Adaptive prescribed performance control of nonlinear systems with unknown dead zone

An alternative adaptive control with prescribed performance is proposed to address the output tracking of nonlinear systems with a nonlinear dead zone input. An appropriate function that characterizes the convergence rate, maximum overshoot, and steady‐state error is adopted and incorporated into an output error transformation, and thus the stabilization of the transformed system is sufficient to achieve original tracking control with prescribed performance. The nonlinear dead zone is represented as a time‐varying system and Nussbaum‐type functions are utilized to deal with the unknown control gain dynamics. A novel high‐order neural network with a scalar adaptive weight is developed to approximate unknown nonlinearities, thus the computational costs can be diminished dramatically. Some restrictive assumptions on the system dynamics and the dead‐zone are circumvented. Simulations are included to validate the effectiveness of the proposed scheme. Copyright © 2012 John Wiley & Sons, Ltd.     

Online data‐driven composite adaptive backstepping control with exact differentiators

This paper presents an online data‐driven composite adaptive backstepping control for a class of parametric strict‐feedback nonlinear systems with mismatched uncertainties, where both tracking errors and prediction errors are utilized to update parametric estimates. Hybrid exact differentiators are applied to obtain the derivatives of virtual control inputs such that the complexity problem of integrator backstepping can be avoided. Closed‐loop tracking error equations are integrated in a moving‐time window to generate prediction errors such that online recorded data can be utilized to improve parameter adaptation. Semiglobal asymptotic stability of the closed‐loop system is rigorously established by the time‐scales separation and Lyapunov synthesis. The proposed composite adaptation can not only avoid the application of identification models and linear filters resulting in a simpler control structure, but also suppress parametric uncertainties and external perturbations via the time‐interval integral. Simulation results have demonstrated that the proposed approach possesses superior control performances under both noise‐free and noisy‐measurement environments. Copyright © 2015 John Wiley & Sons, Ltd.     

Decentralized optimal tracking control for large-scale nonlinear systems with tracking error constraints

In this article, a decentralized optimal tracking control strategy is proposed for a class of nonlinear systems with tracking error constraints by utilizing adaptive dynamic programming (ADP). It should be noted that ADP technology cannot be directly used to solve decentralized optimal tracking problem of large-scale interconnected nonlinear system with nonzero equilibrium points, since that an infinite domain performance index function may result in an unsolvable solution. In addition, by introducing a smooth function, the constrained tracking error is transformed into an unconstrained one. Then, the error dynamics and a new infinite domain performance index function are designed, such that ADP technology can be used. Following the designed performance index function, the tracking error can be ensured within a small neighborhood of zero. Finally, the feasibility and the effectiveness of the proposed decentralized optimal control scheme are verified through two simulation examples.     

Adaptive dynamic programming for model-free tracking of trajectories with time-varying parameters

Recently proposed adaptive dynamic programming (ADP) tracking controllers assume that the reference trajectory follows time-invariant exo-system dynamics—an assumption that does not hold for many applications. In order to overcome this limitation, we propose a new Q-function that explicitly incorporates a parametrized approximation of the reference trajectory. This allows learning to track a general class of trajectories by means of ADP. Once our Q-function has been learned, the associated controller handles time-varying reference trajectories without the need for further training and independent of exo-system dynamics. After proposing this general model-free off-policy tracking method, we provide an analysis of the important special case of linear quadratic tracking. An example demonstrates that our new method successfully learns the optimal tracking controller and outperforms existing approaches in terms of tracking error and cost.     

Indirect adaptive control of a class of marine vehicles

A nonlinear adaptive framework for bounded‐error tracking control of a class of non‐minimum phase marine vehicles is presented. The control algorithm relies on a special set of tracking errors to achieve satisfactory tracking performance while guaranteeing stable internal dynamics. First, the design of a model‐based nonlinear control law, guaranteeing asymptotic stability of the error dynamics, is presented. This control algorithm solves the tracking problem for the considered class of marine vehicles, assuming full knowledge of the system model. Then, the analysis of the zero‐dynamics is carried out, which illustrates the efficacy of the chosen set of tracking errors in stabilizing the internal dynamics. Finally, an indirect adaptive technique, relying on a partial state predictor, is used to address parametric uncertainties in the model. The resulting adaptive control algorithm guarantees Lyapunov stability of the errors and parameter estimates, as well as asymptotic convergence of the errors to zero. Numerical simulations illustrate the performance of the adaptive algorithm. Copyright © 2009 John Wiley & Sons, Ltd.     

Fuzzy adaptive control of a class of nonlinear systems with unmodeled dynamics

This paper investigates the tracking control problem for a class of pure‐feedback systems with unmodeled dynamics. The useful properties of the fuzzy basis functions and membership are explored to be used for stability analysis, and an alternative Lyapunov function depending on both control input and system state is utilized. Then, an adaptive fuzzy controller is designed to ensure that the tracking error is within a small adjustable neighborhood of the origin, where some conventional assumptions imposed on the unmodeled dynamics have been relaxed. Finally, simulation results are given to validate the theoretical results.     

基于 actor-critic 框架的在线积分强化学习算法研究

针对轮式移动机器人动力学系统难以实现无模型的最优跟踪控制问题,提出了一种基于actor-critic框架的在线积分强化学习控制算法。首先,构建RBF评价神经网络并基于近似贝尔曼误差设计该网络的权值更新律,以拟合二次型跟踪控制性能指标函数。其次,构建RBF行为神经网络并以最小化性能指标函数为目标设计权值更新律,补偿动力学系统中的未知项。最后,通过Lyapunov理论证明了所提出的积分强化学习控制算法可以使得价值函数,行为神经网络权值误差与评价神经网络权值误差一致最终有界。仿真和实验结果表明,该算法不仅可以实现对恒定速度以及时变速度的跟踪,还可以在嵌入式平台上进行实现。     

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.     

新型注入式混合有源滤波器的滑模变结构控制

通过对新型注入混合有源滤波器的建模，该文结合传统滑模变结构控制快速性较好和递推积分PI控制无稳态误差的优点，提出了一种新的滑模变结构控制算法作为有源滤波器的电流跟踪控制方法。通过仿真，从算法的跟踪速度和控制精度2个方面，将新的滑模变结构控制方法与传统滑模变结构控制以及单一的递推积分PI控制进行了比较研究。实验结果也证明了该算法的正确性和新型注入式混合有源滤波器滤波的有效性。     

基于Lyapunov稳定性理论的电励磁直线同步电机自适应控制

为了提高直线伺服系统的动态性能,克服电励磁直线同步电机存在的参数时变性,设计模型参考自适应速度控制器。系统内环是基于参考模型的速度跟踪控制器,外环自适应机构在线调整速度跟踪控制器的可调参数并使参考模型输出速度与控制对象输出速度之间的广义速度误差趋近于零。采用基于Lyapunov稳定性理论的模型参考自适应速度控制器设计方法,在保证广义速度跟踪误差收敛至零的同时,还保证了模型参考自适应速度控制系统具有稳定性和收敛性。采用欧拉数值积分方法,经过计算机数值仿真计算得到参考模型与控制对象的速度输出和初级交轴电流状态变量的时域响应曲线,验证了该自适应速度控制系统具有全局收敛性。     

Output‐feedback H∞ quadratic tracking control of linear systems using reinforcement learning

This paper presents an online learning algorithm based on integral reinforcement learning (IRL) to design an output‐feedback (OPFB) H_∞ tracking controller for partially unknown linear continuous‐time systems. Although reinforcement learning techniques have been successfully applied to find optimal state‐feedback controllers, in most control applications, it is not practical to measure the full system states. Therefore, it is desired to design OPFB controllers. To this end, a general bounded L₂ ‐gain tracking problem with a discounted performance function is used for the OPFB H_∞ tracking. A tracking game algebraic Riccati equation is then developed that gives a Nash equilibrium solution to the associated min‐max optimization problem. An IRL algorithm is then developed to solve the game algebraic Riccati equation online without requiring complete knowledge of the system dynamics. The proposed IRL‐based algorithm solves an IRL Bellman equation in each iteration online in real time to evaluate an OPFB policy and updates the OPFB gain using the information given by the evaluated policy. An adaptive observer is used to provide the knowledge of the full states for the IRL Bellman equation during learning. However, the observer is not needed after the learning process is finished. A simulation example is provided to verify the convergence of the proposed algorithm to a suboptimal OPFB solution and the performance of the proposed method.     

A modular adaptive control design with ISS analysis for nonminimum phase hypersonic vehicle models

Air‐breathing hypersonic vehicles typically exhibit a nonminimum phase behavior when altitude is controlled via lift generation. This phenomenon prohibits the use of classical inversion‐based control techniques. Dynamic models of these vehicles are also subject to parametric uncertainties and unmodeled dynamics related to flexible effects of the fuselage. In this paper, we present a modular adaptive control method that achieves asymptotic setpoint tracking in both airspeed and altitude using thrust and elevator deflection as the only control inputs for a generic longitudinal model of a hypersonic cruise. The nonminimum phase problem is overcome through output redefinition, with altitude controlled by pitching moment. The internal dynamics, flight path angle, and altitude are then stabilized by saturating the interconnections and exploiting local stability properties. A new technique for the use of saturation functions in error coordinate is presented. The adaptive controller for altitude uses a pitch rate observer combined with projection. This control augmentation decouples the parameter estimation errors from internal dynamics, allowing for the use of small‐gain arguments. Simulation results from a vehicle model with flexible effects and parametric uncertainty are included to demonstrate control effectiveness.     

Robust model reference adaptive control for multivariable plants

A complete procedure for generating and analysing robust model reference adaptive control schemes for multivariable plants is developed. The procedure consists of two parts: the first part involves the characterization of the integral structure of the modelled part of the plant, and the associated parametrization of the controller structure; and the second part involves the development of a general robust adaptive law for adjusting the controller parameters so that the closed-loop plant is globally stable despite the presence of unmodelled dynamics and bounded disturbances. The use of dominantly rich signals for improving convergence and the bounds for the tracking and parameter error is also analysed.     

新型随动系统的非线性控制和综合方法及实验验证

提出了一种新的电动机随动系统的非线性控制和综合方法.通过引进逆步式设计方法和流程,作者提出了新的随动系统非线性控制的算法和解析解,并且推证了系统的指数式稳定性.进而,作者将非线性控制算法和解析解演化为新的非线性控制系统结构图.在通过数字仿真验证了理论推导后,作者用了单片DSP(TMS320C32)来实现新的电动机随动系统的非线性数字控制.实验和仿真结果验证了新的控制结构和综合方法可以提高随动控制系统的外环频带宽度,减少对于斜坡给定信号的跟踪误差,和提高抗负荷干扰的性能.     

多谐振最优伺服控制器在中频电源中的应用

针对标准线性二次型调节器(LQR)不能无静差跟踪参考以及比例谐振(PR)控制器带非线性负载时输出电压总谐波畸变率过高的问题,提出一种多谐振最优伺服控制算法并应用到中频电源设计中。该方法能够实现闭环系统稳定和渐进跟踪,同时还具有动态响应快、抑制谐波能力强的优点。建立中频电源系统模型,并对最优控制和多谐振伺服控制进行理论分析,通过系统矩阵维数增广的方法将最优跟踪问题转化为最优调节问题,利用Matlab软件的lqr函数求解出状态反馈增益矩阵。通过仿真软件和实验样机对整个系统进行了仿真和实验验证,结果表明:所设计的控制器使闭环系统具有良好的稳态性能和动态性能,稳态电压稳定在114.8~115.2V,动态响应恢复时间在10ms以内,对非线性负载具有较强的谐波抑制能力,输出电压总谐波远少于PR控制和鲁棒控制等方法。     

Robustness and performance of a modified adaptive backstepping controller

Recently, a new class of adaptive control schemes based on non-linear design techniques, have been proposed for minimum phase linear time invariant plants. Under certain assumptions on the plant transfer function, these schemes guarantee uniform signal boundedness and good transient and steady-state response for the regulation or tracking error. In this paper we propose a modification that improves the robustness of these schemes with respect to a class of multiplicative uncertainties and input, output disturbances, without loss of performance. This lack of a trade-off between performance and robustness is a result of restricting the multiplicative unmodelled dynamics to be small in both the low- and high-frequency range. It is also shown that parameter convergence and performance improvement is achieved with a dominantly rich reference input. © 1998 John Wiley & Sons, Ltd.     

Command-filtered-based neuroadaptive control for multi-input multi-output saturated nonstrict-feedback nonlinear systems with prescribed tracking performance

In this article, the prescribed performance control strategy is extended to multi-input multi-output nonstrict-feedback nonlinear systems with asymmetric input saturation, and not only each element in tracking error vector converges to a prescribed small region within preassigned finite time, but also the converging mode during the preset time is prespecifiable and controllable explicitly. By blending the barrier function with novel speed function, a prescribed performance controller using command-filtered-based vector-backstepping design framework is proposed to steer the tracking error vector for the first time, where the boundedness of filter errors is guaranteed by sufficiently small time constant and an error compensator is constructed to handle the effects of filter errors. To attenuate the adverse effects resulted from nondifferentiable input saturation, hyperbolic tangent function is utilized to estimate asymmetric saturation function such that the control input is designed as a new state variable with initial value of zero in augmented system. Nussbaum function is employed to overcome singularity problem caused by the differentiation of hyperbolic tangent function. At each step of backstepping design, the universal approximation property of neural network and the command filter system are utilized to approximate uncertain dynamics and to solve algebraic loop obstacle due to nonstrict-feedback structure, respectively. Moreover, only one parameter needs to be updated online to cope with the lumped uncertain dynamics by virtual parameter technology, rendering a control strategy with low complexity computation. The validity of the presented controller is verified by theoretical analysis and two-link robotic system.     

Filtered adaptive constrained sampled-data control for uncertain multivariable nonlinear systems

A filtered adaptive constrained sampled-data controller for uncertain multivariable nonlinear systems in the presence of various constraints is synthesized in this paper. A piecewise constant adaptive law drives that estimation error dynamics to zero at each sampling time instant yields adaptive parameters. The filtered control scheme consists of two components. Based on an estimation/cancellation strategy, a disturbance rejection control law is designed to compensate the nonlinear uncertainties within the bandwidth of low-pass filters, whereas a constraint violation avoidance control law is designed to solve an online constrained optimization problem. Although a reduced sampling time helps to minimize the estimation error caused by the neglect of unknowns, the resulting aggressive signals put more restrictions on the control law. Greater sacrifice of tracking performance is required to satisfy the constraints. The constraints violation avoidance control law is in favor of a larger sampling time. Sufficient conditions are given to guarantee the stability of the closed-loop system with the sampled-data controller, where the input/output signals are held constant over the sampling period. Numerical examples are provided to validate the theoretical results, comparisons between the constrained sampled-data controller and unconstrained adaptive controller with the implementation of different sampling times are carried out.     

Algorithms of adaptive tracking of unknown multisinusoidal signals in linear systems with arbitrary input delay

The problem of adaptive tracking control is addressed for the class of linear time‐invariant plants with known parameters and arbitrary known input delay. The reference signal is a priori unknown and is represented by a sum of biased harmonics with unknown amplitudes, frequencies, and phases. Asymptotic tracking is provided by predictive adjustable control with parameters generated by one of three designed adaptation algorithms. The first algorithm is based on a gradient scheme and ensures zero steady‐state tracking error with all signals bounded. The other two algorithms additionally involve the scheme with fast parametric convergence improving the closed‐loop system performance. In all the algorithms, the problem of delay compensation is resolved by special augmentation of tracking error. The adjustable control law proposed do not require identification of the reference signal parameters.     

Indirect model reference adaptive control with dynamic adjustment of parameters

The paper discusses in detail a new method for indirect model reference adaptive control (MRAC) of linear time-invariant continuous-time plants with unknown parameters. The method involves not only dynamic adjustment of plant parameter estimates but also those of the controller parameters. Hence the overall system can be described by a set of non-linear differential equations as in the case of direct control. Many of the difficulties encountered in the conventional indirect approach, where an algebraic equation is solved to determine the control parameters, are consequently bypassed in this method. The proof of stability of the equilibrium state of the overall system is found to be different from that used in direct control. Using Lyapunov's theory, it is first shown that the parameter errors between the parameter estimates of the identifier and the true parameters of the plant, as well as those between the actual parameters of the controller and their desired values, are bounded. Following this, using growth rates of signals in the adaptive loop as well as order arguments, it is shown that the error equations are globally uniformly stable and that the tracking (control) error tends to zero asymptotically. This in turn establishes the fact that both direct and indirect model reference adaptive schemes require the same amount of prior information to achieve stable adaptive control.