Iterative identification methods for a class of bilinear systems by using the particle filtering technique

This article mainly studies the iterative parameter estimation problems of a class of nonlinear systems. Based on the auxiliary model identification idea, this article utilizes the estimated parameters to construct an auxiliary model, and uses its outputs to replace the unknown noise-free process outputs, and develops an auxiliary model least squares-based iterative (AM-LSI) identification algorithm. For further improving the parameter estimation accuracy, we use a particle filter to estimate the unknown noise-free process outputs, and derive a particle filtering least squares-based iterative (PF-LSI) identification algorithm. During each iteration, the AM-LSI and PF-LSI algorithms can make full use of the measured input–output data. The simulation results indicate that the proposed algorithms are effective for identifying the nonlinear systems, and can generate more accurate parameter estimates than the auxiliary model-based recursive least squares algorithm.     

Maximum likelihood hierarchical least squares-based iterative identification for dual-rate stochastic systems

For a dual-rate sampled-data stochastic system with additive colored noise, a dual-rate identification model is obtained by using the polynomial transformation technique, which is suitable for the available dual-rate measurement data. Based on the obtained model, a maximum likelihood least squares-based iterative (ML-LSI) algorithm is presented for identifying the parameters of the dual-rate sampled-data stochastic system. In order to improve the computation efficiency of the algorithm, the identification model of a dual-rate sampled-data stochastic system is divided into two subidentification models with smaller dimensions and fewer parameters, and a maximum likelihood hierarchical least squares-based iterative (H-ML-LSI) algorithm is proposed for these subidentification models by using the hierarchical identification principle. The simulation results indicate that the proposed algorithms are effective for identifying dual-rate sampled-data stochastic systems and the H-ML-LSI algorithm has a higher computation efficiency than the ML-LSI algorithm.     

The filtering‐based maximum likelihood iterative estimation algorithms for a special class of nonlinear systems with autoregressive moving average noise using the hierarchical identification principle

For 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.     

State estimation for bilinear systems through minimizing the covariance matrix of the state estimation errors

This paper considers the state estimation problem of bilinear systems in the presence of disturbances. The standard Kalman filter is recognized as the best state estimator for linear systems, but it is not applicable for bilinear systems. It is well known that the extended Kalman filter (EKF) is proposed based on the Taylor expansion to linearize the nonlinear model. In this paper, we show that the EKF method is not suitable for bilinear systems because the linearization method for bilinear systems cannot describe the behavior of the considered system. Therefore, this paper proposes a state filtering method for the single‐input–single‐output bilinear systems by minimizing the covariance matrix of the state estimation errors. Moreover, the state estimation algorithm is extended to multiple‐input–multiple‐output bilinear systems. The performance analysis indicates that the state estimates can track the true states. Finally, the numerical examples illustrate the specific performance of the proposed method.     

Combined state and parameter estimation for Hammerstein systems with time delay using the Kalman filtering

This paper discusses the state and parameter estimation problem for a class of Hammerstein state space systems with time delay. Both the process and the measurement noises are considered in the system. On the basis of the observable canonical state space form and the key term separation, a pseudolinear regressive identification model is obtained. For the unknown states in the information vector, the Kalman filter is used to search for the optimal state estimates. A Kalman filter–based least squares iterative and a recursive least squares algorithms are proposed. Extending the information vector to include the latest information terms, which are missed for the time delay, the Kalman filter–based recursive extended least squares algorithm is derived to obtain the estimates of the unknown time delay, parameters, and states. The numerical simulation results are given to illustrate the effectiveness of the proposed algorithms.     

Estimation algorithm for system with non‐Gaussian multiplicative/additive noises based on variational Bayesian inference

This paper considers estimation algorithms for linear and nonlinear systems contaminated by non‐Gaussian multiplicative and additive noises. Based on the variational idea, in order to derive optimal estimation algorithms, we combine the multiplicative noise with states as the joint parameters to estimate. The application of variational Bayesian inference to joint estimation of the state and the multiplicative noise is established. By treating the states as unknown quantities as well as the multiplicative noise, there are now correlations between the states and multiplicative noise in the posterior distribution. There are two main goals in Bayesian learning. The first is approximating the marginal likelihood (PDF of multiplicative noise) to perform model comparison. The second is approximating the posterior distribution over the states (also called a system model), which can then be used for prediction. The two goals constitute the iterative algorithm. The rules for determining the loop is the Kullback‐Leibler divergence between the true distribution of state and a chosen fixed tractable distribution, which is used to approximate the true one. The iterative algorithm is deduced, which is initialized based on the idea of sampling. Meanwhile, the convergence analysis of the proposed iterative algorithm is presented. The numerical simulation results in a comparison between the proposed method and these existing classic algorithms in the context of nonlinear hidden Markov models, state‐space models, and target‐tracking models with non‐Gaussian multiplicative noise demonstrate the superiorities, not only in speed, precision, and computation load but also in the ability to process non‐Gaussian complex noise.     

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

This 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.     

Optimal model‐free control for a generic MIMO nonlinear system with application to autonomous mobile robots

In this paper, the design procedure for optimal model‐free control algorithm is presented for the tracking problem of completely unknown nonlinear dynamic systems operating under unknown disturbances. The procedure includes a new structure in the context of model‐free control and data‐driven control algorithms. In the new structure, the unknown nonlinear functions are segmented into 1 unknown linear‐in‐states part and another unknown nonlinear part. The adaptive laws proposed for estimating the unknown system dynamics are regressor‐free estimation methods in which there is no need for regressor parameters and, consequently, the persistent excitation condition is not required anymore. Moreover, the main controller gains are updated online, incorporating the adapted values of linear terms in the system dynamics. A comparative study is presented to show that the proposed optimal model‐free control outperforms the state‐of‐the‐art model‐free control algorithms. In addition, the simulation results for the application of the algorithm on autonomous mobile robots are provided.     

Iterative parameter and order identification for fractional-order nonlinear finite impulse response systems using the key term separation

This article considers the parameter estimation for a fractional-order nonlinear finite impulse response system with colored noise. For the fractional-order systems, the challenge and difficulty are to identify the order and parameters of the systems simultaneously under colored noise disturbances. In order to reduce the problem of redundant parameter estimation, the output form of the system can be expressed by a linear combination of unknown parameters through the separation of the key term separation. A key term separation auxiliary model gradient-based iterative algorithm is derived by using the negative gradient search. Meanwhile, to achieve the higher estimation accuracy, we propose a key term separation auxiliary model multiinnovation gradient-based iterative algorithm by utilizing the multiinnovation theory. Finally, the simulation results test the effectiveness of the proposed algorithms.     

Parameter estimation for time-delay systems based on the frequency responses and harmonic balance methods

Two parameter estimation methods for linear time-delay systems are proposed based on the frequency responses and the harmonic balance methods. One is the stochastic gradient gradient-based iterative (SG-GI) algorithm and the other is the recursive least squares gradient-based iterative (RLS-GI) algorithm. These two methods can estimate the unknown parameters and the unknown time delays simultaneously by giving sinusoidal signals with different angular frequencies. Furthermore, a comparative study reveals that the RLS-GI algorithm is more effective than the SG-GI algorithm. The effectiveness of the proposed algorithms is illustrated by a numerical example.     

Combined estimation of the parameters and states for a multivariable state-space system in presence of colored noise

This article addresses the combined estimation issues of parameters and states for multivariable systems in the state-space form disturbed by colored noises. By utilizing the Kalman filtering principle and the coupling identification concept, we derive a Kalman filtering based partially coupled recursive generalized extended least squares (KF-PC-RGELS) algorithm to jointly estimate the parameters and the states. Using the past and the current data in parameter estimation, we propose a Kalman filtering based multi-innovation partially coupled recursive generalized extended least-squares algorithm to enhance the parameter estimation accuracy of the KF-PC-RGELS algorithm. Finally, a simulation example is provided to test and compare the performance of the proposed algorithms.     

Algorithms for state bounding in large-scale systems

A class of linear and time-invariant large-scale systems under unknown input signals and measurement errors is considered. No assumptions about statistical properties of the unknown quantities are made. the uncertainty is modelled through bounds that define ellipsoids in which lie unknown initial conditions and unknown ellipsoidal tubes containing signal trajectories. the state estimation problem consists of determining on-line the smallest sets in the state space in which the unknown system state lies. Two hierarchical estimation algorithms are proposed and compared, namely the ‘completely decentralized with interaction measurements’ (CDIM) algorithm and the two-level hierarchical with interaction measurements (THIM) algorithm. the THIM algorithm leads to smaller estimating sets owing to additional co-ordinator-level information.     

Filtering-based recursive least squares estimation approaches for multivariate equation-error systems by using the multiinnovation theory

This article researches the filtering-based parameter estimation issues for a class of multivariate control systems with colored noise. A filtering-based recursive generalized extended least squares algorithm is derived, in which the data filtering technique is used for transforming the original system into two subidentification systems and the least squares principle is used for estimating parameters of these two subsystems. Furthermore, in order to improve the parameter estimation accuracy, the multiinnovation theory is added for deducing a filtering-based multiinnovation recursive generalized extended least squares algorithm. The numerical example confirms that these two proposed algorithms are effective.     

Separable multi-innovation stochastic gradient estimation algorithm for the nonlinear dynamic responses of systems

This article is concerned with the parameter identification problem of nonlinear dynamic responses for the linear time-invariant system by means of an impulse excitation signal and discrete observation data. Using the impulse signal as the input, the impulse response experiment is carried out and the dynamical moving sampling is designed to generate the measured data for deriving new identification algorithms. By applying the moving window data that contain the dynamical information of the system to be identified, an objective function with respect to the parameters of the systems is constructed according to the impulse response. In accordance with different functional relations between the system parameters and the system output response, the unknown parameter vector of the system is separated into a linear parameter vector and a nonlinear parameter vector. Based on the separated parameter vectors, two subidentification models are constructed and a separable identification algorithm is presented through the gradient search to improve the accuracy. Moreover, for the purpose of enhancing the estimation accuracy and capturing the dynamical feature of the systems, the moving window data are employed to develop the separable identification algorithm. The performance of the proposed separable identification method is illustrated via a numerical example.     

Generalized State Estimation of Flexible AC Power Systems Considering Wind Generators and Primary Frequency Control

This paper proposes a practical approach to incorporate the mathematical models of both fixed-speed and variable-speed wind turbine generators, automatic load frequency controls as well as voltage magnitude and frequency dependent loads into a weighted least squares-based state estimation algorithm suitable for the analysis of flexible alternating current (AC) transmission systems. As opposed to conventional static state estimators, where the inclusion of these electric components has been neglected so far, the proposed approach permits the determination of the steady state operation of a power system in the event of a supply-demand unbalance by estimating the magnitude of the frequency deviation from its nominal value. The state estimation is based on measurements related to those that should be obtained by a supervisory control and data acquisition (SCADA) system and phasor measurement units. For the purpose of this paper, the set of values associated with SCADA measurements (nodal power injections, power flows, and voltage magnitudes) and phasor measurement unit (PMU) measurements (voltage and current phasors) are generated from a power flow analysis of the network under study. Lastly, numerical simulations are reported to demonstrate the effectiveness of the proposed approach.     

改进的零注入约束双线性WLAV状态估计

实际电网中存在许多注入功率严格为零的零注入节点,零注入节点的注入功率量测为绝对准确的量测量,但这些测量并没有得到充分利用。因此,通过注入功率为零的节点建立功率约束方程作为对状态估计的约束条件,再对极坐标系下的电力系统非线性量测方程进行两步线性化,得到计及零注入约束的双线性抗差状态估计方程,用约束方程对第一步线性过程结果进行修正。最终结果证明该算法在提高估计精度的基础上,不会增加系数矩阵的阶数,且改进后的算法仍然拥有较高的计算效率。国内某实际省网以及选取的IEEE标准系统的仿真结果证明了该方法能有效提高计算精度和计算效率。     

An adaptive observer for joint estimation of states and parameters in both state and output equations

An adaptive observer is a recursive algorithm for joint state–parameter estimation of parameterized state‐space systems. Previous works on globally convergent adaptive observers consider unknown parameters either in state equations or in output equations, but not in both of them. In this paper, a new adaptive observer is designed for linear time‐varying systems with unknown parameters in both state and output equations. Its global convergence for simultaneous estimation of states and parameters is formally established under appropriate assumptions. A numerical example is presented to illustrate the performance of this adaptive observer. Copyright © 2011 John Wiley & Sons, Ltd.     

Optimal worst case solutions for state estimation of power system with uncertain network parameters

This paper addresses the problem of power system state estimation under the condition that transmission line network parameters are unknown but bounded. A robust estimation in the sense of an optimal worst case solution is determined. Data collected via remote terminal units, i.e. voltage magnitude, power flow, and power injection, are used as measurement quantities. The state variables are bus voltage phasors expressed in rectangular coordinates. This makes it possible to express the relations between measured data and state variables as quadratic functions. The proposed formulation based on the structured robust least squares optimization yields a minimization problem with bilinear matrix inequality constraints. A solution method based on semidefinite programming is also presented. Some test results on the standard IEEE test systems are given. © 2016 Institute of Electrical Engineers of Japan. Published by John Wiley & Sons, Inc.     

A fast heuristic optimization algorithm for measurement placement in distribution system state estimation

An accurate, real‐time estimation of the states of a power distribution system is highly desirable but hard to achieve because of the complexity of the network and the relative inefficiency of the measuring system. To increase the efficiency, this paper analyzes the mathematical relationship between the measurement errors and estimation errors of the state vector using the classic weighted least square method. Then a heuristic algorithm is proposed to improve the accuracy by optimizing the deployment of the real‐time measuring points, which is based on the deterministic factors of the measuring points/branches over the system state. The basic implementation starts with an initial measurement set and replaces the least important branches in the set with the most important branches outside the set using iterative optimization. The algorithm was tested in the IEEE 14‐bus and 33‐bus distribution systems and achieved 50% increase in accuracy at much lower computational cost compared with exhaustive search. Moreover, the proposed algorithm has also been compared with representative and widely used evolution algorithms such as particle swarm optimization and quantum‐behaved particle swarm optimization. This comparison shows that our method can achieve stable and comparable accuracy with a speed at least 10 times higher. The performance of our method can be even better with increasing network size. © 2017 Institute of Electrical Engineers of Japan. Published by John Wiley & Sons, Inc.     

电力系统状态估计算法的综合分析

简要介绍了电力系统状态估计的基本概念及功能,描述了状态估计的数学模型。介绍了几种电力系统状态估计的基本算法,即加权最小二乘法、快速分解法、基于量测变换及逐次型的状态估计算法等,并对这些算法作了简明的对比,指出各个算法的优缺点。最后,为了满足电力系统状态估计的要求,又提出了几种新型的状态估计算法。并且指出了状态估计算法中值得研究的几个方面。