The residual based interactive stochastic gradient algorithms for controlled moving average models
Introduction
The so-called system identification or parameter estimation [1], [2], [3] is to identify/estimate the unknown parameters of the mathematical models under consideration, e.g., the autoregressive moving average (ARMA) model using the observation data. This paper considers the parameter identification problem of a controlled moving average (CMA) model using the measured input–output data and presents a gradient based identification algorithm using the gradient search principle.
The stochastic gradient (SG) algorithms [4] are a class of important identification methods and have received much attention in many areas, including signal processing [5], system identification and parameter estimation [6], [7], [8], adaptive control [9], [10], [11], [12]. For example, based on the gradient search principle, Ding and Chen presented the SG and hierarchical SG algorithms for multivariable systems [13], [14], the (auxiliary model based) SG algorithms for dual-rate systems [15], [16], the multi-innovation SG identification algorithms for linear regression systems [17], the extended stochastic gradient (ESG) algorithm for Hammerstein nonlinear systems [18] and for Hammerstein–Wiener ARMAX Systems [19].
Compared with the least squares type identification algorithms [20], [21], [22], [23], [24], [25], [26], [27], [28], [29], the stochastic gradient algorithm has less computational load. Therefore, We frame our study in the residual based interactive stochastic gradient (ISG) algorithm of the CMA models and explore the convergence of the proposed algorithm under weak conditions that the noise has unbounded time-varying variance.
Section snippets
The system description
Consider a controlled moving average (CMA) model,where is the observation output, is the control input, is an uncorrelated noise sequence with zero mean and time-varying variance (E denotes the expectation operator) and independent of , represents the unit backward shift operator: , and and are polynomials in with
Assume that the orders
The ESG algorithm
Let be the estimate of and be the estimate of and defineFrom (5), we have
Replacing and in the above equation with their estimates and , respectively, the estimated residual can be computed by
The residual based extended stochastic gradient (ESG) algorithm of estimating is as follows [4]:
The ISG algorithm
In this section, we present the interactive stochastic gradient (ISG) algorithm for identifying and , respectively.
From (5), we have
For two identification models in (4), (11), defining and minimizing two cost functions:and using the gradient search principle, we can get the following identification algorithms of estimating and :
Convergence analysis
The following lemma is required to establish the main convergence result. Lemma 1 For and , assume that , , and . Then .
The proof of Lemma 1 is straightforward and hence omitted. Theorem 1 For the system in (4) and the ISG algorithm in (12), (13), (14) and (18), (19), (20), (21), (22), (23), assume that is an uncorrelated noise withThat is, is a random noise with zero mean and unbounded time-varying
Examples
Two illustrative examples are given in this section, one is for the stationary case and the other is for the non-stationary case. Example 1 The stationary case: Assume that the simulated model takes the following formHere is taken as a persistent excitation pseudo-random binary signal sequence with the amplitude , i.e., the zero mean and unit variance, i.e.,
Conclusions
The paper presents a residual based interactive stochastic gradient algorithm for CMA models and studies the convergence of the proposed algorithm under the weak condition; the analysis method used in this paper can be extended to study the convergence of the residual based stochastic gradient algorithm for finite impulse response Hammerstein nonlinear systems with moving average noise models [23], [18].
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