Elsevier

Automatica

Volume 82, August 2017, Pages 158-164
Automatica

Brief paper
Feedback quadratic filtering

https://doi.org/10.1016/j.automatica.2017.04.046Get rights and content

Abstract

This paper concerns the state estimation problem for linear discrete-time non-Gaussian systems. It is known that filters based on quadratic functions of the measurements processes (Quadratic Filter) improve the estimation accuracy of the optimal linear filter. In order to enlarge the class of systems, which can be processed by a Quadratic Filter, we rewrite the system model by introducing an output injection term. The resulting filter, named the Feedback Quadratic Filter, can be applied also to non asymptotically stable systems. We prove that the performance of the Feedback Quadratic Filter depends on the gain parameter of the output term, which can be chosen so that the estimation error is always less than or equal to the Quadratic Filter.

Introduction

In this paper, we study the state estimation problem for linear discrete-time non-Gaussian systems. In many applications, the widely used Gaussian assumption must be removed (see Spall, 1985, Spall, 2003 and Wu & Chen, 1993). In these cases, the conditional expectation, which gives the optimal minimum variance estimation, is the solution of an infinite dimensional problem (Zakai, 1969). Methods to approximate the state conditional probability density function include Monte Carlo methods (Arulampalam, Maskell, Gordon, & Clapp, 2002), sums of Gaussian densities (Arasaratnam, Haykin, & Elliott, 2007) and weighted sigma points (Julier & Uhlmann, 2004) among others. These general solutions can cope with nonlinearities and/or with the presence of noise outliers (Stojanovic & Nedic, 2015) or unknown parameters (Stojanovic & Nedic, 2016), and they generally have high computational cost. In the context of linear non-Gaussian systems, many research works aim at filtering algorithms that are easily computable (see Afshar, Yang, & Wang, 2012; Bilik & Tabrikian, 2010; Carravetta, Germani, & Raimondi, 1996; Gordon, Salmond, & Smith, 1993; Kassam & Thomas, 1976; Maryak, Spall, & Heydon, 2004;Picinbono & Devaut, 1988; Spall, 1995; Zhang, Kuai, Ren, Luo, & Lin, 2016 and the references therein). In the minimum variance framework a natural development is to use quadratic or polynomial functions of the observations to improve the estimation accuracy while preserving easy computability and recursion (Carravetta et al., 1996, De Santis et al., 1995, Verriest, 1985). The suboptimal polynomial estimate is obtained by applying the KF to a system augmented with the powers of state and observations. A drawback of this approach is that the resulting augmented system is bilinear and the noise variance depends on the state variance of the original system. Thus, if the variance of the state grows unboundedly, so does the equivalent noise. As a consequence the stability of the resulting Quadratic Filter (QF) is guaranteed only for asymptotically stable systems. In this paper, we propose to use an output injection term to overcome this problem and obtain an internally stable QF. Furthermore, we show that any recursively implementable filter based on the use of powers of measurements has an error that depends on the choice of the gain of this output injection term, in contrast with the linear case. Thus, the gain can be chosen to achieve a smaller estimation error than the QF. A preliminary version of this work has been published in Cacace, Conte, Germani, and Palombo (2014), where the theoretical analysis was missing.

Section snippets

Discussion on quadratic filtering

Consider the problem of state estimation for a discrete-time linear system with non-Gaussian noise in the form x(k+1)=Ax(k)+Bu(k)+fk,x(0)=x0y(k)=Cx(k)+gk where x(k)Rn, u(k)Rp, y(k)Rq, ARn×n, BRn×p and CRq×n. {fk} and {gk} are sequences of non-Gaussian random variables with values in Rn and Rq, respectively. The system is assumed fully observable, i.e. rank O(A,C)=n, where O(A,C) is the observability matrix of the pair (A,C).

Throughout the paper we use the following notations. x[i] is the i

The proposed approach

The proposed method, named Feedback Quadratic Filter (FQF), consists of the following steps.

  • (a)

    System (1)–(2) is rewritten as a system with a feedback given by an output injection term.

  • (b)

    The modified system is decomposed in the sum of a deterministic and a stochastic component.

  • (c)

    The augmented quadratic system is derived for the asymptotically stable stochastic component.

  • (d)

    The KF for mutually correlated state and output noises is applied to the augmented quadratic system. The final estimate is the sum

Properties of the feedback quadratic filter

In this section we analyze two properties of the FQF. The first one concerns the internal stability of the filter. The second one is the relationship between the estimation error and the gain of the output injection term.

Theorem 3

If L is such that the eigenvalues of à are in the open unit circle, then the FQF is asymptotically stationary and internally asymptotically stable.

Proof

As stated in Section  3.4, the FQF is the KF for mutually correlated noises applied to the steady-state version of system (27)–

Numerical examples

In order to validate the proposed approach we consider three examples. In the first two examples the non-Gaussian noise sequences {fk} and {gk} are fki(ω)=0.4wfi(ω)1.2(1wfi(ω)),gk(ω)=1.5wg(ω)0.5(1wg(ω)), where wfi, wg are independent Bernoulli random variables with pfi=0.75, pg=0.25.

In all the examples, the estimation accuracy of the FQF is compared with the one obtained with the standard linear KF. We remark that, since the considered systems are linear, the improvement of the estimate

Conclusions

The FQF algorithm presented in this paper extends recursive quadratic filtering to non-Gaussian and not asymptotically stable linear systems. The proposed algorithm has the same size as the plain QF and needs the same information, namely the knowledge of the moments of the noise up to 4th order. An additional advantage is the reduction of the estimation error with respect to the plain QF. The proposed method can be readily extended to polynomial filtering (Carravetta et al., 1996).

Filippo Cacace graduated in electronic engineering at Politecnico di Milano in 1988, where he received the Ph.D. Degree in computer science in 1992. Since 2003 he has been working at the Faculty of Engineering of the University Campus Bio-Medico of Rome, where he is currently an Associate Professor. His current research interests include nonlinear systems and observers, stochastic and delay systems, system identification and applications to systems biology.

References (25)

  • J.C. Spall

    The Kantorovich inequality for error analysis of the Kalman filter with unknown noise distributions

    Automatica

    (1995)
  • P. Afshar et al.

    ILC-based minimum entropy filter design and implementation for non-Gaussian stochastic systems

    IEEE Transactions on Control Systems Technology

    (2012)
  • B.D.O. Anderson et al.

    Optimal filtering

    (1979)
  • I. Arasaratnam et al.

    Discrete-time nonlinear filtering algorithms using Gauss–Hermite quadrature

    Proceedings of the IEEE

    (2007)
  • M.S. Arulampalam et al.

    A tutorial on particle filters for online nonlinear/non-Gaussian Bayesian tracking

    IEEE Transactions on Signal Processing

    (2002)
  • A.V. Balakrishnan

    Kalman filtering theory

    (1984)
  • I. Bilik et al.

    MMSE-Based filtering in presence of non-Gaussian system and measurement noise

    IEEE Transactions on Aerospace and Electronic Systems

    (2010)
  • Cacace, F., Conte, F., Germani, A., & Palombo, G. (2014). Quadratic Filtering for non-Gaussian and not Asymptotically...
  • F. Carravetta et al.

    Polynomial filtering for linear discrete time non-Gaussian systems

    SIAM Journal on Control and Optimization

    (1996)
  • A. De Santis et al.

    Optimal quadratic filtering of linear discrete-time non-Gaussian systems

    IEEE Transactions on Automatic Control

    (1995)
  • N.J. Gordon et al.

    Novel approach to nonlinear/non-Gaussian Bayesian state estimation

    Proceedings of the Institution of Electrical Engineers

    (1993)
  • S.J. Julier et al.

    Unscented filtering and nonlinear estimation

    Proceedings of the IEEE

    (2004)
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      Citation Excerpt :

      This is needed in order to prove that with the output injection the state and the noises of the extended system are second-order asymptotically bounded. A somewhat unexpected known result of the theory of polynomial filter/controllers is that, when the system is rewritten with an output injection gain, the performance depends on the gain [15]. This is clearly not the case for linear filters/controllers.

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    Filippo Cacace graduated in electronic engineering at Politecnico di Milano in 1988, where he received the Ph.D. Degree in computer science in 1992. Since 2003 he has been working at the Faculty of Engineering of the University Campus Bio-Medico of Rome, where he is currently an Associate Professor. His current research interests include nonlinear systems and observers, stochastic and delay systems, system identification and applications to systems biology.

    Francesco Conte was born in Popoli (PE), Italy, in 1984. He received the Master Degree in computer science and automatic engineering and the Ph.D. Degree in electrical and information engineering both from the University of L’Aquila, Italy, in 2009 and 2013, respectively. Currently, he is a Postdoc with the Department of Electrical, Electronics and Telecommunication Engineering and Naval Architecture (DITEN), University of Genoa, Genova, Italy. From December 2008 to April 2009, he was a visiting scholar with the French National Institute for Research in Computer Science and Control (INRIA), Grenoble, France. He is author of more than 30 scientific publications. His research interests include nonlinear and non-Gaussian systems filtering, analysis, estimation and control of delay and nonlinear systems, power system modeling and control with a particular focus on the smart control of distributed generation sources, controllable loads and energy storage systems.

    Alfredo Germani received the Laurea Degree in physics and the Post-doctoral Degree in computer and system engineering from University of Rome La Sapienza, Rome, Italy, in 1972 and 1974, respectively. From 1975 to 1986, he was a Researcher at Istituto di Analisi dei Sistemi e Informatica A. Ruberti of the Italia National Research Council, Rome. In 1978 and 1979, he was a Visiting Scholar with the Department of System Science, University of California at Los Angeles. From 1986 to 1987, he was a Professor of Automatic Control at the University of Calabria, Italy. Since 1987 he is Full Professor of System Theory at Universit of L’Aquila, Italy where from 1989 to 1992 he was the Chairperson of the School of Electronic Engineering. At University of L’Aquila, from 1995 to 2000 he led the Department of Electrical and Computer Sciences Engineering, and from 1996 to 2009 he has been the Coordinator of the Ph.D. School in Electronics. From 2007 to 2010 he has been a member of the Advisory Board of the Engineering School. Since 2012 he has been a member with the Department of Information Engineering, Computer Sciences and Mathematics. He has published more than 160 research papers in books, international journals and refereed conference proceedings in the fields of systems theory, systems identification and data analysis, nonlinear, stochastic and optimal control theory, distributed and delay systems, finitely additive white noise theory, approximation theory, optimal polynomial filtering for non-Gaussian systems, image processing and restoring, and mathematical modeling for biological processes.

    Giovanni Palombo received the Master Degree in Biomedical Engineering in 2012 and the Ph.D. Degree in Automatics and System Theory in 2016, respectively from Università Campus Bio-Medico di Roma and Università degli Studi dell’ Aquila. His current research interests focus mainly on non-Gaussian filtering, delay systems and SDE.

    The material in this paper was presented at the 53rd IEEE Conference on Decision and Control, December 15–17, 2014, Los Angeles, CA, USA. This paper was recommended for publication in revised form by Associate Editor Wei Xing Zheng under the direction of Editor Torsten Söderström.

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