Adaptive backstepping repetitive learning control design for nonlinear discrete‐time systems with periodic uncertainties

This paper addresses a tracking problem for uncertain nonlinear discrete‐time systems in which the uncertainties, including parametric uncertainty and external disturbance, are periodic with known periodicity. Repetitive learning control (RLC) is an effective tool to deal with periodic unknown components. By using the backstepping procedures, an adaptive RLC law with periodic parameter estimation is designed. The overparameterization problem is overcome by postponing the parameter estimation to the last backstepping step, which could not be easily solved in robust adaptive control. It is shown that the proposed adaptive RLC law without overparameterization can guarantee the perfect tracking and boundedness of the states of the whole closed‐loop systems in presence of periodic uncertainties. In addition, the effectiveness of the developed controller is demonstrated by an implementation example on a single‐link flexible‐joint robot. Copyright © 2014 John Wiley & Sons, Ltd.     

Adaptive iterative learning control of discrete‐time varying systems with unknown control direction

Without using Nussbaum gain, a novel method is presented to solve the unknown control direction problem for discrete‐time systems. The underlying idea is to fully exploit the convergence property of parameter estimates in well‐known adaptive algorithms. By incorporating two modifications into the control and the parameter update laws, respectively, we present an adaptive iterative learning control scheme for discrete‐time varying systems without the prior knowledge of the sign of control gain. It is shown that the proposed adaptive iterative learning control can achieve perfect tracking over the finite time interval while all the closed‐loop signals remain bounded. An illustrative example is presented to verify effectiveness of the proposed scheme. Copyright © 2012 John Wiley & Sons, Ltd.     

Composite energy function‐based iterative learning control for systems with nonparametric uncertainties

In this work, we propose new iterative learning control (ILC) schemes that deal with nonlinear multi‐input multi‐output systems under alignment condition with nonparametric uncertainties. A major contribution of this work is to remove the classical resetting condition. Another major contribution of this work is to deal with norm‐bounded nonlinear uncertainties that satisfy local Lipschitz condition, in particular to deal with nonlinear uncertain state‐dependent input gain matrix that could be non‐square left invertible and local Lipschitzian. Two types of composite energy function are proposed to facilitate the ILC design and property analysis. Through rigorous analysis, we show that the new ILC schemes proposed warrant the asymptotical tracking convergence of system states. In the end, an illustrative example is provided to demonstrate the efficacy of the proposed ILC scheme. Copyright © 2013 John Wiley & Sons, Ltd.     

Immersion and invariance adaptive control for discrete‐time systems in strict‐feedback form with input delay and disturbances

This work presents a new adaptive control algorithm for a class of discrete‐time systems in strict‐feedback form with input delay and disturbances. The immersion and invariance formulation is used to estimate the disturbances and to compensate the effect of the input delay, resulting in a recursive control law. The stability of the closed‐loop system is studied using Lyapunov functions, and guidelines for tuning the controller parameters are presented. An explicit expression of the control law in the case of multiple simultaneous disturbances is provided for the tracking problem of a pneumatic drive. The effectiveness of the control algorithm is demonstrated with numerical simulations considering disturbances and input‐delay representative of the application.     

Adaptive control of discrete‐time nonlinear systems by recurrent neural networks in quasi‐sliding mode like regime

The aim of this study was to design an adaptive control strategy based on recurrent neural networks (RNNs). This neural network was designed to obtain a non‐parametric approximation (identification) of discrete‐time uncertain nonlinear systems. A discrete‐time Lyapunov candidate function was proposed to prove the convergence of the identification error. The adaptation laws to adjust the free parameters in the RNN were obtained in the same stability analysis. The control scheme used the states of the identifier, and it was developed fulfilling the necessary conditions to establish a behavior comparable with a quasi‐sliding mode regime. This controller does not use the regular form of the switching function that commonly appears in the sliding mode control designs. The Lyapunov candidate function to design the controller and the identifier simultaneously requires the existence of positive definite solutions of two different matrix inequalities. As consequence, a class of separation principle was proven when the RNN‐based identifier and the controller were designed by the same analysis. Simulations results were designed to show the behavior of the proposed controller solving the tracking problem for the trajectories of a direct current (DC) motor. The performance of the proposed controller was compared with the solution obtained when a classical proportional derivative controller and an adaptive first‐order sliding mode controller assuming poor knowledge of the plant. In both cases, the proposed controller showed superior performance when the relation between the tracking error convergence and the energy used to reach it was evaluated. Copyright © 2016 John Wiley & Sons, Ltd.     

Robust adaptive tracking control for nontriangular time-delay nonlinear systems with input dead-zone and multiple uncertainties

This article studies the robust adaptive tracking control problem of nontriangular nonlinear systems that are affected by multiple state delays rather than the input-delay. Different from the related studies, the considered systems involve input dead-zone and various uncertainties arising in the control coefficients, structure parameters, time delays, and disturbances. A new adaptive control strategy is presented by introducing a dynamic-gain-based Lyapunov-Krasovskii functional and by generalizing the tuning function method in the framework of time-delay system theory. All the states of the closed-loop system are bounded and the tracking error can be adjusted sufficiently small. In the simulation, the delayed chemical system is studied to demonstrate the validity of the strategy.     

Distributed adaptive iterative learning control for nonlinear multiagent systems with state constraints

This paper addresses the consensus problem of nonlinear multiagent system with state constraints. A novel γ‐type barrier Lyapunov function is adopted to handle with the bounded constraints. The iterative learning control strategy is introduced to estimate the unknown parameter and basic control signal. Five control schemes are designed, in turn, to address the consensus problem comprehensively from both theoretical and practical viewpoints. These schemes include the original adaptive scheme, projection‐based scheme, smooth function‐based scheme and its alternative, and dead‐zone–like scheme. The consensus convergence and constraints guarantee are strictly proved for each control scheme by using the barrier composite energy function approach. Illustrative simulations verify the theoretical analysis.     

Model reference adaptive iterative learning control for linear systems

In this paper, we propose a model reference adaptive control (MRAC) strategy for continuous‐time single‐input single‐output (SISO) linear time‐invariant (LTI) systems with unknown parameters, performing repetitive tasks. This is achieved through the introduction of a discrete‐type parametric adaptation law in the ‘iteration domain’, which is directly obtained from the continuous‐time parametric adaptation law used in standard MRAC schemes. In fact, at the first iteration, we apply a standard MRAC to the system under consideration, while for the subsequent iterations, the parameters are appropriately updated along the iteration‐axis, in order to enhance the tracking performance from iteration to iteration. This approach is referred to as the model reference adaptive iterative learning control (MRAILC). In the case of systems with relative degree one, we obtain a pointwise convergence of the tracking error to zero, over the whole finite time interval, when the number of iterations tends to infinity. In the general case, i.e. systems with arbitrary relative degree, we show that the tracking error converges to a prescribed small domain around zero, over the whole finite time interval, when the number of iterations tends to infinity. It is worth noting that this approach allows: (1) to extend existing MRAC schemes, in a straightforward manner, to repetitive systems; (2) to avoid the use of the output time derivatives, which are generally required in traditional iterative learning control (ILC) strategies dealing with systems with high relative degree; (3) to handle systems with multiple tracking objectives (i.e. the desired trajectory can be iteration‐varying). Finally, simulation results are carried out to support the theoretical development. Copyright © 2006 John Wiley & Sons, Ltd.     

具有匹配不确定自适应系统的鲁棒性分析

针对具有结构匹配不确定的奇异摄动系统,利用Lyapunov稳定性理论,通过构造Lyapunov函数方法进行鲁棒性研究。给出了一个理想奇异摄动系统的稳定鲁棒控制,即匹配不确定奇异摄动系统稳定控制的稳定条件,并讨论了稳定条件中的参数变化规律及实现的可能性,推广了现有结果。     

Discrete‐time periodic adaptive control for parametric systems with non‐sector nonlinearities

In this paper, a periodic adaptive control approach is proposed for a class of discrete‐time parametric systems with non‐sector nonlinearities. The proposed periodic adaptive control law is characterized by either one‐period delayed parametric updating or two‐period delayed parametric updating when input gain contains periodic unknowns. Logarithmic‐type discrete Lyapunov function is employed to handle the difficulties caused by the uncertainties that do not satisfy the linear growth condition. Some extensions to nonlinear systems with multiple unknown parameters and time‐varying input gain, tracking tasks, as well as higher‐order systems in canonical form, are also discussed. Copyright © 2013 John Wiley & Sons, Ltd.     

Neural network based terminal iterative learning control for uncertain nonlinear non‐affine systems

In this paper, a novel neural network based terminal iterative learning control method is proposed for a class of uncertain nonlinear non‐affine systems to track run‐varying reference point with initial state variance. In this new control scheme, the non‐affine terminal dynamics are converted affine, and the unrealisable recurrent network is simplified into realisable static network. As a result, the effect of initial state and control signal on terminal output can be estimated by neural network. With this estimation, the proposed control scheme can drive nonlinear non‐affine systems to track run‐varying reference point in the presence of initial state variance. Stability and convergence of this approach are proven, and numerical simulation results are provided to verify its effectiveness.     

Global adaptive finite‐time stabilization of a class of switched nonlinear systems with parametric uncertainty

This paper investigates the global adaptive finite‐time stabilization of a class of switched nonlinear systems, whose subsystems are all in p (p≤1) normal form with unknown control coefficients and parametric uncertainties. The restrictions on the power orders and the nonlinear perturbations are relaxed. By using the parameter separation technique, the uncertain parameters are separated from nonlinear functions. A systematic design procedure for a common state feedback controller and a switching adaptive law is presented by employing the backstepping methodology. It is proved that the closed‐loop system is finite‐time stable under arbitrary switching by utilizing the common Lyapunov function. Finally, with the application to finite‐time control of chemical reactor systems, the effectiveness of the proposed method is demonstrated. Copyright © 2015 John Wiley & Sons, Ltd.     

Robust learning controller design for MIMO stochastic discrete‐time systems: An H∞‐based approach

This paper is devoted to designing iterative learning control (ILC) for multiple‐input multiple‐output discrete‐time systems that are subject to random disturbances varying from iteration to iteration. Using the super‐vector approach to ILC, statistical expressions are presented for both expectation and variance of the tracking error, and time‐domain conditions are developed to ensure their asymptotic stability and monotonic convergence. It shows that time‐domain conditions can be tied together with an H_∞‐based condition in the frequency domain by considering the properties of block Toeplitz matrices. This makes it possible to apply the linear matrix inequality technique to describe the convergence conditions and to obtain formulas for the control law design. Furthermore, the H_∞‐based approach is shown applicable to ILC design regardless of the system relative degree, which can also be used to address issues of model uncertainty. For a class of systems with a relative degree of one, simulation tests are provided to illustrate the effectiveness of the H_∞‐based approach to robust ILC design. Copyright © 2011 John Wiley & Sons, Ltd.     

Adaptive set‐point regulation of discrete‐time nonlinear systems

In this paper, adaptive set‐point regulation controllers for discrete‐time nonlinear systems are constructed. The system to be controlled is assumed to have a parametric uncertainty, and an excitation signal is used in order to obtain the parameter estimate. The proposed controller belongs to the category of indirect adaptive controllers, and its construction is based on the policy of calculating the control input rather than that of obtaining a control law. The proposed method solves the adaptive set‐point regulation problem under the assumption that the target state is reachable for each fixed parameter value. Additional feature of the proposed method is that Lyapunov‐like functions have not been used in the construction of the controllers. Copyright © 2015 John Wiley & Sons, Ltd.     

Real‐time iterative learning control‐two applications with time scales between years and nanoseconds

Iterative learning control (ILC) is a family of digital control concepts, which can be used for a large variety of different applications. Each application has its own properties like sampling time and storage needs. This paper shows two real‐time ILC applications with different time scales and storage demands. First, the cavities of one of the world's leading pulsed free‐electron laser are controlled by a norm‐optimal ILC using only the information about the last pulse but with sample times below microseconds. Second, a heating system is controlled by a data‐driven ILC with a sample time in the range of minutes but using all available historic data sets of past trials. Tensor decomposition methods for storage demand and complexity reduction are applied to both applications, which results in a norm‐optimal tensor ILC, as well as, a data‐driven tensor ILC, although the time constants for the two applications vary by eight orders of magnitude.     

Closed‐loop iterative learning control for non‐linear systems with initial shifts

This paper is concerned with the problem of the iterative learning control with current cycle feedback for a class of non‐linear systems with well‐defined relative degree. The tracking error caused by a non‐zero initial shift is detected as extended D‐type learning algorithm is applied. The defect is overcome by adding terms including the output error, its derivatives as well as integrals. Asymptotic tracking of the final output to the desired trajectory is guaranteed. As an alternative approach, an initial rectifying action is introduced in the extended D‐type learning algorithm and shown effective to achieve the desired trajectory jointed smoothly with a transitional trajectory from the starting position. Also these algorithms with adjustable tracking interval ensure better robustness performance in the presence of initial shifts. Numerical simulation is conducted to demonstrate the theoretical results. Copyright © 2002 John Wiley & Sons, Ltd.     

Stability analysis and adaptive control for nonlinear discrete‐time systems with time delays and dead zone via state‐feedback technique

This research addresses the stability analysis and adaptive state‐feedback control for a class of nonlinear discrete‐time systems with multiple interval time‐varying delays and symmetry dead zone. The multiple interval time‐varying delays and symmetry dead zone are considered in the nonlinear discrete‐time system. The multiple interval time‐varying delays are bounded by the nonlinear function with unknown coefficients, and the symmetry dead zone is considered without the knowledge of the dead zone parameters. The adaptive state‐feedback controller is designed for the nonlinear discrete‐time systems with multiple interval time‐varying delays and dead zone. The discrete Lyapunov‐Krasovskii functional is introduced, such that the solutions of the closed‐loop error system converge to an adjustable bounded region and the state errors can be rendered arbitrarily small by adjusting the adaptive parameters. The designed adaptive state‐feedback controller does not require the knowledge of maximum and minimum values for the characteristic slopes of the dead zone. Finally, three simulation examples are given to show the effectiveness of the proposed methods.     

Robust H∞ filtering for switched linear discrete‐time systems with polytopic uncertainties

In this paper, the problem of robust H_∞ filtering for switched linear discrete‐time systems with polytopic uncertainties is investigated. Based on the mode‐switching idea and parameter‐dependent stability result, a robust switched linear filter is designed such that the corresponding filtering error system achieves robust asymptotic stability and guarantees a prescribed H_∞ performance index for all admissible uncertainties. The existence condition of such filter is derived and formulated in terms of a set of linear matrix inequalities (LMIs) by the introduction of slack variables to eliminate the cross coupling of system matrices and Lyapunov matrices among different subsystems. The desired filter can be constructed by solving the corresponding convex optimization problem, which also provides an optimal H_∞ noise‐attenuation level bound for the resultant filtering error system. A numerical example is given to show the effectiveness and the potential of the proposed techniques. Copyright © 2006 John Wiley & Sons, Ltd.     

基于自适应迭代学习的多智能体系统编队控制

针对带未知时变参数的非线性多智能体系统的编队问题,提出一种分布式自适应迭代学习控制策略。首先,通过傅里叶级数对系统的不确定参数进行展开,采用一个收敛级数序列处理傅里叶级数展开产生的截断误差,结合多智能体运行过程中的编队误差推导自适应迭代学习控制律和参数更新律;其次,针对领导者动态对大部分智能体都是未知的情况,设计新的辅助控制来补偿未知动态和避免未知有界干扰;然后,基于李亚普诺夫能量函数证明了在所设计控制律作用下多智能体系统编队误差随着迭代次数的增加在有限时间内趋于0;最后,将该控制策略运用到多无人机编队系统中,并通过搭建半物理实验平台,验证了控制方法的有效性。实验结果表明该控制方法可以确保多智能体快速形成所需编队,并且每个智能体在有限时间内可以精确跟踪期望轨迹。所提方法充分考虑了多智能体系统的参数不确定性以及抗干扰的能力,为实际应用中复杂多智能体系统的精确控制提供了有效的方法。     

Robust reliable dissipative filtering for Markovian jump nonlinear systems with uncertainties

This paper investigates the problem of robust reliable dissipative filtering for a class of Markovian jump nonlinear systems with uncertainties and time‐varying transition probability matrix described by a polytope. Our main attention is focused on the design of a reliable dissipative filter performance for the filtering error system such that the resulting error system is stochastically stable and strictly dissipative. By introducing a novel augmented Lyapunov–Krasovskii functional, a new set of sufficient conditions is obtained for the existence of reliable dissipative filter design in terms of linear matrix inequalities (LMIs). More precisely, a sufficient LMI condition is derived for reliable dissipative filtering that unifies the conditions for filtering with passivity and H_∞ performances. Moreover, the filter gains are characterized in terms of solution to a set of linear matrix inequalities. Finally, two numerical examples are provided to demonstrate the effectiveness and potential of the proposed design technique. Copyright © 2016 John Wiley & Sons, Ltd.