The residual based extended least squares identification method for dual-rate systems

In this paper, we focus on a class of dual-rate sampled-data systems in which all the inputs u(t) are available at each instant while only scarce outputs y(qt) can be measured (q being an integer more than unity). To estimate the parameters of such dual-rate systems, we derive amathematicalmodel by using the polynomial transformation technique, and apply the extended least squares algorithm to identify the dual-rate systems directly from the available input–output data {u(t), y(qt)}. Then, we study the convergence properties of the algorithm in details. Finally, we give an example to test and illustrate the algorithm involved. © 2008 Elsevier Ltd. All rights reserved. 1. Problem formulation Let us consider the discrete-time system described by a controlled auto-regression model [1,2], y(t)+ a1y(t − 1)+ a2y(t − 2)+ · · · + any(t − n) = b0u(t)+ b1u(t − 1)+ b2u(t − 2)+ · · · + bnu(t − n)+ v(t), (1) where u(t) and y(t) are the system input and output, {v(t)} is a random noise sequence with zero mean and unknown timevarying variance, ai and bi are the unknown parameters, and n is the known system order. Let z−1 be the unit backward shift operator [z−1u(t) = u(t − 1), zy(t) = y(t − q)], and A(z) and B(z) be polynomials in z−1, A(z) = 1+ a1z + a2z + · · · + anz, B(z) = b0 + b1z −1 + b2z −2 + · · · + bnz . Then (1) can be written into a compact form, A(z)y(t) = B(z)u(t)+ v(t). (2) Define the information vector φ(t) and parameter vector θ as φ(t) = [−y(t − 1),−y(t − 2), . . . ,−y(t − n), u(t), u(t − 1), . . . , u(t − n)]T, (3) θ = [a1, a2, . . . , an, b0, b1, b2, . . . , bn] T. (4) I This research was supported by the National Natural Science Foundation of China (No. 60574051) and the Natural Science Foundation of Jiangsu Province, China (BK2007017) and by Program for Innovative Research Team of Jiangnan University. ∗ Corresponding author. E-mail addresses: [email protected] (J. Ding), [email protected] (F. Ding). 0898-1221/$ – see front matter© 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.camwa.2008.02.047 1480 J. Ding, F. Ding / Computers and Mathematics with Applications 56 (2008) 1479–1487 Thus y(t) = φT(t)θ + v(t), (5) where the superscript T denotes the matrix transpose. The so-called dual-rate system identification or parameter estimation is to identify/estimate the system parameters with the available dual-rate input–output data {u(t), y(qt) : t = 0, 1, 2, . . .}, q ≥ 2 being an integer [3]. Some traditional system identification methods assume that the system input–output data {u(t), y(t)} are available at each sampling instant; such approaches are here called single-rate system identificationmethods [1,2]. This paper considers the dual-rate sampled-data systems [4,5], in which the input and output sampling rate are different, namely, all the inputs {u(t) : t = 0, 1, 2, . . .} are available at each instant, while only scarce outputs {y(qt) : t = 0, 1, 2, . . .} are available (q ≥ 2 being a positive integer). In such systems, the input–output data available are {u(t), y(qt) : t = 0, 1, 2, . . .}. Thus, the intersample outputs (or missing outputs), y(qt + i), i = 1, 2, . . . , q − 1, are not available. Therefore, the traditional parameter estimation algorithms are not applicable for dual-rate systems due to the missing outputs, so this paper focuses on identification problems for the dual-rate systems with missing outputs. The basic idea is to employ a polynomial transformation technique to derive a dual-ratemodel, and to extend the least squares for single-rate systems to estimate the parameters of the obtained dual-ratemodel, and further to study the convergence properties of the algorithm proposed. The approach here differs from the ones in [4,6] which used the auxiliary model technique to identify/estimate the parameters and missing outputs of dual-rate sampled-data systems. The paper is organized as follows: Section 2 discusses the modeling issues related to the dual-rate systems and derives a dual-rate model by using a polynomial transformation technique. Based on this model, Section 3 proposes a parameter estimation algorithm and introduces some preliminary backgrounds for performance analysis of the estimation algorithm to be used later. Section 4 proves the convergence of the parameter estimation and the intersample output estimation given by the proposed algorithm in Section 3. Section 5 presents an illustrative example for the results in Section 3, and shows the effectiveness of the algorithms proposed in the paper. Finally, we offer some concluding remarks in Section 6. 2. A polynomial transformation technique The model in (5) needs to be transformed into a form that we can use directly on the dual-rate data. A polynomial transformation technique is employed here to do this. The details are as follows. Let the roots of A(z) be zi (i = 1, 2, . . . , n) to get A(z) = (1− z1z)(1− z2z) · · · (1− znz). Define a polynomial,