A Discrete-Time Periodic Adaptive Control Approach for Time-Varying Parameters With Known Periodicity

A periodic adaptive control approach is proposed for a class of nonlinear discrete-time systems with time-varying parametric uncertainties that are periodic, and the only prior knowledge is the periodicity. The new adaptive controller updates the parameters and the control signal periodically in a point-wise manner over one entire period, and in the sequel, achieves the asymptotic tracking convergence. The result is further extended to a scenario with mixed time-varying and time-invariant parameters, and a hybrid classical and a periodic adaptation law is proposed to handle the scenario more appropriately. The extension of the periodic adaptation to systems with unknown input gain, higher order dynamics, and tracking problems is also discussed.     

周期时变时滞非线性参数化系统的自适应学习控制

针对一阶未知非线性参数化周期时变时滞系统, 设计了一种自适应学习控制方案. 假设未知时变参数, 时变时滞和参考信号的共同周期是已知的, 通过重构系统方程, 将包含时变时滞在内的所有未知时变项合并成为一个周期时变向量, 采用周期自适应律估计该向量. 通过构造一个Lyapunov-Krasovskii型复合能量函数证明了所有信号有界并且跟踪误差收敛. 结果被推广到一类含有混合参数的高阶非线性系统. 通过两个仿真例子说明本文所提出的控制算法的有效性.     

Persistent identification and adaptation: stabilization of slowlyvarying systems in H∞

In this paper, the problem of persistent identification and adaptive stabilization of time-varying systems is studied within the framework of deterministic worst case identification and slow H^∞ adaptation. The plants under consideration are unstable and time-varying and cannot be stabilized by a fixed robust controller. Starting from an initial well-designed operating point, the controller must persistently adapt to the time-varying plant to maintain uniform stability over all future time. A key property which guarantees uniform stability is that the identification-adaptation iteration satisfies a certain invariance principle. We demonstrate that the adaptive design using periodic external inputs, least squares identification, and slow H^∞ adaptation possesses such an invariance property, leading to a successful adaptive stabilization methodology. Generic natures of our findings are discussed     

Adaptation of diagonal recurrent neural network model

An adaptive direct recurrent neural network model is developed for nonlinear dynamic system modelling in this paper. The model adaptation is achieved with the extended Kalman filter (EKF). A novel recursive algorithm is proposed to calculate the Jacobian matrix in the model adaptation so that the algorithm is simple and converges fast. The effectiveness of the developed adaptive model is demonstrated by applying to modelling a simulated continuous stirred tank reactor (CSTR). The model converges to the new process dynamics very quickly after a constant disturbance is added, and therefore can be used as an adaptive model in the adaptive model predictive control or internal model control for time-varying systems or fault tolerant control of nonlinear systems.     

Partial-period adaptive repetitive control by symmetry

This paper presents a partial-period adaptive repetitive control method for a class of periodically time-varying nonlinear systems. To reduce the size of memory usage, the symmetric feature of periodic parameters is explored to form partial-period adaptation mechanisms. Both Half- and quarter-period adaptation strategies are proposed, and characterized analytically. The stability of the closed-loop system with each repetitive control is established, along with tracking error convergence to zero. In addition, the saturated-adaptation is suggested for providing bounded estimation.     

一类高阶非线性参数化系统自适应重复学习控制

针对一类高阶非线性参数化系统,利用参数重组技巧,提出了一种自适应重复学习控制方法.该方法结合反馈线性化,可以处理参数在一个未知紧集内周期性、快时变的非线性系统.通过引进微分-差分参数自适应律,设计了一种自适应控制策略,使广义跟踪误差在误差平方范数意义下渐近收敛于零.通过构造Lyapunov泛函,给出了闭环系统收敛的一个充分条件.实例仿真结果说明了该方法的可行性和有效性.     

On the stability of continuous-time adaptive controllers for pure delay systems

We analyse the stability properties of the linear retarded differential-difference equation (RDDE) that arises in the study of adaptive control of pure delay systems. Three different results are given. Firstly, using Floquet theory necessary and sufficient conditions for stability are established for the case of periodic signals of specified frequencies. Secondly, the theory of averaging is used to derive stability-instability conditions under slow adaptation. Finally, a globally stable modified adaptive regulator for systems with known, possibly time-varying, time delay is presented. Two alternative proofs of the latter results are given. One based on the method of Lyapunov functionals and the second one using a Razumikhin-type theorem.     

输入饱和非线性系统的周期自适应补偿学习控制

针对一类输入饱和不确定Brunovsky标准型非线性时滞系统,提出一种周期自适应跟踪补偿学习算法. 利用信号置换思想重组系统,基于最小公倍周期函数变换,将时滞时变项和不确定项合并为辅助参数,进而设计周期自适应学习律估计该辅助量,并利用饱和补偿器逼近和补偿超出饱和限的部分,由此构成综合控制器,以保证系统状态对有界期望值的跟踪,解决了饱和输入周期系统的重复迭代学习控制问题. 最后通过构造Lyapunov-Krasovskii复合能量函数的差分,计算证明了系统跟踪误差的收敛性和闭环信号值的有界性. 常见耦合非线性机械臂系统的力矩控制仿真,进一步验证了该算法的有效性.     

Adaptive learning tracking control of robotic manipulators with uncertainties

An adaptive learning tracking control scheme is developed for robotic manipulators by a synthesis of adaptive control and learning control approaches. The proposed controller possesses both adaptive and learning properties and thereby is able to handle robotic systems with both time-varying periodic uncertainties and time invariant parameters. Theoretical proofs are established to show that proposed controllers ensure asymptotical tracking performance. The effectiveness of the proposed approaches is validated through extensive numerical simulation results.     

周期时变系统的鲁棒自适应重复控制

针对周期时变系统,提出一种鲁棒自适应重复控制方法.该方法利用周期学习律估计周期时变参数,并结合鲁棒自适应方法处理非周期不确定性.与现有重复控制不同的是,在控制器设计中引入了新变量—周期数,利用周期系统的重复特性,使界的逼近误差随周期数的增加而逐渐减少,保证了系统的全局渐近稳定性.同时将该方法应用于一类非线性参数化系统,使系统在非参数化扰动的情形下,输出误差仍能收敛于0,倒立摆模型的仿真验证了此结果.该设计方法适用于消除神经网络逼近误差对重复控制系统的影响,理论证明了基于神经网络的鲁棒自适应重复控制系统中所有变量的有界性和输出误差的渐近收敛性,关于机械臂模型的仿真结果验证了受控系统具有良好的跟踪性能.     

Robust adaptive state-feedback tracking for nonlinear systems

For a class of singie-input/single-output uncertain nonlinear systems, affected both by uncertain time-varying parameters (with known bounds) and unknown time-varying bounded disturbances, a new robust adaptive state-feedback control algorithm is presented. It guarantees: boundedness of all signals and arbitrary disturbance attenuation when both disturbances and time-varying parameters are present, and asymptotic tracking with arbitrary transient performance when no disturbance is acting on the system and parameters are constant. The adaptation may be switched off, still guaranteeing bounded signals and disturbance attenuation     

Finite-time H∞ control of periodic piecewise linear systems

In this paper, the finite-time stability, stabilisation, L₂-gain and H_∞ control problems for a class of continuous-time periodic piecewise linear systems are addressed. By employing a time-varying control scheme in which the time interval of each subsystem constitutes a number of basic time segments, the finite-time controllers can be developed with periodically time-varying control gains. Based on a piecewise time-varying Lyapunov-like function, a sufficient condition of finite-time stability and the relevant time-varying controller are proposed. Considering the finite-time boundedness of the closed-loop periodic system, the L₂-gain criterion with continuous time-varying Lyapunov-like matrix function is studied. A finite-time H_∞ controller is proposed based on the L₂-gain analysis. Finally, numerical simulations are presented to illustrate the effectiveness of the proposed criteria.     

Reduction model approach for linear time-varying systems with input delays based on extensions of Floquet theory

We solve stabilization problems for linear time-varying systems under input delays. We show how changes of coordinates lead to systems with time invariant drifts, which are covered by the reduction model method and which lead to the problem of stabilizing a time-varying system without delay. For continuous time periodic systems, we can use Floquet theory to find the changes of coordinates. We also prove an analogue for discrete time systems, through a discrete time extension of Floquet theory.     

Adaptive control with optimal robust tracking

The problem of optimal robust tracking in two-parameter adaptive control systems under non-linear time-varying unmodelled dynamics is examined. A new robust stability criterion is derived for analysing the robustness of adaptive control systems with non-linear time-varying model errors. Based on the concept of excess robustness and the theory of the minimum H^∞norm, a simple and feasible design algorithm is presented to synthesize a two-parameter adaptive controller which ensures that adaptive control systems can achieve the object of optimal robust tracking in the presence of non-linear time-varying unmodelled dynamics. Simulation results that demonstrate features of the two-parameter adaptive controller with optimal robust tracking in the light of the design algorithm are included.     

Adaptive control by predictive identification and optimization

A model-reference adaptive control system is described where extrapolation techniques are used for identification and for error-prediction at discrete time intervals. The system employs rectangular adaptation pulses of finite duration to minimize a cost-functional of predicted square errors. Weighted squares of the error rate-of-change are included in the cost-functional to be minimized and a number of constraints are considered. Simulation resuits for systems consisting of linear, time-varying, and nonlinear second- to fifth-order processes with linear second-order reference-models are given, where satisfactory adaptation is accomplished.     

Global output-feedback stabilization for a class of uncertain time-varying nonlinear systems

This paper investigates the global output-feedback stabilization for a class of uncertain time-varying nonlinear systems. The remarkable structure of the systems is the presence of uncertain control coefficients and unmeasured states dependent growth whose rate is inherently time-varying and of unknown polynomial-of-output, and consequently the systems have heavy nonlinearities, serious uncertainties/unknowns and serious time-variations. This forces us to explore a time-varying plus adaptive methodology to realize the task of output-feedback stabilization, rather than a purely adaptive one. Detailedly, based on a time-varying observer and transformation, an output-feedback controller is designed by skillfully combining adaptive technique, time-varying technique and well-known backstepping method. It is shown that, with the appropriate choice of the design parameters/functions, all the signals of the closed-loop system are bounded, and furthermore, the original system states globally converge to zero. It is worth mentioning that, the heavy nonlinearities are compensated by an updating law, while the serious unknowns and time-variations are compensated by a time-varying function. The designed controller is still valid when the system has an additive input disturbance which, essentially different from those studied previously, may not be periodic or bounded by any known constant.     

Application of iterative suboptimal strategies for improving adaptation transients in adaptive systems

Some solutions which involve iterative optimization techniques are given to improve adaptation transients in adaptive systems. Two suboptimal controllers are developed and applied to an equivalent discrete-time system valid for describing the behaviour of a recent adaptive control scheme when one of the (time-varying) parameters entering the adaptation algorithm varies within a closed domain admissible from a convergence point of view. The resulting suboptimal control strategies are translated into corrections of this parameter in order to recompute the adaptation algorithm over a finite horizon. Some numerical simulations illustrate the usefulness of the proposed scheme.     

Sliding control of non-linear systems containing time-varying uncertainties with unknown bounds

The sliding controller is very effective in dealing with system uncertainties defined in compact sets. If the bounds of the uncertainties are not available, the adaptive sliding controller might be designed. One restriction for the adaptive sliding scheme is that the unknown parameter should be constant, which is not always satisfied in practice. For a non-linear system with general uncertainties (i.e. time varying with unknown bounds), both the traditional sliding control and adaptive sliding control do not work properly. This paper proposes a new sliding control scheme for non-linear systems containing time-varying uncertainties with unknown bounds. The uncertainties are assumed to be piecewise continuous functions of time and satisfy the Dirichlet conditions. By representing these uncertainties in finite-term Fourier series, they can be estimated by updating the Fourier coefficients. Since the coefficients are time-invariant, update laws are easily obtained from the Lyapunov approach to guarantee output error convergence. Computer simulations are performed to show efficacy of the proposed schemes.     

一种新的时变系统极点配置自适应控制方法*

本文针对连续快时变系统，提出不要求系统参数未知部分上界的先验信息的自适应算法，得到间接自适应控制方案并且证明闭环系统稳定，仿真结果说明本文的方法有效．     

不确定非完整动力学系统的时变自适应控制

研究了具有未知惯性参数非完整动力学系统的镇定问题,对于一类非完整系统,给出了一种新的时变自适应律镇定律,不同于其它文中的控制律,该镇定律不是高增益的,文中也讨论了一般不确定非完整动力学系统的镇定问题,证明了时变周期镇定律的存在性,一个简单的例子说明了如何应用文中结果设计镇定律,仿真结果表明了本文所提设计方法的有效性。