Parameter Identiﬁcation of Nonlinear Systems with Time-delay Based on the Multi-innovation Stochastic Gradient Algorithm

This paper considers the parameter identiﬁcation problem of block-oriented Hammerstein nonlinear systems with time-delay. Firstly, we adopt the data ﬁltering technique to transform the identiﬁcation model so that all the parameters will be separated in the resulting identiﬁcation model which has no redundant parameters. Secondly, a multi-innovation stochastic gradient algorithm is used to estimate the system parameters. The proposed method has high computational efﬁciency and good accuracy. Simulation results are presented to demonstrate the effectiveness of the proposed algorithm.


Instruction
Parameter identification means estimating the parameters of partially unknown systems based on noisy observations, which is the foundation of many issues such as signal processing, system identification and system control [1][2][3][4][5]. In the last few decades, the system identification theory has been well studied and many system identification methods e.g., the instrumental variable methods [6,7], the bias compensation methods [8,9], the least squares methods [10,11], the multi-innovation identification methods [12,13] have been developed to model the dynamical systems.
In reality, many exist systems exhibit nonlinear phenomena and nonlinear system identification is currently an active research topic in parameter identification. Typical nonlinear systems include the Hammerstein system, the Wiener systems and their combinations, etc. The Hammerstein system consists of a nonlinear block plus a linear block, while the Wiener systems consists of a linear dynamic system block followed by a static nonlinearity block. The above systems have been well studied and there are a lot of parameter identification algorithms about them have been proposed. In practice, there is a class of block-oriented multi-input single-output Hammerstein nonlinear systems which have the following form: where u(t) and y(t) are the system input and output, and v(t) is a white noise with zero mean, u f (t) is a nonlinear function, A(z) and B(z) are the polynomials in the unit backward shift operator z −1 y(t) = y(t − 1) which contains the parameter vectors to be identified.
Many studies have been done on the identification of nonlinear systems which include the products of the original system parameters such as (1). For example, Wang et al. presented an extended stochastic gradient identification algorithm for Hammerstein-Wiener system with colored noises and gave two methods of separating the parameters of the system [14]. Xiao et al. presented a filtering based recursive least squares identification algorithm for a input nonlinear dynamical adjustment models with memoryless nonlinear blocks followed by linear dynamical blocks [15]. However, the over-parameterized methods have many redundant estimates and the the dimensions of parameter vectors will be larger. As a result, the accuracy and speed of the parameters evaluation will be affected. Due to that time delay exists in many nonlinear systems, identification and construction of delay system models from experimental data have great value in control area. In this paper, on the basis of the work in [16], a multi-innovation stochastic gradient algorithm is proposed to identify the parameters of nonlinear systems with nonlinear systems which include the products of the original system parameters and time-delay.
The rest of this paper is organized as follows. The problem formulation and identification model are performed in section 2. Section 3 derives the parameters identification process of the nonlinear systems with time-delay. Numerical simulations are performed in section 4 to verify the effectiveness of the presented schemes, and concluding remarks are made in the final section.
2Parameter Identification of Nonlinear Systems with Time-delay Based on the Multi-innovation Stochastic Gradient Algorithm

System description and identification model
Consider the following nonlinear system with time-delay: where u f (t) is a nonlinear function which can be approximable by a polynomial: is a random white noise with zero mean and variance δ 2 , y(t) is the measured system output, and A(z) and B(z) are the polynomials in the unit backward shift operator z −1 y(t) = y(t − 1) with: Assume that the order n a and n b are known and u(t) = 0, v(t) = 0 and y(t) = 0 as t ≤ 0. Multiplying both sides of the equation (2) Define the filtered output and the filtered noise as Then we can have On the other hand, Eq.(5) can be rewritten as Then we can get the following expression Substituting (8) and (9) into (11) gives Define the parameter vectors and the information vectors Eq.(20) is the identification model.

The stochastic gradient (SG) algorithm
In the following, we derive a stochastic gradient algorithm for the nonlinear systems with time-delay to identify the parameter vector θ. Define the cost functions: Letθ(t) be the estimate of θ at time t. Using the gradient search and minimizing J(θ), we can obtain the following stochastic gradient algorithm of computing the estimateθ(t): However, the information vector Ψ(t) contains unknown inner variables y f (t−i), i = 1, 2, ..., n c , v f (t−i), i = 1, 2, ..., n d and v(t − i), i = 1, 2, ..., n d , the stochastic gradient algorithm cannot be applied directly. By means of the auxiliary model identification idea, we can replace the unknown variables by their corresponding estimatesŷ f (t−i), i = 1, 2, ..., n a , v f (t − i), i = 1, 2, ..., n b andv(t − i), i = 1, 2, ..., n b . Definê (24) whereŷ Thus the stochastic gradient algorithm can be described as follows: The steps of computing the parameter estimate θ(t) as t increases using the SG algorithm is summarized as follows: 1. Let t = 1, set the initial valuesθ(0) = 1 n /p 0 , where p 0 is a large number (e.g., p 0 = 10 6 ).

The multi-innovation stochastic gradient (MI-SG) algorithm
Let p be the data length p >> N 0 , and define the stacked vectors or matrix . . .
From (4) and (5), we have Form a cost function where the norm of the matrix X is defined as X 2 = tr[XX T ]. Based on the multi-innovation identification theory, the multi-innovation stochastic gradient algorithm can be described as follows: . . .

Example
This section provides an example to show the effectiveness of the MI-SG algorithm for the nonlinear systems with timedelay, compared with the SG algorithm. Consider the following nonlinear system: In simulation, the inputs u(t) is taken as an uncorrelated persistent excitation signal sequence with zero mean and unit variance, and v(t) as a white noise sequence with zero mean and variance δ 2 v = 0.2 2 . We apply the SG algorithm and MI-SG algorithm(p=4 and 8) to estimate the parameters of this system.
Numerical results are displayed in tables 1, 2, 3 and figure 1. The parameter estimates and their errors with different innovation length are shown in tables 1, 2 and 3, and the parameter estimation errors δ = θ (t) − θ / θ vs. t are shown in figure  1. In addition, figure 2 shows the time evolution curves of the system parameters vs. t of the MI-SG method with p=8.
4Parameter Identification of Nonlinear Systems with Time-delay Based on the Multi-innovation Stochastic Gradient Algorithm     The experiments has been conducted to demonstrate the properties of the proposed algorithm. Some attractive points can be summarized as follows: (1) The parameter estimation errors of the three algorithms become smaller and smaller as t increasing.
(2) MI-SG performers better than SG because it can consider more data information.
(3) When the innovation length p becomes longer, the better identification results will be obtained.

Conclusions
This paper investigates the parameter identification of the nonlinear system with time-delay using the data filtering technique and the multi-innovation stochastic gradient algorithm. Based on the data filtering technique, the original identification model is converted into a new model which has no cross-products of the parameters and is easier to be identified. The simulation results indicate that the proposed method is effective. The presented method can also be used to solve other relevant system identification problems.