Extended Robust $\mathcal{H}_{2}$ and $\mathcal{H}_{\infty}$ Filter Design for Discrete Time-Invariant Linear Systems with Polytopic Uncertainty

This paper is concerned with the problem of robust $\mathcal{H}_{2}$ and $\mathcal{H}_{\infty}$ filter design for discrete-time linear time-invariant systems with polytopic parameter uncertainties. Less conservative robust $\mathcal{H}_{2}$ and $\mathcal{H}_{\infty}$ filter design procedures are proposed in terms of single-parameter minimization problems with linear matrix inequality constraints. To this end, we generalize the filter structures available in the literature to date in such a way that the filter’s next state is built by summing the filter’s states over several samples from the past to the present. For stability of the filtering error system, the homogeneous polynomial parameter-dependent Lyapunov functions are employed. Finally, illustrative examples are given to demonstrate the merits of the proposed methods.

[1]  Emilia Fridman,et al.  A new H∞ filter design for linear time delay systems , 2001, IEEE Trans. Signal Process..

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

[3]  Truong Q. Nguyen,et al.  Robust and reduced-order filtering: new LMI-based characterizations and methods , 2001, IEEE Trans. Signal Process..

[4]  Karolos M. Grigoriadis,et al.  A unified algebraic approach to linear control design , 1998 .

[5]  J.C. Geromel,et al.  H/sub 2/ and H/sub /spl infin// robust filtering for convex bounded uncertain systems , 1998, Proceedings of the 37th IEEE Conference on Decision and Control (Cat. No.98CH36171).

[6]  Alexandre Trofino,et al.  Robust H2 filtering for uncertain linear systems: LMI based methods with parametric Lyapunov functions , 2005, Syst. Control. Lett..

[7]  Edoardo Mosca,et al.  Robust H2 and Hinfinity filtering for uncertain linear systems , 2006, Autom..

[8]  Magdi S. Mahmoud,et al.  New filter design for linear time-delay systems , 2011 .

[9]  J. Bernussou,et al.  A new robust D-stability condition for real convex polytopic uncertainty , 2000 .

[10]  Dimitri Peaucelle,et al.  Periodic FIR controller synthesis for discrete-time uncertain linear systems , 2011, IEEE Conference on Decision and Control and European Control Conference.

[11]  Stephen P. Boyd,et al.  Linear Matrix Inequalities in Systems and Control Theory , 1994 .

[12]  David Zhang,et al.  Improved robust H2 and Hinfinity filtering for uncertain discrete-time systems , 2004, Autom..

[13]  J. Geromel,et al.  A new discrete-time robust stability condition , 1999 .

[14]  Maurício C. de Oliveira,et al.  H2 and H∞ robust filtering for convex bounded uncertain systems , 2001, IEEE Trans. Autom. Control..

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

[16]  José Claudio Geromel,et al.  Optimal linear filtering under parameter uncertainty , 1999, IEEE Trans. Signal Process..

[17]  Ricardo C. L. F. Oliveira,et al.  Robust H2 and H∞ filter design for uncertain linear systems via LMIs and polynomial matrices , 2011, Signal Process..

[18]  Huijun Gao,et al.  Finite Frequency H∞ Control for Vehicle Active Suspension Systems , 2011 .

[19]  R.H.C. Takahashi,et al.  H/sub 2//H/sub /spl infin// filter design for systems with polytope-bounded uncertainty , 2006, IEEE Transactions on Signal Processing.

[20]  Johan Efberg,et al.  YALMIP : A toolbox for modeling and optimization in MATLAB , 2004 .

[21]  Ricardo H. C. Takahashi,et al.  H2/H∞ filter design for systems with polytope-bounded uncertainty , 2006 .

[22]  Maurício C. de Oliveira,et al.  H[sub 2] and Hinfinity Robust Filtering for Discrete-Time Linear Systems , 2000, SIAM J. Control. Optim..

[23]  Keith J. Burnham,et al.  Robust filtering for a class of stochastic uncertain nonlinear time-delay systems via exponential state estimation , 2001, IEEE Trans. Signal Process..

[24]  Arash Sadeghzadeh,et al.  Robust H 2 and H ∞ filtering for discrete-time uncertain Linear fractional transform systems , 2015 .

[25]  Dimitri Peaucelle,et al.  Periodically time-varying memory state-feedback controller synthesis for discrete-time linear systems , 2011, Autom..

[26]  Ricardo C. L. F. Oliveira,et al.  Parameter-dependent H2 and Hinfinity filter design for linear systems with arbitrarily time-varying parameters in polytopic domains , 2008, Signal Process..

[27]  Huijun Gao,et al.  Finite Frequency $H_{\infty }$ Control for Vehicle Active Suspension Systems , 2011, IEEE Transactions on Control Systems Technology.

[28]  Pierre-Alexandre Bliman,et al.  A Convex Approach to Robust Stability for Linear Systems with Uncertain Scalar Parameters , 2003, SIAM J. Control. Optim..

[29]  Huijun Gao,et al.  A Parameter-Dependent Approach to Robust $H_{\infty }$ Filtering for Time-Delay Systems , 2008, IEEE Transactions on Automatic Control.

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

[31]  Ricardo C. L. F. Oliveira,et al.  Parameter-Dependent LMIs in Robust Analysis: Characterization of Homogeneous Polynomially Parameter-Dependent Solutions Via LMI Relaxations , 2007, IEEE Transactions on Automatic Control.

[32]  Johan Löfberg,et al.  YALMIP : a toolbox for modeling and optimization in MATLAB , 2004 .

[33]  Graziano Chesi,et al.  Polynomially parameter-dependent Lyapunov functions for robust stability of polytopic systems: an LMI approach , 2005, IEEE Transactions on Automatic Control.

[34]  J. Geromel,et al.  Robust filtering of discrete-time linear systems with parameter dependent Lyapunov functions , 1999, Proceedings of the 38th IEEE Conference on Decision and Control (Cat. No.99CH36304).

[35]  Brian Birge,et al.  PSOt - a particle swarm optimization toolbox for use with Matlab , 2003, Proceedings of the 2003 IEEE Swarm Intelligence Symposium. SIS'03 (Cat. No.03EX706).

[36]  José Claudio Geromel,et al.  I robust filter design with performance certificate via convex programming , 2008, Autom..

[37]  Lihua Xie,et al.  New approaches to robust minimum variance filter design , 2001, IEEE Trans. Signal Process..

[38]  M. C. D. Oliveiraa,et al.  A new discrete-time robust stability condition ( , 1999 .

[39]  Arkadi Nemirovski,et al.  Lmi Control Toolbox For Use With Matlab , 2014 .

[40]  J. Stoustrup,et al.  Robust stability and performance of uncertain systems in state space , 1992, [1992] Proceedings of the 31st IEEE Conference on Decision and Control.

[41]  Huijun Gao,et al.  A new design of robust H2 filters for uncertain systems , 2008, Syst. Control. Lett..

[42]  WangChanghong,et al.  ROBUST H∞ FILTERING FOR UNCERTAIN DISCRETE-TIME STATE-DELAYED SYSTEMS： APARAMETER-DEPENDENT LYAPUNOV APPROACH , 2004 .

[43]  Ricardo H. C. Takahashi,et al.  H 2 /H infinity filter design for systems with polytope-bounded uncertainty. , 2006 .