Optimal instrumental-variable methods for identification of multivariable linear systems

The identification of multivariable linear systems using instrumental-variable (IV) methods is discussed. The emphasis is on accuracy properties. The IV estimates of multi-input-multi-output system parameters are proved to be asymptotically Gaussian distributed under weak conditions. An explicit expression for the covariance matrix of the parameter estimates is given. It is then shown how this matrix can be optimized by appropriately choosing the IV variant. The optimal accuracy so obtained is for some model structures equal to that corresponding to a prediction error method.     

多变量系统的耦合梯度辨识算法与性能分析

针对多变量系统维数大、参数多、一般的辨识算法计算量大的问题, 基于耦合辨识概念, 推导多变量系统的耦合随机梯度算法, 利用鞅收敛定理分析算法的收敛性能. 算法的主要思想是将系统模型分解为多个单输出子系统,在子系统的递推辨识过程中, 将每个子系统的参数估计值耦合起来. 所提出算法与最小二乘算法和耦合最小二乘算法相比, 具有较少的计算量, 收敛速度可以通过引入遗忘因子得到改善. 性能分析表明了所提出算法收敛, 仿真实例验证了算法的有效性.

     

Parameter identification technique for multivariate stochastic systems

This paper considers the problem of obtaining accurate estimates of multivariate systems with reasonable computations. To avoid the structural identification problem which is associated with multivariate systems, we observe the system by a linear combination of the outputs. The two stage least square method is employed to estimate the model parameters. An optimum combination of the outputs is obtained such that the parameter estimates have the least asymptotic generalized variance. Computer simulations are provided to illustrate the usefulness of the proposed method.     

反馈非线性系统随机梯度辨识算法及其收敛性

反馈非线性受控自回归系统是由前向通道的受控自回归模型和反馈通道的静态非线性构成, 这类系统经过参数化后得到双线性参数辨识模型. 本文通过对辨识模型中双线性参数乘积项进行分解, 基于梯度搜索原理, 提 出了反馈非线性系统的随机梯度辨识算法. 为了改善随机梯度算法的收敛速度, 引入遗忘因子, 文章给出了遗忘因子随机梯度算法, 利用随机过程理论, 建立了随机梯度算法的参数估计收敛定理, 证明了算法的收敛性. 最后, 通过数值仿真验证了算法的有效性.     

Mean-square filter design for stochastic polynomial systems with Gaussian and Poisson noises

This paper addresses the mean-square finite-dimensional filtering problem for polynomial system states with both, Gaussian and Poisson, white noises over linear observations. A constructive procedure is established to design the mean-square filtering equations for system states described by polynomial equations of an arbitrary finite degree. An explicit closed form of the designed filter is obtained in case of a third-order polynomial system. The theoretical result is complemented with an illustrative example verifying performance of the designed filter.     

Identification of multi-input systems based on correlation techniques

This article proposes a correlation analysis-based identification method for multi-input single-output systems. The basic idea is to estimate the equivalent FIR model parameters with the orders increasing, and to compute the parameter estimates of the original systems (i.e. each fictitious subsystem) using the system inputs and the outputs of the estimated FIR models and using the least squares optimisation. Simulation results indicate that the proposed algorithm can work well.     

Self-tuning regulators for a class of multivariable systems

Control of a class of multivariable systems described by linear vector difference equations with constant but unknown parameters is discussed. A multivariable minimum variance strategy is first presented. This gives a generalization of the minimum variance strategy for single-input single-output systems. A multivariable self-tuning regulator based on the minimum variance strategy is then proposed. It uses a recursive least squares estimator and a linear controller obtained directly from the current estimates. The asymptotic properties of the algorithm are discussed. If the estimated parameters converge, the resulting controller will under certain conditions give the minimum variance strategy. The analysis also gives insight into the case when several single-input single-output self-tuning regulators are operating in cascade mode.     

Performance analysis of stochastic gradient algorithms under weak conditions

By using the stochastic martingale theory, convergence properties of stochastic gradient （SG） identification algorithms are studied under weak conditions. The analysis indicates that the parameter estimates by the SG algorithms consistently converge to the true parameters, as long as the information vector is persistently exciting （i.e., the data product moment matrix has a bounded condition number） and that the process noises are zero mean and uncorrelated. These results remove the strict assumptions, made in existing references, that the noise variances and high-order moments exist, and the processes are stationary and ergodic and the strong persis- tent excitation condition holds. This contribution greatly relaxes the convergence conditions of stochastic gradient algorithms. The simulation results with bounded and unbounded noise variances confirm the convergence conclusions proposed.     

耦合多新息随机梯度辨识方法性能分析

针对一类耦合参数多变量系统, 提出一种耦合多新息随机梯度方法. 通过该方法进行参数辨识并对该方法进行性能分析. 该方法的基本思路在于利用历史新息中包含的信息, 将耦合随机梯度算法中的新息项扩展为多新息向量, 从而提升耦合随机梯度算法中单个子系统的辨识效果. 仿真结果表明, 通过增加新息长度可以提升辨识结果的收敛速度和精度.

     

On identification and adaptive estimation for systems with interrupted observations

A linear discrete-time system with interrupted observations is considered. The interrupted observation mechanism is expressed in terms of a finite-state ergodic Markov chain. The transition probability matrix of the Markov chain and some system parameters may be unknown, but constant, and are assumed to belong to a compact set. A novel scheme, called truncated maximum likelihood estimation, is proposed for consistent estimation of the unknown parameters. Sufficient conditions for strong consistency are investigated. The truncated maximum likelihood procedure is computationally feasible whereas the standard maximum likelihood procedure is not, given large observation records. Finally, using the truncated ML algorithm, a suboptimal adaptive state estimator is proposed and its asymptotic behavior is analyzed.     

Subspace-based methods for the identification of linear time-invariant systems

Subspace-based methods for system identification have attracted much attention during the past few years. This interest is due to the ability of providing accurate state-space models for multivariable linear systems directly from input-output data. The methods have their origin in classical state-space realization theory as developed in the 1960s. The main computational tools are the QR and the singular-value decompositions. Here, an overview of existing subspace-based techniques for system identification is given. The methods are grouped into the classes of realization-based and direct techniques. Similarities between different algorithms are pointed out, and their applicability is commented upon. We also discuss some recent ideas for improving and extending the methods. A simulation example is included for comparing different algorithms. The subspace-based approach is found to perform competitive with respect to prediction-error methods, provided the system is properly excited.     

多变量输出误差系统的辅助模型随机梯度辨识算法

考虑了多变量输出误差系统的辨识问题. 使用系统可得到的输入输出数据构造一个辅助模型, 用辅助模型的输出代替信息向量中的未知变量, 提出了一个基于辅助模型的随机梯度辨识算法. 使用鞅收敛定理的收敛性分析表明: 提出的算法给出的参数估计收敛于它们的真值. 给出了带遗忘因子的辅助模型随机梯度算法来改进参数估计精度, 仿真结果证实了提出的结论.     

Invariants and canonical forms for systems structural and parametric identification

The basic definitions regarding invariant functions and canonical forms for an equivalence relation on a generic set are first recalled.With reference to observable state space models and to the equivalence relation induced by a change of basis it is then shown how the image of a complete set of independent invariants for the considered equivalence relation can be used to parametrize a subset of canonical forms in the given set.Then the set of polynomial input-output models of the type P(z)y(t)=Q(z)u(t) and the equivalence relation induced by the premultiplication of P and Q by a unimodular matrix are considered and canonical forms parametrized by a complete set of independent invariants introduced.Since the two sets of canonical forms share common sets of complete independent invariants, very simple algebraical links between state space and input-output canonical forms can be deduced.The previous results are used to design efficient algorithms solving the problem of the canonical structural and parametric realization and identification of generic input-output sequences generated by a linear, discrete, time-invariant multivariable system.The results obtained in the identification of a real process are then reported.     

Dedicated microprocessors for realtime identification of multivariable systems

In this paper, an algorithm is proposed for identifying multivariable systems in state-space form from noisy data, which is suitable for implementation on dedicated microprocessor systems. The proposed algorithm uses the normalized stochastic approximation criterion which reduces the computational complexity and memory requirements. It is shown that the overall performance of the proposed stochastic approximation algorithm when using a dedicated microprocessor with fixed point arithmetic is superior to the extended least-squares method in terms of memory requirements, execution speed per iteration, and the estimation results.     

具有乘性噪声的线性离散时间随机控制系统综述

具有乘性噪声的随机不确定系统的控制问题有着广泛的应用背景. 本文概述了具有乘性噪声的线性离散时间随机系统的稳定性分析、均方镇定、最优控制以及最优估计问题和相关结论. 同时, 本文研究了具有状态与控制乘性噪声的线性多变量离散时间系统的均方镇定和最优控制问题, 分析了这两个问题之间的联系, 并讨论了最优状态反馈控制器的设计算法.     

Two-stage multi-innovation stochastic gradient algorithm for multivariate output-error ARMA systems based on the auxiliary model

ABSTRACT

This paper investigates the parameter estimation problem for multivariate output-error systems perturbed by autoregressive moving average noises. Since the identification model has two different kinds of parameters, a vector and a matrix, the gradient algorithm cannot be used directly. Therefore, we decompose the original system model into two sub-models and proceed the identification problem by the collaboration between the two sub-models. By employing the gradient search and determining the optimal step-sizes, we present an auxiliary model based two-stage projection algorithm. However, in order to alleviate the sensitivity to the noise, we reselect the step-sizes and derive the auxiliary model based two-stage stochastic gradient (AM-2S-SG) algorithm. Based on the AM-2S-SG algorithm, an auxiliary model based two-stage multi-innovation stochastic gradient algorithm is proposed to generate more accurate estimates. Finally, numerical simulations are provided to demonstrate the effectiveness of the proposed algorithms.     

Pseudo-canonical forms,identifiable parametrizations and simple parameter estimation for linear multivariable systems: Parameter estimation

In this paper the estimation of parameters for the pseudo-canonical, input-output models introduced in Corrêa, G. O. and Glover, K. (Automatica, 20, 429–442, 1984) is considered. Modifications of three equation-error type estimators (linear least-squares, ‘compensated’ and instrumental variable estimator) are derived, which incorporate the constraint on parameter values associated with these models. The consistency of the proposed estimators is analysed. Finally, results are presented on the construction of parametrization-independent instrumental variables.     

Least-squares-based iterative identification algorithm for Hammerstein nonlinear systems with non-uniform sampling

This paper focuses on identification problems for Hammerstein systems with non-uniform sampling. By using the over-parameterization technique, we derive a linear regressive identification model with different input updating rates. To solve the identification problem of Hammerstein output error systems with the unmeasurable variables in the information vector, the least-squares-based iterative algorithm is presented by replacing the unmeasurable variables with their corresponding iterative estimates. The performances of the proposed algorithm are analysed and compared by using a numerical example.     

On the adaptive control of a class of systems with random parameters and disturbances

The adaptive control of discrete time parameter linear stochastic systems with random parameters is investigated. It is shown that systems whose (unknown) autoregressive parameters undergo bounded martingale difference disturbances may be stabilized by the application of the so-called Modified Least Squares adaptive control algorithm. Asymptotically, the sample mean square performance criterion is equal to the one step ahead minimum variance control loss (which equals the prediction error variance when the system parameters are known) plus a term which is bounded by a quantity proportional to the square of the bound on the parameter disturbance. This latter term may be interpreted as the increase in the prediction error variance due to the random parameter variation.