Modeling the system of suboptimal robust tracking under unknown upper bounds on the uncertainties and external disturbances

This paper addresses a problem of suboptimal robust tracking for a discrete-time plant under unknown upper bounds on external disturbances. The transfer function of the nominal plant model is assumed to be known, whereas the upper bounds on the external disturbance, measurement noise, and coprime factor perturbations are assumed to be unknown. The results of numerical modeling of the algorithm of suboptimal robust tracking based on relaxed verification of the estimates of upper bound in the closed loop and the use of control criterion associated with such verification as the identification criterion are presented and analyzed. The results of numerical modeling illustrate efficiency of the proposed method of control design.     

Control-oriented model validation and errors quantification in the /spl lscr//sub 1/ setup

A priori information required for robust synthesis includes a nominal model and a model of uncertainty. The latter is typically in the form of additive exogenous disturbance and plant perturbations with assumed bounds. If these bounds are unknown or too conservative, they have to be estimated from measurement data. In this paper, the problem of errors quantification is considered in the framework of the /spl lscr//sub 1/ optimal robust control theory associated with the /spl lscr//sub /spl infin// signal space. The optimal errors quantification is to find errors bounds that are not falsified by measurement data and provide the minimum value of a given control criterion. For model with unstructured uncertainty entering the system in a linear fractional manner, the optimal errors quantification is reduced to quadratic fractional programming. For system under coprime factor perturbations, the optimal errors quantification is reduced to linear fractional programming.     

Worst case identification of continuous time systems via interpolation

We consider a worst case robust control oriented identification problem recently studied by several authors. This problem is one of identification in the continuous time setting. We give a more general formulation of this problem. The available a priori information in this paper consists of a lower bound on the relative stability of the plant, a frequency dependent upper bound on a certain gain associated with the plant, and an upper bound on the noise level. The available experimental information consists of a finite number of noisy plant point frequency response samples. The objective is to identify, from the given a priori and experimental information, an uncertain model that includes a stable nominal plant model and a bound on the modeling error measured in norm. Our main contributions include both a new identification algorithm and several new ‘explicit’ lower and upper bounds on the identification error. The proposed algorithm belongs to the class of ‘interpolatory algorithms’ which are known to possess a desirable optimality property under a certain criterion. The error bounds presented improve upon the previously available ones in the aspects of both providing a more accurate estimate of the identification error as well as establishing a faster convergence rate for the proposed algorithm.     

Error quantification of the normalised right graph symbol for an errors-in-variables system

This paper proposes a novel method to quantify the error of a nominal normalized right graph symbol (NRGS) for an errors-in-variables (EIV) system corrupted with bounded noise. Following an identification framework for estimation of a perturbation model set, a worst-case v-gap error bound for the estimated nominal NRGS can be first determined from \textit{a priori} and \textit{a posteriori} information on the underlying EIV system. Then, an NRGS perturbation model set can be derived from a close relation between the v-gap metric of two models and ${\rm H}_\infty$-norm of their NRGSs' difference. The obtained NRGS perturbation model set paves the way for robust controller design using an ${\rm H}_\infty$ loop-shaping method because it is a standard form of the well-known NCF (normalized coprime factor) perturbation model set. Finally, a numerical simulation is used to demonstrate the effectiveness of the proposed identification method.     

MODEL SET IDENTIFICATION IN FREQUENCY-DOMAIN AND ITS APPLICATION TO JOINT DESIGN WITH ROBUST CONTROL

In this paper, we propose a new model set identification method for robust control, which determines both nominal models and uncertainty bounds in frequency-domain using periodgrams obtained from experimental data. This method also gives less conservative model sets when we have more experimental data, which is one of the distinguished features compared with the existing model set identification methods. To this end, first, we construct a new noise model set in terms of periodgrams, which consists of hard-bounded (or deterministic) noises but takes account of a low correlation property of noise signals, simultaneously. Then, based on the noise model, we show how to compute the nominal models and the upper bounds of modeling error via convex optimization, which minimize given cost functions. Furthermore, by introducing a weighting function compatible with control performance criterion into the identification cost function, we consider a joint design method of the proposed model set identification and H_∞ control. Numerical examples show the effectiveness of the proposed method.     

基于无偏置项LSSVM的稳健在线过程建模方法

针对直接利用最小二乘支持向量机(LSSVM)对动态过程在线建模时预测精度易受过程输出测量值上的粗大误差和噪声影响的问题,在分析样本序列结构特征和噪声作用特征基础上,提出一种基于无偏置项LSSVM的稳健在线过程建模方法。该方法在每一预测周期中根据预测误差与设定阈值之间的关系来识别和恢复异常测量值、识别和修正含噪声测量值,从而降低样本中的噪声,使得出的LSSVM较好地跟踪过程的动态特性。这种在线过程建模方法具有稳健性,能减少输出值上粗大误差和高斯白噪声对LSSVM预测精度的影响,提高预测精度。数字仿真显示该方法的有效性和优越性。     

On the time complexity of worst-case system identification

In this paper we treat a general worst-case system identification problem. This problem is worst-case with respect to both noise and system modeling uncertainty. We consider this problem under various a priori information structures. We determine bounds on the minimum duration identification experiment that must be run to identify the plant to within a specified guaranteed worst-case error bound. Our results are algorithm independent. We show that this minimum duration is prohibitively long. Based on our results, we suggest that worst-case (with respect to noise) system identification requires unrealistic amounts of experimental data     

A parametric statistical approach to FDI for the industrial actuator benchmark

A parametric statistical approach to the industrial actuator fault-detection and isolation benchmark is presented. An algorithm for detecting a change in the dynamics of a linear system is formulated as a set of sequential probability ratio tests of the innovations from a bank of Kalman filters. The algorithm is extended to allow estimation of a disturbance using a generalised likelihood ratio test. Modifications are proposed for when the model is nonlinear and the modeling error is significant. The algorithm is evaluated using the benchmark test data and is shown to provide low detection delays while being robust to noise, disturbances and model error.     

递推鲁棒频域辨识算法误差界计算的改进*

文［１］研究了带未建模动态系统的频域辨识问题，辨识的结果为传递函数在单位圆上的有限个点估计及其误差界。本文指出文［１］定理的错误，并对其进行了更正，数字仿真表明，更正的误差界比文［１］中的误差界更精确。     

Robust centralized and weighted measurement fusion Kalman estimators for uncertain multisensor systems with linearly correlated white noises

This paper addresses the design of robust centralized fusion (CF) and weighted measurement fusion (WMF) Kalman estimators for a class of uncertain multisensor systems with linearly correlated white noises. The uncertainties of the systems include multiplicative noises, missing measurements, and uncertain noise variances. By introducing the fictitious noises, the considered system is converted into one with only uncertain noise variances. According to the minimax robust estimation principle, based on the worst-case system with the conservative upper bounds of uncertain noise variances, the robust CF and WMF time-varying Kalman estimators (predictor, filter, and smoother) are presented in a unified framework. Applying the Lyapunov equation approach, their robustness is proved in the sense that their actual estimation error variances are guaranteed to have the corresponding minimal upper bounds for all admissible uncertainties. Using the information filter, their equivalence is proved. Their accuracy relations are proved. The computational complexities of their algorithms are analyzed and compared. Compared with CF algorithm, the WMF algorithm can significantly reduce the computational burden when the number of sensors is larger. A robust weighted least squares (WLS) measurement fusion filter is also presented only based on the measurement equation, and it is proved that the robust accuracy of the robust CF or WMF Kalman filter is higher than that of robust WLS filter. The corresponding robust fused steady-state estimators are also presented, and the convergence in a realization between the time-varying and steady-state robust fused estimators is proved by the dynamic error system analysis (DESA) method. A simulation example shows the effectiveness and correctness of the proposed results.     

Coprime factors reduction methods for linear parameter varying and uncertain systems

We present a generalization of the coprime factors model reduction method of Meyer and propose a balanced truncation reduction algorithm for a class of systems containing linear parameter varying and uncertain system models. A complete derivation of coprime factorizations for this class of systems is also given. The reduction method proposed is thus applicable to linear parameter varying and uncertain system realizations that do not satisfy the structured ℓ₂-induced stability constraint required in the standard nonfactored case. Reduction error bounds in the ℓ₂-induced norm of the factorized mapping are given.     

Quality evaluation for a coprime factor perturbed model set basedon frequency-domain data

Quality assessment is investigated under a probabilistic framework for a prescribed model set. The results on unfalsified probability estimation are extended from additive modeling errors to normalized coprime factor perturbations. An analytic formula has been derived for the sample unfalsified probability. It is shown that with increasing the data length, the sample unfalsified probability converges in probability to a number which is independent of experimental data. Numerical simulations show that the proposed sample unfalsified probability is appropriate in the evaluation of the quality of a model set     

Normalized H∞ controller reduction with a priorierror bounds

This paper describes an approach to the reduction of controllers for the normalized coprime factor robustness problem as well as the normalized H_∞ problem. It is shown that a relative error approximation of a coprime factor representation of any suboptimal controller leads to a stability guarantee and an upper bound on the performance degradation when the reduced order controller is implemented. When the approximation is performed on the controller generator, guaranteed a priori stability and performance bounds are obtained in terms of the synthesis Riccati equation solutions of the normalized H_∞ control problems     

Model quality evaluation in H2 identification

Model quality evaluation in set-membership identification is investigated, In the recent literature, two main approaches have been used to investigate this problem, based on the concepts of n-width and of radius of information. In this paper it is shown that the n-width is related to the asymptotic value of the conditional radius of information of the identification problem with noise free measurements. Upper and lower bounds of the conditional radius of information are derived for the H₂ identification of exponentially stable systems using approximating n-dimensional models linear in the parameters in the presence of power bounded measurement errors. The derived bounds are shown to be convergent to the radius for a large number of data and model dimensions. Moreover, a formula for computing the worst case identification error for any linear algorithm is given. In particular, it is shown that the identification error of the least square algorithm may be increasing with respect to the model dimension (“peaking effect”), An almost-optimal linear algorithm is presented, that is not affected by this peaking effect, and indeed is asymptotically optimal     

L1鲁棒辨识中的一般中心估计算法及其特例

本文分析具了Ｌ１误差的线性时不变系统的最不利情况的辨识问题，对系统假设的先验信息未知系统的脉冲响应函数控制稳定，并且假定实验数据具有噪声干扰的，本文提出了一般中心估计算法并分析了最不利情况的误差界限，并进一步研究了在某些特殊情况下，中心估计的简单求法及其性质，本文所给算法和辨识结果是面向鲁棒控制的。     

Nonparametric estimation for normalized coprime factors of a MIMO system

This paper extends to multi-input multi-output (MIMO) systems the results on frequency response estimation for normalized coprime factors (NCF) under a stochastic framework from closed-loop frequency domain experimental data. Under the condition that the covariance matrices are available for the external disturbances and measurement errors, an analytic solution has been obtained for the maximum likelihood estimate (MLE), on the basis of a linear fractional transformation (LFT) representation for all the plant possible normalized right coprime factors (NRCF). It is proved that the estimate can be expressed by a linear combination of a normalized random matrix with all its columns having independent complex normal distributions. Some methods are suggested to reduce the estimation bias when high-quality experimental data can be obtained. A numerical example is included that confirms the theoretical results.     

A measurement fusion method for nonlinear system identification using a cooperative learning algorithm

Identification of a general nonlinear noisy system viewed as an estimation of a predictor function is studied in this article. A measurement fusion method for the predictor function estimate is proposed. In the proposed scheme, observed data are first fused by using an optimal fusion technique, and then the optimal fused data are incorporated in a nonlinear function estimator based on a robust least squares support vector machine (LS-SVM). A cooperative learning algorithm is proposed to implement the proposed measurement fusion method. Compared with related identification methods, the proposed method can minimize both the approximation error and the noise error. The performance analysis shows that the proposed optimal measurement fusion function estimate has a smaller mean square error than the LS-SVM function estimate. Moreover, the proposed cooperative learning algorithm can converge globally to the optimal measurement fusion function estimate. Finally, the proposed measurement fusion method is applied to ARMA signal and spatial temporal signal modeling. Experimental results show that the proposed measurement fusion method can provide a more accurate model.     

A least third-order cumulants objective function

A novel Least Cumulants Method is proposed to tackle the problem of fitting to underlying function in small data sets with high noise level because higher-order statistics provide an unique feature of suppressing Gaussian noise processes of unknown spectral characteristics. The current backpropagation algorithm is actually the Least Square Method based algorithm which does not perform very well in noisy data set. Instead, the proposed method is more robust to the noise because a complete new objective function based on higher-order statistics is introduced. The proposed objective function was validated by applying to predict benchmark sunspot data and excellent results are obtained. The proposed objective function enables the network to provide a very low training error and excellent generalization property. Our results indicate that the network trained by the proposed objective function can, at most, provide 73% reduction of normalized test error in the benchmark test.     

Synthesis of decoupling controller for non-minimum phase plants of different pole numbers on RHP within uncertainties

This article mainly studies the decoupling controller design for non-minimum phase plants of different pole numbers on RHP within uncertainties. The normalised coprime factorisation is considered to achieve the robustness requirements. The pole-zero cancellations on RHP should be averted for the sake of robustness. For convenience, the H ^∞ sub-optimal controller is utilised to meet the robust criterion of the plant. Some necessary state space formulae are also provided to facilitate the synthesis of the decoupling controller. The configuration of the two-parameter compensation is employed. The Bezout identity makes the feedforward controller easy to determine. A brief algorithm is presented. In addition, the proposed synthesis is illustrated with a numerical example. The robust bounds of the feedback controller can be assessed for both the additive uncertainty and the coprime factor uncertainties. The result shows that the compensated system is decoupled and is guaranteed to be internally stable within the specified robust bound although the pole number varies on RHP.