Auxiliary model based least squares parameter estimation algorithm for feedback nonlinear systems using the hierarchical identification principle☆
Introduction
System modelling is important for studying the motion laws of dynamical systems [1], [2], [3]. The parameters of a system can be estimated through an identification algorithm from the measurement data [4], [5], [6]. Recently, parameter estimation for feedback closed-loop control systems has received much attention on system identification [7], [8]. For example, Marion et al. considered the instrumental variable and instrumental variable-related identification methods for closed-loop systems [9].
The auxiliary model can handle the identification problems in the presence of the unmeasurable variables in the information vector. In the area of parameter estimation, Ding and Chen proposed an auxiliary model based recursive least squares algorithm for a dual-rate sampled-data system [10]. They used an auxiliary model to estimate the unknown noise-free outputs of the system and to identify the parameters of the underlying fast single-rate model from available dual-rate input–output data directly. Wang et al. discussed the auxiliary model based recursive generalized least squares algorithm for Hammerstein output error autoregressive systems by using the key-term separation principle [11]. Liu et al. proposed a multi-innovation stochastic gradient algorithm for multiple-input single-output systems using the auxiliary model [12].
The iterative numerical algorithms can be used to solve matrix equations [13], [14], [15], [16], [17], and to estimate the parameters of systems from input–output data [18], [19], [20]. In the area of system identification, Wang presented a least squares-based recursive and iterative estimation for output error moving average (OEMA) systems using the data filtering, transforming an output error moving average system into two identification models [21]; Wang and Ding derived an extended stochastic gradient identification algorithm for Hammerstein–Wiener ARMAX systems [5], presented the auxiliary model based RELS and MI-ELS algorithms for Hammerstein OEMA systems [22], and developed a least squares based and a gradient based iterative identification algorithms for Wiener nonlinear systems [23].
Recently, Duan et al. proposed a two-stage recursive least squares parameter estimation algorithm for output error models [24]; Yao et al. proposed a two-stage least squares based iterative identification algorithm for controlled autoregressive moving average (CARMA) systems [25]; Hu and Ding proposed a multi-stage least squares based iterative estimation algorithm for feedback nonlinear systems with moving average noises using the hierarchical identification principle [26]. The main contributions of this paper are as follows. On the basis of the decomposition identification technique in [26], [27], [28], [29], this paper transforms a complex identification problem into small subproblems which are easier to solve and presents an auxiliary model based least squares parameter estimation algorithm for feedback nonlinear systems.
This paper is organized as follows. Section 2 gives the system description and identification model for feedback nonlinear systems. Section 3 derives an auxiliary model and decomposition based recursive least squares algorithm for feedback nonlinear systems. Section 4 offers a comparison between the proposed algorithm and the least squares based auxiliary model identification algorithm. Section 5 gives an example for the proposed algorithms. Finally, we offer some concluding remarks in Section 6.
Section snippets
System description and identification model
Let us introduce some notation [30], [31]. The symbol stands for an identity matrix of order n; the superscript T denotes the matrix transpose; represents an n-dimensional column vector whose elements are 1; the norm of a matrix is defined as ; denotes the estimate of at time t.
Consider the feedback nonlinear system in Fig. 1, where the and are the input–output sequences of the system, is a stochastic white noise sequence with zero mean and
The auxiliary model based and decomposition based recursive least squares algorithm
Note that the internal variable x(t) is the true output and unknown, y(t) is corrupted by the white noise v(t). On the basis of the work [26], [32], [36], system (5) can be decomposed into two subsystems.
Define the information vectors and parameter vectorsand the intermediate variableEq. (5) can be equivalently written asThese equations contain the
The auxiliary model based recursive least squares algorithm
In the following, we derive the auxiliary model based recursive least squares algorithm to identify the feedback nonlinear system using auxiliary model identification idea.
Define the information vector and the parameter vector as Eq. (5) can be rewritten as Observing that the unknown inner variables x(i) in the vector are unmeasurable, we replace the variables x(i) with its estimates , and with its estimate
Example
Consider the following feedback nonlinear system:In simulation, the input is taken as a persistent excitation signal sequence with zero mean and unit variance, as a white noise sequence with zero mean and variances and is independent of . Using the proposed AM-TS-RLS and AM-RLS algorithms to estimate the parameters
Conclusions
Based on the auxiliary model identification idea and the hierarchical identification principle, a two-stage recursive least squares algorithm and a recursive least squares algorithm are derived for feedback nonlinear systems. The simulation results show that the proposed algorithms can effectively estimate the parameters of nonlinear output error systems. The convergence properties of the proposed algorithm are worth further studying. The proposed two-stage identification method can combine
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Hierarchical least squares identification for feedback nonlinear equation-error systems
2020, Journal of the Franklin InstituteCitation Excerpt :For feedback nonlinear output-error systems, a three-stage extended least squares based iterative estimation algorithm and an extended least squares based iterative estimation algorithm have been reported for the forward channel being an equation-error moving average model (i.e., controlled autoregressive moving average model) and for the feedback channel being a static nonlinear function with known basis [33]. Based on these previous work in [32,33], this paper studies the hierarchical least squares algorithm for feedback nonlinear controlled autoregressive systems. The initial values of the RLS algorithm can be set according to the D-RLS algorithm.
Hierarchical least squares algorithms for nonlinear feedback system modeling
2016, Journal of the Franklin InstituteCitation Excerpt :Hu et al. derived an auxiliary model based least squares parameter estimation algorithm [31], however, this method is still a two-stage algorithm and can only be used in the online identification. Compared with the works in [30,31], the main contribution of this paper is to propose a hierarchical recursive least squares algorithm that avoids the complex steps of filtering algorithm and to derive a hierarchical least squares iterative algorithm which is computationally efficient and can be used in the offline identification field, such as the waste water treatment process and other industry circle. The rest of this paper is organized as follows.
The recursive least squares identification algorithm for a class of Wiener nonlinear systems
2016, Journal of the Franklin InstituteMaximum likelihood based recursive parameter estimation for controlled autoregressive ARMA systems using the data filtering technique
2015, Journal of the Franklin InstituteReduced-rank gradient-based algorithms for generalized coupled Sylvester matrix equations and its applications
2015, Computers and Mathematics with ApplicationsIterative identification methods for input nonlinear multivariable systems using the key-term separation principle
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