Hierarchical Identification Principle Research Articles

Highly efficient maximum-likelihood identification methods for bilinear systems with colored noises

This paper mainly discussed the highly efficient iterative identification methods for bilinear systems with autoregressive moving average noise. Firstly, the input-output representation of the bilinear systems is derived through eliminating the unknown state variables in the model. Then based on the maximum-likelihood principle, a maximum-likelihood gradient-based iterative (ML-GI) algorithm is proposed to identify the parameters of the bilinear systems with colored noises. For improving the computational efficiency, the original identification model is divided into three sub-identification models with smaller dimensions and fewer parameters, and a hierarchical maximum-likelihood gradient-based iterative (H-ML-GI) algorithm is derived by using the hierarchical identification principle. A gradient-based iterative (GI) algorithm is given for comparison. Finally, the algorithms are verified by a simulation example and a practical continuous stirred tank reactor (CSTR) example. The results show that the proposed algorithms are effective for identifying bilinear systems with colored noises and the H-ML-GI algorithm has a higher computational efficiency and a faster convergence rate than the ML-GI algorithm and the GI algorithm.

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.

Highly-efficient filtered hierarchical identification algorithms for multiple-input multiple-output systems with colored noises

Multiple-input multiple-output (MIMO) systems have extensive applications in industrial processes and systems engineering. This letter applies the filtering identification idea to establish a filtered identification model and investigate a filtered auxiliary model-based recursive least squares (F-AM-RLS) algorithm for parameter identification of MIMO systems with colored noises. To improve the computational efficiency, this work further proposes a four-stage filtered auxiliary model-based recursive least squares (4S-F-AM-RLS) algorithm by means of the hierarchical identification principle. Then, by incorporating the forgetting factor, a four-stage filtered auxiliary model-based forgetting factor recursive least squares (4S-F-AM-FF-RLS) algorithm is given to improve the convergence speed and the estimation accuracy. Additionally, the computational complexity analysis of the proposed algorithms indicates that the 4S-F-AM-RLS algorithm effectively reduces the computational burden and improves computational efficiency. Finally, the effectiveness of the F-AM-RLS, 4S-F-AM-RLS and 4S-F-AM-FF-RLS algorithms is validated through a numerical example.

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

This paper presents a hierarchical identification model for multiple-input multiple-output (MIMO) systems. An auxiliary model-based hierarchical stochastic gradient (AM-HSG) algorithm is derived by means of the auxiliary model identification idea and the hierarchical identification principle. Furthermore, an auxiliary model-based hierarchical multi-innovation stochastic gradient (AM-HMISG) algorithm is derived by utilizing the multi-innovation identification theory. In order to compare the computational efficiency of the AM-HSG and AM-HMISG algorithms for identifying MIMO systems, this paper gives the existing traditional gradient algorithms and discusses the complexity of these algorithms in detail. Finally, the simulation example tests the effectiveness of all four algorithms.

Research on partition control of direct air‐cooling condenser based on multivariable Hammerstein CARMA model predictive control

AbstractPartition control of the fan array within a direct air‐cooling condenser (DACC) is recognized as a highly effective energy‐saving measure and a means to mitigate the adverse effects of ambient winds. This study focuses on the development of a dynamic partition model, which establishes the relationship between mass flow rate and fan array speed under windy conditions. The model is grounded in multivariable Hammerstein controlled autoregressive moving average (H‐CARMA) systems. Additionally, an innovative extended stochastic gradient algorithm, founded on the hierarchical identification principle, is introduced to estimate the model's unknown parameters. The recursive least square is employed as a benchmark to verify the accuracy and efficiency of the approach. Subsequently, leveraging the identified model, the model predictive controller based on the differential evolution method is designed to derive the optimal control law. Finally, simulation test and analysis are carried out to testify the effectiveness of our control strategy. The experiment result shows that the dynamic characteristics of the system have been significantly improved and the fan power consumption is reduced by 3.5% compared to the PID controller.

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.

Decomposition and composition modeling algorithms for control systems with colored noises

SummaryThis article proposes a novel identification framework for estimating the parameters of the controlled autoregressive autoregressive moving average (CARARMA) models with colored noise. By means of building an auxiliary model and using the hierarchical identification principle, this article investigates a highly‐efficient parameter estimation algorithm. In order to meet the need for identifying the systems with large‐scale parameters, the whole parameters of the CARARMA system is separated into two parameter sets and a decomposition and composition recursive identification algorithm (i.e., hierarchical generalized extended least squares algorithm or decomposition‐based recursive generalized extended least squares algorithm) is presented. Moreover, this article analyzes the convergence of the proposed decomposition and composition recursive identification algorithm. The performance analysis shows that the proposed decomposition and composition identification algorithm can reduce the complexity of the identification algorithm compared with the algorithm without decomposition.

Hierarchical gradient‐ and least‐squares‐based iterative estimation algorithms for input‐nonlinear output‐error systems from measurement information by using the over‐parameterization

AbstractThis article investigates the parameter identification problems of the stochastic systems described by the input‐nonlinear output‐error (IN‐OE) model. This IN‐OE model consists of two submodels, one is an input nonlinear model and the other is a linear output‐error model. The difficulty in the parameter identification of the IN‐OE model is that the information vector contains the unknown variables, which are the noise‐free (true) outputs of the system, the approach taken here is to replace the unknown terms with the outputs of the auxiliary model. Based on the over‐parameterization model and the hierarchical identification principle, an over‐parameterization auxiliary model hierarchical gradient‐based iterative algorithm and an over‐parameterization auxiliary model hierarchical least‐squares‐based iterative algorithm are proposed to estimate the unknown parameters of the IN‐OE systems. Finally, two numerical simulation examples are given to demonstrate the effectiveness of the proposed algorithms.

Two‐stage and three‐stage recursive gradient identification of Hammerstein nonlinear systems based on the key term separation

AbstractThis article explores recursive algorithms for parameter identification issues of Hammerstein output‐error systems. The proposed approach includes the key term separation auxiliary model recursive gradient algorithm, which utilizes the gradient search and the key term separation. To enhance computational efficiency, the system is decomposed into two or three subsystems through the hierarchical identification principle. Based on this, a key term separation based auxiliary model two‐stage recursive gradient algorithm and a key term separation based auxiliary model three‐stage recursive gradient algorithm are presented. The simulation results verify the validity of the obtained algorithms.

Hierarchical recursive least squares parameter estimation methods for multiple‐input multiple‐output systems by using the auxiliary models

SummaryMultiple‐input multiple‐output (MIMO) models are widely used in practical engineering. This article derives a new identification model of the MIMO system by decomposing the MIMO system into several multiple‐input single‐output subsystems. By means of the auxiliary model identification idea, an auxiliary model‐based recursive least squares (AM‐RLS) algorithm is derived for identifying the MIMO systems. In order to reduce the computational burden for identifying MIMO systems, this article presents a hierarchical identification model for the MIMO systems. By applying the hierarchical identification principle, an auxiliary model‐based hierarchical least squares (AM‐HLS) algorithm is proposed for improving the computational efficiency. The computational efficiency analysis indicates that the AM‐HLS algorithm is effective in reducing the calculation amount compared with the AM‐RLS algorithm. Moreover, this article analyzes the convergence of the AM‐HLS algorithm. The simulation example shows that the AM‐RLS and AM‐HLS algorithms studied in this article are effective.

Highly‐computational hierarchical iterative identification methods for multiple‐input multiple‐output systems by using the auxiliary models

AbstractThe identification of multiple‐input multiple‐output (MIMO) systems is an important part of designing complex control systems. This article studies an auxiliary model least squares iterative (AM‐LSI) algorithm for MIMO systems. With the expansion of the system scale and limitations of the computer resources, there is an urgent need for an identification algorithm that provides higher computational efficiency. To address this issue, this article further derives a hierarchical identification model and proposes a new auxiliary model hierarchical least squares iterative (AM‐HLSI) algorithm for MIMO systems by applying the hierarchical identification principle. Through the analysis of the computational efficiency, the AM‐HLSI algorithm has higher computational efficiency than the AM‐LSI algorithm. Additionally, the feasibility of the AM‐LSI and AM‐HLSI algorithms is validated by a simulation example.

Parameter estimation of fractional‐order Hammerstein state space system based on the extended Kalman filter

SummaryThis paper addresses the combined estimation issues of the parameters and states for fractional‐order Hammerstein state space systems with colored noises. An extended state estimator is derived by using the parameter estimates to replace the unknown system parameters in Kalman filter. The hierarchical identification principle is introduced to solve the unknown parameters of measurement noises. By introducing the forgetting factor, an extended Kalman filtering‐based hierarchical forgetting factor stochastic gradient algorithm is presented to estimate the unknown states, parameters and fractional‐order. A numerical example is respectively presented to demonstrate the feasibility of the proposed identification algorithm. It can be seen that the estimation errors are relatively small, which reflects the proposed algorithms have good identification effect.

Data filtering‐based parameter estimation algorithms for a class of nonlinear systems with colored noises

AbstractThis article studies the data filtering‐based identification algorithms for a class of nonlinear system with autoregressive noise. By means of the data filtering technique and the hierarchical identification principle, the identification model is transformed into two sub‐identification models, and a filtering hierarchical gradient‐based iterative algorithm is proposed for improving parameter estimation accuracy and reducing computational burden. Meanwhile, to further improve the identification performance, the multi‐innovation identification theory is used to derived the filtering hierarchical multi‐innovation gradient‐based iterative algorithm. The gradient‐based iterative algorithm is given for comparison. The analysis shows that the filtering hierarchical gradient‐based iterative algorithm has smaller computational burden and can give more accurate parameter estimates than the gradient‐based iterative algorithm, and the filtering hierarchical multi‐innovation gradient‐based iterative algorithm can track time‐varying parameters based on the dynamical window data. Finally, the example part is provided to verify the effectiveness of the proposed algorithms.

Least squares parameter estimation and multi-innovation least squares methods for linear fitting problems from noisy data

Least squares is an important method for solving linear fitting problems and quadratic optimization problems. This paper explores the properties of the least squares methods and the multi-innovation least squares methods. It demonstrates lemmas and theorems about the least squares and multi-innovation least squares parameter estimation algorithms after reviewing and surveying some important contributions in the area of system identification, such as the auxiliary model identification idea, the multi-innovation identification theory, the hierarchical identification principle, the coupling identification concept and the filtering identification idea. The results of the least squares and multi-innovation least squares algorithms for linear regressive systems with white noises can be extended to other systems with colored noises.

A NOVEL ITERATIVE METHOD FOR SOLVING THE COUPLED SYLVESTER-CONJUGATE MATRIX EQUATIONS AND ITS APPLICATION IN ANTILINEAR SYSTEM

This paper is devoted to constructing a modified relaxed gradient based iterative (MRGI) algorithm to solve the coupled Sylvester-conjugate matrix equations (CSCMEs) based on the hierarchical identification principle. Convergence analysis shows that the proposed algorithm is effective for arbitrary initial matrices. Further, we apply the MRGI algorithm to study a more general coupled Sylvester conjugate matrix equations and give a sufficient condition to guarantee that the iterative solution converges to the exact solution. Two numerical experiments are provided to demonstrate that the MRGI algorithm has better efficiency and accuracy than the three existing algorithms, which are presented by Wu et al. (2010) and Huang and Ma (2018). Finally, we derive an application of MRGI algorithm in discrete-time antilinear system.

An iterative method based on ADMM for solving generalized Sylvester matrix equations

As is well known, the alternating direction method of multipliers (ADMM) is one of the most famous distributed algorithms for solving the convex optimization problems. In this paper, by applying the hierarchical identification principle, we propose an iterative algorithm based on matrix form of ADMM for finding the least squares solution of the generalized Sylvester matrix equation AXB+CXD=E. We prove that the proposed method is convergent under some common conditions for any initial matrices. The simplicity and effectiveness of the new method are shown by the reported numerical results.

Modeling nonlinear systems using the tensor network B‐spline and the multi‐innovation identification theory

AbstractThe nonlinear autoregressive exogenous (NARX) model shows a strong expression capacity for nonlinear systems since these systems have limited information about their structures. However, it is difficult to model the NARX system with a relatively high dimension by using the popular polynomial NARX and the neural network efficiently. This article uses the tensor network B‐spline (TNBS) to model the NARX system, whose representation of the multivariate B‐spline weight tensor can alleviate the computation and store burden for processing high‐dimensional systems. Furthermore, applying the multi‐innovation identification theory and the hierarchical identification principle, the recursive algorithm by combining the ‐norm is proposed to the NARX system with Gaussian noise. Because of the local adjustability of the B‐spline curve, the TNBS can fit nonlinear systems with strong nonlinearity by the meaning of setting a proper degree and knots number. Finally, a numerical experiment is given to demonstrate the effectiveness of the proposed algorithm.

Auxiliary model‐based recursive least squares algorithm for two‐input single‐output Hammerstein output‐error moving average systems by using the hierarchical identification principle

AbstractThis article considers the parameter estimation problems of two‐input single‐output Hammerstein output‐error moving average systems. The system is decomposed into two subsystems based on the hierarchical principle. The first model is used to identify the linear parameters and the parameters of the unknown measurable information vector. The second model is for identifying non‐linear parameters. By using the auxiliary model, we introduce a forgetting factor to improve the parameter estimation accuracy. The auxiliary model‐based forgetting factor recursive least squares algorithm and the auxiliary model‐based forgetting factor multi‐innovation recursive least squares algorithm are presented. The simulation results indicate that the proposed algorithms are effective.

Two‐stage gradient‐based iterative algorithms for the fractional‐order nonlinear systems by using the hierarchical identification principle

SummaryThis article focuses on the parameter estimation issues for a fractional‐order nonlinear system with autoregressive noise. In the process, the challenge and difficulty are to identify the parameters of the system as well as the order. To reduce the complexity of the structure, we split the system into two subsystems by utilizing the hierarchical identification principle and derive a two‐stage gradient‐based iterative (2S‐GI) algorithm by minimizing two criterion functions. Compared with the calculation amount of the gradient‐based iterative algorithm, the computation of the 2S‐GI algorithm is significantly reduced. Moreover, in order to improve the identification accuracy, we propose a two‐stage moving‐data‐window gradient‐based iterative algorithm. Finally, the simulation examples test the effectiveness of the proposed algorithms.

Parameter Estimation Algorithms for Hammerstein Finite Impulse Response Moving Average Systems Using the Data Filtering Theory

This paper considers the parameter estimation problems of Hammerstein finite impulse response moving average (FIR–MA) systems. Based on the matrix transformation and the hierarchical identification principle, the Hammerstein FIR–MA system is recast into two models, and a decomposition-based recursive least-squares algorithm is deduced for estimating the parameters of these two models. In order to further improve the accuracy of the parameter estimation, a multi-innovation hierarchical least-squares algorithm based on the data filtering theory proposed. Finally, a simulation example demonstrates the effectiveness of the proposed scheme.