The Generalized Moment-Based Filter

Can we solve the filtering problem from the only knowledge of few moments of the noise terms? In this technical note, by exploiting set of distributions based filtering, we solve this problem without introducing additional assumptions on the distributions of the noises (e.g., Gaussianity) or on the final form of the estimator (e.g., linear estimator). Given the moments (e.g., mean and variance) of random variable X, it is possible to define the set of all distributions that are compatible with the moments information. This set can be equivalently characterized by its extreme distributions: a family of mixtures of Dirac's deltas. The lower and upper expectation of any function g of X are obtained in correspondence of these extremes and can be computed by solving a linear programming problem. The filtering problem can then be solved by running iteratively this linear programming problem. In this technical note, we discuss theoretical properties of this filter, we show the connection with set-membership estimation and its practical applications.

[1]  Luigi Chisci,et al.  Recursive state bounding by parallelotopes , 1996, Autom..

[2]  Anders Lindquist,et al.  Important moments in systems, control and optimization , 2009, Proceedings of the 48h IEEE Conference on Decision and Control (CDC) held jointly with 2009 28th Chinese Control Conference.

[3]  James E. Smith,et al.  Generalized Chebychev Inequalities: Theory and Applications in Decision Analysis , 1995, Oper. Res..

[4]  Alan F. Karr,et al.  Extreme Points of Certain Sets of Probability Measures, with Applications , 1983, Math. Oper. Res..

[5]  J. L. Maryak,et al.  Use of the Kalman filter for inference in state-space models with unknown noise distributions , 1997, Proceedings of the 1997 American Control Conference (Cat. No.97CH36041).

[6]  M. Kreĭn,et al.  The Markov Moment Problem and Extremal Problems , 1977 .

[7]  X. Mao,et al.  Robust stability of uncertain stochastic differential delay equations , 1998 .

[8]  U. Shaked,et al.  Stability and guaranteed cost control of uncertain discrete delay systems , 2005 .

[9]  Zhong-Ping Jiang,et al.  A converse Lyapunov theorem for discrete-time systems with disturbances , 2002, Syst. Control. Lett..

[10]  Michele Pavon,et al.  Hellinger Versus Kullback–Leibler Multivariable Spectrum Approximation , 2007, IEEE Transactions on Automatic Control.

[11]  A. Shapiro.  ON DUALITY THEORY OF CONIC LINEAR PROBLEMS , 2001 .

[12]  Horacio J. Marquez,et al.  Razumikhin-type stability theorems for discrete delay systems , 2007, Autom..

[13]  Andrew R. Teel,et al.  Tractable Razumikhin-type conditions for input-to-state stability analysis of delay difference inclusions , 2013, Autom..

[14]  P. Walley.  Statistical Reasoning with Imprecise Probabilities , 1990 .

[15]  Ioana Popescu,et al.  Optimal Inequalities in Probability Theory: A Convex Optimization Approach , 2005, SIAM J. Optim..

[16]  Alessio Benavoli,et al.  Pushing Kalman's idea to the extremes , 2012, 2012 15th International Conference on Information Fusion.

[17]  James C. Spall.  The Kantorovich inequality for error analysis of the Kalman filter with unknown noise distributions , 1995, Autom..

[18]  J. L. Maryak,et al.  Use of the Kalman filter for inference in state-space models with unknown noise distributions , 1996, IEEE Transactions on Automatic Control.

[19]  Michele Pavon,et al.  On the Georgiou-Lindquist approach to constrained Kullback-Leibler approximation of spectral densities , 2006, IEEE Transactions on Automatic Control.

[20]  橋本 浩一,et al.  Control and modeling of complex systems : cybernetics in the 21st century : festschrift in honor of Hidenori Kimura on the occasion of his 60th birthday , 2003 .

[21]  Marco Zaffalon,et al.  Robust Filtering Through Coherent Lower Previsions , 2011, IEEE Transactions on Automatic Control.

[22]  J. Shohat,et al.  The problem of moments , 1943 .

[23]  Tryphon T. Georgiou,et al.  Kullback-Leibler approximation of spectral density functions , 2003, IEEE Trans. Inf. Theory.

[24]  Anders Lindquist,et al.  A Convex Optimization Approach to Generalized Moment Problems , 2003 .