Elsevier

Automatica

Volume 48, Issue 11, November 2012, Pages 2965-2970
Automatica

Brief paper
Practical stability of approximating discrete-time filters with respect to model mismatch

https://doi.org/10.1016/j.automatica.2012.08.006Get rights and content

Abstract

This paper establishes practical stability results for an important range of approximate discrete-time filtering problems involving mismatch between the true system and the approximating filter model. Practical stability is established in the sense of an asymptotic bound on the amount of bias introduced by the model approximation. Our analysis applies to a wide range of estimation problems and justifies the common practice of approximating intractable infinite dimensional nonlinear filters by simpler computationally tractable filters.

Introduction

Many filtering problems involve estimation of system quantities from noisy measurements in situations where the exact (or true) model of the system is either unknown or is more complicated than can be handled using standard techniques. In these types of filtering problems, tractable filters are often proposed on the ad hoc basis of an approximating system that reasonably represents the true dynamics. For example, using this informal idea, hidden Markov model (HMM) filters and Kalman filters have been exploited in a wide range of signal and image processing applications, see Barniv and Kella (1987), Lai and Ford (2010) and Rabiner (1989). Despite the successful application of approximate filters in a large number of applications, conditions that ensure reasonable filter behaviour have not been completely established in many situations.

When considering filtering behaviour, there are two basic types of stability properties of interest: asymptotic stability with respect to initialisation errors and stability properties in the presence of modelling errors. The first type of stability properties is important because initial conditions are rarely known perfectly, whilst the second type of property is important because system models are usually not completely known (or are too complex). Fortunately, asymptotic stability of filters with respect to erroneous initial conditions has been established in many situations including Kalman filter (Anderson and Moore, 1979, Jazwinski, 1970, Ocone and Pardoux, 1996), risk-sensitive filters (Dey & Charalambous, 2000), as well as some general asymptotic stability results provided in (Budhiraja, 2001, Budhiraja and Ocone, 1999, Clark et al., 1999). In comparison, only a small number of stability type results for situations involving model mismatch have been established. These include stability with respect to model mismatch for Kalman filters (Anderson and Moore, 1979, Jazwinski, 1970), HMM filters (Le Gland & Mevel, 2000), and particle filters (Azimi-Sadjadi and Krishnaprasad, 2005, Crisan and Heine, 2008, Heine and Crisan, 2008). Further, some convergence and stochastic stability type results for extended Kalman filters are presented in (Reif, Gunther, & Yaz, 1999). Beyond these results, the expected performance properties of approximating filters can be indirectly characterised through performance limits provided by optimal filters (for example, lower error bounds of optimal filters are established in Barak (1986), Bobrovsky and Zakai (1976), Zakai and Ziv (1972), Zeitouni (1984) for the purpose of characterising the possible performance of approximating filters). However, this type of analysis only provides lower error bounds for approximating filters and, of course, any specific filter might perform considerably worse than the lower bound.

In this paper, we investigate stability of general approximating filters in the presence of modelling errors. In this situation, the recursive nature of the filtering process might suggest that the inclusion of modelling error, at each filtering step, could lead to an unbounded growth in estimation error. However, if the approximating filter exhibits some initial condition forgetting properties, and the error introduced by the model approximation is bounded on a finite time interval, then we can show that the filtering error is bounded forever. Moreover, under some additional multi-step consistency assumptions, practical stability of the approximating filters can be established. The stability proofs used here are similar in nature to the proofs used in the important nonlinear control stability results established in Nešić, Teel, and Kokotović (1999).

This paper is structured as follows: In Section 2, we introduce our filter approximation problem. In Section 3, we establish a preliminary bound result for approximating filters, before our main practical stability result is established in Section 4. In Section 5, an example is presented and some conclusions are then provided in Section 6.

Section snippets

Dynamics

For the time step k0, we will consider the following state process xkRn and measurement process ykRm, xk+1=f(xk)+wk+1yk+1=c(xk)+vk+1 where x0 has a priori distribution σ0, f():RnRn, and c():RnRm. Here, wkRn and vkRm are sequences of independent and identically distributed i.i.d. random variables with strictly positive densities ϕw() and ϕv(), respectively. The random variables wk, vk, and x0 are assumed to be mutually independent for all k. We will use the shorthand y[,m] to denote

Bounded error of an approximating filter

A function ψ is said to be of class-K if it is continuous, strictly increasing, and ψ(0)=0. Moreover, function β is of class-KL if β(,t) is of class-K for each t0 and β(s,) is decreasing to zero for each s>0 (see Khalil, 2002, Chapter 4 for descriptions of system stability involving such functions).

Consider a given set Γ and πh() operator. We will now introduce some important definitions.

Definition 3.1 Asymptotic Stability of an Approximating Filter with Respect to Initial Conditions

Consider an approximating filter σk|[1,k],σ0h() (some fixed h>0). For a given set Γ, the approximating

Practical stability of approximating filters

Let us first introduce some assumptions on the class of approximating filters.

Definition 4.1 Asymptotic Stability of a Class of Approximating Filters with Respect to Initial Conditions

For a given set Γ, the class of approximating filters σk|[1,k],σ0h() is said to be asymptotically stable with respect to initial conditions if there exists a H>0 and a β(,)KL such that, for all h(0,H], all σ0,σ̄0Gh(Rn), all ωΓ, and all k0, we have that σk|[1,k],σ0h()σk|[1,k],σ̄0h()1β(σ0σ̄01,k).

Definition 4.2 Multi-step Consistency

For a given set Γ and projection operator πh(), the class of approximating filters σk|[1,k],σ0h() is said

Example

For k=0,,T1, consider a true and an approximating system of the form xk+1=axk+vk+1yk+1=cxk+wk+1 where wk and vk are zero-mean Gaussian noises, and T=100. Consider a true system with ae=0.95,ce=1, and process and measurement noises with covariances Qe=1 and Re=1, respectively. Also consider an approximating system with ah=0.99,ch=1, and process and measurement noises with covariances Qh=1.2 and Rh=1.2, respectively.

Let us consider the set Γ={ω:|yk|Bm for all k[1,T]}, where Bm=10. Note that

Conclusion

This paper establishes an asymptotic error bound on filtering performance in the situations involving model mismatch. We also present results on practical stability of a filter with respect to modelling errors. The results are established using forgetting and consistency properties, and are illustrated for a Kalman filter mismatch example.

Onvaree Techakesari was born in Bangkok, Thailand in 1985. She received her B.Eng degree from Queensland University of Technology, Australia, in 2008 and she is currently working towards the Ph.D. degree at Queensland University of Technology. Her research is in the area of filter stability and control with aerospace applications.

References (27)

  • B. Bobrovsky et al.

    A lower bound on the estimation error for certain diffusion processes

    IEEE Trans. Inform. Theory

    (1976)
  • R.G. Brown et al.

    Introduction to Random Signals and Applied Kalman Filtering with Matlab Exercises and Solutions

    (1997)
  • J.M.C. Clark et al.

    Relative entropy and error bounds for filtering of Markov processes

    Mathematics of Control, Signals, and Systems

    (1999)
  • Cited by (0)

    Onvaree Techakesari was born in Bangkok, Thailand in 1985. She received her B.Eng degree from Queensland University of Technology, Australia, in 2008 and she is currently working towards the Ph.D. degree at Queensland University of Technology. Her research is in the area of filter stability and control with aerospace applications.

    Jason J. Ford was born in Canberra, Australia. He received the B.Sc., B.E., and Ph.D. degrees from the Australian National University, Canberra, Australia, in 1995 and 1998. He joined the Australian Defence Science and Technology Organisation as a Research Scientist in 1998. He was a research fellow at the University of New South Wales, Australian Defence Force Academy in 2004. In 2005 he joined the Queensland University of Technology, where he is currently a Senior Lecturer. His research interests include signal processing and control for aerospace.

    Dragan Nešić is a Professor in the Department of Electrical and Electronic Engineering (DEEE) at The University of Melbourne, Australia. He received his B.E. degree in Mechanical Engineering from The University of Belgrade, Yugoslavia in 1990, and his Ph.D. degree from Systems Engineering, RSISE, Australian National University, Canberra, Australia in 1997. Since February 1999 he has been with The University of Melbourne. His research interests include networked control systems, discrete-time, sampled-data and continuous-time nonlinear control systems, input-to-state stability, extremum seeking control, applications of symbolic computation in control theory, hybrid control systems, and so on. He was awarded a Humboldt Research Fellowship (2003) by the Alexander von Humboldt Foundation, an Australian Professorial Fellowship (2004–2009) and Future Fellowship (2010–2014) by the Australian Research Council. He is a Fellow of IEEE and a Fellow of IEAust. He is currently a Distinguished Lecturer of CSS, IEEE (2008–present). He served as an Associate Editor for the journals Automatica, IEEE Transactions on Automatic Control, Systems and Control Letters and European Journal of Control.

    This research was supported under Australian Research Council’s Linkage Projects funding scheme (project number LP100100302) and the Smart Skies Project, which is funded, in part, by the Queensland State Government Smart State Funding Scheme. The work of the last author is supported by the Australian Research Council under the Future Fellow and Discovery Project schemes. The material in this paper was partially presented at 50th IEEE Conference on Decision and Control (CDC 2011), December 12–15, 2011, Orlando, Florida, USA. This paper was recommended for publication in revised form by Associate Editor Valery Ugrinovskii under the direction of Editor Ian R. Petersen.

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