Generalized forgetting functions for on‐line least‐squares identification of time‐varying systems

The problem of on‐line identification of a parametric model for continuous‐time, time‐varying systems is considered via the minimization of a least‐squares criterion with a forgetting function. The proposed forgetting function depends on two time‐varying parameters which play crucial roles in the stability analysis of the method. The analysis leads to the consideration of a Lyapunov function for the identification algorithm that incorporates both prediction error and parameter convergence measures. A theorem is proved showing finite time convergence of the Lyapunov function to a neighbourhood of zero, the size of which depends on the evolution of the time‐varying error terms in the parametric model representation. Copyright © 2001 John Wiley & Sons, Ltd.     

Output‐feedback co‐ordinated decentralized adaptive tracking: The case of MIMO subsystems with delayed interconnections

Exact decentralized output‐feedback Lyapunov‐based designs of direct model reference adaptive control (MRAC) for linear interconnected delay systems with MIMO subsystems are introduced. The design process uses a co‐ordinated decentralized structure of adaptive control with reference model co‐ordination which requires an exchange of signals between the different reference models. It is shown that in the framework of the reference model co‐ordination zero residual tracking error is possible, exactly as in the case with SISO subsystems. We develop decentralized MRAC on the base of a priori information about only the local subsystems gain frequency matrices without additional a priori knowledge about the full system gain frequency matrix. To achieve a better adaptation performance we propose proportional, integral time‐delayed adaptation laws. The appropriate Lyapunov–Krasovskii type functional is suggested to design the update mechanism for the controller parameters, and in order to prove stability. Two different adaptive DMRAC schemes are proposed, being the first asymptotic exact zero tracking results for linear interconnected delay systems with MIMO subsystems. Copyright © 2005 John Wiley & Sons, Ltd.     

平面开关磁阻电机模型参考自适应位置控制

为了提高平面开关磁阻电机的位置精确度,研究一种基于模型参考自适应控制理论的平面开关磁阻电机控制方法。采用最小二乘法辨识了平面开关磁阻电机的线性化模型参数,根据李亚普若夫稳定性理论,以力指令为控制量并采用输入输出变量设计了平面开关磁阻电机模型参考自适应位置控制器,基于dSPACE半实物实时仿真系统,构建了实时在线控制实验平台,进行了平面开关磁阻电机的模型参考自适应位置控制实验。研究表明:基于模型参考自适应控制的平面开关磁阻电机系统能平稳、准确地跟随给定位置,提高了电机位置精确度,验证了提出的平面开关磁阻电机模型参考自适应控制方法的可行性和有效性。     

Adaptive‐optimal control under time‐varying stochastic uncertainty using past learning

An adaptive‐optimal control architecture is presented for adaptive control of constrained aerospace systems with matched uncertainties that are subject to dynamic stochastic change. The architecture brings together three key elements, ie, model predictive control–based reference command shaping, Gaussian process (GP)–based Bayesian nonparametric model reference adaptive control (MRAC), and online GP clustering over nonstationary GPs. Model predictive control optimizes reference model and its shaped output is passed into GP–based MRAC, which is used to learn the model in presence of significant time‐varying stochastic uncertainty while maintaining stability. Based on a likelihood ratio test, the changepoints are detected and learned. Lastly, the models are created and clustered by non‐Bayesian clustering algorithm. The key salient feature of our architecture is that not only can it detect changes but also it uses online GP clustering to enable the controller to utilize past learning of similar models to significantly reduce learning transients. Furthermore, persistence of excitation conditions are significantly relaxed due to the use of GP‐MRAC. Stability of the architecture is argued theoretically and performance is validated empirically on different scenarios for wing rock dynamics.     

The innovation algorithms for multivariable state‐space models

This paper derives the input‐output representation of the dynamical system described by a linear multivariable state‐space model and the corresponding multivariate linear regressive model (ie, multivariate equation‐error model). A projection identification algorithm, a multivariate stochastic gradient identification algorithm, and a multi‐innovation stochastic gradient (MISG) identification algorithm are proposed for multivariate equation‐error systems by using the negative gradient search and the multi‐innovation identification theory. The convergence analysis of the MISG algorithm indicates that the parameter estimation errors converge to zero under the persistent excitation condition. Finally, a numerical example illustrates the effectiveness of the proposed algorithms.     

Adaptive control for a class of uncertain linear parameter‐varying flight aircraft systems

The concept of linear parameter‐varying (LPV) model has been developed as a convenient framework to describe a special class of uncertain LPV flight aircraft systems. In this paper, an adaptive control method for a class of uncertain LPV systems whose state‐space matrix elements are unknown affine functions of a set of measurable scalar parameters is presented. Firstly, the scalar parameters are separated from the state matrices such that the LPV model is rewritten as general unknown parameter model, then state feedback adaptive control laws, in both cases: the matched uncertainty and the unmatched uncertainty, are designed with the aim of controlling the system state to follow a desired trajectory. The sufficient condition of stability is derived using a Lyapunov equation, not a parameterized Lyapunov equation. Simulation tests based on a simple example and a nonlinear model of a transport aircraft are given to illustrate the effectiveness of the control algorithm and to demonstrate that the adaptive controller satisfies the performance requirement for an aircraft control system. Copyright © 2016 John Wiley & Sons, Ltd.     

A clarifying analysis of feedback error learning in an LTI framework

Feedback error learning (FEL) is a proposed technique for reference‐feedforward adaptive control. FEL in a linear and time‐invariant (LTI) framework has been studied recently; the studies can be seen as proposed solutions to a ‘feedforward MRAC’ problem. This paper reanalyzes two suggested schemes with new interpretations and conclusions. It motivates the suggestion of an alternative scheme for reference‐feedforward adaptive control, based on a certainty‐equivalence approach. The suggested scheme differs from the analyzed ones by a slight change in both the estimator and the control law. Boundedness and error convergence are then guaranteed when the estimator uses normalization combined with parameter projection onto a convex set where stability of the estimated closed‐loop system holds. Copyright © 2008 John Wiley & Sons, Ltd.     

Indirect adaptive control of nonlinear system via dynamic multilayer neural networks with multi‐time scales

This paper deals with adaptive nonlinear identification and trajectory tracking problem via dynamic multilayer neural network with different time scales. By means of a Lyapunov‐like analysis, we determine stability conditions for the on‐line identification. Then, a sliding mode controller is designed for trajectory tracking with consideration of the modeling error and disturbance. The main contributions of the paper lie in the following aspects. First, we extend our prior identification results of single‐layer dynamic neural networks with multi‐time scales to those of multilayer case. Second, the e‐modification in standard use in adaptive control is introduced in the on‐line update laws to guarantee bounded weights and bounded identification errors. Third, the potential singularity problem in controller design is solved by using new update laws for the NN weights so that the control signal is guaranteed bounded. The stability of proposed controller is proved by using Lyapunov function. Simulation results demonstrate the effectiveness of the proposed algorithm. Copyright © 2014 John Wiley & Sons, Ltd.     

Real‐time identification of nonlinear multiple‐input–multiple‐output systems with unknown input time delay using Wiener model with Neuro‐Laguerre structure

In this article, a real‐time block‐oriented identification method for nonlinear multiple‐input–multiple‐output systems with input time delay is proposed. The proposed method uses the Wiener structure, which consists of a linear dynamic block (LDB) followed by a nonlinear static block (NSB). The LDB is described by the Laguerre filter lattice, whereas the NSB is characterized using the neural networks. Due to the online adaptation of the parameters, the proposed method can cope with the changes in the system parameters. Moreover, the convergence and bounded modeling error are shown using the Lyapunov direct method. Four practical case studies show the effectiveness of the proposed algorithm in the open‐loop and closed‐loop identification scenarios. The proposed method is compared with the recently published methods in the literature in terms of the modeling accuracy, parameter initialization, and required information from the system.     

Novel model reference adaptive control architecture using semi‐initial excitation‐based switched parameter estimator

This paper is a generalization of the recently developed techniques of initial excitation (IE)–based adaptive control with an introduction to the definition of semi‐initial excitation (semi‐IE), a still more relaxed notion than IE. Classical adaptive controllers typically ensure Lyapunov stability of the extended error dynamics (tracking error + parameter estimation error) and asymptotic tracking, while requiring a stringent condition of persistence of excitation (PE) for parameter convergence. Of late, the authors have proposed a new adaptive control architecture, which guarantees parameter convergence under the online‐verifiable IE condition leading to exponential stability of the extended error dynamics. In earlier works, it has been established that the IE condition is significantly milder than the classical PE condition. The current work further slackens the excitation condition by proposing the concept of semi‐IE. The proposed adaptive controller is proved to ensure convergence of the parameter estimation error to a lower‐dimensional manifold under the weaker semi‐IE condition, while the stronger condition of IE guarantees convergence of the parameter estimation error to zero. The designed algorithm is shown to improve transient response of tracking error sufficiently in contrast to conventional adaptive controllers.     

Set‐theoretic adaptive filtering based on data‐driven sparsification

In this article, we propose a fast and efficient algorithm named the adaptive parallel Krylov‐metric projection algorithm. The proposed algorithm is derived from the variable‐metric adaptive projected subgradient method, which has recently been presented as a unified analytic tool for various adaptive filtering algorithms. The proposed algorithm features parallel projection—in a variable‐metric sense—onto multiple closed convex sets containing the optimal filter with high probability. The metric is designed based on (i) sparsification by means of a certain data‐dependent Krylov subspace and (ii) maximal use of the obtained sparse structure for fast convergence. The numerical examples show the advantages of the proposed algorithm over the existing ones in stationary/nonstationary environments. Copyright © 2011 John Wiley & Sons, Ltd.     

基于模型参考自适应的异步电动机转动惯量的辨识

概述了基于模型参考自适应理论的异步电动机模型辨识及参数自测定方法,介绍基于该理论的异步电动机转速和转动惯量的辨识算法,给出仿真结果及有关结论。     

An adaptive sliding‐mode observer for a class of uncertain nonlinear systems

In this paper, the problem of simultaneous state and parameter estimation is studied for a class of uncertain nonlinear systems. A nonlinear adaptive sliding‐mode observer is proposed based on a nonlinear parameter estimation algorithm. It is shown that such a nonlinear algorithm provides a rate of convergence faster than exponential, ie, faster than the classic linear algorithm. Then, the proposed parameter estimation algorithm is included in the structure of a sliding‐mode state observer, providing an ultimate bound for the full estimation error and attenuating the effects of the external disturbances. Moreover, the synthesis of the observer is given in terms of linear matrix inequalities. The corresponding proofs of convergence are developed based on the Lyapunov function approach and input‐to‐state stability theory. Some simulation results illustrate the efficiency of the proposed adaptive sliding‐mode observer.     

Robust adaptive control for unmatched systems with guaranteed parameter estimation convergence

This paper provides a modified model reference adaptive control (MRAC) scheme to achieve better transient control performance for systems with unknown unmatched dynamics, where an adaptive law with guaranteed convergence is introduced. We first revisit the standard MRAC system and analyze the tracking error bound by using L₂‐norm and Cauchy‐Schwartz inequality. Based on this analysis, we suggest a feasible way to compensate the undesired transient dynamics induced by the gradient descent–based adaptive laws subject to sluggish convergence or even parameter drift. Then, a modified adaptive law with an alternative leakage term containing the parameter estimation error is developed. With this adaptive law, the convergence of both the estimation error and tracking error can be proved simultaneously. This enhanced convergence property can contribute to deriving smoother control signal and improved control response. Moreover, this paper provides a simple and numerically feasible approach to online verify the well‐known persistent excitation condition by testing the positive definiteness of an introduced auxiliary matrix. Comparative simulations based on a benchmark 3‐DOF helicopter model are given to validate the effectiveness of the proposed MRAC approach and show the improved performance over several other MRAC schemes.     

一种具有积分误差约束的模型参考自适应辨识算法

提出了一种具有积分误差目标函数约束的模型参考自适应辨识算法，给出了切实体现积分约束作用的参数辨识算法的实现形式。该算法不仅可辨识包括时滞在内的过程模型参数，而且可有效地抑制建模误差和随机干扰对参数辨识精度的影响。基于Lyapunov稳定性理论证明了辨识算法全局渐近收敛性，仿真结果表明了算法的有效性。     

Model reference adaptive control system design for linear time‐varying systems with unstructured and bounded varying functions

In recent years, many methods of model reference adaptive control system (MRACS) for a linear time‐varying (LTV) plant have been proposed. These methods assumed that the structure of plant parameters is known in advance. However, it is difficult to get a priori information of plant parameters. In this paper, an MRACS design for an LTV system based on high‐order estimator (HOE) is proposed. By applying dynamic certainty equivalence (DyCE) to LTV plants, a new MRAC law of LTV system is derived without knowing the structure of the plant parameters. The MRACS law is generated by using high‐order derivatives of an estimated parameter, so that robust HOE with a normalization signal and σ modification for the system introduced. Our proposed method can attain better performance than conventional methods, such as estimation with variable forgetting factor (VF) and the gradient projection method (GPM). The robust HOE establishes the boundedness of all of the estimated parameters under the condition that the estimated parameter and the first derivative of the parameter are bounded. It is shown that all signals in the adaptive loop are bounded and the output error converges to a closed set. The proposed method is compared to the familiar schemes, the gradient projection method and the estimation based on forgetting factor through numerical simulations, and the effectiveness of our proposed method is shown. © 2000 Scripta Technica, Electr Eng Jpn, 130(4): 87–98, 2000     

A new adaptive backstepping control algorithm for motion control systems — an implicit and symbolic computation approach

A new adaptive integral backstepping control algorithm for motion control systems is proposed. The design approach is non‐conventional in that it only implicitly incorporates error states into a design Lyapunov function and computes time derivative of the Lyapunov function symbolically to the point of getting parameter estimate update laws on which desirable structures are imposed. The closed‐loop error system turns out to be linear and time‐varying and can be reformulated as a linear–non‐linear cascade system. Uniform exponential stability or absolute stability can be achieved. The design is robust against system parameter variations. The effectiveness and robustness of the control algorithm are verified by numerical examples. Copyright © 2002 John Wiley & Sons, Ltd.     

Online concurrent reinforcement learning algorithm to solve two‐player zero‐sum games for partially unknown nonlinear continuous‐time systems

Online adaptive optimal control methods based on reinforcement learning algorithms typically need to check for the persistence of excitation condition, which is necessary to be known a priori for convergence of the algorithm. However, this condition is often infeasible to implement or monitor online. This paper proposes an online concurrent reinforcement learning algorithm (CRLA) based on neural networks (NNs) to solve the H _∞ control problem of partially unknown continuous‐time systems, in which the need for persistence of excitation condition is relaxed by using the idea of concurrent learning. First, H _∞ control problem is formulated as a two‐player zero‐sum game, and then, online CRLA is employed to obtain the approximation of the optimal value and the Nash equilibrium of the game. The proposed algorithm is implemented on actor–critic–disturbance NN approximator structure to obtain the solution of the Hamilton–Jacobi–Isaacs equation online forward in time. During the implementation of the algorithm, the control input that acts as one player attempts to make the optimal control while the other player, that is, disturbance, tries to make the worst‐case possible disturbance. Novel update laws are derived for adaptation of the critic and actor NN weights. The stability of the closed‐loop system is guaranteed using Lyapunov technique, and the convergence to the Nash solution of the game is obtained. Simulation results show the effectiveness of the proposed method. Copyright © 2014 John Wiley & Sons, Ltd.     

Widely convergent method for finding solutions of simultaneous nonlinear equations

The authors propose a new approach for solving the nonlinear problem, based on a model-reference adaptive control (MRAC). The algorithm is composed of three parts: a plant containing unknown parameters, a reference model for compactly specifying the desired output and an adaptation mechanism for updating the adjustable parameters. The structure of the adaptation mechanism is explicitly designed so that the asymptotical stability is guaranteed according to Lyapunov theorem. The method reformulates the algebraic nonlinear equations into a set of ordinary differential equations, whose equilibrium point corresponds to the solution of the original nonlinear problems. The stability and region of attraction coincide with entire state space no matter whether the parameters reside in solvable region or unsolvable region.     

Least‐squares parameter estimation for systems with irregularly missing data

This paper considers the problems of parameter identification and output estimation with possibly irregularly missing output data, using output error models. By means of an auxiliary model (or reference model) approach, we present a recursive least‐squares algorithm to estimate the parameters of missing data systems, and establish convergence properties for the parameter and missing output estimation in the stochastic framework. The basic idea is to replace the unmeasurable inner variables with the output of an auxiliary model. Finally, we test the effectiveness of the algorithm with an example system. Copyright © 2009 John Wiley & Sons, Ltd.