Robust observed-state feedback design for discrete-time systems rational in the uncertainties

Design of controllers in the form of a state-feedback coupled to a state observer is studied in the context of uncertain systems. The classical approach by Luenberger is revisited. Results provide a heuristic design procedure that mimics the independent state-feedback / observer gains design by minimizing the coupling of observation error dynamics on the ideal state-feedback dynamics. The proposed design and analysis conditions apply to linear systems rationally-dependent on uncertainties defined in the cross-product of polytopes. Convex linear matrix inequality results are given thanks to the combination of a new descriptor multi-affine representations of systems and the S-variable approach. Stability and H∞ performances are assessed by multi-affine parameter-dependent Lyapunov matrices for both cases of constant and time-varying uncertainties (or combinations of the two). Numerical complexity issues and ways to keep it as limited as possible are discussed and illustrated on an academic example.

[1]  B. Barmish.  Necessary and sufficient conditions for quadratic stabilizability of an uncertain system , 1985 .

[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]  Chang-Hua Lien,et al.  Robust observer-based control of systems with state perturbations via LMI approach , 2004, IEEE Transactions on Automatic Control.

[4]  C. Scherer,et al.  Multiobjective output-feedback control via LMI optimization , 1997, IEEE Trans. Autom. Control..

[5]  Masayuki Sato,et al.  LMI Tests for Positive Definite Polynomials: Slack Variable Approach , 2009, IEEE Transactions on Automatic Control.

[6]  Carsten W. Scherer,et al.  Robustness with dynamic IQCs: An exact state-space characterization of nominal stability with applications to robust estimation , 2008, Autom..

[7]  E. Yaz.  Linear Matrix Inequalities In System And Control Theory , 1998, Proceedings of the IEEE.

[8]  Minyue Fu,et al.  Guaranteed cost control of uncertain nonlinear systems via polynomial lyapunov functions , 2002, IEEE Trans. Autom. Control..

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

[10]  Graziano Chesi,et al.  Sufficient and Necessary LMI Conditions for Robust Stability of Rationally Time-Varying Uncertain Systems , 2013, IEEE Transactions on Automatic Control.

[11]  Hassan K. Khalil,et al.  High-gain observers in nonlinear feedback control , 2009, 2009 IEEE International Conference on Control and Automation.

[12]  Valter J. S. Leite,et al.  LMI based robust stability conditions for linear uncertain systems: a numerical comparison , 2002, Proceedings of the 41st IEEE Conference on Decision and Control, 2002..

[13]  Michel Verhaegen,et al.  Robust output-feedback controller design via local BMI optimization , 2004, Autom..

[14]  Dimitri Peaucelle,et al.  Robust H 2 performance analysis and synthesis of linear polytopic discrete-time periodic systems via LMIs , 2005 .

[15]  A. Trofino.  Robust stability and domain of attraction of uncertain nonlinear systems , 2000, Proceedings of the 2000 American Control Conference. ACC (IEEE Cat. No.00CH36334).

[16]  Pierre Apkarian,et al.  Continuous-time analysis, eigenstructure assignment, and H2 synthesis with enhanced linear matrix inequalities (LMI) characterizations , 2001, IEEE Trans. Autom. Control..

[17]  Dimitri Peaucelle,et al.  Multi-objective H2/H∞/Impulse-to-Peak Control of a Space Launch Vehicle , 2006, Eur. J. Control.

[18]  Jamal Daafouz,et al.  Stability analysis and control synthesis for switched systems: a switched Lyapunov function approach , 2002, IEEE Trans. Autom. Control..

[19]  Chang-Hua Lien,et al.  LMI optimization approach on robustness and H∞ control analysis for observer-based control of uncertain systems , 2008 .

[20]  Simon Hecker,et al.  Generalized LFT-based representation of parametric uncertain models , 2003, 2003 European Control Conference (ECC).

[21]  Carsten W. Scherer,et al.  LMI Relaxations in Robust Control , 2006, Eur. J. Control.

[22]  Eric Walter,et al.  Ellipsoidal parameter or state estimation under model uncertainty , 2004, Autom..

[23]  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).

[24]  S. O. Reza Moheimani,et al.  Perspectives in robust control , 2001 .

[25]  Dimitri Peaucelle,et al.  LMI results for robust control design of observer-based controllers, the discrete-time case with polytopic uncertainties , 2014 .

[26]  D. Luenberger.  An introduction to observers , 1971 .

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

[28]  J. Lofberg,et al.  YALMIP : a toolbox for modeling and optimization in MATLAB , 2004, 2004 IEEE International Conference on Robotics and Automation (IEEE Cat. No.04CH37508).

[29]  Kim-Chuan Toh,et al.  SDPT3 -- A Matlab Software Package for Semidefinite Programming , 1996 .

[30]  J. Bernussou,et al.  H/sub 2/ and H/sub /spl infin// robust output feedback control for continuous time polytopic systems , 2007 .

[31]  Masami Saeki,et al.  Synthesis of output feedback gain-scheduling controllers based on descriptor LPV system representation , 2003, 42nd IEEE International Conference on Decision and Control (IEEE Cat. No.03CH37475).

[32]  Robert E. Skelton,et al.  Stability tests for constrained linear systems , 2001 .

[33]  Dimitri Peaucelle,et al.  Discussion on: “Parameter-Dependent Lyapunov Function Approach to Stability Analysis and Design for Uncertain Systems with Time-Varying Delay” , 2005 .

[34]  Goutam Chakraborty,et al.  LMI approach to robust unknown input observer design for continuous systems with noise and uncertainties , 2010 .

[35]  Dimitri Peaucelle,et al.  S-Variable Approach to LMI-Based Robust Control , 2014 .

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

[37]  Dimitri Peaucelle,et al.  Slack variable approach for robust stability analysis of switching discrete-time systems , 2013 .

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

[39]  J. Doyle,et al.  Review of LFTs, LMIs, and mu , 1991, [1991] Proceedings of the 30th IEEE Conference on Decision and Control.

[40]  Tomomichi Hagiwara,et al.  A dilated LMI approach to robust performance analysis of linear time-invariant uncertain systems , 2005, Autom..

[41]  Emilia Fridman,et al.  A descriptor system approach to H∞ control of linear time-delay systems , 2002, IEEE Trans. Autom. Control..

[42]  Masoud Abbaszadeh,et al.  LMI optimization approach to robust H∞ observer design and static output feedback stabilization for discrete‐time nonlinear uncertain systems , 2009 .

[43]  Ricardo C. L. F. Oliveira,et al.  Robust LMIs with parameters in multi-simplex: Existence of solutions and applications , 2008, 2008 47th IEEE Conference on Decision and Control.

[44]  Denis Arzelier,et al.  Robust analysis of Demeter benchmark via quadratic separation , 2010 .

[45]  Jamal Daafouz,et al.  Parameter dependent Lyapunov functions for discrete time systems with time varying parametric uncertainties , 2001, Syst. Control. Lett..

[46]  L. Vandenberghe,et al.  Extended LMI characterizations for stability and performance of linear systems , 2009, Syst. Control. Lett..

[47]  Graziano Chesi.  Robust Static Output Feedback Controllers via Robust Stabilizability Functions , 2014, IEEE Trans. Autom. Control..

[48]  D. Peaucelle,et al.  Robust disk pole assignment by state and output feedback for generalised uncertainty models , 1998, Proceedings of the 37th IEEE Conference on Decision and Control (Cat. No.98CH36171).