On parameter convergence in least squares identification and adaptive control

Least squares estimation is appealing in performance and robustness improvements of adaptive control. A strict condition termed persistent excitation (PE) needs to be satisfied to achieve parameter convergence in least squares estimation. This paper proposes a least squares identification and adaptive control strategy to achieve parameter convergence without the PE condition. A modified modeling error that utilizes online historical data together with instant data is constructed as additional feedback to update parameter estimates, and an integral transformation is introduced to avoid the time derivation of plant states in the modified modeling error. On the basis of these results, a regressor filtering–free least squares estimation law is proposed to guarantee exponential parameter convergence by an interval excitation condition, which is much weaker than the PE condition. And then, an identification‐based indirect adaptive control law is proposed to establish exponential stability of the closed‐loop system under the interval excitation condition. Illustrative results considering both identification and control problems have verified the effectiveness and superiority of the proposed approach.     

Improved adaptive fuzzy control for MIMO nonlinear time-delay systems

This paper presents an improved observer-based indirect adaptive fuzzy control scheme for multiinput-multioutput (MIMO) nonlinear time-delay systems.The control scheme synthesizes adaptive fuzzy control with adaptive fuzzy identification.An observer is designed to observe the system state,and an identifier is developed to identify the unknown parts of the system.The update laws for parameters utilize two types of errors in the adaptive time-delay fuzzy logic systems,the observation error and the identification error.Performance analysis proves the superiority of the update laws in terms of faster and improved tracking and parameter convergence.Simulation results of two-link manipulator demonstrate the effectiveness of the improved control scheme.     

Redesign of adaptive observers for improved parameter identification in nonlinear systems

We propose a method for redesigning adaptive observers for nonlinear systems. The redesign uses an adaptive law that is based on delayed observers. This increases the computational burden, but gives significantly better parameter identification and robustness properties. In particular, given that a special persistency of excitation condition is satisfied, we prove uniform global asymptotic stability and semi-global exponential stability of the origin of the state and parameter estimation error, and give explicit lower bounds on the convergence rate of both the state and parameter estimation error dynamics. For initial conditions with a known upper bound, we prove tunable exponential convergence rate. To illustrate the use of the proposed method, we apply it to estimate the unmeasured flow rate and the uncertain friction parameters in a model of a managed pressure drilling system. The simulation results clearly show the improved performance of the redesigned adaptive observer compared to a traditional design.     

Hybrid adaptive fuzzy identification and control of nonlinearsystems

We present a combined direct and indirect adaptive control scheme for adjusting an adaptive fuzzy controller, and adaptive fuzzy identification model parameters. First, using adaptive fuzzy building blocks, with a common set of parameters, we design and study an adaptive controller and an adaptive identification model that have been proposed for a general class of uncertain structure nonlinear dynamic systems. We then propose a hybrid adaptive (HA) law for adjusting the parameters. The HA law utilizes two types of errors in the adaptive system, the tracking error and the modeling error. Performance analysis using a Lyapunov synthesis approach proves the superiority of the HA law over the direct adaptive (DA) method in terms of faster and improved tracking and parameter convergence. Furthermore, this is achieved at negligible increased implementation cost or computational complexity. We prove a theorem that shows the properties of this hybrid adaptive fuzzy control system, i.e., bounds for the integral of the squared errors, and the conditions under which these errors converge asymptotically to zero are obtained. Finally, we apply the hybrid adaptive fuzzy controller to control a chaotic system, and the inverted pendulum system     

Efficient learning from adaptive control under sufficient excitation

Parameter convergence is desirable in adaptive control as it enhances the overall stability and robustness properties of the closed‐loop system. In existing online historical data (OHD)–driven parameter learning schemes, all OHD are exploited to update parameter estimates such that parameter convergence is guaranteed under a sufficient excitation (SE) condition which is strictly weaker than the classical persistent excitation condition. Nevertheless, the exploitation of all OHD not only results in possible unbounded adaptation but also loses the flexibility of handling slowly time‐varying uncertainties. This paper presents an efficient OHD‐driven parameter learning scheme for adaptive control, where a variable forgetting factor is specifically designed and is equipped with an estimation error feedback such that exponential parameter convergence is achieved under the SE condition without the aforesaid drawbacks. The proposed parameter learning scheme is incorporated with direct adaptive control to construct an OHD‐based composite learning control strategy. Numerical results have verified the effectiveness of the proposed approach.     

离散时间间接型模型参考自适应控制的一种系统化的分析方法

This paper presents the design and analysis of indirect model reference adaptive control(MRAC)with normalized adaptive law for a class of discrete-time systems.The main work includes three parts.Firstly,the constructed plant parameter estimation algorithm not only possesses the same properties as those of traditional estimation algorithms but also avoids the possibility of division by zero.Secondly,by finding the relationship between the plant parameter estimate and controller parameter estimate and using the properties of plant parameter estimate,the similar properties of controller parameter estimate are also established. Thirdly,based on the relationship properties between the normalizing signal and all the signals in the closed-loop system and on some important mathematical tools on discrete-time systems,as in the continuous-time case,a systematic stability and convergence analysis approach to the discrete indirect MRAC scheme is developed rigorously.     

Stochastic adaptive prediction and model reference control

Guo and Chen (1991) have recently shown how to establish the self-optimality and mean square stability of a self-tuning regulator. The idea allows us to proceed with the development of a more comprehensive theory of stochastic adaptive filtering, control and identification. In adaptive filtering, we examine both indirect and noninterlaced direct schemes for prediction, using both least-squares and gradient parameter estimation algorithms. In addition to analyzing similar direct adaptive control algorithms, we propose new generalized certainty equivalence adaptive model reference control laws with simultaneous disturbance rejection. We also establish that the parameters converge to the null space of a certain matrix. From this one may deduce the convergence of several adaptive controllers     

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.     

Parameter convergence and stability in continuous-time indirectadaptive control

The author considers an indirect adaptive unity feedback controller consisting of an mth-order SISO (single input, single output) compensator controlling an nth-order strictly proper SISO plant. It is shown that exponential convergence of the plant parameter estimation error as well as asymptotic time invariance and global exponential stability of the controlled closed-loop system can be guaranteed by requiring that the reference input has at least 2n+m points of spectral support     

随机系统的多模型直接自适应解耦控制器

针对多变量离散时间随机系统, 提出了一种采用广义最小方差性能指标的多模型直接自适应解耦控制器. 该多模型控制器由多个固定控制器和两个自适应控制器构成. 固定控制器用以覆盖系统参数的可能变化范围, 自适应控制器用以保证系统的稳定性和提高暂态性能. 该多模型控制器利用矩阵的伪交换性和拟Diophantine方程性质, 基于广义最小方差性能指标, 将随机系统辨识算法和最优控制器设计相结合, 直接辨识出控制器的参数, 通过广义最小方差性能指标中加权多项式的选取,不但实现了多变量系统的动态解耦控制, 而且消除了稳态误差、配置了闭环极点. 文末给出了全局收敛性分析. 仿真结果表明该方法明显优于常规自适应控制器.     

Rate of convergence in model reference adaptive control

A new adaptive scheme for continuous-time model reference adaptive control systems is proposed. It is shown that this scheme allows us to increase the exponential rate of parameter convergence to an arbitrary, desired level subject to a sufficiently rich reference input and a sufficiently large adaptive gain, while retaining global stability of the overall system. For simplicity and brevity the results are presented for plants with known high-frequency gain kp.     

An exponentially convergent adaptive sliding mode control of robot manipulators

A new robust adaptive algorithm for control of robot manipulators is proposed to account for a desired transient response with global exponential convergence of tracking errors without any persistent excitating assumption on the regressor. Its novelty lies in a new dynamic sliding surface that allows a systematic combination of adaptive control and variable structure control to yield a sliding mode inside an adaptive control loop. During sliding mode, parameter uncertainty appears in terms of known variables in such a manner that a new robust parameter estimator with enhanced stability properties is established. On the other hand, if the regressor meets the persistent exciting condition, the global uniform exponential stability of the equilibrium concerning the adaptive closed-loop error equation is easily established. The proposed controller from the VSS viewpoint relaxes the longstanding condition on a priori knowledge of the size of the parametric uncertainty to induce a sliding mode. On the other hand, from the adaptive control viewpoint it relaxes the standard assumption of the persistent excitation on the regressor to obtain the exponential convergence of tracking errors. Also, the stability against time-varying parameters is briefly discussed. Concluding remarks concerning its structural behaviour are given, and computer simulation data show a robust performance.     

多变量周期系统的多模型二阶段自适应控制

对于一类参数未知的多变量周期系统,传统自适应控制方法存在参数收敛慢的问题,导致系统暂态响应差、控制效果不理想.因此,本文针对多变量周期系统设计了多模型二阶段自适应控制器.首先根据先验知识,确定不确定区域范围,并在不确定区域内建立多个自适应模型.然后根据李雅普诺夫理论得到第一阶段辨识方程;在第二阶段中,充分考虑辨识误差并...     

Contribution to the indirect decentralized adaptive control of manipulation robots

In this paper, efficient approaches to the synthesis of indirect decentralized adaptive control for manipulation robots are presented. The first part of control synthesis consists of the estimation of unknown dynamic robot parameters using the methods of recursive identification and fast dynamic as well as identification models in a symbolic form. The second part of synthesis includes the self-tuning control strategy which is a basis for adaptive control synthesis according to the estimates of the unknown dynamic parameters. Using the theory of decentralized systems, a new robust algorithm for adaptive control with the ability of adaptation in the feedforward or feedback loop are proposed. A complete stability and convergence analysis is presented. A special part of the paper represents an analysis of practical implementation of the proposed control algorithms on modern microprocessor-based robot controllers. Based on this analysis, an efficient application of indirect adaptive algorithms in real time with high-quality system performance is shown. Adaptive algorithms are verified through simulation of trajectory tracking for an industrial robot with unknown dynamic parameters of payload.     

Exponential parameter and tracking error convergence guarantees for adaptive controllers without persistency of excitation

In model reference adaptive control (MRAC) the modelling uncertainty is often assumed to be parameterised with time-invariant unknown ideal parameters. The convergence of parameters of the adaptive element to these ideal parameters is beneficial, as it guarantees exponential stability, and makes an online learned model of the system available. Most MRAC methods, however, require persistent excitation of the states to guarantee that the adaptive parameters converge to the ideal values. Enforcing PE may be resource intensive and often infeasible in practice. This paper presents theoretical analysis and illustrative examples of an adaptive control method that leverages the increasing ability to record and process data online by using specifically selected and online recorded data concurrently with instantaneous data for adaptation. It is shown that when the system uncertainty can be modelled as a combination of known nonlinear bases, simultaneous exponential tracking and parameter error convergence can be guaranteed if the system states are exciting over finite intervals such that rich data can be recorded online; PE is not required. Furthermore, the rate of convergence is directly proportional to the minimum singular value of the matrix containing online recorded data. Consequently, an online algorithm to record and forget data is presented and its effects on the resulting switched closed-loop dynamics are analysed. It is also shown that when radial basis function neural networks (NNs) are used as adaptive elements, the method guarantees exponential convergence of the NN parameters to a compact neighbourhood of their ideal values without requiring PE. Flight test results on a fixed-wing unmanned aerial vehicle demonstrate the effectiveness of the method.     

非线性不确定系统的自适应观测器设计

非线性状态观测器可改善过程控制性能和故障诊断,针对一类参数不确定非线性系统提出了自适应观测器设计方法。通过微分同胚变换,将非线性系统转换为仅依赖原系统输入、输出的自适应观测器规范形式。利用自适应调节器估计未知参数,用构造的观测器实现状态的重构。Lyapunov稳定性理论分析了状态观测误差动态方程的稳定性,用来证明所设计的自适应观测器为全局渐近收敛的,既实现了系统状态的渐近重构又确保了在持续激励条件下未知参数估计以指数快速收敛到真值,并通过仿真试验。仿真结果表明提出方法的有效性。     

An exponentially stable adaptive friction compensator

This note presents a novel adaptive compensation scheme for Coulomb friction in a servocontrol system. An adaptive observer for estimating the unknown Coulomb friction coefficient is also derived on the basis of the Lyapunov technique. In addition, a linearizing control law is developed to compensate for the friction force and obtain the tracking objective. The proposed adaptive compensation guarantees an exponential convergence for state errors and parameter error, and known adaptive schemes guarantee only an asymptotic (or stable) convergence. Simulation results demonstrate the effectiveness of the proposed method for a single-mass servocontrol system     

基于参数估计误差的鲁棒自适应律设计及验证

针对传统自适应控制系统设计的自适应律参数收敛慢进而影响控制系统瞬态性能的问题,研究一类新的基于参数估计误差修正的鲁棒自适应律设计.首先引入滤波操作给出参数估计误差的提取方法,构建出含参数估计误差修正项的自适应律,进而将该自适应律用于控制器设计和分析中,可同时实现控制误差和参数估计误差指数收敛.对比分析了几类传统自适应律和所提出自适应律的收敛性和鲁棒性,并给出了保证参数收敛所需持续激励条件的一种直观、简便的在线判别方法.数值仿真及基于自制三自由度直升机系统俯仰轴实验结果表明,基于参数误差修正的自适应律及控制器可得到优于传统自适应方法的跟踪控制和参数估计性能.     

Pole placement direct adaptive control for time-varying ill-modeledplants

The problem is discussed of preserving the stability of pole placement direct adaptive control in spite of output bounded disturbances, time varying plant model parameters, and unmodeled dynamics, assumed to be small in the mean. The controller parameter estimates are shown to track, in the mean, their true (time varying) parameter values. Such a convergence property is achieved using an ad hoc, internally generated, excitation sequence that ensures persistent excitation. In the ideal case the convergence of the parameter estimates is exponential, avoiding, in particular, possible chaotic phenomena