An LMI approach for H2 and H∞ reduced-order filtering of uncertain discrete-time Markov and Bernoulli jump linear systems

Abstract This paper proposes a new approach based on parameter-dependent linear matrix inequality (LMI) conditions associated with a scalar parameter that are sufficient to provide robust H 2  and H ∞  reduced-order mode-dependent, partially mode-dependent or mode-independent filters for discrete-time Markov jump linear systems (MJLS) with time-invariant uncertain transition probabilities. Time-invariant uncertainties in the state–space matrices of the modes can be handled as well. As main difference with respect to the existing approaches in the literature, the filter matrices are obtained directly from the slack variables introduced in the conditions. Moreover, the proposed conditions become also necessary for a particular choice of the scalar parameter when mode-dependent full-order filters are designed for systems without uncertainties. Additionally, for precisely known generalized Bernoulli jump systems (i.e., the case where all the rows of the transition probability matrix are equal), optimal solutions are obtained for both mode-dependent and mode-independent full-order filters. Examples (including one motivated by a practical application) are presented to illustrate the proposed approach.

[1]  J. D. Val,et al.  Full InformationH∞-Control for Discrete-Time Infinite Markov Jump Parameter Systems , 1996 .

[2]  Márcio F. Braga,et al.  Brief Paper - H 2 control of discrete-time Markov jump linear systems with uncertain transition probability matrix: improved linear matrix inequality relaxations and multi-simplex modelling , 2013 .

[3]  Pierre-Alexandre Bliman,et al.  An existence result for polynomial solutions of parameter-dependent LMIs , 2004, Syst. Control. Lett..

[4]  Luiz Affonso Guedes,et al.  A Discrete Wavelet Transform (DWT)-Based Energy-Efficient Selective Retransmission Mechanism for Wireless Image Sensor Networks , 2012, J. Sens. Actuator Networks.

[5]  Oswaldo Luiz V. Costa,et al.  H2-Control and the Separation Principle for Discrete-Time Markovian Jump Linear Systems , 2004, Math. Control. Signals Syst..

[6]  Alim P. C. Gonçalves,et al.  Filtering of discrete‐time Markov jump linear systems with uncertain transition probabilities , 2011 .

[7]  Ricardo C. L. F. Oliveira,et al.  H∞ state feedback control for MJLS with uncertain probabilities , 2015, Autom..

[8]  Jos F. Sturm,et al.  A Matlab toolbox for optimization over symmetric cones , 1999 .

[9]  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.

[10]  O. Costa,et al.  Robust linear filtering for discrete-time hybrid Markov linear systems , 2002 .

[11]  Alim P. C. Gonçalves,et al.  Dynamic Output Feedback Control of Discrete-Time Markov Jump Linear Systems through Linear Matrix Inequalities , 2009, SIAM J. Control. Optim..

[12]  Matteo Bertocco,et al.  Experimental Study of Coexistence Issues Between IEEE 802.11b and IEEE 802.15.4 Wireless Networks , 2008, IEEE Transactions on Instrumentation and Measurement.

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

[14]  Raja Sengupta,et al.  A bounded real lemma for jump systems , 2003, IEEE Trans. Autom. Control..

[15]  Lixian Zhang,et al.  Mode-dependent Hinfinity filtering for discrete-time Markovian jump linear systems with partly unknown transition probabilities , 2009, Autom..

[16]  Oswaldo Luiz V. Costa,et al.  Stationary filter for linear minimum mean square error estimator of discrete-time Markovian jump systems , 2002, IEEE Trans. Autom. Control..

[17]  Alim P. C. Gonçalves,et al.  Optimal and mode-independent filters for generalised Bernoulli jump systems , 2015, Int. J. Syst. Sci..

[18]  Alim P. C. Gonçalves,et al.  ℋ 2 filtering of discrete-time Markov jump linear systems through linear matrix inequalities , 2008, Int. J. Control.

[19]  Lixian Zhang,et al.  Stability and stabilization of Markovian jump linear systems with partly unknown transition probabilities , 2009, Autom..

[20]  Alim P. C. Gonçalves,et al.  H∞ robust and networked control of discrete-time MJLS through LMIs , 2012, J. Frankl. Inst..

[21]  M. Fragoso,et al.  ℋ︁∞ filtering for discrete‐time linear systems with Markovian jumping parameters† , 2003 .

[22]  James Lam,et al.  H∞ and H2 filtering for linear systems with uncertain Markov transitions , 2016, Autom..

[23]  M. Fragoso,et al.  Stability Results for Discrete-Time Linear Systems with Markovian Jumping Parameters , 1993 .