Auxiliary model-based least-squares identification methods for Hammerstein output-error systems

The difficulty in identification of a Hammerstein (a linear dynamical block following a memoryless nonlinear block) nonlinear output-error model is that the information vector in the identification model contains unknown variables—the noise-free (true) outputs of the system. In this paper, an auxiliary model-based least-squares identification algorithm is developed. The basic idea is to replace the unknown variables by the output of an auxiliary model. Convergence analysis of the algorithm indicates that the parameter estimation error consistently converges to zero under a generalized persistent excitation condition. The simulation results show the effectiveness of the proposed algorithms.     

Comments on “Identification of Hammerstein nonlinear ARMAX systems”

This note points out some errors in the proof of the main result of Ding and Chen [2005. Identification of Hammerstein nonlinear ARMAX systems. Automatica 41(9), 1479-1489] and mentions the well-known results that have solved the problem discussed in Theorem 1 of the above paper by Ding and Chen.     

Least squares based iterative identification for a class of multirate systems

This paper studies modeling and identification problems for multi-input multirate systems with colored noises. The state-space models are derived for the systems with different input updating periods and furthermore the corresponding transfer functions are obtained. To solve the difficulty of identification models with unmeasurable noises terms, the least squares based iterative algorithm is presented by replacing the unmeasurable variables with their iterative estimates. Finally, the simulation results indicate that the proposed iterative algorithm has advantages over the recursive algorithms.     

Identification of nonlinear systems using a piecewise-linear Hammerstein model

In the paper a method for nonlinear system identification is proposed. It is based on a piecewise-linear Hammerstein model, which is linear in the parameters. The model and the identification algorithm are adapted to allow the parameter identification in the presence of a special form of the excitation signal. The identification method is derived from a recursive least-squares algorithm, which is properly adapted to take into account the proposed model structure and the properties of the identification signal. The applicability of the approach is illustrated by an example in which a discontinuous nonlinear static function is connected to a dynamic block.     

Convergence of normalized iterative identification of Hammerstein systems

An iterative identification algorithm of Hammerstein systems needs a proper initial condition to guarantee its convergence. In this paper, we propose a new algorithm by fixing the norm of the parameter estimates. The normalized algorithm ensures the convergence property under arbitrary nonzero initial conditions. The proofs of the property also give a geometrical explanation on why the normalization guarantees the convergence. An additional contribution is that the static function in the Hammerstein system is extended to square-integrable functions.     

Extended stochastic gradient identification algorithms for Hammerstein–Wiener ARMAX systems

An extended stochastic gradient algorithm is developed to estimate the parameters of Hammerstein–Wiener ARMAX models. The basic idea is to replace the unmeasurable noise terms in the information vector of the pseudo-linear regression identification model with the corresponding noise estimates which are computed by the obtained parameter estimates. The obtained parameter estimates of the identification model include the product terms of the parameters of the original systems. Two methods of separating the parameter estimates of the original parameters from the product terms are discussed: the average method and the singular value decomposition method. To improve the identification accuracy, an extended stochastic gradient algorithm with a forgetting factor is presented. The simulation results indicate that the parameter estimation errors become small by introducing the forgetting factor.     

Iterative identification of Hammerstein systems

Convergences of iterative algorithms have been established for identification of Hammerstein systems in the case that the unknown nonlinearities are odd. Then, the results are further extended to nonsmooth nonlinearities.     

Maximum likelihood stochastic gradient estimation for Hammerstein systems with colored noise based on the key term separation technique

This paper considers the identification problems of Hammerstein finite impulse response moving average (FIR-MA) systems using the maximum likelihood principle and stochastic gradient method based on the key term separation technique. In order to improve the convergence rate, a maximum likelihood multi-innovation stochastic gradient algorithm is presented. The simulation results show that the proposed algorithms can effectively estimate the parameters of the Hammerstein FIR-MA systems.     

Performance analysis of multi-innovation gradient type identification methods

It is well-known that the stochastic gradient (SG) identification algorithm has poor convergence rate. In order to improve the convergence rate, we extend the SG algorithm from the viewpoint of innovation modification and present multi-innovation gradient type identification algorithms, including a multi-innovation stochastic gradient (MISG) algorithm and a multi-innovation forgetting gradient (MIFG) algorithm. Because the multi-innovation gradient type algorithms use not only the current data but also the past data at each iteration, parameter estimation accuracy can be improved. Finally, the performance analysis and simulation results show that the proposed MISG and MIFG algorithms have faster convergence rates and better tracking performance than their corresponding SG algorithms.     

Quantifying the accuracy of Hammerstein model estimation

This paper investigates the accuracy of the linear component that forms part of an overall Hammerstein model-structure estimate, and a key finding is that the process of estimating the non-linear element can have a strong effect on the associated estimate of the linear dynamics. Furthermore, this effect is not explained simply by way of considering how the input spectrum is changed by the non-linearity. Instead, it arises that the linear model-estimate variability may be dominated by a term that depends on the frequency response of the linear system itself. Amongst other things, the main results derived here have experiment design implications for Hammerstein system estimation.     

Adaptive tracking and recursive identification for Hammerstein systems

A weighted least squares (WLS) based adaptive tracker is designed for a class of Hammerstein systems. It is proved that the tracking error is asymptotically minimized. Incorporating with the diminishing excitation technique, the minimality of the tracking error and strong consistency of the estimates for parameters of the system are simultaneously achieved. Numerical examples are given and the simulation results are consistent with the theoretical analysis.     

Hierarchical gradient-based identification of multivariable discrete-time systems

In this paper, we use a hierarchical identification principle to study identification problems for multivariable discrete-time systems. We propose a hierarchical gradient iterative algorithm and a hierarchical stochastic gradient algorithm and prove that the parameter estimation errors given by the algorithms converge to zero for any initial values under persistent excitation. The proposed algorithms can be applied to identification of systems involving non-stationary signals and have significant computational advantage over existing identification algorithms. Finally, we test the proposed algorithms by simulation and show their effectiveness.     

Decoupling the linear and nonlinear parts in Hammerstein model identification

The main result of this paper is to show that the linear part can be made decoupled from the nonlinear part in Hammerstein model identification. Therefore, identification of the linear part for a Hammerstein model becomes a linear problem and accordingly enjoys the same convergence and consistency results as if the unknown nonlinearity is absent.     

An efficient hierarchical identification method for general dual-rate sampled-data systems

For the lifted input–output representation of general dual-rate sampled-data systems, this paper presents a decomposition based recursive least squares (D-LS) identification algorithm using the hierarchical identification principle. Compared with the recursive least squares (RLS) algorithm, the proposed D-LS algorithm does not require computing the covariance matrices with large sizes and matrix inverses in each recursion step, and thus has a higher computational efficiency than the RLS algorithm. The performance analysis of the D-LS algorithm indicates that the parameter estimates can converge to their true values. A simulation example is given to confirm the convergence results.     

Recursive identification of Hammerstein systems with application to electrically stimulated muscle

Modeling of electrically stimulated muscle is considered in this paper where a Hammerstein structure is selected to represent the isometric response. Motivated by the slowly time-varying properties of the muscle system, recursive identification of Hammerstein structures is investigated. A recursive algorithm is then developed to address limitations in the approaches currently available. The linear and nonlinear parameters are separated and estimated recursively in a parallel manner, with each updating algorithm using the most up-to-date estimation produced by the other algorithm at each time instant. Hence the procedure is termed the alternately recursive least square (ARLS) algorithm. When compared with the leading approach in this application area, ARLS exhibits superior performance in both numerical simulations and experimental tests with electrically stimulated muscle.     

Rate of convergence of the LMS method

It is shown under reasonable assumptions that for any q the L_q-norm of the estimation error generated by the LMS algorithm is of the order of magnitude , where T is the length of the observation period.     

基于辅助模型的多新息广义增广随机梯度算法

将辅助模型辨识思想与多新息辨识理论相结合,利用系统可测信忠建立一个辅助模型.分别用辅助模型输出和噪声估计值代替辨识模型信忠向量中未知真实输出变量和不可测噪声项,并引入新忠长度扩展标量新息为新息向量,提出了Box-lenkins模型的辅助模型多新忠广义增广随机梯度辨识方法.所提出方法重复使用系统数据,能够改善参数估计精度,加快算法的收敛速度.     

Parameter estimation with scarce measurements

In this paper, the problems of parameter estimation are addressed for systems with scarce measurements. A gradient-based algorithm is derived to estimate the parameters of the input–output representation with scarce measurements, and the convergence properties of the parameter estimation and unavailable output estimation are established using the Kronecker lemma and the deterministic version of the martingale convergence theorem. Finally, an example is provided to demonstrate the effectiveness of the proposed algorithm.     

Computational complexity analysis of set membership identification of Hammerstein and Wiener systems

This paper analyzes the computational complexity of set membership identification of Hammerstein and Wiener systems. Its main results show that, even in cases where a portion of the plant is known, the problems are generically NP-hard both in the number of experimental data points and in the number of inputs (Wiener) or outputs (Hammerstein) of the nonlinearity. These results provide new insight into the reasons underlying the high computational complexity of several recently proposed algorithms and point out the need for developing computationally tractable relaxations.