A new kernel-based approach for linear system identification

This paper describes a new kernel-based approach for linear system identification of stable systems. We model the impulse response as the realization of a Gaussian process whose statistics, differently from previously adopted priors, include information not only on smoothness but also on BIBO-stability. The associated autocovariance defines what we call a stable spline kernel. The corresponding minimum variance estimate belongs to a reproducing kernel Hilbert space which is spectrally characterized. Compared to parametric identification techniques, the impulse response of the system is searched for within an infinite-dimensional space, dense in the space of continuous functions. Overparametrization is avoided by tuning few hyperparameters via marginal likelihood maximization. The proposed approach may prove particularly useful in the context of robust identification in order to obtain reduced order models by exploiting a two-step procedure that projects the nonparametric estimate onto the space of nominal models. The continuous-time derivation immediately extends to the discrete-time case. On several continuous- and discrete-time benchmarks taken from the literature the proposed approach compares very favorably with the existing parametric and nonparametric techniques.     

基于脉冲响应的输出误差模型的辨识

基于系统脉冲响应参数, 利用相关分析方法, 提出了一种辨识输出误差模型参数的方法. 该方法是利用有限脉冲响应模型逼近输出误差模型, 通过依次递增脉冲响应参数的数目N来提高逼近精度. 理论分析表明, 只要N足够大, 模型的辨识精度可以满足实际要求. 提出的辨识方法可以在假设阶次N =1的条件下, 依次递增计算N较大时的脉冲响应参数和目标函数值, 从而根据脉冲响应确定系统的参数. 仿真试验说明提出的方法估计输出误差模型的参数是有效的.     

A measurement-based approach for designing reduced-order controllers with guaranteed bounded error

The objective of this paper is to present a measurement-based control-design approach for single-input single-output linear systems with guaranteed bounded error. A wide range of control-design approaches available in the literature are based on parametric models. These models can be obtained analytically using physical laws or via system identification using a set of measured data. However, due to the complex properties of real systems, an identified model is only an approximation of the plant based on simplifying assumptions. Thus, the controller designed based on a simplified model can seriously degrade the closed-loop performance of the system. In this paper, an alternative approach is proposed to develop fixed-order controllers based on measured data without the need for model identification. The proposed control technique is based on computing a suitable set of fixed-order controller parameters for which the closed-loop frequency response fits a desired frequency response that meets the desired closed-loop performance specifications. The control-design problem is formulated as a nonlinear programming problem using the concept of bounded error. The main advantages of our proposed approach are: (1) it guarantees that the error between the computed and the desired frequency responses is less than a small value; (2) the difficulty of finding the globally optimal solution in the error minimisation problem is avoided; (3) the controller can be designed without the use of any analytical model to avoid errors associated with the identification process; and (4) low-order controllers can be designed by selecting a fixed low-order controller structure. To experimentally validate and illustrate the efficacy of the proposed approach, proportional-integral measurement-based controllers are designed for a DC (direct current) servomotor.     

Nonlinear adaptive control system design with asymptotically stable parameter estimation error

The paper presents a new general method for nonlinear adaptive system design with asymptotic stability of the parameter estimation error. The advantages of the approach include asymptotic unknown parameter estimation without persistent excitation and capability to directly control the estimates transient response time. The method proposed modifies the basic parameter estimation dynamics designed via a known nonlinear adaptive control approach. The modification is based on the generalised prediction error, a priori constraints with a hierarchical parameter projection algorithm, and the stable data accumulation concepts. The data accumulation principle is the main tool for achieving asymptotic unknown parameter estimation. It relies on the parametric identifiability system property introduced. Necessary and sufficient conditions for exponential stability of the data accumulation dynamics are derived. The approach is applied in a nonlinear adaptive speed tracking vector control of a three-phase induction motor.     

Identification of linear systems via spectral analysis giventime-domain data: consistency, reduced-order approximation, andperformance analysis

Estimation of a parametric input-output (I/O) infinite impulse response transfer function given time-domain I/O data is considered. Some of the desirable properties of any approach to this problem are: unimodality of the performance surface, consistent identification in the sufficient-order case, and stability of the fitted model under undermodeling. We first consider a frequency-domain solution to the least squares equation error identification problem using the power spectrum and the cross-spectrum of the I/O data to estimate the I/O parametric transfer function. The proposed approach is shown to yield a unimodal performance surface, consistent identification in colored noise and sufficient-order case, and stable fitted models under undermodeling for arbitrary stationary inputs so long as they are persistently exciting of sufficiently high order. Asymptotic performance analysis is carried out for both sufficient-order and reduced-order cases. These asymptotic results are then used to derive statistics on the corresponding estimated transfer function. We also investigate an iterative pseudomaximum likelihood approach and analyze its performance under sufficient-order modeling. Finally, computer simulation examples are provided to illustrate the two approaches     

Identification/reduction to a balanced realization via the extendedimpulse response gramian

A new identification/reduction algorithm for linear, discrete time-invariant (LDTI) systems is proposed which is based on the extended impulse response gramian first defined here for LDTI systems. The reduction algorithm employs singular value decomposition to retain states corresponding to dominant singular values of these gramians. The proven properties of the reduced order models include convergence to a balanced realization with conditional controllability, observability, and asymptotic stability. A suboptimal property of the model (in minimizing an impulse response error norm) is also demonstrated. The proposed technique can handle impulse response data of deterministic or stochastic nature for system identification application     

Bi-Objective Optimal Control Modification Adaptive Control for Systems with Input Uncertainty

This paper presents a new model-reference adaptive control method based on a bi-objective optimal control formulation for systems with input uncertainty. A parallel predictor model is constructed to relate the predictor error to the estimation error of the control effectiveness matrix. In this work, we develop an optimal control modification adaptive control approach that seeks to minimize a bi-objective linear quadratic cost function of both the tracking error norm and the predictor error norm simultaneously. The resulting adaptive laws for the parametric uncertainty and control effectiveness uncertainty are dependent on both the tracking error and the predictor error, while the adaptive laws for the feedback gain and command feedforward gain are only dependent on the tracking error. The optimal control modification term provides robustness to the adaptive laws naturally from the optimal control framework. Simulations demonstrate the effectiveness of the proposed adaptive control approach.      

Experimental Estimation of Model Error Bounds Based on Modified Stochastic Approximation

A modification of the estimation algorithm stochastic approximation is presented. With assumptions to the statistical distribution of the training data it becomes possible, to estimate not only the mean value but also well directed deviating values of the data distribution. Thus, detailed error models can be identified by means of parameter-linear formulation of the new algorithm. By definition of suitable probabilities, these parametric error models are estimating soft error bounds. That way, an experimental identification method is provided that is able to support a robust controller design. The method was applied at an industrial robot, which is controlled by feedback linearisation. Based on a dynamic model realised by a neural network, the presented approach is utilised for the robust design of the stabilising decentral controllers.     

某随动系统动态模型相关——最小二乘辨识

本文应用相关仪对某随动系统进在线辨识,得到系统的非参数模型——脉冲涵数,在此基础上利用最小二乘法拟合成参数模型—系统的差分方程。通过双线性z变换,变换为连续数学模型——传递函数,并做阶跃响应检验,辨识得到的系统动态模型的阶跃响应与实际系统的阶跃响应基本一致。在随机干扰作用下,由此方法得到的动态模型较频率测试法或理论分析的结果更为精确。     

A system identification problem motivated by robust control

Linear, dynamic model set estimation based on noisy, input-output data is addressed here from a confidence set standpoint. Following the usual robust control perspective a model set estimate for the ‘true’, but unknown, impulse response (truncated at the data length due to causality) is sought via a nominal model belonging to a pre-specified parametric class of approximating models (of ‘low order’) plus some quantitative information on the mismatch between the approximating model and the underlying (possibly, ‘high-order’) one. The solution proposed is based on the asymptotic, parameter estimation theory of Ljung (1978), and Ljung and Caines (1979). It hinges upon the characterization of a joint confidence set for an optimal approximation for the underlying system in a parametric class and the corresponding approximation error, which is shown to be consistent. This is done under relatively weak conditions on the system input (stationarity conditions are not imposed) and observation noise (a specific distribution form is not assumed), and without assuming that the approximating model class contains the underlying model.     

Joint DOA and channel estimation with data detection based on 2D unitary ESPRIT in massive MIMO systems

We propose a novel method for joint two-dimensional (2D) direction-of-arrival (DOA) and channel estimation with data detection for uniform rectangular arrays (URAs) for the massive multiple-input multiple-output (MIMO) systems. The conventional DOA estimation algorithms usually assume that the channel impulse responses are known exactly. However, the large number of antennas in a massive MIMO system can lead to a challenge in estimating accurate corresponding channel impulse responses. In contrast, a joint DOA and channel estimation scheme is proposed, which first estimates the channel impulse responses for the links between the transmitters and antenna elements using training sequences. After that, the DOAs of the waves are estimated based on a unitary ESPRIT algorithm using previous channel impulse response estimates instead of accurate channel impulse responses and then, the enhanced channel impulse response estimates can be obtained. The proposed estimator enjoys closed-form expressions, and thus it bypasses the search and pairing processes. In addition, a low-complexity approach toward data detection is presented by reducing the dimension of the inversion matrix in massive MIMO systems. Different cases for the proposed method are analyzed by changing the number of antennas. Experimental results demonstrate the validity of the proposed method.     

On the estimation of transfer functions,regularizations and Gaussian processes—Revisited

Intrigued by some recent results on impulse response estimation by kernel and nonparametric techniques, we revisit the old problem of transfer function estimation from input–output measurements. We formulate a classical regularization approach, focused on finite impulse response (FIR) models, and find that regularization is necessary to cope with the high variance problem. This basic, regularized least squares approach is then a focal point for interpreting other techniques, like Bayesian inference and Gaussian process regression. The main issue is how to determine a suitable regularization matrix (Bayesian prior or kernel). Several regularization matrices are provided and numerically evaluated on a data bank of test systems and data sets. Our findings based on the data bank are as follows. The classical regularization approach with carefully chosen regularization matrices shows slightly better accuracy and clearly better robustness in estimating the impulse response than the standard approach–the prediction error method/maximum likelihood (PEM/ML) approach. If the goal is to estimate a model of given order as well as possible, a low order model is often better estimated by the PEM/ML approach, and a higher order model is often better estimated by model reduction on a high order regularized FIR model estimated with careful regularization. Moreover, an optimal regularization matrix that minimizes the mean square error matrix is derived and studied. The importance of this result lies in that it gives the theoretical upper bound on the accuracy that can be achieved for this classical regularization approach.     

An algorithm for nonparametric GARCH modelling

A simple iterative algorithm for nonparametric first-order GARCH modelling is proposed. This method offers an alternative to fitting one of the many different parametric GARCH specifications that have been proposed in the literature. A theoretical justification for the algorithm is provided and examples of its application to simulated data from various stationary processes showing stochastic volatility, as well as empirical financial return data, are given. The nonparametric procedure is found to often give better estimates of the unobserved latent volatility process than parametric modelling with the standard GARCH(1,1) model, particularly in the presence of asymmetry and other departures from the standard GARCH specification. Extensions of the basic iterative idea to more complex time series models combining ARMA or GARCH features of possibly higher order are suggested.     

一种频率域相关性分布式扩散最小均方算法

以均方误差为代价函数的最小均方（LMS）自适应滤波算法具有结构简单、易于实现、计算复杂度低、稳定性好等优点,然而在对未知系统的脉冲响应进行估计时,传统的分布式扩散最小均方（DLMS）算法易受到噪声的干扰,从而降低估计精度。针对该问题,提出一种频率域相关性分布式扩散最小均方（FCDLMS）算法。利用不相关信号的相关函数值趋近于零的性质,在DLMS算法基础上分别将输入信号的自相关函数以及输入和期望信号的互相关函数作为新的观测数据,消除噪声干扰,从而给出相关性DLMS （CDLMS）算法,并将算法扩展至频率域,在频率域中使用乘法运算而非卷积运算来更新抽头系数,减少计算复杂度。实验结果表明,与传统DLMS算法相比,频率域相关性分布式扩散最小均方算法在噪声环境下对分布式自适应网络中的未知系统脉冲响应具有更好的估计结果,算法性能更优,同时也能较好地适应多抽头数、多节点数、强噪声的复杂环境。     

Subspace like identification incorporating prior information

The subspace identification methods have proved to be a powerful tool, which can further benefit from the prior information incorporation algorithm proposed in this note. In the industrial environment, there is often some knowledge about the identified system (known static gains, dominant time constants, low frequency character, etc.), which can be used to improve model quality and its compliance with first principles. The proposed algorithm has two stages. The first one is similar to the subspace methods as it uses their interpretation as an optimization problem of finding parameters of an optimal multi-step linear predictor for the experimental data. This problem is reformulated in the Bayesian framework allowing prior information incorporation in the form of the mean value and the covariance of the impulse response, which is shown to be useful for the incorporation of several prior information types. The second stage with state space model realization from the posterior impulse response estimate is different from the standard subspace methods as it is based on the structured weighted lower rank approximation, which is necessary to preserve the prior information incorporated in the first stage.     

Identification of sparse impulse responses – design and implementation using the partial Haar block wavelet transform

This paper proposes an implementation for identifying sparse impulse responses. The new scheme follows the approach in which the location of the channel response peak is estimated in the wavelet domain. A short time-domain adaptive filter is then located about the estimated peak to identify the sparse response. The primary purpose of this paper is to present an efficient design of such a system. The use of a new block wavelet transform results in up to 70% less computational complexity and improved peak detection, as compared to previous solutions. A new robust time-domain adaptive filtering location and update scheme is also proposed that significantly reduces the occurrence of jitter problems and leads to improved residual mean-square error performance. The behavior of the transform-domain adaptive filter is analyzed, the Wiener solution is determined, and an accurate analytical model is obtained for the mean-square deviation of the adaptive coefficients. Monte Carlo simulations show excellent echo cancellation performance for typical ITU-T echo channels.     

Robust model reference adaptive control for transient performance enhancement

To circumvent the potentially poor transient response induced by nonlinear uncertain dynamics in the adaptive control system, this article proposes a new model reference adaptive control design scheme to improve its transient control response. We first construct a compensator to online extract the undesired dynamics in the online learning, which is incorporated into the reference model and control simultaneously. Then, an error feedback term is incorporated into the reference model to speed up the convergence of both the compensator and tracking error. Moreover, a new leakage term containing the estimation error is constructed and then added in the adaptive law to guarantee the convergence of both the estimation error and tracking error. To further reveal the mechanisms behind these proposed methods, a new methodology to analyze the transient error bounds based on L₂‐norm and Cauchy‐Schwartz inequality is also developed. Based on the analysis results, we find that the proposed methods can effectively reduce the bound of the tracking error and thus achieve an improved transient control performance without violating the system stability even with high‐gain adaptation. In addition, the frequency‐domain analysis is resorted to show the comparative responses of different adaptive laws, which indicate that the proposed adaptive law can maintain the stability margin even with a high‐gain learning rate. A numerical example is given to demonstrate improved control responses of these proposed schemes.     

Adaptive backstepping control for an n-degree of freedom robotic manipulator based on combined state augmentation

The aim of this paper is to improve the tracking performance of a robotic manipulator by designing an adaptive controller and implementing it on the system. The proposed controller guarantees the system stability as well as good tracking performance in existence of nonlinearity and parameter uncertainties. The requirement to decrease the system response overshoot and steady state error as well as increasing speed of tracking for manipulators is essential to many manufacturers. To this mean, in this paper, the tracking error equations for an n-DOF manipulator are derived and the response characteristics are improved by augmenting a new state to the system equations. The stability of the closed-loop system is guaranteed based on the Lyapunov theory via backstepping control approach. The robotic manipulator model contains parametric uncertainties and many of the parameter values are unknown. To solve the problem, an adaption law is proposed via adaptive backstepping mechanism. Different experiments are carried out for a 2-DOF manipulator to show the effectiveness of the proposed approach and the results are compared with four of the recently revealed researches on control. Experimental results present the superiority of the state augmented adaptive backstepping in tracking the desired joint angles. Moreover, in order to present the industrial application of the proposed control method, it is simulated for a large industrial Scara manipulator.     

频率选择性衰落信道下STBC-SC-FDE系统的信道频率响应估计

本文针对频率选择性衰落信道下2发1收STBC-SC-FDE系统收发两端的信号编码与处理结构,应用系统的频率域输入—输出模型,设计了一种最小二乘信道频率响应（CFR）估计算法。对算法均方误差的分析表明,采用Chu序列作为最优训练序列不仅能够实现CFR估计的最小均方误差,而且能保证系统具有比较低的峰均功率比（PAPR）。该算法相比那些先估计信道时域响应（CIR）再转换为CFR的方法能够节省更多的计算资源。最后,Monte Carlo 仿真验证了该算法的性能。     

Distributed moving horizon state estimation: Simultaneously handling communication delays and data losses

In this work, we consider distributed moving horizon state estimation of nonlinear systems subject to communication delays and data losses. In the proposed design, a local estimator is designed for each subsystem and the distributed estimators communicate to collaborate. To handle the delays and data losses simultaneously, a predictor is designed for each subsystem estimator. A two-step prediction-update strategy is used in the predictor design in order to get a reliable prediction of the system state. In the design of each subsystem estimator, an auxiliary nonlinear observer is also taken advantage of to calculate a reference subsystem state estimate. In the local estimator, the reference state estimate is used to generate a confidence region within which the local estimator optimizes its subsystem state estimate. Sufficient conditions under which the proposed design gives decreasing and ultimately bounded estimation error are provided. The effectiveness of the proposed approach is illustrated via the application to a chemical process example.