Joint identification of plant rational models and noise distribution functions using binary-valued observations

doi:10.1016/j.automatica.2005.12.004

Automatica

Volume 42, Issue 4, April 2006, Pages 535-547

https://doi.org/10.1016/j.automatica.2005.12.004 Get rights and content

Abstract

System identification of plants with binary-valued output observations is of importance in understanding modeling capability and limitations for systems with limited sensor information, establishing relationships between communication resource limitations and identification complexity, and studying sensor networks. This paper resolves two issues arising in such system identification problems. First, regression structures for identifying a rational model contain non-smooth nonlinearities, leading to a difficult nonlinear filtering problem. By introducing a two-step identification procedure that employs periodic signals, empirical measures, and identifiability features, rational models can be identified without resorting to complicated nonlinear searching algorithms. Second, by formulating a joint identification problem, we are able to accommodate scenarios in which noise distribution functions are unknown. Convergence of parameter estimates is established. Recursive algorithms for joint identification and their key properties are further developed.

Introduction

System identification of plants with binary-valued output observations is of importance in understanding modeling capability for systems with limited sensor information, establishing relationships between communication resource limitations and identification complexity, and studying sensor networks. The authors introduced in Wang, Zhang, and Yin (2003) a framework in which such identification problems can be rigorously pursued either in a stochastic setting or a worst-case scenario. This paper extends the results in two key directions: (a) systems in Wang et al. (2003) are finite impulse response models. Due to nonlinearity in output observations, switching (non-smooth) nonlinearity enters the regressor for rational models, leading to a difficult problem of nonlinear filtering. By introducing a two-step identification procedure that employs periodic signals, empirical measures, and identifiability features, rational models can be identified without resorting to complicated nonlinear searching algorithms. (b) Identification algorithms in Wang et al. (2003) assume knowledge of noise distribution functions. Unlike traditional identification problems where actual noise distribution functions are usually not used in the algorithms, the identification algorithms for binary-valued observations use explicitly the noise distribution functions. Consequently, they do not apply if noise distribution functions are unknown. Since in practice noise distribution functions are either unknown or only estimated with limited prior information, removing this condition is of essential importance. By formulating a joint identification problem, we are able to accommodate the situations in which noise distribution functions are unknown. Identification errors and input design are examined in a stochastic information framework. Convergence of parameter estimates is established. Recursive algorithms for joint identification and their basic properties are further derived.

This work is based on the premise that the order of the system is finite and known. For infinite dimensional systems that are approximated by finite dimensional models, the problem of unmodeled dynamics and model complexity becomes an essential issue. This has been studied in Wang, 1997, Wang and Yin, 1999, Wang and Yin, 2000, Wang and Yin, 2002 for system identification with regular sensors, and in Wang et al. (2003) for system identification with binary-valued sensors. Estimating the order of the system is a worthwhile direction, but beyond the scope of this paper.

The paper is organized as follows. The main problem is formulated in Section 2. Our development starts in Section 3 with estimation of plant outputs when noise distribution functions are known. The main tool is empirical measures and their convergence. Section 4 establishes the main results on identifiability of plant parameters. A basic property of rational systems is established. It shows that if the input is periodic and full rank, system parameters are uniquely determined by its periodic outputs. Consequently, under such inputs, convergence of parameter estimates can be established when the convergence results of Section 3 are utilized. Section 5 is devoted to the general scenario where noise distribution functions are unknown and must be estimated. Identification of distribution functions and system parameters are intimately intertwined. Together, they form a nonlinear identification problem. Algorithms for identifying jointly plant parameters and distribution functions are introduced. It is shown that under some mild conditions, convergence of both estimates can be established when one uses signal scaling and threshold shifting to leverage on providing excitation for parameter estimation. A simple application example is given in Section 6 to summarize the main steps of identification experimental design, identification algorithms, and accuracy evaluation developed in this paper. For computational efficiency, recursive algorithms for joint identification are presented in Section 7. Some brief concluding remarks are made in Section 8.

For some related but different identification algorithms such as binary reinforcement and some applications, the reader is referred to Caianiello and de Luca (1966), Chen and Yin (2003), Elvitch, Sethares, Rey, and Johnson (1989), Eweda (1995), Gersho (1984), Pakdaman and Malta (1998), Yin, Krishnamurthy, and Ion (2003). The main tools for stochastic analysis and identification methodologies can be found in Billingsley (1968), Chen and Guo (1991), Feller, 1968, Feller, 1971, Kushner and Yin (2003), Ljung (1987), Pollard (1984), Serfling (1980). This paper is a continuation of the authors’ early work in Wang et al. (2003), Wang, 1997, Wang and Yin, 1999, Wang and Yin, 2000, Wang and Yin, 2002.

Section snippets

Problem formulation

Consider the following system: $y_{k} = G (q) u_{k} + d_{k} = x_{k} + d_{k},$ which is in an output error form. Here, q is the one-step shift operator ${qu}_{k} = u_{k - 1}$ ; ${d_{k}}$ is a sequence of random noise (sensor noise); $x_{k} = G (q) u_{k}$ is the noise-free output of the system; $G (q)$ is a stable rational function of q $G (q) = \frac{B (q)}{1 - A (q)} = \frac{b_{1} q + \dots + b_{n} q^{n}}{1 - (a_{1} q + \dots + a_{n} q^{n})} .$ The input ${u_{k}}$ is uniformly bounded by $| u_{k} | ⩽ u_{\max}$ and can be selected by the designer otherwise. The observation ${y_{k}}$ is measured by a binary-valued sensor of threshold $C > 0$ , $s_{k} = I_{{y_{k} ⩽ C}} = \{\begin{matrix} 1, & y_{k} \end{matrix}$

Estimation of $x_{k}$ : known noise distribution

To estimate $x_{k}$ , select $u_{k}$ to be $2 n$ -periodic. Then the noise-free output $x_{k} = G (q) u_{k}$ is also $2 n$ -periodic, after a short transient duration.¹Hence, $x_{j + 2 \ln} = x_{j}$ , for any positive integer l. ${x_{k}}$ will be determined entirely by $2 n$ unknown real numbers $γ^{j}, j = 1, \dots, 2 n$ , $x_{j} = γ^{j}, j = 1, \dots, 2 n .$ $Γ = [γ^{1},$

Estimation of parameter $θ$

Under a periodic input u, the one-to-one mapping between $θ$ and the periodic output x of the system G will be first established. This relationship will be used to derive an estimate of $θ$ from that of x.

Joint identification of distribution functions and system parameters

The developments above rely on the knowledge of the distribution function $F (\cdot)$ or its inverse. However, in most applications, the noise distributions are not known, or only limited information is available. On the other hand, input/output data from the system contain information about the noise distribution. By viewing unknown distributions and system parameters jointly as uncertainties, we develop a methodology of joint identification.

To estimate the distribution function $ξ = F (λ)$ , one needs

Algorithm flowcharts and an illustrative example

Our algorithms for joint identification of system parameters and noise distributions are summarized in Fig. 3.

We now use an example to demonstrate the identification algorithms presented so far.

Suppose that the true plant is a first-order system $x_{k} = - a_{0} x_{k - 1} + b_{0} u_{k - 1}, y_{k} = x_{k} + d_{k},$ where $a_{0} = 0.4; b_{0} = 1.6$ . ${d_{k}$ } is an i.i.d. sequence, uniformly distributed on $[- 1.2, 1.2]$ . Hence, the true distribution function is $ξ = F (z) = (1 / 2.4) z + 0.5$ for $z \in [- 1.2, 1.2]$ . The true system parameters and the distribution function

Recursive algorithms

In this section, we develop a class of recursive algorithms for estimating $α$ and $γ$ . In lieu of the line search (21) and least-squares procedure (22), the estimate ${\hat{α}}_{m}$ will be constructed via an adaptive filtering algorithm to reduce the computational complexity, and estimate ${\hat{γ}}_{m}$ will be recursified. This section is divided into three parts: first, we present the algorithms. Then, we establish the convergence of the schemes. Finally, we make some additional remarks on alternatives.

The

Conclusions

When sensors are nonlinear and non-smooth such as the switching sensors investigated in this paper, system identification for plants in ARMA structures usually becomes difficult, due to lack of constructive and convergent identification algorithms. This paper introduces a two-step approach to resolve this complicated problem. This approach is further extended to accommodate the common scenarios in which noise distribution functions are unknown. Convergence properties of all the algorithms are

Acknowledgements

The research of L.Y. Wang was supported in part by the National Science Foundation under ECS-0329597, and in part by Wayne State University Research Enhancement Program. The research of G. Yin was supported in part by the National Science Foundation under DMS-0304928, and in part by Wayne State University Research Enhancement Program. The research of J.F. Zhang was supported by the National Natural Science Foundation of China.

Le Yi Wang received the Ph.D. degree in electrical engineering from McGill University, Montreal, Canada, in 1990. Since 1990, he has been with Wayne State University, Detroit, MI, where he is currently a Professor in the Department of Electrical and Computer Engineering. His research interests are in the areas of H-infinity optimization, complexity and information, robust control, system identification, adaptive systems, hybrid and nonlinear systems, information processing and learning, as well

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Dr. Wang was awarded the Research Initiation Award in 1992 from the National Science Foundation. He also received the Faculty Research Award from Wayne State University, in 1992, and the College Outstanding Teaching Award from the College of Engineering, Wayne State University, in 1995. He was a keynote speaker in two international conferences. He serves on the IFAC Technical Committee on Modeling, Identification and Signal Processing. He served as an Associate Editor of the IEEE Transactions on Automatic Control, and currently is an Editor of the Journal of System Sciences and Complexity, an Associate Editor of Journal of Control Theory and Applications and International Journal of Control and Intelligent Systems.

G. George Yin received his B.S. in Mathematics from the University of Delaware in 1983, M.S. in Electrical Engineering and Ph.D. in Applied Mathematics from Brown University in 1987. Subsequently, he joined the Department of Mathematics, Wayne State University, and became a professor in 1996. He is a fellow of IEEE. He served on the Mathematical Review Date Base Committee, IFAC Technical Committee on Modeling, Identification and Signal Processing, and various conference program committees; he was the editor of SIAM Activity Group on Control and Systems Theory Newsletters, the SIAM Representative to the 34th CDC, Co-Chair of 1996 AMS–SIAM Summer Seminar in Applied Mathematics, and Co-Chair of 2003 AMS–IMS–SIAM Summer Research Conference: Mathematics of Finance, Co-organizer of 2005 IMA Workshop on Wireless Communications. He is an Associate Editor of Automatica and SIAM Journal on Control and Optimization, and was also an Associate Editor of IEEE Transactions on Automatic Control from 1994 to 1998, and is (or was) on the editorial board of five other journals.

Ji-Feng Zhang received his B.S. degree in Mathematics from Shandong University in 1985, and the Ph.D. degree from the Institute of Systems Science (ISS), Chinese Academy of Sciences (CAS) in 1991. Since 1985 he has been with the ISS, CAS, where he is now a Professor of the Academy of Mathematics and Systems Science, the Vice-Director of the ISS. His current research interests are system modeling and identification, adaptive control, stochastic systems, and descriptor systems. He received the Science Fund for Distinguished Young Scholars from NSFC in 1997, the First Prize of the Young Scientist Award of CAS in 1995, and now is a Vice-General Secretary of the Chinese Association of Automation (CAA), Vice-Director of the Control Theory Committee of CAA, Deputy Editor-in-Chief of the journals “Acta Automatica Sinica” and “Journal of Systems Science and Mathematical Sciences”.

^☆: This paper was not presented at any IFAC meeting. This paper was recommended for publication in revised form by Associate Editor Antonio Vicino under the direction of Editor Torsten Soederstroem.

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Joint identification of plant rational models and noise distribution functions using binary-valued observations☆

Abstract

Introduction

Section snippets

Problem formulation

Estimation of xk: known noise distribution

Estimation of parameter θ

Joint identification of distribution functions and system parameters

Algorithm flowcharts and an illustrative example

Recursive algorithms

Conclusions

Acknowledgements

Automatica

Convergence of probability measures

Decision equation for binary systems: application to neural behavior

Kybernetik

Identification and stochastic adaptive control

Asymptotic properties of sign algorithms for adaptive filtering

IEEE Transactions on Automatic Control

Quiver diagrams and signed adaptive fiters

IEEE Transactions on Acoustics Speech and Signal Processing

Convergence analysis of an adaptive filter equipped with the sign-sign algorithm

IEEE Transactions on Automatic Control

Adaptive filtering with binary reinforcement

IEEE Transactions on Information Theory

Estimation of $x_{k}$ : known noise distribution

Estimation of parameter $θ$