Elsevier

Automatica

Volume 36, Issue 7, July 2000, Pages 1001-1008
Automatica

Brief Paper
Variable gain parameter estimation algorithms for fast tracking and smooth steady state

https://doi.org/10.1016/S0005-1098(00)00009-1Get rights and content

Abstract

Based on the ideas of the set-membership identification, we present in this paper variable gain least mean-square (LMS) and weighted recursive least-squares (WRLS) algorithms. It is shown that the proposed algorithms possess a fast tracking ability to the variations of the parameter and at the same time are less sensitive to noise leading to a smooth steady state.

Introduction

Based on the ideas of the set-membership identification, we present in this paper variable gain least mean-square (LMS) and weighted recursive least-squares (WRLS) algorithms that possess a fast tracking ability and a smooth steady state.

Consider a discrete-time systemyiiTθi+vi,i=1,2,…,where yiR is the system output, φiRn the measurable regressor, θiRn the parameter vector and viR the noise. The subscript i on θi emphasizes that the parameter vector can be in general time varying.

The purpose of the adaptive parameter identification algorithms is to estimate the true but unknown parameter vector θi from the observed input–output data. Let θ̂i be the estimate of θi, the general form of many parameter identification algorithms may be described byθ̂i=θ̂i−1+Kiφi(yi−φiTθ̂i−1),where Ki is the gain matrix. The celebrated least mean-square (LMS) algorithm (Benveniste, 1987; Treichler, Johnson & Larimore, 1987) and the weighted recursive least-squares (WRLS) algorithm (Goodwin & Sin, 1984) for parameter estimation take this form.

Although based on the idea of minimizing the “prediction” error assuming that the parameter vector is time invariant, both the LMS and the WRLS algorithms are often used in applications where the parameter vector θi is slowly time varying and the noise vi is not negligible. In such applications, the choice of the gain becomes crucial. In general, a large gain makes the LMS and the WRLS to have a better ability for tracking the variation of θi but also make them sensitive to noise leading especially to a non-smooth steady-state performance. On the other hand, a small gain makes the algorithms less sensitive to noise leading to a smooth steady-state performance but at the same time results in a poor tracking ability for slowly time-varying systems, for instance, the system with some infrequent abrupt parameter changes.

Based on the above discussion, it is intuitively clear that to satisfy both a fast tracking ability and a smooth steady-state requirement, a variable gain is needed. In particular, when there is a large parameter estimation error including those due to parameter variations, the gain should be large so that the algorithm has a fast tracking ability and when there is no variation in the parameter and/or the algorithm approaches the steady state, the gain should be small. The difficulty is that the variation is unknown to us. Then, the question is what would be an indication that a large variation has occurred? There has been a great deal of research along this direction in the setting of stochastic identification (Benveniste, 1987; Ljung, 1987; Treichler et al., 1987), especially in the case when the parameter vector has some abrupt but infrequent changes, see excellent books (Bar-Shalom & Li, 1993; Basseville & Benveniste, 1980) where the basic idea is multiple hypotheses testing. Based on the a priori statistical knowledge of the system/noise and the obtained measurement information, hypotheses are tested to determine if the parameter vector has experienced a change. To use stochastic approaches, much a priori statistical information needs to be known. Our work reported here, however, does not follow this stochastic direction. Instead, we follow the so-called set-membership approach. To this end, let us review the concepts of the set-membership identification (Norton 1994, Norton 1995; Special issue on parameter estimation with bounded error). Consider system (1.1) and assume that θiθ for simplicity. If there is no information other than that the bound εi on the unknown noise is available, i.e., εi≥|vi|, i=1,2,…, then any estimate θ̂ satisfying |yi−φiTθ̂|≤εi is compatible with the input–output measurement at time i and the noise bound εi. Only when |yi−φiTθ̂|>εi, we can say that this estimate θ̂ is definitely not the true parameter θ which generates the data and an update is needed. Roughly speaking, in terms of the parameter estimation algorithm, |yi−φiTθ̂|≤εi calls for a zero gain while |yi−φiTθ̂|>εi calls for a non-zero gain. The difficulty of this approach is that a hard bound on the noise must be available. To overcome this difficulty, we propose the following: Let the positive constant δ>0 be a design variable, called the tolerance level. If ei=|yi−φiTθ̂i−1|≤δ, i.e., the output prediction error of the estimate θ̂i−1 is within the tolerance level, a small gain is called for. When ei=|yi−φiTθ̂i−1|≥δ, i.e., the output prediction error of the estimate θ̂i−1 is outside the tolerance level, a large gain is called for.

The idea of variable gain algorithms based on the set-membership setting is not new (see, e.g., Norton 1994, Norton 1995; Special issue on parameter estimation with bounded error, 1990; Kurzhanski, 1996; Milanese, Norton, Piet-Lahanier & Walter, 1996) and those dealing with potentially time-varying parameters (Dasgupta & Huang, 1987; Deller, Nayeri & Odeh, 1993; Gollamudi, Nagaraj, Kapoor & Huang, 1998; Norton & Mo, 1990; Piet-Lahanier & Walter, 1994; Rao & Huang, 1993). However, the work reported in this paper extends the traditional set-membership identification methods in two significant ways. The first one is that contrary to every bounded approach reported in literature including those dealing with potentially time-varying parameters (Dasgupta & Huang, 1987; Deller et al., 1993; Gollamudi et al., 1998; Norton & Mo, 1990; Piet-Lahanier & Walter, 1994; Rao & Huang, 1993), our proposed algorithms do not require explicit knowledge of the noise bound. To the best knowledge of the authors, this is new. Note that the results of Bai, Nagpal and Tempo (1996), Dasgupta and Huang (1987), Deller et al. (1993), Gollamudi et al. (1998), Norton and Mo (1990), Piet-Lahanier and Walter (1994) and Rao and Huang (1993) require an upper bound on the unknown noise vi. If the assumed bound is smaller than the actual noise bound, the resulting membership set can be empty. Of course, the bound information can be very beneficial, should it become available. The other difference is the choice of the gain. Most bounded approach algorithms including our previous work (Bai et al., 1996) render zero gain when the estimate does not violate the noise bound. This choice resembles the idea of a dead zone and is very efficient in terms of computation. The set-membership algorithms reported in Dasgupta and Huang (1987), Deller et al. (1993), Gollamudi et al. (1998) and Rao and Huang (1993) all feature selective update on the parameter estimates. In some cases (see, e.g., Dasgupta & Huang, 1987; Deller et al., 1993; Rao & Huang, 1993) only less than 10% of data points were used to obtain parameter estimates while achieving comparable mean-square error performance to RLS. On the other hand, however, this zero gain can slow down the algorithm substantially. In our proposed algorithms, a non-zero gain is assigned even when the estimate is within the tolerance level and therefore provides a fast tracking ability as shown in simulations.

The organization of the paper is as follows. Section 2 presents a variable gain LMS algorithm along with its properties. A variable forgetting factor WRLS algorithm is proposed in Section 3. Simulation examples are provided in Section 4 to demonstrate effectiveness of the proposed algorithms. Proofs are given in the appendix.

Section snippets

A variable gain LMS

Before presenting the algorithm, we would like to make an observation. The LMS and the WRLS are basically derived from the assumption that the parameter vector is time invariant or slowly time varying. Therefore, it is not expected that a LMS with a variable gain and a WRLS with a variable forgetting factor can solve all the tracking problems for time-varying systems. We hope that the developed algorithm can deal with slowly time-varying systems or systems with infrequent abrupt parameter

A variable forgetting factor WRLS

Based on the same idea as in the variable gain LMS algorithm, we present in this section a variable forgetting factor WRLS algorithm with fast tracking ability and smooth steady-state performance.

The Variable Forgetting Factor WRLS Algorithm

Consider system (1.1). Let δ>0 be any positive constant and 0<ᾱ<ᾱ<1. Defineθ̂i=θ̂i−1+PiφiαiiTPiφi(yi−φiTθ̂i−1),Pi=1αi−1Pi−1Pi−1φi−1φi−1TPi−1αi−1i−1TPi−1φi−1,P0>0withαi=ᾱif|yi−φiTθ̂i−1|>δ,ᾱif|yi−φiTθ̂i−1|≤δ.

Theorem 3.1

Consider system (1.1) withδθi=θiθi−1and the variable forgetting factor WRLS. Then,

Simulation

To illustrate the performance of the variable gain LMS and the variable forgetting factor WRLS, we consider the following numerical example:yi+1=(yi,ui)aibi+viiTθi+vi,where the input ui is a sequence of random variable uniformly in [0,1] and the noise vi is a sequence of random variable uniformly in [−1,1]. The unknown ai≡0.5 and bi is time varying. The slack variable δ=0.8. For simulation, we consider two types of time varying bi. In the first one, bi is with some abrupt but infrequently

Concluding remark

Two variable gain parameter identification algorithms are proposed. It is important to note that although derived based on the ideas of set-membership identification, these algorithms are quite different from set-membership identification algorithms, in particular no explicit noise bound is needed. Their deterministic properties have been studied in the paper and are shown to be favorable compared to the existing algorithms. It will be interesting to investigate their stochastic properties.

Dr. Er-Wei Bai received his education from Fudan University, Shanghai Jiaotong University and the University of California at Berkeley. He is currently associate professor of electrical engineering at the University of Iowa, Iowa City, where he teaches and conducts research in the area of identifications and signal processing.

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Dr. Er-Wei Bai received his education from Fudan University, Shanghai Jiaotong University and the University of California at Berkeley. He is currently associate professor of electrical engineering at the University of Iowa, Iowa City, where he teaches and conducts research in the area of identifications and signal processing.

Yih-Fang Huang received his Ph.D. degree in Electrical Engineering from Princeton University in 1982. Since August 1982, he has been on the Faculty at University of Notre Dame where he is currently Professor and Chair of the Electrical Engineering Department. In Spring 1993, he received the Toshiba Fellowship and was Toshiba Visiting Professor at Waseda University, Tokyo, Japan, in the Department of Electrical Engineering.

Dr. Huang's research interests are in the general areas of adaptive signal processing with applications to communications. He has been one of the leading contributors to the field of set-membership parameter estimation and has made fundamental contributions to this field with his OBE algorithms which led to a novel adaptive equalization paradigm, namely, U-SHAPE (Updator-SHaring Adaptive Parallel Equalizer). With the collaborations of his students, he also developed SMART (Set-Membership Adaptive Recursive Techniques), a toolbox for set-membership filtering. Currently, his primary interests are in applications of SMART and other adaptive techniques to interference mitigation in wireless communications.

Dr. Huang has served as Associate Editor for the IEEE Transactions on Circuits and Systems (1989–1991, 1992–1993). He also served as Vice President — Publications for the IEEE Circuits and Systems Society from January, 1997 to December, 1998. Currently, he is an Associate Editor for the Journal of Franklin Institute and a member of the Steering Committee of the IEEE Transactions on Multimedia. Dr. Huang is a Fellow of the IEEE.

This paper was not presented at any IFAC meeting. This paper was recommended for publication in revised form by Associate Editor B. Ninness under the direction of Editor T. Söderström. The work was supported in part by funds of NSF.

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