Elsevier

Automatica

Volume 41, Issue 8, August 2005, Pages 1455-1461
Automatica

Brief paper
Kalman filtering for multiple time-delay systems

https://doi.org/10.1016/j.automatica.2005.03.018Get rights and content

Abstract

This paper is to study the linear minimum variance estimation for discrete-time systems with instantaneous and l-time delayed measurements by using re-organized innovation analysis. A simple approach to the problem is presented in this paper. It is shown that the derived estimator involves solving l+1 different standard Kalman filtering with the same dimension as the original system.

Introduction

The problem of estimation, which includes filtering, prediction and smoothness, has been one of the key research topics of control community since the seminal paper by Wiener (1950). The Kalman filter, which addresses the minimization of filtering error covariance (termed as H2 estimation), emerged as a major tool of state estimation in the 1960s, see Kailath, Sayed, and Hassibi (1999), Anderson and Moore (1979) and references therein. In the past decades, the Kalman filtering has been well studied via a Riccati equation approach. It has been a classical tool in signal processing, communication and control applications. Note that the Kalman filtering formulation is only applicable to the standard systems without delays. In the time delays context, a common approach is the PDE (partial differential equation), see Kwakernaak (1967), Richard (2003), Zhang et al., 2003, Zhang et al., 2004 and references therein. This approach is usually related to solving a partial differential equation and boundary condition equations which do not have an explicit solution in general. For the case of discrete-time systems, the problem has been investigated via system augmentation and standard Kalman filtering, see Kailath et al. (1999) and Anderson and Moore (1979) or the polynomial approach (Kucera, 1979). Note that the augmented Kalman filtering approach is computationally expensive, especially when the dimension of the system is high and the measurement lags are large. On the other hand, the polynomial approach only addresses the steady-state filtering problem and it requires solving a much higher order of spectral factorization for systems with delays.

In this paper, we are concerned with the minimum mean square error (MMSE) estimation problem for the systems with instantaneous and multiple-time delayed measurement systems. Such problem has important applications in many engineering problems such as in communications and multiple-sensor fusion (Klein, 1999). Moreover, the problem studied in this paper is related to some complicated problems such as H fixed-lag smoothing (Zhang, Xie, & Soh, 2001; Zhang et al., 2004), preview control (Kojima & Ishijima, 2001) and H control with control input signal delays (Tadmor, 2002). using the re-organized innovation sequences (Zhang et al., 2001), we shall present a simple Kalman filtering formulation to the systems with l-delay measurements. It will be shown that the solution consists of l standard Kalman filters with the same dimension as the original system.

This paper is organized as follows. The problem to be addressed is stated in Section 2. Section 3 presents the main results. The comparison of the computational cost between the presented algorithm and the traditional Kalman filtering with augmentation is given in Section 4. Section 5 gives an example to show the computation procedure of the new method. The conclusions are given in Section 6.

Section snippets

Problem statement

We consider the linear discrete-time systemx(t+1)=Φ(t)x(t)+Γ(t)u(t),where x(t)Rn is the state, and u(t)Rr is the input noise. The state is observed by l different systems with delays which are described asyi(t)=Hi(t)x(ti)+vi(t),i=0,1,,l,where ti=ti-1-di, with d0=0,di>0 for i>0 and t0=t. yi(t)Rpi are delayed measurements, vi(t)Rpi are the measurement noises. The initial state x(0), u(t) and vi(t)(i=0,1,,l) are uncorrelated white noises with zero means and known covariance matrices, E[x(0)xT

Re-organized measurements

In this subsection, the instantaneous and l-delayed measurements will be re-organized as the delay free measurements that are from l+1 different observation equations.

It is well known, given the measurement sequence {Y(i)}i=0t, the optimal state estimator x^(t|t) is the projection of x(t) onto the linear space spanned by the measurement sequence, denoted by L{{Y(i)}i=0t} (Kailath et al., 1999, Anderson and Moore, 1979). Note that the linear space L{{Y(i)}i=0T} is equivalent toL{Yl+1(0),,Yl+1(tl

Comparison

The section is to give the comparison of the computation cost between the traditional augmentation method and the presented approach in the paper.

Note the Problem P in the paper can be solved by introducing the additional delayed states asxaT(t)=[xT(t)xT(t-1)xT(t-d¯1)xT(t-d¯l-1-1)xT(t-d¯l)].The augmented state space model isxa(t+1)=Φa(t)xa(t)+Γa(t)u(t),Y(t)=Ha(t)xa(t)+vs(t),where Y(t)=[y0T(t)ylT(t)]T,vs(t)=[v0T(t)vlT(t)]T,Γa(t)=[ΓT(t)0]T, andΦa(t)=Φ(t)Ip1Ip1IplIpl0,Ha(t)=H0(t)00H

Numerical example

A numerical example is given below to show the computation procedure of the presented method.

Example 5.1

Consider system (2.1)–(2.2) with l=2, d1=d2=20 andΦ(t)=0.800.90.5,Γ(t)=0.60.5,H0(t)=[12],H1(t)=[20.5],H2(t)=[31].The initial state value x(0), and noises u(t), v0(t), v1(t) and v2(t) are uncorrelated white noises with zero means and unity covariance matrices, i.e., P0=I2,Qu(k)=1,Qv0(k)=Qv1(k)=Qv2(k)=1. Our aim is to calculate the optimal estimate x^(t|t) of the signal x(t) based on observation {{Y(i)}i=

Conclusion

We have studied the optimal filtering for systems with instantaneous and l-time delayed measurements. By applying the so-called re-organized innovation approach (Zhang et al., 2001), a simple solution is derived. It includes solving l+1 different Kalman filters with the same dimension as the original system. As compared with the state augmentation and polynomial method, the presented approach is much more computational attractive, especially when the delays are large. The proposed results in

Xiao Lu received the B.S. degree from College of Electron and Information, Dalian Jiaotong University in 1998. He is now toward a doctor degree of Dalian University of Technology. His interests include optimal estimation and control, robust filtering and control, time-delay systems.

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Xiao Lu received the B.S. degree from College of Electron and Information, Dalian Jiaotong University in 1998. He is now toward a doctor degree of Dalian University of Technology. His interests include optimal estimation and control, robust filtering and control, time-delay systems.

Huanshui Zhang graduated in mathematics from the Qufu Normal University in 1986 and received his M.Sc. and Ph.D. degrees in control theory and signal processing from the Heilongjiang University, PR China, and Northeastern University, PR China, in 1991 and 1997, respectively. He worked as a postdoctoral fellow at the Nanyang Technological University from 1998 to 2001 and Research Fellow at Hong Kong Polytechnic University from 2001 to 2003. He joined Shandong Taishan College in 1986 as an Assistant Professor and became an Associate Professor in 1994. He joined Shandong University in 1999 as a Professor. Currently he is a professor of Shenzhen Graduate School of Harbin Institute of Technology.

His interests include optimal estimation, robust filtering and control, time delay systems, singular systems, wireless communication and signal processing.

Professor Wei Wang obtained the Bachelor, Master Degree and Ph.D. in Industrial Automation from Northeastern University, China, in 1982, 1986 and 1988 respectively. He is presently professor and director of Research Center of Information and Control, Dalian University of Technology, China. Previously he was a post-doctor at the Division of Engineering Cybernetics, Norwegian Science and Technology University (1990–1992), professor and vice director of Research Center of Automation, Northeastern University, China (1995–1999), vice director of the National Engineering Research Center of Metallurgical Automation (1995–1999), and a research fellow at the Department of Engineering Science, University of Oxford (1998–1999).

His research interests are in adaptive control, predictive control, robotics, computer integrated manufacturing systems, and computer control of industrial process. He is the author of the book entitled Generalized Predictive Control Theory and its Applications published by Science Publishing House, China. He has published over 100 papers in international and domestic journals and conferences. He is now a member of IFAC Technical Committee of Cost Oriented Automation and a member of IFAC Technical Committee of Mining, Mineral and Metal Processing.

Kok Lay Teo received his Ph.D. degree in electrical engineering from the University of Ottawa, Ottawa, ON, Canada. He was with the Department of Applied Mathematics, University of New South Wales, Australia, from 1974 to 1985, and then with the Department of Industrial and Systems Engineering, National University of Singapore, Singapore, from 1985 to 1987. He returned to Australia as an Associate Professor with the Department of Mathematics, the University of Western Australia, from 1988 to 1996. He was Professor of Applied Mathematics at Curtin University of Technology, Australia, from 1996 to 1998. He then took up the position as the Chair Professor of Applied Mathematics and Head of Department of Applied Mathematics at The Hong Kong Polytechnic University from 1999 to 2004. He is currently Professor of Applied Mathematics and Head of Department of Mathematics and Statistics at Curtin University of Technology. He has delivered several keynote and fully funded invited lectures, and published 5 books and over 250 journal papers and a number of conference papers. The software package, MISER3.3, for solving general constrained optimal control problems was developed by the research team under his leadership. He is Editor-in-Chief of the Journal of Industrial and Management Optimization. He also serves as an associate editor of a number of international journals, including Automatica; Nonlinear Dynamics and Systems Theory, Journal of Global Optimization; Engineering and Optimization; Discrete and Continuous Dynamic Systems (Series A and Series B); Dynamics of Continuous, Discrete and Impulsive Systems (Series A and Series B). He has also edited special issues for several Journals, including Annals of Operations Research, Journal of Global Optimization, Dynamics of Continuous, Discrete and Impulsive Systems (both Series A and Series B). His research interests include both the theoretical and practical aspects of optimal control and optimization, and their practical applications such as in signal processing in telecommunications, and financial portfolio optimization.

This paper was not presented at any IFAC meeting. This paper was recommended for publication in revised form by Associate Editor Kenko Uchida under the direction of Editor Ian Petersen. Supported by the National Nature Science Foundation of China (60174017).

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