Direct and indirect least squares methods in continuous-time parameter estimation

The discrete-time least squares approach is extended to the estimation of parameters in continuous nonlinear models. The resulting direct integral least squares (DILS) method is both simple and numerically efficient and it usually improves the mean-squared error of the estimates compared with the conventional indirect least squares (ILS) method. The biasedness of the DILS estimates may become serious if the sample points are widely spaced in time and/or the signal-to-noise ratio is low and so a continuous-time symmetric bootstrap (SB) estimator which removes this problem is described. The DILS, SB and ILS methods form a three-stage procedure combining the robustness and numerical efficiency of direct methods with the asymptotic unbiasedness of ILS procedures.     

On strong consistency of least squares identification algorithms

In this paper almost sure convergence results are derived for least squares identification algorithms. The convergence conditions expressed in terms of the measurable signal model states derived for asymptotically stable signal models and possibly nonstationary processes are in essence the same as those previously given, but are derived more directly. Strong consistency results are derived for the case of signal models with unstable modes and exponential rates of convergence to the unstable modes are demonstrated. These latter convergence results are stronger than those earlier ones in which weak consistency conditions are given and there is also less restriction on the noise disturbances than in earlier theories. The derivations in the paper appeal to martingale convergence theorems and the Toeplitz lemma.     

Exponential convergence of adaptive identification and control algorithms

Output and equation error adaptive identification algorithms are shown to be exponentially convergent under a deterministic or stochastic persistently exciting (or spanning) condition on the system inputs together with several other standard conditions. An adaptive control algorithm is shown to be exponentially convergent under a deterministic or stochastic persistently exciting condition on the reference trajectory together with some standard conditions.     

A new method of parameter estimation and adaptive control of nonlinear systems with filterless least-squares

This paper is concerned with the parameter estimation and adaptive control problem for a sort of strict-feedback nonlinear systems with unknown parameters. In order to deal with the nonlinear functions, we develop a kind of least-squares estimator in the parameter estimation part. With such an estimator, we design the state-feedback control law and analyze the properties of the closed-loop system. To be concrete, the parameter estimator can guarantee the boundedness of parameter estimate error. At the same time, the adaptive control law is proposed to make the system states converge to zero and the equilibrium stable globally under some common assumptions. Finally, simulations are given to demonstrate the theoretical results.     

Stochastic adaptive control using a modified least squares algorithm

Recent papers on stochastic adaptive control have established global convergence for algorithms using a stochastic approximation iteration. However, to date, global convergence has not been established for algorithms incorporating a least squares iteration. This paper establishes global convergence for a slightly modified least squares stochastic adaptive control algorithm. It is shown that, with probability one, the algorithm will ensure that the system inputs and outputs are sample mean square bounded and the mean square output tracking error achieves its global minimum possible value for linear feedback control.     

The least squares algorithm, parametric system identification and bounded noise

The least squares parametric system identification algorithm is analyzed assuming that the noise is a bounded signal. A bound on the worst-case parameter estimation error is derived. This bound shows that the worst-case parameter estimation error decreases to zero as the bound on the noise is decreased to zero.     

一类非线性系统基于最小二乘支持向量机的自适应H∞控制

提出一类非线性不确定系统基于最小二乘支持向量机的白适应H∞控制方法．该方法基于最小二乘支持向量机估计对象的未知非线性函数,并给出了最小二乘支持向量机权向量和偏移值的在线学习规则．引入H∞控制器用于减弱外部干扰及最小二乘支持向量机近似误差对输出误差的影响．利用李亚普诺夫理论证明了整个闭环系统一致最终有界稳定．仿真结果表明了该方法的有效性．     

On parameter convergence in adaptive control

It is well known that the parameter error as well as the model-plant mismatch error in a model reference adaptive scheme tends exponentially to zero iff a certain sufficient richness condition holds for signals inside the time-varying plant control loop. In this paper we give conditions on the reference signal (the exogenous input to the adaptive loop) — namely, that it have as many spectral lines as there are unknown parameters, in order to guarantee parameter convergence.     

Tracking and parameter identification for model reference adaptive control

We provide barrier Lyapunov functions for model reference adaptive control algorithms, allowing us to prove robustness in the input‐to‐state stability framework and to compute rates of exponential convergence of the tracking and parameter identification errors to zero. Our results ensure identification of all entries of the unknown weight and control effectiveness matrices. We provide easily checked sufficient conditions for our relaxed persistency of excitation conditions to hold. Our illustrative numerical example demonstrates the performance of the control methods.     

基于辅助模型的递推增广最小二乘辨识方法

针对有色噪声干扰的输出误差滑动平均系统, 将辅助模型与递推增广最小二乘算法相结合: 用辅助模型的输出代替辨识模型信息向量中的未知真实输出项, 用估计残差代替信息向量中的不可测噪声项, 从而提出了基于辅助模型的递推增广最小二乘辨识方法. 为了展示所提方法的特点, 文中还给出了经过模型变换的递推增广最小二乘算法. 理论分析和仿真研究表明, 提出的方法原理简单、计算量小, 可以给出高精度参数估计, 且能够用于在线辨识.     

Outliers, inliers and the generalized least trinuned squares estimator in system identification

The least trimmed squares estimator (LTS) is a well known robust estimator in terms of protecting the estimate from the outliers. Its high computational complexity is however a problem in practice. We show that the LTS estimate can be obtained by a simple algorithm with the complexity O( N In N) for large N, where N is the number of measurements. We also show that though the LTS is robust in terms of the outliers, it is sensitive to the inliers. The concept of the inliers is introduced. Moreover, the Generalized Least Trimmed Squares estimator (GLTS) together with its solution are presented that reduces the effect of both the outliers and the inliers.     

Outliers, inliers and the generalized least trimmed squares estimator in system identification

The least trimmed squares estimator (LTS) is a well known robust estinaator in terms of protecting the estimatefrom the outliers. Its high computational complexity is however a problem in practice. We show that the LTS estimate can be obtained by a simple algorithm with the complexity O( N In N) for large N, where N is the number of measurements. We also showthat though the LTS is robust in terms of the outliers, it is sensitive to the inliers. The concept of the inliers is introduced. Moreover, the Generalized Least Trimmed Squares estimator (GLTS) together with its solution are presented that reduces the effect of both the outliers and the inliers.     

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.     

同参数估计对偶的自适应控制算法

本文把线性和非线性系统统一处理。从自适应控制算法与参数估计算法的对偶性出发,提出了自适应控制算法的一种统一格式。这种格式算法简单,并在一定的条件下,能使控制误差一致的足够小。     

Detection techniques in least squares identification

Parameter estimation schemes based on least squares identification and detection ideas are proposed for ease of computation, reduced numerical difficulties, and bias reduction in the presence of colored noise correlated with the states of the signal generating system. The algorithms are simpler because in the calculations, the state vector is at one point replaced by a quantized version. This technique avoids to some extent numerical difficulties associated with ill-conditioning in least squares schemes and thus obviates the need for square root algorithms and the need for high order precision calculations. In recursive form, the schemes are designed to yield parameter estimates with negligible bias without the additional computational effort or instability risks associated with generalized and extended least squares, recursive maximum likelihood schemes, or the method of instrumental variables. Nonrecursive schemes are designed to minimize computational effort in a batch processing situation while at the same time giving some reduction of bias in the state dependent colored noise situation.

The novel algorithms have the limitation that they are suboptimal and there is thus a consequent reduction in the speed of convergence for some applications. The merits of the proposed schemes are assessed via simulation studies in this paper and an adaptive equalization application in a companion paper.     

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.     

一种基于最小二乘支持向量机的预测控制算法

针对工业过程中普遍存在的非线性被控对象，提出一种基于最小二乘支持向量机建模的预测控制算法．首先，用具有RBF核函数的LS-SVM离线建立被控对象的非线性模型；然后，在系统运行过程中，将离线模型在每一个采样周期关于当前采样点进行线性化，并用广义预测算法实现对被控系统的预测控制．仿真结果表明了该算法的有效性和优越性．     

A perspective on convergence of adaptive control algorithms

This paper presents an overview of the current status of convergence theory for adaptive control algorithms. Rather than giving a comprehensive survey, the paper aims to emphasize the conceptual common ground between different approaches. Possible areas for future research are also discussed.