeer,et al.  Linear Robust Control , 1994 .

[36]  Isaac Yaesh,et al.  Robust H∞ filtering of stationary continuous-time linear systems with stochastic uncertainties , 2001, IEEE Trans. Autom. Control..

[37]  Weihai Zhang,et al.  State Feedback HINFINITY Control for a Class of Nonlinear Stochastic Systems , 2006, SIAM J. Control. Optim..

[38]  K. Loparo,et al.  Stochastic stability properties of jump linear systems , 1992 .

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

[40]  Hassan K. Khalil,et al.  A separation principle for the control of a class of nonlinear systems , 2001, IEEE Trans. Autom. Control..

[41]  A. Schaft,et al.  Robust Filtering of Stationary Continuous-Time Linear Systems With Stochastic Uncertainties , 2001 .

[42]  P. Khargonekar,et al.  Mixed H/sub 2//H/sub infinity / control: a convex optimization approach , 1991 .

[43]  Petar V. Kokotovic,et al.  Nonlinear observers: a circle criterion design and robustness analysis , 2001, Autom..