Elsevier

Automatica

Volume 50, Issue 10, October 2014, Pages 2494-2503
Automatica

Convex saturated particle filter

https://doi.org/10.1016/j.automatica.2014.08.026Get rights and content

Abstract

In many systems the state variables are defined on a compact set of the state space. To estimate the states of such systems, the constrained particle filters have been used with some success. The performance of the standard particle filters can be improved if the measurement information is used during the importance sampling of the filtering phase. It has been shown that the particles obtained in such a way approximate the true state of the system more accurately. The measurement is incorporated into the filtering algorithm through a user-specified detection function, which aims to detect the saturation as it occurs. The algorithm derived from the aforementioned principle is called the Saturated Particle Filter (SPF). In our previous work we have derived a complete SPF framework for the class of systems with one-dimensional constraints. In this paper we derive a novel Convex SPF that extends our method to multidimensional systems with convex constraints. The effectiveness of the new method is demonstrated using an illustrative example.

Introduction

Dynamic filters are commonly used in various engineering problems that are modeled by a Stochastic Dynamical System (SDS). When an SDS has linear dynamics and additive Gaussian noises it is well known that the optimal estimator, in the mean square error (MSE) sense, is the Kalman Filter (KF) (Kalman, 1960, Ristic et al., 2004). In case of nonlinear and/or non-Gaussian noises, in general, the optimal solution is computationally intractable. Thus, one needs to rely on suboptimal solutions. Several versions of the KF that give a suboptimal solution have been developed to address the nonlinear filtering problem (Patwardhan, Prakash, Narasimhan, Gopaluni, & Shah, 2012). These include, among others, the Extended Kalman Filter (Ristic et al., 2004), the Unscented Kalman Filter (Julier and Uhlmann, 1996, Julier and Uhlmann, 2004, Ristic et al., 2004), and the Gauss–Hermite Filter (Arasaratnam et al., 2007, Ito and Xiong, 2000). All of these methods are classified as parametric, i.e., they solve a finite dimensional estimation problem. Parametric filters are simple to implement and very effective when applied to stochastic processes that can be accurately approximated by Gaussian processes. However, in case of highly nonlinear and non-Gaussian dynamical systems, their performance deteriorates.

As an alternative to parametric methods, the non-parametric Particle Filter (PF) has been proposed (Arulampalam et al., 2002, Ristic et al., 2004) as a tool to solve a general filtering problem. This algorithm aims to estimate a Probability Density Function (PDF) of the state rather than a point statistic of the state. Thus, the estimation problem becomes infinite dimensional.

The PF approximates a PDF of the state of the system by a set of points which are obtained by utilizing the Importance Sampling method (Arulampalam et al., 2002), and then weighted according to the Bayes rule. The PF is based on a Monte Carlo approximation and it has been proven (Cristian & Doucet, 2002) that under mild technical assumptions the PF-based PDF converges to the true posterior PDF as the number of samples grows. However, for highly nonlinear and non-Gaussian systems the PF might require a large number of samples to achieve an accurate estimate. This makes the algorithm computationally expensive, especially in high-dimensional systems, and as a consequence, it limits its on-line applicability. It has been noted (Arulampalam et al., 2002, Carlin et al., 1992, Gilks and Berzuini, 2001, Ristic et al., 2004) that the choice of the importance sampling density is a crucial step towards reducing the computational costs of PFs, and therefore making the method more feasible for on-line applications.

The properties of the PF have been extensively studied in recent years (Arulampalam et al., 2002, Cristian and Doucet, 2002, Ristic et al., 2004), and many versions of the PF have been developed for specific types of problems (Arulampalam et al., 2002, Lang et al., 2007, Prakash et al., 2010, Shao et al., 2010). In particular, state estimation of a Constrained Stochastic Dynamical System (CSDS) attracted much attention (Kyriakides et al., 2005, Lang et al., 2007, Prakash et al., 2010, Shao et al., 2010, Stano et al., 2011, Straka et al., 2011). The CSDS is a system for which, at each time step k, at least one of the state variables is restricted to a compact set. These systems are frequently met both in industrial applications (Stano et al., 2010, Vachhani et al., 2006), and in theoretical research (Shao et al., 2010, Stadje, 1997, Straka et al., 2011). To estimate the states of the CSDS one can use the constrained PF (Kyriakides et al., 2005, Lang et al., 2007, Prakash et al., 2010, Shao et al., 2010). This method produces a state estimate that does not violate the physical constraints of the system that is achieved either by discarding unsuitable particles (Kyriakides et al., 2005, Lang et al., 2007), or by projecting them on a boundary of the constraint region (Prakash et al., 2010, Shao et al., 2010). The latter approach is specially suitable for systems characterized by PDFs that are singular (discontinuous) at the boundary of the constraint region. The boundary of such a set is denoted by the saturation region and the particles located at this boundary are called the saturated particles. The system defined by this type of PDF is called (Stano et al., 2011) the Saturated Stochastic Dynamical System (SSDS), which is a special class of the CSDS.

An efficient method for estimating the states of SSDSs, the Saturated Particle Filter (SPF), has been proposed in Stano et al. (2011). The SPF combines the projection approach of Shao et al. (2010) with a novel sampling method that effectively detects the saturation moment, and forces the particles to rapidly jump to that part of the state space which is close to the saturation region. Such sampling is obtained by designing an importance density function that makes use of both the measurement and the knowledge of the system constraints. Several resampling methods suitable for SPF have been derived in Stano, Lendek, and Babuška (2013). Furthermore, also in Stano et al. (2013), it has been shown that the SPF asymptotically converges to the true posterior PDF.

The SPF proposed in Stano et al. (2011) has been derived for a special class of SSDSs, namely for systems that allow only one-dimensional saturation, i.e., if xk=[xk(1)xk(n)]T is an n-dimensional state variable, then only one of the variables xk(1),,xk(n) can be saturated. The framework of Stano et al. (2011) can be easily extended to systems with multidimensional saturations provided that the saturated variables are independent. However, the extension for general multidimensional SSDSs is not straightforward. In this paper we aim to fill this gap in the SPF framework by deriving the Convex Saturated Particle Filter (CSPF), which is applicable to multidimensional systems with convex constraints imposed on the states. The assumption of the convexity of the constraints, from the practical perspective, is not very restrictive. In fact a stronger condition of linear constraints is commonly met in the literature (Dantzig, 1998, Prakash et al., 2010, Straka et al., 2011, Vachhani et al., 2006).

The paper is organized as follows: in Section  2.1 the mathematical framework of the Convex Saturated Stochastic Dynamical System is defined and the estimation problem is stated. In Section  2.2 the standard solution to the previously formulated problem is given. The novel Convex Saturated Particle Filter is derived in Section  3. In Section  4 the new filter is compared with the benchmark method. Section  5 concludes the paper.

Section snippets

Preliminaries

This section contains preliminaries and basic motivations for the development of the Saturated Particle Filter. First, the mathematical framework that we use to model saturated processes is defined. Next, the basic facts about Particle Filtering are recalled.

Convex saturated particle filter

In this section we propose a new SPF that is designed for CSSDSs. The CSPF is capable of quickly detecting whether or not saturation occurred by comparing the measurements with the state constraints. This information is used to forcibly move the particles to the region of higher probability, which leads to improved accuracy of the estimate. This procedure renders possible the reduction of the number of particles used by PF, thus reducing the computational load of the algorithm. The detection of

Numerical simulations

To illustrate the estimation abilities of the newly proposed CSPF we compare it with the CBPF applied to a simple CSSDS that models the motion of a two-dimensional object under random disturbance. We assume a static sensor placed at the origin that measures the distance and the bearing of the moving object. This model is a version of a classical nonlinear tracking problem discussed, e.g., in Arulampalam, Ristic, Gordon, and Mansell (2004), Gilks and Berzuini (2001) and Gordon et al. (1993). In

Conclusions

In this paper we extended the previously proposed estimation method, the SPF, which makes an effective use of the measurements during the importance sampling, to multidimensional SSDS. Such extension requires an extra condition to be imposed on the system, namely the constraints of the system need to be convex sets in Rn. With the convexity assumption satisfied, the multidimensional detection function can be properly defined. This function is then used to derive the multidimensional CSPF that

Acknowledgment

This research is funded by the dredging company IHC Systems B. V., P. O. Box 41, 3360 AA Sliedrecht, the Netherlands.

Paweł Mirosław Stano received the M.Sc. degree (cum laude) in applied mathematics from Jagiellonian University, Cracow, Poland in 2008, the M.Sc. degree (cum laude) in stochastic mathematics from VU University Amsterdam, the Netherlands in 2008, and the Ph.D. from Delft University of Technology, the Netherlands in 2013. In 2012 he was a visiting scholar at the Coordinated Science Laboratory, University of Illinois, Urbana-Champaign and in 2013 he worked as an R&D software engineer at IHC

References (33)

  • D. Cristian et al.

    A survey of convergence results on particle filtering methods for practitioners

    IEEE Transactions on Signal Processing

    (2002)
  • G. Dantzig

    Linear programming and extensions

    (1998)
  • A. Doucet et al.

    On sequential Monte Carlo sampling methods for Bayesian filtering

    Statistics and Computing

    (2000)
  • G.B. Folland
  • X. Fu et al.

    An improvement on resampling algorithms of particle filters

    IEEE Transactions on Signal Processing

    (2010)
  • W.R. Gilks et al.

    Following a moving target—Monte Carlo inference for dynamic Bayesian models

    Journal of the Royal Statistical Society

    (2001)
  • Paweł Mirosław Stano received the M.Sc. degree (cum laude) in applied mathematics from Jagiellonian University, Cracow, Poland in 2008, the M.Sc. degree (cum laude) in stochastic mathematics from VU University Amsterdam, the Netherlands in 2008, and the Ph.D. from Delft University of Technology, the Netherlands in 2013. In 2012 he was a visiting scholar at the Coordinated Science Laboratory, University of Illinois, Urbana-Champaign and in 2013 he worked as an R&D software engineer at IHC Systems. He is currently a postdoc researcher at the European Commission’s Joint Research Centre. His research interests include estimation of nonlinear and non-Gaussian systems, statistics, and numerical methods.

    Arnold J. den Dekker received the M.Sc. degree (cum laude) in applied physics and the Ph.D. degree (cum laude) from Delft University of Technology (DUT), Delft, The Netherlands, in 1992 and 1997, respectively. From 1997 to 1999, he was a Research Fellow with the Department of Physics, University of Antwerp (UAntwerp), Antwerp, Belgium. In 1999 he was awarded a prestigious talent research fellowship of the Royal Netherlands Academy of Arts and Sciences (KNAW), which funded his position as a KNAW research fellow at DUT from 1999 to 2004. Currently he is an Assistant Professor at DUT’s Delft Center for Systems and Control and a visiting professor at UAntwerp’s Vision Lab. His research interests include statistical signal processing and inverse problems, with applications in physical imaging systems.

    Zsófia Lendek received the M.Sc. degree in control engineering from the Technical University of Cluj-Napoca, Romania, in 2003, and the Ph.D. degree from the Delft University of Technology, the Netherlands, in 2009. She is currently an associate professor at the Technical University of Cluj-Napoca, Romania. Her research interests include observer and controller design for nonlinear systems, in particular Takagi–Sugeno fuzzy systems.

    Robert Babuška received the M.Sc. degree (cum laude) in control engineering from the Czech Technical University in Prague, in 1990, and the Ph.D. degree (cum laude) from the Delft University of Technology, the Netherlands, in 1997.

    He has had faculty appointments at the Technical Cybernetics Department of the Czech Technical University Prague and at the Electrical Engineering Faculty of the Delft University of Technology. Currently, he is a Professor at the Delft Center for Systems and Control, Faculty of Mechanical Engineering, Delft University of Technology. His research interests include neural and fuzzy systems for modeling, identification and state-estimation, fault-tolerant control, learning and adaptive control and dynamic multi-agent systems. He is involved in several projects in which these techniques are being applied in the fields of robotics, mechatronics, and aerospace.

    Robert Babuška has co-authored over 390 publications, including three research monographs, three edited books, 31 invited chapters in books, 85 journal papers and more than 270 conference papers. He has been serving as an associate editor of the IEEE Transactions on Fuzzy Systems, Engineering Applications of Artificial Intelligence, and as an area editor of Fuzzy Sets and Systems. He was the chairman of the IFAC Technical Committee on Cognition and Control.

    The material in this paper was not presented at any conference. This paper was recommended for publication in revised form by Associate Editor Giancarlo Ferrari-Trecate under the direction of Editor Ian R. Petersen.

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