Key Term Separation Technique Research Articles

Iterative parameter identification for Hammerstein systems with ARMA noises by using the filtering identification idea

SummaryIn practical applications, many processes have nonlinear characteristics that require nonlinear models for accurate description. However, constructing such models and determining their parameters are a challenging task. This article explores filtered identification methods for estimating the parameters of a particular type of nonlinear Hammerstein systems with ARMA noise. An auxiliary model‐based least squares algorithm is developed for such systems based on the auxiliary model identification idea. A hierarchical least squares algorithm that utilizes the hierarchical identification principle is proposed to enhance computational efficiency. Additionally, a key term separation technique is employed to express the system output as a linear combination of parameters, allowing the system to be decomposed into smaller subsystems for more efficient estimation of parameters. Simulation results demonstrate the effectiveness of these proposed algorithms.

Auxiliary model maximum likelihood gradient‐based iterative identification for feedback nonlinear systems

AbstractThis article considers the iterative identification problems for a class of feedback nonlinear systems with moving average noise. The model contains both the dynamic linear module and the static nonlinear module, which brings challenges to the identification. By utilizing the key term separation technique, the unknown parameters from both linear and nonlinear modules are included in a parameter vector. Furthermore, an auxiliary model maximum likelihood gradient‐based iterative algorithm is derived to estimate the unknown parameters. In addition, an auxiliary model maximum likelihood stochastic gradient algorithm is derived as a comparison. The numerical simulation results indicate that the auxiliary model maximum likelihood gradient‐based iterative algorithm can effectively estimate the parameters of the feedback nonlinear systems and get more accurate parameter estimates than the auxiliary model maximum likelihood stochastic gradient algorithm.

Maximum likelihood gradient‐based iterative estimation for closed‐loop Hammerstein nonlinear systems

AbstractThis article studies a new iterative method for a class of closed‐loop Hammerstein systems. The new iterative method solves the crossproducts between the parameters of the linear block and the nonlinear block by using the key term separation technique, decomposes a system into two subidentification models by utilizing the hierarchical identification principle for reduced computational complexity, and maximizes the maximum likelihood cost function by using the input and output data with a data window for improved parameter estimation accuracy. A numerical simulation example and a continuous stirred tank reactor experiment are presented to demonstrate that the proposed algorithm can work effectively.

Nonlinear Hammerstein System Identification: A Novel Application of Marine Predator Optimization Using the Key Term Separation Technique

The mathematical modelling and optimization of nonlinear problems arising in diversified engineering applications is an area of great interest. The Hammerstein structure is widely used in the modelling of various nonlinear processes found in a range of applications. This study investigates the parameter optimization of the nonlinear Hammerstein model using the abilities of the marine predator algorithm (MPA) and the key term separation technique. MPA is a population-based metaheuristic inspired by the behavior of predators for catching prey, and utilizes Brownian/Levy movement for predicting the optimal interaction between predator and prey. A detailed analysis of MPA is conducted to verify the accurate and robust behavior of the optimization scheme for nonlinear Hammerstein model identification.

Novel Fractional Swarming with Key Term Separation for Input Nonlinear Control Autoregressive Systems

In recent decades, fractional order calculus has become an important mathematical tool for effectively solving complex problems through better modeling with the introduction of fractional differential/integral operators; fractional order swarming heuristics are also introduced and applied for better performance in different optimization tasks. This study investigates the nonlinear system identification problem of the input nonlinear control autoregressive (IN-CAR) model through the novel implementation of fractional order particle swarm optimization (FO-PSO) heuristics; further, the key term separation technique (KTST) is introduced in the FO-PSO to solve the over-parameterization issue involved in the parameter estimation of the IN-CAR model. The proposed KTST-based FO-PSO, i.e., KTST-FOPSO accurately estimates the parameters of an unknown IN-CAR system with robust performance in cases of different noise scenarios. The performance of the KTST-FOPSO is investigated exhaustively for different fractional orders as well as in comparison with the standard counterpart. The results of statistical indices through Monte Carlo simulations endorse the reliability and stability of the KTST-FOPSO for IN-CAR identification.

Data filtering-based multi-innovation forgetting gradient algorithms for input nonlinear FIR-MA systems with piecewise-linear characteristics

The piecewise-linear characteristics often appear in the nonlinear systems that operate in different ways in different input regions. This paper studies the identification issue of a class of block-oriented systems with piecewise-linear characteristics. The asymmetric piecewise-linear nonlinearity is expressed as a linear parametric representation through introducing an appropriate switching function, then the identification model of the system is derived by using the key term separation technique. On this model basis, a multi-innovation forgetting gradient algorithm is presented to estimate the unknown parameters. To further enhance the identification accuracy, the filtering identification model of the system is derived by changing the structure of the system without changing the relationship between the input and output. Further, a data filtering-based multi-innovation forgetting gradient algorithm is proposed through the use of the data filtering technique. A simulation example is employed to illustrate that the proposed approaches are effective for parameter estimation and the data filtering-based multi-innovation forgetting gradient algorithm has better estimation performance.

Hierarchical recursive least squares algorithms for Hammerstein nonlinear autoregressive output‐error systems

SummaryThis article considers the parameter estimation problem of Hammerstein nonlinear autoregressive output‐error systems with autoregressive moving average noises. Applying the key term separation technique, the original system is decomposed into three subsystems: the first subsystem contains the unknown parameters related to the output, the second subsystem contains the unknown parameters related to the input, and the third subsystem contains the unknown parameters related to the noise model. A hierarchical recursive least squares algorithm is proposed based on the hierarchical identification principle for interactively identifying each subsystem. The simulation results confirm that the proposed algorithm is effective in estimating the parameters of Hammerstein nonlinear autoregressive output‐error systems.

Three‐stage forgetting factor stochastic gradient parameter estimation methods for a class of nonlinear systems

AbstractThis article focuses on the parameter estimation for a class of nonlinear systems, that is, multi‐input single‐output or two‐input single‐output Hammerstein finite impulse response systems with autoregressive moving average noise. The key is to investigate new estimation methods for on‐line parameter estimation of the considered system. By using the gradient search and introducing the forgetting factor, the forgetting factor stochastic gradient estimation method is developed. For the purpose of improving the parameter estimation accuracy, the system is decomposed into three subsystems with fewer variables applying the key term separation technique: the first two subsystems contain the unknown parameters related to the input and the third subsystem contains the unknown parameters related to the noise. Then a three‐stage forgetting factor stochastic gradient algorithm is proposed based on the hierarchical identification principle for interactively identifying each subsystem. The simulation results show the effectiveness of the presented algorithm.

Maximum likelihood‐based gradient estimation for multivariable nonlinear systems using the multiinnovation identification theory

SummaryThis article considers the identification problems of multivariable input nonlinear systems with unmeasured disturbances. For the identification difficulty caused by the crossproducts between the parameters of the linear block and the nonlinear block, the key term separation technique is adopted to separate the parameters of the nonlinear block from the parameters of the linear block. By combining the model decomposition technique and the hierarchical identification principle, a key term separation‐based maximum likelihood recursive extended stochastic gradient algorithm with reduced computational complexity is presented to estimate all the parameters directly. By introducing the multiinnovation identification theory, a key term separation‐based maximum likelihood multiinnovation extended stochastic gradient algorithm is proposed to improve the parameter estimation accuracy. The simulation results illustrate the effectiveness of the proposed methods.

Maximum Likelihood Iterative Algorithm for Hammerstein Systems with Hard Nonlinearities

In this paper, we consider several iterative algorithms for Hammerstein systems with hard nonlinearities. The Hammerstein system is first simplified as a polynomial identification model through the key term separation technique, and then the parameters are estimated by using the maximum likelihood (ML) based gradient-based iterative algorithm. Furthermore, an ML least squares auxiliary variable algorithm and an ML bias compensation gradient-based iterative algorithm are developed to identify the saturation system with colored noise. Simulation results are included to illustrate the effectiveness of the proposed algorithms.

Improved least‐squares identification for multiple‐output non‐linear stochastic systems

This study considers the identification problems of multiple-output non-linear equation-error moving average systems. There exist the product items of the parameters between the non-linear and linear parts. To solve this difficulty, the key term separation technique is adopted. By using the model decomposition technique and the hierarchical identification principle, a maximum likelihood-based recursive extended least-squares estimation algorithm with reduced computational complexity is presented to estimate the parameters of the non-linear part and the linear part interactively. The simulation results demonstrate the effectiveness of the proposed method.

A hierarchical least squares identification algorithm for Hammerstein nonlinear systems using the key term separation

Mathematical models are basic for designing controller and system identification is the theory and methods for establishing the mathematical models of practical systems. This paper considers the parameter identification for Hammerstein controlled autoregressive systems. Using the key term separation technique to express the system output as a linear combination of the system parameters, the system is decomposed into several subsystems with fewer variables, and then a hierarchical least squares (HLS) algorithm is developed for estimating all parameters involving in the subsystems. The HLS algorithm requires less computation than the recursive least squares algorithm. The computational efficiency comparison and simulation results both confirm the effectiveness of the proposed algorithms.

Data filtering based forgetting factor stochastic gradient algorithm for Hammerstein systems with saturation and preload nonlinearities

This paper considers the parameter estimation problem for Hammerstein systems with saturation and preload nonlinearities. Based on the key term separation technique, the output of the system is expressed as a linear combination of all the system parameters. By introducing the forgetting factors and using the data filtering technique, a data filtering based forgetting factor stochastic gradient (F-FF-SG) algorithm is presented. The simulation examples illustrate that the F-FF-SG algorithm has faster convergence rates and better parameter estimation accuracies than the stochastic gradient algorithm and the data filtering based stochastic gradient algorithm.

Combined state and multi‐innovation parameter estimation for an input non‐linear state‐space system using the key term separation

In this study, the authors study the state and parameter estimation problem for an input non-linear system consisting of a static non-linear block and a linear time-invariant state space subsystem. Using the filtering technique, a filtering based multi-innovation generalised stochastic gradient (SG) algorithm is proposed for avoiding estimating the redundant parameters based on the key term separation technique. Compared with the multi-innovation generalised SG algorithm, the proposed algorithm has higher parameter estimation accuracy. Two simulation examples are provided to show that the proposed algorithm works well.

Filtering-Based Multistage Recursive Identification Algorithm for an Input Nonlinear Output-Error Autoregressive System by Using the Key Term Separation Technique

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.

Least Squares Identification for Hammerstein Multi-input Multi-output Systems Based on the Key-Term Separation Technique

System modeling and parameter estimation are basic for system analysis and controller design. This paper considers the parameter identification problem of a Hammerstein multi-input multi-output (H-MIMO) system. In order to avoid the product terms in the identification model, we derive a pseudo-linear identification model of the H-MIMO system through separating a key term from the output equation of the system and present a hierarchical generalized least squares (LS) algorithm for estimating the parameters of the system. Moreover, we present a new LS algorithm to reduce the computational burden. The proposed algorithms are simple in principle and can achieve a higher computational efficiency than the over-parameterization-based LS estimation algorithm. Finally, we test the proposed algorithms by the simulation example and show their effectiveness.

Decomposition-based least squares parameter estimation algorithm for input nonlinear systems using the key term separation technique

A decomposition-based recursive least squares algorithm is developed for estimating the parameters of the input nonlinear systems composed of a dynamic controlled autoregressive block following a static nonlinear function with the known basis. The basic idea is that introducing the key term separation technique to parameterize the input nonlinear system and employing the hierarchical identification principle to decompose the parameterized system into two fictitious subsystems whose parameter vectors are recursively estimated using the recursive least squares methods. The proposed algorithm avoids estimating the redundant parameters compared with the over-parameterization identification method and improves computational efficiency. The simulation results confirm the effectiveness of the proposed algorithm.

Parameter estimation of piecewise Hammerstein systems

The system is often described as a series of blocks linked together in non-linear system identification. Such block-oriented models are built with static non-linear subsystems and linear dynamic systems. This paper deals with the parameter estimation of Hammerstein systems with piecewise non-linearities, which is a blocked-oriented model where a static non-linear blocking is followed by a linear dynamic system. The basic idea is as follows. The key term separation technique is applied initially, and then a corresponding auxiliary model is constructed. Hence, the identification problem of the system is converted to a non-linear function optimization problem over parameter space. Once again, the estimates of all the parameters are obtained by a proposed particle swarm optimization algorithm. Finally, compared with the existing methods, the simulation results confirm that the presented method is valid. Moreover, the presented method is further extended to estimate Hammerstein systems with discontinuity non-linearities.

Identification of Hammerstein systems using key‐term separation principle, auxiliary model and improved particle swarm optimisation algorithm

The dynamic behaviour of many systems can be approximated by a static non-linearity in series with a linear dynamic part. Systems with static input or output non-linearities are very common in many engineering applications. Such models are known as block-oriented models in the existing literature. The Hammerstein model is a special kind of block-oriented model, where a non-linear block is followed by a linear system. This study investigates the identification of Hammerstein systems with asymmetric two-segment piecewise-linear non-linearities. The basic idea is to employ a key-term separation technique and a corresponding auxiliary model initially. Then, the identification problem of non-linear system is changed into a non-linear function optimisation problem over parameter space. Further, the estimates of all the parameters of the non-linear block, the linear subsystem and the noise part are obtained based on an improved particle swarm optimisation algorithm. Finally, simulation examples are included to demonstrate the effectiveness and robustness of the proposed identification scheme.

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.