State estimation for Markovian Jump Linear Systems with bounded disturbances

In this paper, we investigate the state estimation problem for a class of Markovian Jump Linear Systems (MJLSs) in the presence of bounded polyhedral disturbances. A set-membership estimation algorithm is first proposed to find the smallest consistent set of all possible states, which is shown to be expressed by a union of multiple polytopes. The posterior probabilities of the system jumping modes are then estimated by introducing the Lebesgue measure, based on which the optimal point estimate is further provided. Moreover, a similarity relationship for polytopes is defined and an approximate method is presented to calculate the Minkowski sum of polytopes, which can help reduce the computational complexity of the overall estimation algorithm.

[1]  Donald Goldfarb,et al.  An O(n3L) primal interior point algorithm for convex quadratic programming , 1991, Math. Program..

[2]  E. Walter,et al.  Exact recursive polyhedral description of the feasible parameter set for bounded-error models , 1989 .

[3]  A. Vicino,et al.  Sequential approximation of feasible parameter sets for identification with set membership uncertainty , 1996, IEEE Trans. Autom. Control..

[4]  Christophe Andrieu,et al.  Efficient particle filtering for jump Markov systems. Application to time-varying autoregressions , 2003, IEEE Trans. Signal Process..

[5]  Mübeccel Demirekler,et al.  Risk-sensitive filtering for jump Markov linear systems , 2008, Autom..

[6]  Tan-Jan Ho,et al.  A switched IMM-Extended Viterbi estimator-based algorithm for maneuvering target tracking , 2011, Autom..

[7]  Antonio Vicino,et al.  Optimal estimation theory for dynamic systems with set membership uncertainty: An overview , 1991, Autom..

[8]  James Renegar,et al.  A polynomial-time algorithm, based on Newton's method, for linear programming , 1988, Math. Program..

[9]  Inseok Hwang,et al.  Stochastic Linear Hybrid Systems: Modeling, Estimation, and Application in Air Traffic Control , 2009, IEEE Transactions on Control Systems Technology.

[10]  K. Fukuda,et al.  Computing All Faces of the Minkowski Sum of V-Polytopes , 2005 .

[11]  Eugenio Cinquemani,et al.  State estimation in stochastic hybrid Systems with sparse observations , 2006, IEEE Transactions on Automatic Control.

[12]  F. Lewis,et al.  Optimal and Robust Estimation: With an Introduction to Stochastic Control Theory, Second Edition , 2007 .

[13]  Narendra Karmarkar,et al.  A new polynomial-time algorithm for linear programming , 1984, Comb..

[14]  H. Balakrishnan,et al.  State estimation for hybrid systems: applications to aircraft tracking , 2006 .

[15]  Luc Jaulin,et al.  Robust set-membership state estimation; application to underwater robotics , 2009, Autom..

[16]  J. P. Norton,et al.  Fast and robust algorithm to compute exact polytope parameter bounds , 1990 .

[17]  V. L. Turova-Botkina,et al.  An algorithm for finding the Chebyshev center of a convex polyhedron , 1994, System Modelling and Optimization.

[18]  T. Başar,et al.  H∞-Control of Markovian Jump Systems and Solutions to Associated Piecewise-Deterministic Differential Games , 1995 .

[19]  Alexandre Trofino,et al.  Mode-Independent ${\cal H}_{\infty}$ Filters for Markovian Jump Linear Systems , 2006, IEEE Transactions on Automatic Control.

[20]  F. Schweppe.  Recursive state estimation: Unknown but bounded errors and system inputs , 1967 .

[21]  V. Broman,et al.  A compact algorithm for the intersection and approximation of N -dimensional polytopes , 1990 .

[22]  Christian P. Robert,et al.  Monte Carlo Statistical Methods , 2005, Springer Texts in Statistics.

[23]  Fuchun Sun,et al.  Design of Hinfinity filter for Markov jumping linear systems with non-accessible mode information , 2008, Autom..

[24]  Erchin Serpedin,et al.  Mathematical Foundations for Signal Processing, Communications, and Networking , 2011 .

[25]  C. F. Long,et al.  Influence of the manufacturing process on the scheduling problem , 1976 .

[26]  Dan Halperin,et al.  Exact and efficient construction of Minkowski sums of convex polyhedra with applications , 2007, Comput. Aided Des..

[27]  Stephen P. Boyd,et al.  Convex Optimization , 2004, Algorithms and Theory of Computation Handbook.

[28]  Arnaud Doucet,et al.  Stochastic sampling algorithms for state estimation of jump Markov linear systems , 2000, IEEE Trans. Autom. Control..

[29]  Y. Bar-Shalom,et al.  The interacting multiple model algorithm for systems with Markovian switching coefficients , 1988 .

[30]  T. Alamo,et al.  A set-membership state estimation algorithm based on DC programming , 2008, Autom..

[31]  Graham C. Goodwin,et al.  A moving horizon approach to Networked Control system design , 2004, IEEE Transactions on Automatic Control.

[32]  O. Costa.  Linear minimum mean square error estimation for discrete-time Markovian jump linear systems , 1994, IEEE Trans. Autom. Control..

[33]  V. A. Yemelicher,et al.  Polytopes, Graphs and Optimisation , 1984 .

[34]  Florence Denis,et al.  Contributing vertices-based Minkowski sum computation of convex polyhedra , 2009, Comput. Aided Des..

[35]  G. Ziegler.  Lectures on Polytopes , 1994 .

[36]  Elmer G. Gilbert,et al.  The minimal disturbance invariant set: Outer approximations via its partial sums , 2006, Autom..

[37]  R. P. Marques,et al.  Discrete-Time Markov Jump Linear Systems , 2004, IEEE Transactions on Automatic Control.

[38]  El Kebir Boukas,et al.  Stability of discrete-time linear systems with Markovian jumping parameters , 1995, Math. Control. Signals Syst..

[39]  Hao Wu,et al.  State estimation for networked systems: an extended IMM algorithm , 2013, Int. J. Syst. Sci..

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

[41]  Komei Fukuda,et al.  From the zonotope construction to the Minkowski addition of convex polytopes , 2004, J. Symb. Comput..

[42]  Yugang Niu,et al.  Robust Filtering Design for Stochastic System With Mode-Dependent Output Quantization , 2010, IEEE Transactions on Signal Processing.