Robust control oriented identification of errors-in-variables models based on normalised coprime factors

A robust control oriented identification approach is proposed to deal with the identification of errors-in-variables models (EIVMs), which are corrupted with input and output noises. Based on normalised coprime factor model (NCFM) representations, a frequency-domain perturbed NCFM for an EIVM is derived according to a geometrical explanation for the v-gap metric. As a result, identification of the EIVM is converted into that of the NCFM. Besides an identified nominal NCFM, its worst case error has to be quantified. Unlike other traditional control-oriented identification methods, the v-gap metric is employed to measure the uncertainties including a priori information on the disturbing noises and the worst case error for the resulting nominal NCFM. Since this metric is also used as an optimisation criterion, the associate parameter estimation problem can be effectively solved by linear matrix inequalities. Finally, a numerical simulation shows the effectiveness of the proposed method.     

The least-squares identification of FIR systems subject to worst-case noise

The least-squares identification of FIR systems is analyzed assuming that the noise is a bounded signal and the input signal is a pseudo-random binary sequence. A lower bound on the worst-case transfer function error shows that the least-square estimate of the transfer function diverges as the order of the FIR system is increased. This implies that, in the presence of the worst-case noise, the trade-off between the estimation error due to the disturbance and the bias error (due to unmodeled dynamics) is significantly different from the corresponding trade-off in the random error case: with a worst-case formulation, the model complexity should not increase indefinitely as the size of the data set increases.     

多输入多输出变量带误差模型的最坏情况频域辨识

本文将单输入单输出(SISO)变量带误差(EIV)模型的频域最坏情况辨识方法推广应用于多输入多输出 (MIMO)情况. 类似于SISO情况, 多输入多输出变量带误差(MIMO EIV)模型的辨识模型集合由估计的系统名义模型及 其最坏情况误差界描述. 所估计的系统名义模型表征为正规右图符号, 其最坏情况误差界具有可能的更少保守性, 可利 用EIV 模型的先验信息和后验信息由v-gap度量量化得到. 因此, 这种模型集合非常适合于后期利用Vinnicombe提出 的H1回路成形法设计鲁棒控制器. 最后, 利用一数值仿真实例验证所提出辨识方法的有效性.     

UD分解与偏差补偿结合用于变量带误差模型辨识

本文提出一种基于UD(upper-diagonal)分解与偏差补偿结合的辨识方法,用于变量带误差(errors-in-variables,EIV)模型辨识.考虑单输入单输出(single input and single output,SISO)线性动态系统,当输入和输出含有零均值、方差未知的高斯测量白噪声时,该类系统的模型参数估计是一种典型的EIV模型辨识问题.为了获得这种EIV模型参数的无偏估计,本文先推导出最小二乘模型参数估计偏差量与输入输出噪声方差以及最小二乘损失函数与输入输出噪声方差的关系,然后采用UD分解方法递推获得模型参数估计值,再利用输入输出噪声方差估计值补偿模型参数估计偏差,以此获得模型参数的无偏估计.本文还讨论了算法实现过程中遇到的一些问题及修补方法,并通过仿真例验证了所提辨识方法的有效性.     

大工业过程的稳态模型分散辨识及其强一致性分析

针对动态线性大工业过程,提出了获得其可分稳态模型强一致性估计的分散辨识方法.该方法仅使用设定点的阶跃信号作为输入激励信号,并且每个子过程的输入输出稳态模型辨识是在相应的局部单元完成的,因而大大减少了对过程的干扰和信息的交换量.所提出的方法简洁,并且辨识精度高,仿真结果说明了该辨识方法的有效性和实用性.     

Algorithms for recursive/semi-recursive bias-compensating least squares system identification within the errors-in-variables framework

Algorithms for the recursive/semi-recursive estimation of the system parameters as well as the measurement noise variances for linear single-input single-output errors-in-variables systems are considered. Approaches based on three offline techniques are presented: namely, the bias eliminating least squares, the Frisch scheme and the extended bias compensating the least squares method. Whilst the underlying equations used within these approaches are identical under certain design choices, the performances of the recursive/semi-recursive algorithms are investigated via simulation, in order to determine the most suitable technique for practical applications.     

Consistent dynamic PCA based on errors-in-variables subspace identification

In this paper, we make a comparison between dynamic principal component analysis (PCA) and errors-in-variables (EIV) subspace model identification (SMI) and establish consistency conditions for the two approaches. We first demonstrate the relationship between dynamic PCA and SMI. Then we show that when process variables are corrupted by measurement noise dynamic PCA fails to give a consistent estimate of the process model in general whether or not process noise is present. We then propose an indirect dynamic PCA approach for the consistent estimate of the process model resorting to EIV SMI algorithms. Consistent dynamic PCA models are obtained with and without process disturbances. Additional features of the indirect approach include (i) easy determination of the number of lagged variables in the model; (ii) determination of the number of significant process disturbances; and (iii) consistent estimate of the dynamic PCA models with and without process disturbances. We conduct two simulation examples and an industrial case study to support our theoretical results, where the relationship between dynamic PCA and EIV SMI is numerically verified.     

L 2-optimal identification of MIMO errors-in-variables models from the v-gap geometrical interpretation

An L ₂-optimal identification method is extended to cope with MIMO errors-in-variables (EIV) model estimation based on a geometrical interpretation for the v-gap metric. The L ₂-optimal approximate models are composed of system and noise models and characterised by a normalised right graph symbol (NRGS) and its complementary inner factor (CIF), respectively. This metric can be evaluated as the supreme of sine values of the maximal principal angles between NRGS frequency responses of two concerned models. In order to make full use of the angular cosine formula for complex vectors to reduce computational loads, a CIF of the NRGS of the perturbed model is introduced and thus, the system parameter optimisation can be efficiently solved by sequential quadratic programming methods. With the estimated system model, the associated noise model can be built by right multiplication of an inner matrix. Finally, a simulation example demonstrates the effectiveness of the proposed identification method.     

基于行列式比的阶次和参数同步递推辨识算法

针对输出误差模型,结合辅助模型的思想对原有阶次辨识和参数估计的方法进行融合和扩展,推导出基于辅助模型的行列式比定阶法,同时得出模型的阶次和参数,不仅减少了辨识过程的计算量,也节约了辨识时间。考虑到原有行列式比定阶法可能存在的不准确性,提出了一种系统模型的确认方法,增强了阶次辨识能力。仿真实验也充分表明,对行列式比定阶法的扩展不仅可以准确地辨识出系统的阶次,得出的参数估计值也有较高的精度。     

一种基于Pade近似的频域辨识与频域模型降阶新方法

研究了基于积分最小二乘指标的SISO时滞系统频域辨识与频域模型降阶问题.通过采用Pade近似,将积分最小二乘指标推广到可以处理时滞系统的情形.深入分析了Pade近似所引入误差,揭示了采用Pade近似的可行性与有效性.所提出的方法能够用最小二乘类算法高效求解,无须困难的非线性优化.仿真验证了所提出方法的有效性.     

Performance analysis of the auxiliary model-based least-squares identification algorithm for one-step state-delay systems

Based on the input–output representation of one-step state-delay systems, we use the auxiliary model-based recursive least-squares algorithm to estimate the parameters of the systems and study the convergence of the proposed algorithm by using the stochastic process theory. A simulation example is provided.     

随机线性系统参数估计的偏差补偿方法

将偏差补偿最小二乘法（ＢＥＬＳ）推广到一般单输入单输出系统的辨识。结果表明：这种推广的偏差补偿最小二乘法可以在有色噪声下获得系统参数的无偏估计而不需对噪声建模。仿真例子验证了理论分析的正确性。     

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.     

On exponential convergence and robustness of least-squares identification

Robustness and convergence properties of exponentially weighted least-squares identification are studied. It is shown that exponential convergence in the noiseless case can be obtained for a class of increasing or decreasing regression vectors. The rate of change of the limits in the regressors affect the convergence rates, which are explicitly given. It is demonstrated that for a sub-class of regressors decreasing-in-the-norm exponential convergence without the noise does not guarantee robustness subject to a bounded noise. Instead, exponential divergence of the estimate is shown in a specific case.     

Parameter identification for a traffic flow model

In this paper, a macroscopic model is presented which describes the traffic flow on a freeway by a set of nonlinear, deterministic difference equations. The model is deduced from simple physical and empirical considerations and contains a set of free parameters which have to be estimated using real traffic data. This identification procedure is formulated here as a parameter optimization problem which is solved by nonlinear programming. In addition, the sensitivity of the model with respect to parameter changes and structural changes is investigated. Although stochastic events play a role in traffic dynamics, the results demonstrate that the validated model copes surprisingly well with real traffic behaviour.     

Accuracy analysis of time domain maximum likelihood method and sample maximum likelihood method for errors-in-variables and output error identification

For identifying errors-in-variables models, the time domain maximum likelihood (TML) method and the sample maximum likelihood (SML) method are two approaches. Both methods give optimal estimation accuracy but under different assumptions. In the TML method, an important assumption is that the noise-free input signal is modelled as a stationary process with rational spectrum. For SML, the noise-free input needs to be periodic. It is interesting to know which of these assumptions contain more information to boost the estimation performance. In this paper, the estimation accuracy of the two methods is analyzed statistically for both errors-in-variables (EIV) and output error models (OEM). Numerical comparisons between these two estimates are also done under different signal-to-noise ratios (SNRs). The results suggest that TML and SML have similar estimation accuracy at moderate or high SNR for EIV. For OEM identification, these two methods have the same accuracy at any SNR.     

Indirect closed-loop identification by optimal instrumental variable method

Recently, a new bias-compensating least-squares (BCLS) method was proposed for the identification of a closed-loop system with high-order controller. The major feature of this method is that it can achieve consistent parameter estimation without modelling the coloured noises acting on the system. This paper studies the connection between the BCLS method and the instrumental variable (IV) family. It is shown that the BCLS method is a kind of weighted instrumental variable (WIV) method whose results do not depend upon the order of the controller.     

递推最小二乘参数辨识与仿真实例

介绍了基于递推最小二乘法进行系统辨识的基本原理,对给定的实际输入输出数据运用MATLAB的M语言编写递推最小二乘算法,最后给出相应的仿真结果和分析,并对得到的模型进行验证。     

An interative learning control scheme using the weighted least-squares method

An iterative learning control scheme is described for linear discrete-time systems. A weighted least-squares criterion of learning error is optimized to obtain a unique control gain for a case when the number of sampling is relatively small. It is then shown that algorithmic convergence can be readily guaranteed, because the present learning rule consists of a steady-state Kalman filter. By paying attention to the sparse system structure for the system's impulse response model, we further derive a suboptimal iterative learning control for a practical case when the number of sampling is large.