Robust stability, ℋ₂ analysis and stabilisation of discrete-time Markov jump linear systems with uncertain probability matrix

References

• Blair , WP and Sworder , DD . 1975 . Feedback Control of a Class of Linear Discrete Systems with Jump Parameters and Quadratic Cost Criteria . International Journal of Control , 21 : 833 – 841 .
   Web of Science ®Google Scholar
• Bliman , P-A . 2004 . An Existence Result for Polynomial Solutions of Parameter-dependent LMIs . Systems & Control Letters , 51 : 165 – 169 .
   Web of Science ®Google Scholar
• Bliman , P-A , Oliveira , RCLF , Montagner , VF and Peres , PLD . Existence of Homogeneous Polynomial Solutions for Parameter-dependent Linear Matrix Inequalities with Parameters in the Simplex . Proceedings of the 45th IEEE Conference on Decision and Control . San Diego, CA. pp. 1486 – 1491 .
   Google Scholar
• Boyd , S and Vandenberghe , L . 2004 . Convex Optimization , Cambridge, , UK : Cambridge University Press .
   Google Scholar
• Choi , M , Lam , TY and Reznick , B . Sum of Squares of Real Polynomials . Proceedings of Symposia in Pure Mathematics, 58.2 . Providence, RI. pp. 103 – 126 .
   Google Scholar
• Chesi , G , Garulli , A , Tesi , A and Vicino , A . 2005 . Polynomially Parameter-dependent Lyapunov Functions for Robust Stability of Polytopic Systems: An LMI Approach . IEEE Transactions on Automatic Control , 50 : 365 – 370 .
   Web of Science ®Google Scholar
• Chesi , G , Garulli , A , Tesi , A and Vicino , A . 2007 . Robust Stability of Time-varying Polytopic Systems via Parameter-dependent Homogeneous Lyapunov Functions . Automatica , 43 : 309 – 316 .
   Web of Science ®Google Scholar
• Costa , OLV , Assumpção Filho , EO , Boukas , EK and Marques , RP . 1999 . Constrained Quadratic State Feedback Control of Discrete-time Markovian Jump Linear Systems . Automatica , 35 : 617 – 626 .
   Web of Science ®Google Scholar
• Costa , OLV and de Paulo , WL . 2007 . Indefinite Quadratic with Linear Costs Optimal Control of Markov Jump with Multiplicative Noise Systems . Automatica , 43 : 587 – 597 .
   Web of Science ®Google Scholar
• Costa , OLV , do Val , JBR and Geromel , JC . 1997 . A Convex Programming Approach to ℋ2-control of Discrete-time Markovian Jump Linear Systems . International Journal of Control , 66 : 557 – 579 .
   Web of Science ®Google Scholar
• Costa , EF , do Val , JBR and Fragoso , MD . 2005 . A New Approach to Detectability of Discrete-time Infinite Markov Jump Linear Systems . SIAM Journal on Control and Optimization , 43 : 2132 – 2156 .
   Web of Science ®Google Scholar
• Costa , OLV and Fragoso , MD . 1993 . Stability Results for Discrete-time Linear Systems with Markovian Jumping Parameters . Journal of Mathematical Analysis and Applications , 179 : 154 – 178 .
   Web of Science ®Google Scholar
• Costa , OLV and Fragoso , MD . 1995 . Discrete-time LQ-optimal Control Problems for Infinite Markov Jump Parameter Systems . IEEE Transactions on Automatic Control , 40 : 2076 – 2088 .
   Web of Science ®Google Scholar
• Costa , OLV , Fragoso , MD and Marques , RP . 2005 . Discrete-Time Markovian Jump Linear Systems , New York : Springer-Verlag .
   Google Scholar
• de Oliveira , MC , Bernussou , J and Geromel , JC . 1999 . A New Discrete-time Robust Stability Condition . Systems & Control Letters , 37 : 261 – 265 .
   Web of Science ®Google Scholar
• de Souza , CE . 2006 . Robust Stability and Stabilisation of Uncertain Discrete-time Markovian Jump Linear Systems . IEEE Transactions on Automatic Control , 51 : 836 – 841 .
   Web of Science ®Google Scholar
• do Val , JBR , Geromel , JC and Gonçalves , AP . 2002 . The ℋ2-control for Jump Linear Systems: Cluster Observations of the Markov State . Automatica , 38 : 343 – 349 .
   Web of Science ®Google Scholar
• El Bouhtouri , A and El Hadri , K . 2003 . Robust Stabilisation of Jump Linear Systems with Multiplicative Noise . IMA Journal of Mathematical Control and Information , 20 : 1 – 19 .
   Google Scholar
• El Ghaoui , L and Ait Rami , M . 1996 . Robust State-feedback Stabilisation of Jump Linear Systems via LMIs . International Journal of Robust and Nonlinear Control , 6 : 1015 – 1022 .
   Google Scholar
• Feng , X , Loparo , KA , Ji , Y and Chizeck , HJ . 1992 . Stochastic Stability Properties of Jump Linear Systems . IEEE Transactions on Automatic Control , 37 : 38 – 53 .
   Web of Science ®Google Scholar
• Henrion , D and Lasserre , JB . 2006 . Convergent Relaxations of Polynomial Matrix Inequalities and Static Output Feedback . IEEE Transactions on Automatic Control , 51 : 192 – 202 .
   Web of Science ®Google Scholar
• Ji , Y and Chizeck , HJ . 1990a . Controllability and Stabilisability and Continuous-time Jump Linear-quadratic Control . IEEE Transactions on Automatic Control , 35 : 777 – 788 .
   Web of Science ®Google Scholar
• Ji , Y and Chizeck , HJ . 1990b . Jump Linear Quadratic Gaussian Control: Steady-state and Testable Conditions . Control Theory and Advanced Technology , 6 : 289 – 319 .
   Web of Science ®Google Scholar
• Khanbaghi , M , Malhame , RP and Perrier , M . 2002 . Optimal White Water and Broke Recirculation Policies in Paper Mills via Jump Linear Quadratic Control . IEEE Transactions on Automatic Control , 10 : 578 – 588 .
   Google Scholar
• Kojima , M . 2003 . “ Sums of Squares Relaxations of Polynomial Semidefinite Programs ” . In Research Report B-397, Department of Mathematical and Computing Sciences , Tokyo, , Japan : Tokyo Institute of Technology .
   Google Scholar
• Kojima , M and Muramatsu , M . 2007 . An Extension of Sums of Squares Relaxations to Polynomial Optimisation Problems Over Symmetric Cones . Mathematical Programming Series B , 110 : 315 – 336 .
   Google Scholar
• Kuhn , KD and Madanat , SM . 2006 . A New Approach to Detectability of Discrete-time Infinite Markov Jump Linear Systems . Computer-Aided Civil and Infrastructure Engineering , 21 : 171 – 178 .
   Web of Science ®Google Scholar
• Lasserre , JB . 2001 . Global Optimisation with Polynomials and the Problem of Moments . SIAM Journal on Control and Optimization , 11 : 796 – 817 .
   Google Scholar
• Löfberg , J . YALMIP: A Toolbox for Modelling and Optimisation in MATLAB . Proceedings of the IEEE CCA/ISIC/CACSD Multiconference . Taipei, Taiwan. http://control.ee.ethz.ch/~joloef/yalmip.php
   Google Scholar
• Maronna , RA , Martin , DR and Yohai , VJ . 2006 . Robust Statistics: Theory and Methods , Chichester : John Wiley & Sons .
   Google Scholar
• Nilim , A and El Ghaoui , L . 2005 . Robust Control of Markov Decision Processes with Uncertain Transition Matrices . Operations Research , 53 : 780 – 798 .
   Web of Science ®Google Scholar
• Oliveira , RCFL and Peres , PLD . 2007 . Parameter-dependent LMIs in Robust Analysis: Characterisation of Homogeneous Polynomially Parameter-dependent Solutions via LMI Relaxations . IEEE Transactions on Automatic Control , 52 : 1334 – 1340 .
   Web of Science ®Google Scholar
• Parrilo , PA . Exploiting Structure in Sum of Squares Programs . Proceedings of the 42nd IEEE Conference on Decision and Control . December , Maui, HI, USA. pp. 4664 – 4669 .
   Google Scholar
• Scherer , CW . 2003 . “ Higher-order Relaxations for Robust LMI Problems with Verifications for Exactness ” . In in Proceedings of the 42nd IEEE Conference on Decision and Control 4652 – 4657 . Maui, HI, , USA
   Google Scholar
• Scherer , CW . 2005 . Relaxations for Robust Linear Matrix Inequality Problems with Verifications for Exactness . SIAM Journal on Matrix Analysis and Applications , 27 : 365 – 395 .
   Web of Science ®Google Scholar
• Scherer , CW and Hol , CWJ . 2006 . Matrix Sum-of-squares Relaxations for Robust Semi-definite Programs . Mathematical Programming Series B , 107 : 189 – 211 .
   Web of Science ®Google Scholar
• Seiler , P and Sengupta , R . 2005 . An ℋ∞ Approach to Networked Control . IEEE Transactions on Automatic Control , 50 : 356 – 364 .
   Web of Science ®Google Scholar
• Sturm , JF . 1999 . Using SeDuMi 1.02, a MATLAB Toolbox for Optimisation Over Symmetric Cones . Optimization Methods and Software , 11–12 : 625 – 653 . http://sedumi.mcmaster.ca/
   Web of Science ®Google Scholar
• Wang , Z , Yang , F , Ho , DWC and Liu , X . 2005 . Robust Finite-horizon Filtering for Stochastic Systems with Missing Measurements . IEEE Signal Processing Letters , 12 : 437 – 440 .
   Web of Science ®Google Scholar
• Zampolli , F . 2006 . Optimal Monetary Policy in a regime-switching Economy: The Response to Abrupt Shifts in Exchange Rate Dynamics . Journal of Economic Dynamics and Control , 30 : 1527 – 1567 .
   Web of Science ®Google Scholar