A set-valued filter for discrete time polynomial systems using sum of squares programming

In this paper, we propose a novel approach for robust state reconstruction of multi-output discrete-time systems with polynomial nonlinear dynamics and measurements in presence of unknown but bounded disturbances corrupting both the state and measurement equations. The proposed recursive algorithm is based on ellipsoidal state bounding techniques and consists of a two-step prediction-correction procedure, with each step requiring the solution of a convex optimization problem under sum of squares rather than linear matrix inequality constraints. The presented approach reduces the conservatism in the calculation of the state bounding ellipsoid when compared to existing robust filtering approaches. The applicability and effectiveness of the new algorithm is exemplarily shown in numerical simulations of a benchmark system.

[1]  A. Papachristodoulou,et al.  Nonlinear control synthesis by sum of squares optimization: a Lyapunov-based approach , 2004, 2004 5th Asian Control Conference (IEEE Cat. No.04EX904).

[2]  M. Darouach,et al.  A stable recursive state estimation filter for models with nonlinear dynamics subject to bounded disturbances , 2006, Proceedings of the 45th IEEE Conference on Decision and Control.

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

[4]  Ian R. Petersen,et al.  Robust Kalman Filtering for Signals and Systems with Large Uncertainties , 1999 .

[5]  Ashutosh Sabharwal,et al.  Set estimation via ellipsoidal approximations , 1997, IEEE Trans. Signal Process..

[6]  Marie-Françoise Roy,et al.  Real algebraic geometry , 1992 .

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

[8]  Giuseppe Carlo Calafiore,et al.  Reliable localization using set-valued nonlinear filters , 2005, IEEE Transactions on Systems, Man, and Cybernetics - Part A: Systems and Humans.

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

[10]  B. Reznick,et al.  Sums of squares of real polynomials , 1995 .

[11]  Daniel Ferreira Coutinho,et al.  Set-membership filtering of uncertain discrete-time rational systems through Recursive Algebraic Representations and LMIs , 2008, 2008 47th IEEE Conference on Decision and Control.

[12]  Kim-Chuan Toh,et al.  Solving semidefinite-quadratic-linear programs using SDPT3 , 2003, Math. Program..

[13]  Giuseppe Carlo Calafiore,et al.  Robust filtering for discrete-time systems with bounded noise and parametric uncertainty , 2001, IEEE Trans. Autom. Control..

[14]  Von der Fakult.  Robust Fault Detection and Isolation of Nonlinear Systems with Augmented State Models , 2009 .

[15]  J. Shamma,et al.  Approximate set-valued observers for nonlinear systems , 1997, IEEE Trans. Autom. Control..

[16]  Mark Campbell,et al.  A nonlinear set‐membership filter for on‐line applications , 2003 .

[17]  Eduardo F. Camacho,et al.  Guaranteed state estimation by zonotopes , 2005, Autom..

[18]  M. Kojima.  Sums of Squares Relaxations of Polynomial Semidefinite Programs , 2003 .

[19]  A. Kurzhanski,et al.  Ellipsoidal Calculus for Estimation and Control , 1996 .

[20]  Mark Campbell,et al.  Online nonlinear guaranteed estimation with application to a high performance aircraft , 2002, Proceedings of the 2002 American Control Conference (IEEE Cat. No.CH37301).

[21]  Carsten W. Scherer,et al.  Matrix Sum-of-Squares Relaxations for Robust Semi-Definite Programs , 2006, Math. Program..

[22]  E. Walter,et al.  Guaranteed recursive nonlinear state estimation using interval analysis , 1998, Proceedings of the 37th IEEE Conference on Decision and Control (Cat. No.98CH36171).

[23]  Graziano Chesi,et al.  On the Gap Between Positive Polynomials and SOS of Polynomials , 2007, IEEE Transactions on Automatic Control.

[24]  P. Parrilo.  Structured semidefinite programs and semialgebraic geometry methods in robustness and optimization , 2000 .

[25]  Eric Walter,et al.  Trace Versus Determinant in Ellipsoidal Outer-Bounding, with Application to State Estimation , 1996 .

[26]  Fred C. Schweppe,et al.  Uncertain dynamic systems , 1973 .

[27]  D. Hilbert.  Über die Darstellung definiter Formen als Summe von Formenquadraten , 1888 .

[28]  S. Kapoor,et al.  Set-Membership State Estimation with Optimal Bounding Ellipsoids , 1996 .

[29]  Yurii Nesterov,et al.  Interior-point polynomial algorithms in convex programming , 1994, Siam studies in applied mathematics.

[30]  Stephen P. Boyd,et al.  Semidefinite Programming , 1996, SIAM Rev..

[31]  F. Chernousko.  State Estimation for Dynamic Systems , 1993 .

[32]  Petros G. Voulgaris,et al.  On optimal ℓ∞ to ℓ∞ filtering , 1995, Autom..

[33]  D. Bertsekas,et al.  Recursive state estimation for a set-membership description of uncertainty , 1971 .

[34]  Vicenç Puig,et al.  Passive Robust Fault Detection Using Non-Linear Interval Observers: Application to the DAMADICS Benchmark Problem , 2003 .

[35]  Pablo A. Parrilo,et al.  Introducing SOSTOOLS: a general purpose sum of squares programming solver , 2002, Proceedings of the 41st IEEE Conference on Decision and Control, 2002..

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

[37]  Z. Jarvis-Wloszek,et al.  Lyapunov Based Analysis and Controller Synthesis for Polynomial Systems using Sum-of-Squares Optimization , 2003 .