Hierarchical extended parameter identification methods and convergence for finite impulse response moving average models based on the hierarchical identification principle

SummaryFor a special class of nonlinear systems (ie, bilinear systems) with autoregressive moving average noise, this paper gives the input‐output representation of the bilinear systems through eliminating the state variables in the model. Based on the obtained model and the maximum likelihood principle, a filtering‐based maximum likelihood hierarchical gradient iterative algorithm and a filtering‐based maximum likelihood hierarchical least squares iterative algorithm are developed for identifying the parameters of bilinear systems with colored noises. The original bilinear systems are divided into three subsystems by using the data filtering technique and the hierarchical identification principle, and they are identified respectively. Compared with the gradient‐based iterative algorithm and the multi‐innovation stochastic gradient algorithm, the proposed algorithms have higher computational efficiency and parameter estimation accuracy. The simulation results indicate that the proposed algorithms are effective for identifying bilinear systems.

Recursive identification methods for general stochastic systems with colored noises by using the hierarchical identification principle and the filtering identification idea

Parameter estimation algorithms for multivariable Hammerstein CARMA systems

The identification problem of closed-loop or feedback nonlinear systems is a hot topic. Based on the hierarchical identification principle, this paper presents a hierarchical stochastic gradient algorithm and a hierarchical multi-innovation stochastic gradient algorithm for feedback nonlinear systems. The simulation results show that the hierarchical multi-innovation stochastic gradient can more effectively estimate the parameters of the feedback nonlinear systems than the hierarchical stochastic gradient algorithm.

ABSTRACTThis article investigates the parameter identification problem of the stochastic systems described by the multiple‐input controlled autoregressive moving average model. By employing the data filtering technique, the input‐output data is filtered and the original system with colored noise is changed into the system with white noise. Then, based on the hierarchical identification principle, the filtered system is decomposed into two subsystems, one containing the parameters of the linear dynamic block and the other containing the parameters of the nonlinear static block, leading to a filtering‐based hierarchical stochastic gradient algorithm. To enhance the parameter accuracy, a filtering‐based multi‐innovation stochastic gradient algorithm is presented for the multiple‐input nonlinear stochastic systems. Furthermore, the convergence of the filtering‐based hierarchical stochastic gradient algorithm is analyzed, which can guarantee the stability of the algorithm. Finally, a simulation example is provided to verify the effectiveness of the proposed algorithms.

SummaryModeling an exponential autoregressive (ExpAR) time series is the basis of solving the corresponding prediction and control problems. This paper investigates the hierarchical parameter estimation methods for the ExpAR model. By the hierarchical identification principle, the original nonlinear optimization problem is transformed into the combination of a linear and nonlinear optimization problem, and then, we derive a hierarchical least squares and stochastic gradient (LS‐SG) algorithm. Given the difficulty of determining the step‐size in the hierarchical LS‐SG algorithm, an approach is proposed to obtain the optimal step‐size. To improve the parameter estimation accuracy, the multi‐innovation identification theory is employed to develop a hierarchical least squares and multi‐innovation stochastic gradient algorithm for the ExpAR model. Two simulation examples are provided to test the effectiveness of the proposed algorithms.

Recursive parameter and state estimation for an input nonlinear state space system using the hierarchical identification principle

Auxiliary model-based hierarchical stochastic gradient methods for multiple-input multiple-output systems

This paper considers identification problems for a multivariable controlled autoregressive system with autoregressive noises. A hierarchical generalized stochastic gradient algorithm and a filtering based hierarchical stochastic gradient algorithm are presented to estimate the parameter vectors and parameter matrix of such multivariable colored noise systems, by using the hierarchical identification principle. The simulation results show that the proposed hierarchical gradient estimation algorithms are effective.

SummaryThis paper studies the data filtering‐based identification algorithms for an exponential autoregressive time‐series model with moving average noise. By means of the data filtering technique and the hierarchical identification principle, the identification model is transformed into three sub‐identification (Sub‐ID) models, and a filtering‐based three‐stage extended stochastic gradient algorithm is derived for identifying these Sub‐ID models. In order to improve the parameter estimation accuracy, a filtering‐based three‐stage multi‐innovation extended stochastic gradient (F‐3S‐MIESG) algorithm is developed by using the multi‐innovation identification theory. The simulation results indicate that the proposed F‐3S‐MIESG algorithm can work well.

Parameter estimation algorithm for multivariable controlled autoregressive autoregressive moving average systems

This paper focuses on the identification of input nonlinear multivariable systems (i.e., Hammerstein multi-input multi-output nonlinear systems) described by output-error moving average models. Based on the key-term separation principle, we separate a proper key term in the nonlinear system and transform a complex nonlinear optimization problem into a pseudo-linear optimization problem which does not involve the products of the parameters between the linear parts and the nonlinear parts. Thus, a hierarchical extended stochastic gradient (H-ESG) algorithm is given and a hierarchical multi-innovation extended stochastic gradient (H-MI-ESG) algorithm is derived for the nonlinear systems. The proposed H-MI-ESG algorithm is an extension of the H-ESG algorithms. Compared with the over-parameterization-based least-squares identification algorithm, the H-ESG and H-MI-ESG algorithms have high computational efficiency. The simulation results show the effectiveness of the proposed algorithms.

For the identification of a class of nonlinear multi‐input multi‐output (MIMO) Hammerstein systems with different types of coefficients: a matrix coefficient and scalar coefficients, it is difficult to parameterise such Hammerstein systems into an identification model to which the standard identification method can be easily applied to implement parameter estimation. By the matrix transformation and the over‐parametrisation idea, this study transforms an MIMO Hammerstein system with different types of coefficients into an over‐parametrisation regression identification model, and points out the aroused large computation problem. To overcome the large computational load of the over‐parametrisation method, by the matrix transformation and the hierarchical identification principle, this study recasts the MIMO Hammerstein system into two models, each of which is expressed as a regression form in the parameters of the nonlinear part or in the parameters of the linear part. Then a hierarchical extended stochastic gradient algorithm is presented to alternatively estimate the parameters of the nonlinear part and the parameters of the linear part. The simulation results indicate that the proposed algorithm can effectively identify the nonlinear MIMO Hammerstein system.

This paper derives a data filtering-based two-stage stochastic gradient algorithm and a data filtering-based multistage recursive least-squares algorithm for input nonlinear output-error autoregressive systems (i.e., Hammerstein systems). The output of the system is expressed as a linear combination of all system parameters based on the key term separation technique. The basic idea of the proposed algorithm is to filter the input---output data and to separate the parameter vector into several vectors and to interactively identify each parameter vector. The data filtering-based two-stage stochastic gradient algorithm has higher convergence rate than the stochastic gradient algorithm. Compared with the recursive generalized least-squares algorithm, the dimensions of the involved covariance matrices in the data filtering-based multistage recursive least-squares algorithm become small, and thus the data filtering-based multistage recursive least-squares algorithm has a higher computational efficiency. The numerical simulation results indicate that the proposed algorithms are effective.

This paper considers the parameter identification for a special class of nonlinear systems, i.e., bilinear-in-parameter systems. Based on the hierarchical identification principle, a hierarchical stochastic gradient (HSG) estimation algorithm is presented. The basic idea is to decompose a bilinear-in-parameter system into two subsystems and to derive the HSG identification algorithm for estimating the system parameters by replacing the unknown variables in the information vectors with their estimates obtained at the previous time. The convergence analysis of the proposed algorithm indicates that the parameter estimation errors converge to zero under persistent excitation conditions. The simulation results show that the proposed algorithm is effective.