Elsevier

Automatica

Volume 41, Issue 2, February 2005, Pages 315-325
Automatica

Brief paper
Hierarchical gradient-based identification of multivariable discrete-time systems

https://doi.org/10.1016/j.automatica.2004.10.010Get rights and content

Abstract

In this paper, we use a hierarchical identification principle to study identification problems for multivariable discrete-time systems. We propose a hierarchical gradient iterative algorithm and a hierarchical stochastic gradient algorithm and prove that the parameter estimation errors given by the algorithms converge to zero for any initial values under persistent excitation. The proposed algorithms can be applied to identification of systems involving non-stationary signals and have significant computational advantage over existing identification algorithms. Finally, we test the proposed algorithms by simulation and show their effectiveness.

Introduction

For decades, a great deal of work has been published on the identification of multivariable, or multi-input multi-output (MIMO), systems (see, e.g., Gauthier & Landau, 1978; El-Sherief & Sinha, 1979; Sinha & Kwong, 1979; El-Sherief, 1981; Verhaegen and Dewilde, 1992a, Verhaegen and Dewilde, 1992b; Verhaegen, 1993, Verhaegen, 1994; Overshee and De Moor, 1994, Overshee and De Moor, 1996; Chou & Verhaegen, 1997; McKelvey, Akcay, & Ljung, 1996; Pintelon, 2002); however, further research in this area is still required for the following reasons:

  • In the area of MIMO system identification based on difference equations, most existing identification algorithms using transfer matrices employ the idea of decomposing a MIMO system into several subsystems, depending on the number of outputs, and then of estimating parameters of the subsystems one by one (Gauthier and Landau, 1978, El-Sherief and Sinha, 1979, Sinha and Kwong, 1979, El-Sherief, 1981). Since such identification algorithms require computing many covariance matrices (one for each subsystem), they have the drawback of having heavy computational load. The simultaneous identification of all parameters of a system can reduce the computational burden, e.g., Sen and Sinha (1976) suggested to use a matrix pseudo-inverse approach; however, the computational load is large due to large number of zero entries in the information matrix in the estimation algorithm. Moreover, the algorithm in Sen and Sinha (1976) handles noise-free data only. Recently, Pintelon (2002) studied the stochastic properties (strong convergence, asymptotic normality, strong consistency) of the frequency-domain subspace algorithms described in McKelvey et al. (1996) and Van Overschee and De Moor (1996), where the true noise covariance matrix was replaced by the sample noise covariance matrix obtained from a small number of independent repeated experiments.

  • In the off-line state-space model identification literature, subspace state-space identification (4SID for short) methods based on the RQ factorization and singular value decomposition (SVD) have been developed for MIMO systems (Verhaegen and Dewilde, 1992a, Verhaegen and Dewilde, 1992b; Verhaegen, 1993, Verhaegen, 1994, Overshee and De Moor, 1994, Overshee and De Moor, 1996, Chou and Verhaegen, 1997). The basic idea is to determine the extended observability matrix from the SVD or RQ factorization of an information matrix consisting of given input/output (I/O) data, and then to compute the system parameter matrices. But, as the size of the information matrix grows, the difficulty and complexity in computation increase.

  • In the recursive 4SID area, some methods (e.g., Gustafsson, 1998, Oku and Kimura, 2002) are based on the idea of directly updating an estimate of the extended observability matrix by using subspace tracking; other methods (e.g., Verhaegen & Deprettere, 1991; Lovera, Gustafsson, & Verhaegen, 2000) are based on subspace tracking ideas for the recursive update of the RQ factorizations or the SVD by using array signal processing algorithms; see Comon and Golub (1990) for a review of subspace tracking algorithms and Yang, 1995, Yang, 1996 for projection approximation subspace tracking. Finally, Cho, Xu, and Kailath (1994) presented a recursive identification method of state-space models using the generalized Schur algorithm for updating the noise covariance matrix. However, in order to obtain system parameter estimates, these 4SID methods also require some additional computation, e.g., computing the A-matrix using the shift invariant structure of the extended observability matrix (obtained by use of SVD) and computing B- and D-matrices using least squares methods (Lovera et al., 2000). Our approach in this work is to update directly parameter estimates as in the prediction error method based on difference equation descriptions (Ljung, 1999); moreover, we do not assume that the problems are stationary and/or ergodic, which is different from those mentioned above. The algorithms proposed are simple and easy to implement, and have less computational effort than existing algorithms.

Therefore, developing computationally efficient and recursive system identification algorithms is the goal in this paper. We will frame our study in the identification of transfer matrix models. The key idea is the so-called hierarchical identification, and is inspired by the hierarchical control based on the decomposition-coordination principle for large-scale systems (Singh, 1980, Tamura and Yoshikawa, 1990; Drouin, Abou-Kandil, & Mariton, 1991). Hierarchical identification uses subsystem decomposition in identification, and is also called bootstrap identification.

The principle of hierarchical identification is as follows. A system is decomposed into several subsystems with smaller dimension and fewer variables, and then the parameter vector and/or parameter matrix of each subsystem is identified, respectively. Because of such decomposition, difficulties arise in that there exist common unknown parameters/quantities among subsystems, which normally requires difficult iterative calculation. In order to overcome such difficulties, when recursively computing the parameter estimate of the ith subsystem, the hierarchical identification principle implies that the unknown parameters of other subsystems which appeared in the ith subsystem are replaced with their estimates. Using this idea, we present the hierarchical gradient iterative algorithm and hierarchical stochastic gradient algorithm for MIMO systems. The main advantage of such algorithms is that they require less computational effort than existing identification algorithms, e.g., the Sen and Sinha's algorithm and the 4SID methods mentioned above.

The hierarchical identification methods have important applications in parameter identification of multirate systems (Chen and Qiu, 1994, Qiu and Chen, 1994, Qiu and Chen, 1999; Li et al., 2001, Li et al., 2002; Li, Shah, Chen, & Qi, 2003; Tangirala, Li, Patwardhan, Shah, & Chen, 2001; Sheng, Chen, & Shah, 2002), because lifting converts a multirate time-varying system into a time-invariant MIMO system.

The paper is organized as follows. In Section 2, we discuss modeling issues related to MIMO systems. In Sections 3 and 4, we develop the hierarchical gradient iterative algorithm and hierarchical stochastic gradient identification algorithm, and analyze the performance of the proposed algorithms. In Section 5, we compare computational efficiency of our algorithm with several existing ones, establishing a clear advantage. Section 6 presents an illustrative example for the results in this paper. Finally, concluding remarks are given in Section 7.

Section snippets

The problem formulation

Consider a linear discrete-time, multivariable system described by the following state-space model:x(t+1)=Ax(t)+Bu(t),y(t)=Cx(t)+Du(t),whose input/output relationship can be represented asy(t)=G(z)u(t).Here, x(t)Rn is the state vector, u(t)Rr the system input vector, y(t)Rm the system output vector, (A,B,C,D) the system matrices of appropriate sizes, and G(z)Rm×r the transfer matrix (TM) which relates to the state-space data as follows:G(z)=C(zI-A)-1B+D=Cadj[zI-A]Bdet[zI-A]+D=Cadj[I-Az-1]Bz-

The hierarchical gradient iterative algorithm

In this section, according to the hierarchical identification principle, we decompose the MIMO system in (3) into two subsystems: one containing the parameter vector α, and the other containing the parameter matrix θ; and then the iterative solutions of the parameter vector and parameter matrix of the two subsystems are established by application of the steepest descent principle. The details are as follows.

Define two vectorsb1(t)-y(t)+θTϕ(t),b2(t)y(t)+ψ(t)α.Then, we can decompose the system

The hierarchical stochastic gradient algorithm

In this section, we derive a hierarchical gradient parameter estimation algorithm based on the model discussed in (3) in the stochastic framework, and establish the convergence properties of the algorithm.

Based on the model in (3) and introducing a noise term w(t), we havey(t)+ψ(t)α=θTϕ(t)+w(t).We assume that {w(t),Ft} is a martingale difference vector sequence defined on a probability space {Ω,F,P}, where {Ft} is the σ algebra sequence generated by {w(t)}, i.e.,Ft=σ(w(t),w(t-1),w(t-2),)orFt=σ(

Comparing computational efficiency

In this section, we compare in detail the computational efficiency of our HSG algorithm with several existing ones: the recursive LS and stochastic gradient algorithms based on the model in (4), the stochastic gradient algorithm based on m subsystems.

The LS algorithm of identifying θs in (4), in the stochastic framework, namely, the Sen and Sinha's algorithm, can be expressed as I:θ^s(t)=θ^s(t-1)+L(t)[y(t)-H(t)θ^s(t-1)],θ^s(t)Rmn0+n,L(t)=P(t)HT(t)=P(t-1)HT(t)[Im+H(t)P(t-1)HT(t)]-1,P(t)=[Imn0+n-

Example

In this section, we present an example to illustrate the performance of the proposed algorithms.

Consider the following simulated plant: α(z)y(t)=Q(z)u(t)+w(t),whereα(z)=1+α1z-1+α2z-2+α3z-3,Q(z)=Q1z-1+Q2z-2+Q3z-3,α=[α1α2α3]T=[-1.150.425-0.05]T,θT=[Q1Q2Q3]=11-0.9-0.750.20.1251.21.2-1.08-0.780.240.12.Here u(t)=[u1(t),u2(t)]T is taken as a persistent excitation vector sequence with zero mean and unit variances, and w(t)=[w1(t),w2(t)]T as a white noise vector sequence with zero mean and variances [σw

Conclusions

According to a hierarchical identification principle, a HGI algorithm and HSG are developed for MIMO systems. The analysis indicates that the algorithms proposed can achieve good performance properties (i.e., the parameter estimation errors are uniformly bounded, and consistently converges to zero under persistent excitation), and require less computational efforts than the existing algorithms.

Although the algorithms are proposed for MIMO stochastic systems with an additive white noise

Acknowledgements

The authors are grateful to the Associate Editor and anonymous reviewers for their helpful comments and suggestions.

Feng Ding was born in Guangshui, Hubei Province. He received the B.Sc. degree in Electrical Engineering from the Information Engineering College, Hubei University of Technology (Wuhan, P.R. China) in 1984, and the M.Sc. and Ph.D. degrees in automatic control both from the Department of Automation, Tsinghua University in 1990 and 1994, respectively. From 1984 to 1988, he was an Electrical Engineer at the Hubei Pharmaceutical Factory. Since 1994 he is with Department of Automation, Tsinghua

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    Feng Ding was born in Guangshui, Hubei Province. He received the B.Sc. degree in Electrical Engineering from the Information Engineering College, Hubei University of Technology (Wuhan, P.R. China) in 1984, and the M.Sc. and Ph.D. degrees in automatic control both from the Department of Automation, Tsinghua University in 1990 and 1994, respectively. From 1984 to 1988, he was an Electrical Engineer at the Hubei Pharmaceutical Factory. Since 1994 he is with Department of Automation, Tsinghua University. He is now a Professor in the Control Science and Engineering Research Center at the Southern Yangtze University, Wuxi, China, and has been a Research Associate at the University of Alberta, Edmonton, Canada since 2002. His current research interests include model identification and adaptive control. He co-authored the book Adaptive Control Systems (Tsinghua University Press, Beijing, 2002), and published over eighty papers on modelling and identification as the first author.

    Tongwen Chen received the B.Sc. degree from Tsinghua University (Beijing) in 1984, and the MASc and PhD degrees from the University of Toronto in 1988 and 1991, respectively, all in Electrical Engineering.

    From 1991 to 1997, he was an Assistant/Associate Professor in the Department of Electrical and Computer Engineering at the University of Calgary, Canada. Since 1997, he has been with the Department of Electrical and Computer Engineering at the University of Alberta, Edmonton, Canada, and is presently a Professor of Electrical Engineering. He held visiting positions at the Hong Kong University of Science and Technology, Tsinghua University, and Kumamoto University.

    His current research interests include process control, multirate systems, robust control, network based control, digital signal processing, and their applications to industrial problems. He co-authored with B.A. Francis the book Optimal Sampled-Data Control Systems (Springer, 1995).

    Dr. Chen received a University of Alberta McCalla Professorship for 2000/2001, and a Fellowship from the Japan Society for the Promotion of Science for 2004. He was an Associate Editor for IEEE Transactions on Automatic Control during 1998-2000. Currently he is an Associate Editor for Automatica, Systems and Control Letters, and DCDIS Series B. He is a registered Professional Engineer in Alberta, Canada.

    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. Soederstrorm. This research was supported by the Natural Sciences and Engineering Research Council of Canada.

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    Feng Ding is currently a Research Associate at the University of Alberta, Edmonton, Canada.

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