A new approach to constrained state estimation for discrete‐time linear systems with unknown inputs

Summary This paper addresses the problem of estimating the state for a class of uncertain discrete-time linear systems with constraints by using an optimization-based approach. The proposed scheme uses the moving horizon estimation philosophy together with the game theoretical approach to the H∞ filtering to obtain a robust filter with constraint handling. The used approach is constructive since the proposed moving horizon estimator (MHE) results from an approximation of a type of full information estimator for uncertain discrete-time linear systems, named in short H∞-MHE and H∞–full information estimator, respectively. Sufficient conditions for the stability of the H∞-MHE are discussed for a class of uncertain discrete-time linear systems with constraints. Finally, since the H∞-MHE needs the solution of a complex minimax optimization problem at each sampling time, we propose an approximation to relax the optimization problem and hence to obtain a feasible numerical solution of the proposed filter. Simulation results show the effectiveness of the robust filter proposed.

[1]  E. Gilbert,et al.  Optimal infinite-horizon feedback laws for a general class of constrained discrete-time systems: Stability and moving-horizon approximations , 1988 .

[2]  Giorgio Battistelli,et al.  Robust receding-horizon state estimation for uncertain discrete-time linear systems , 2005, Syst. Control. Lett..

[3]  A. Alessandri,et al.  Robust receding-horizon estimation for uncertain discrete-time linear systems , 2003, ECC.

[4]  Mustapha Ait Rami,et al.  A robust estimation approach for uncertain systems with perturbed measurements , 2016 .

[5]  Yuan Cao,et al.  Reference variable methods of solving min–max optimization problems , 2008, J. Glob. Optim..

[6]  Pramod P. Khargonekar,et al.  FILTERING AND SMOOTHING IN AN H" SETTING , 1991 .

[7]  Lihua Xie,et al.  H∞ estimation for discrete-time linear uncertain systems , 1991 .

[8]  P. Khargonekar,et al.  State-space solutions to standard H/sub 2/ and H/sub infinity / control problems , 1989 .

[9]  Jay H. Lee,et al.  Constrained linear state estimation - a moving horizon approach , 2001, Autom..

[10]  G. Goodwin,et al.  Riccati equations in optimal filtering of nonstabilizable systems having singular state transition matrices , 1986 .

[11]  Tor Arne Johansen,et al.  Linear constrained moving horizon estimator with pre-estimating observer , 2014, Syst. Control. Lett..

[12]  T. S. Chandar,et al.  Robust control of robot manipulators based on uncertainty and disturbance estimation , 2013 .

[13]  H. Fang,et al.  Recursive state estimation for discrete‐time nonlinear systems with event‐triggered data transmission, norm‐bounded uncertainties and multiple missing measurements , 2016 .

[14]  J. Speyer,et al.  A Linear-Quadratic Game Approach to Estimation and Smoothing , 1991, 1991 American Control Conference.

[15]  F. Angulo,et al.  A new approach to state estimation for uncertain linear systems in a moving horizon estimation setting , 2016, Int. J. Autom. Comput..

[16]  James B. Rawlings,et al.  Application of MHE to large-scale nonlinear processes with delayed lab measurements , 2015, Comput. Chem. Eng..

[17]  Beatriz Ricarte,et al.  Real-time estimation of plasma insulin concentration from continuous glucose monitor measurements , 2016, Computer methods in biomechanics and biomedical engineering.

[18]  Florian Nadel,et al.  Stochastic Processes And Filtering Theory , 2016 .

[19]  Eduardo F. Camacho,et al.  Computational burden reduction in min-max MPC , 2011, J. Frankl. Inst..

[20]  Uri Shaked,et al.  A game theory approach to robust discrete-time H∞-estimation , 1994, IEEE Trans. Signal Process..

[21]  Giorgio Battistelli,et al.  Moving horizon state estimation for discrete-time linear systems with binary sensors , 2015, 2015 54th IEEE Conference on Decision and Control (CDC).

[22]  G. Battistelli,et al.  A minimax receding-horizon estimator for uncertain discrete-time linear systems , 2004, Proceedings of the 2004 American Control Conference.

[23]  Brent David Appleby.  Robust estimator design using the H[infinity] norm and [mu] synthesis , 1991 .

[24]  P. Khargonekar,et al.  Filtering and smoothing in an H/sup infinity / setting , 1989, Proceedings of the 28th IEEE Conference on Decision and Control,.

[25]  João Yoshiyuki Ishihara,et al.  Optimal robust filtering for systems subject to uncertainties , 2015, Autom..

[26]  Junmin Wang,et al.  $\mathcal{H}_{\infty}$ Observer Design for LPV Systems With Uncertain Measurements on Scheduling Variables: Application to an Electric Ground Vehicle , 2016, IEEE/ASME Transactions on Mechatronics.

[27]  J. Rawlings,et al.  Nonlinear Moving Horizon State Estimation , 1995 .

[28]  Itzik Klein,et al.  A joint filter for formation tracking , 2015, IEEE Transactions on Aerospace and Electronic Systems.

[29]  P. Khargonekar,et al.  State-space solutions to standard H2 and H∞ control problems , 1988, 1988 American Control Conference.

[30]  Giorgio Battistelli,et al.  Min-Max Moving-Horizon Estimation for Uncertain Discrete-Time Linear Systems , 2012, SIAM J. Control. Optim..

[31]  Hui Zhang,et al.  Active Steering Actuator Fault Detection for an Automatically-Steered Electric Ground Vehicle , 2017, IEEE Transactions on Vehicular Technology.

[32]  G. Battistelli,et al.  Robust Receding-Horizon Estimation for Discrete-time Linear Systems in the Presence of Bounded Uncertainties , 2005, Proceedings of the 44th IEEE Conference on Decision and Control.

[33]  Luo Ji.  MPC Performance monitoring with improved nonlinear state estimation , 2014 .

[34]  Sebastian Engell,et al.  Simulation Study of the Particle Filter and the EKF for State Estimation of a Large-scale DAE-system with Multi-rate Sampling , 2016 .

[35]  Gabriela Cembrano,et al.  Output‐feedback control of combined sewer networks through receding horizon control with moving horizon estimation , 2015 .

[36]  U. Shaked,et al.  H/sub infinity /-optimal estimation: a tutorial , 1992, [1992] Proceedings of the 31st IEEE Conference on Decision and Control.

[37]  Rami Mangoubi.  Robust estimation and failure detection for linear systems , 1995 .

[38]  R. González,et al.  On a system of Hamilton-Jacobi-Bellman inequalities associated to a minimax problem with additive final cost. , 2003 .

[39]  Jay H. Lee,et al.  Receding Horizon Recursive State Estimation , 1993, 1993 American Control Conference.

[40]  Fernando GarcTirado,et al.  Robust Estimation of Uncertain Linear Systems using a Moving Horizon Approach , 2012 .

[41]  Lihua Xie,et al.  Robust filtering for a class of discrete-time uncertain nonlinear systems: AnH∞ approach , 1996 .