Hierarchical Quasi-Fractional Gradient Descent Method for Parameter Estimation of Nonlinear ARX Systems Using Key Term Separation Principle

Recently, a quasi-fractional order gradient descent (QFGD) algorithm was proposed and successfully applied to solve system identification problem. The QFGD suffers from the overparameterization problem and results in estimating the redundant parameters instead of identifying only the actual parameters of the system. This study develops a novel hierarchical QFDS (HQFGD) algorithm by introducing the concepts of hierarchical identification principle and key term separation idea. The proposed HQFGD is effectively applied to solve the parameter estimation problem of input nonlinear autoregressive with exogeneous noise (INARX) system. A detailed investigation about the performance of HQFGD is conducted under different disturbance conditions considering different fractional orders and learning rate variations. The simulation results validate the better performance of the HQFGD over the standard counterpart in terms of estimation accuracy, convergence speed and robustness.

Iterative Parameter Identification for Fractional-Order Block-Oriented Nonlinear Systems with Application to a Battery Model

This paper investigates parameter and order identification of a class of block-oriented nonlinear systems. By using the hierarchical identification principle, the system is divided into two subsystems, which are a linear block system and a nonlinear block system. For the purpose of solving the difficulty of estimating two sets of parameter vectors, the over-parameterization method and the key item separation technique are used, respectively. Therefore, a two-stage over-parameterization gradient-based iterative algorithm and a key term separation two-stage gradient-based iterative algorithm are derived. The simulation results indicate that the proposed algorithms are effective. Finally, the proposed method is evaluated through a battery model. The results show well agreement with the real system outputs.

Efficient Iterative Algorithms for a Class of Nonlinear Systems using the Block Matrix Inversion and Hierarchical Principle

This paper is concerned with the identification of the bilinear systems in the state-space form. The parameters to be identified of the considered system are coupled with the unknown states, which makes the identification problem difficult. To deal with the trouble, the iterative estimation theory is considered to derive the joint parameter and state estimation algorithm. Specifically, a moving data window least squares-based iterative (MDW-LSI) algorithm is derived to estimate the parameters by using the window data. Then, the unknown states are estimated by a bilinear state estimator. Moreover, for the purpose of improving the computational efficiency, a matrix decomposition-based MDW-LSI algorithm and a hierarchical MDW-LSI algorithm are developed according to the block matrix and the hierarchical identification principle. Finally, the computational efficiency is discussed and the numerical simulation is employed to test the proposed approaches.

Modified Jacobi-Gradient Iterative Method for Generalized Sylvester Matrix Equation

We propose a new iterative method for solving a generalized Sylvester matrix equation A1XA2+A3XA4=E with given square matrices A1,A2,A3,A4 and an unknown rectangular matrix X. The method aims to construct a sequence of approximated solutions converging to the exact solution, no matter the initial value is. We decompose the coefficient matrices to be the sum of its diagonal part and others. The recursive formula for the iteration is derived from the gradients of quadratic norm-error functions, together with the hierarchical identification principle. We find equivalent conditions on a convergent factor, relied on eigenvalues of the associated iteration matrix, so that the method is applicable as desired. The convergence rate and error estimation of the method are governed by the spectral norm of the related iteration matrix. Furthermore, we illustrate numerical examples of the proposed method to show its capability and efficacy, compared to recent gradient-based iterative methods.