An open-closed-loop iterative learning control approach for nonlinear switched systems with application to freeway traffic control

For nonlinear switched discrete-time systems with input constraints, this paper presents an open-closed-loop iterative learning control (ILC) approach, which includes a feedforward ILC part and a feedback control part. Under a given switching rule, the mathematical induction is used to prove the convergence of ILC tracking error in each subsystem. It is demonstrated that the convergence of ILC tracking error is dependent on the feedforward control gain, but the feedback control can speed up the convergence process of ILC by a suitable selection of feedback control gain. A switched freeway traffic system is used to illustrate the effectiveness of the proposed ILC law.     

数据丢失对迭代学习控制的影响分析

针对一类线性系统,分析数据丢失对迭代学习控制算法的影响.首先基于lifting方法给出跟踪误差渐近收敛和单调收敛的条件,并分析收敛速度与数据丢失率的关系,结果表明收敛速度随着数据丢失程度的增加而变慢.其次,为抑制迭代变化扰动的影响,给出一种存在数据丢失时的鲁棒迭代学习控制器设计方法,并将控制器设计问题转化为求取线性矩阵不等式的可行解.仿真示例验证了理论分析的结果以及鲁棒迭代学习控制算法的有效性.

     

分数阶迭代学习控制的收敛性分析

本文将传统的迭代学习控制时域和频域分析方法扩展到一类针对分数阶非线性系统的分数阶迭代学习控制时域分析方法.提出了一类新的分数阶迭代学习控制框架并简化了收敛条件,且证明了常增益情况下两类分数阶迭代学习控制收敛条件的等价性问题.该讨论进一步引出了如下两个结果:分数阶不确定系统的分数阶自适应迭代学习控制的可学习区域以及理想带阻型分数阶迭代学习控制的框架.上述结果均得到了仿真验证.     

Monotonically convergent ILC systems designed using bounded real lemma

This article is devoted to iterative learning control (ILC) systems design for multiple-input multiple-output (MIMO), linear time-invariant (LTI) plants. With the bounded real lemma (BRL) applied, a linear matrix inequality (LMI) design approach is presented to develop sufficient conditions for the monotonic convergence of the ILC process. It is shown that regardless of a system relative degree, the convergence conditions can be expressed in terms of LMIs, and formulas can be derived for the learning gain matrices design. For ILC determined in this way, two illustrative examples are provided to verify its effectiveness and robustness against structured and polytopic-type uncertainties.     

A Study On Iterative Learning Control With Adjustment Of Learning Interval For Monotone Convergence In The Sense Of Sup‐Norm

It has been found that some huge overshoot in the sense of sup‐norm may be observed when typical iterative learning control (ILC) algorithms are applied to LTI systems, even though monotone convergence in the sense of λ‐norm is guaranteed. In this paper, a new ILC algorithm with adjustment of learning interval is proposed to resolve such an undesirable phenomenon, and it is shown that the output error can be monotonically converged to zero in the sense of sup‐norm when the proposed ILC algorithm is applied. A numerical example is given to show the effectiveness of the proposed algorithm.     

Formation control for multi‐agent systems through an iterative learning design approach

This paper deals with formation control problems for multi‐agent systems by using iterative learning control (ILC) design approaches. Distributed formation ILC algorithms are presented to enable all agents in directed graphs to achieve the desired relative formations perfectly over a finite‐time interval. It is shown that not only asymptotic stability but also monotonic convergence of multi‐agent formation ILC can be accomplished, and the convergence conditions in terms of linear matrix inequalities can be simultaneously established. The derived results are also applicable to multi‐agent systems that are subject to stochastic disturbances and model uncertainties. Furthermore, the feasibility of convergence conditions and the effect of communication delays are discussed for the proposed multi‐agent formation ILC algorithms. Simulation results are given for uncertain multi‐agent systems to verify the theoretical study. Copyright © 2012 John Wiley & Sons, Ltd.     

一类线性离散切换系统的迭代学习控制

考虑具有任意切换序列线性离散切换系统的迭代学习控制问题. 假设切换系统在有限时间区间内重复运行, P型ILC算法可实现该类系统在整个时间区间内的完全跟踪控制. 采用超向量方法给出了算法在迭代域内收敛的条件, 并在理论上分析了的收敛性. 仿真示例验证了理论的结果.     

Stability analysis of discrete-time iterative learning control systems with interval uncertainty

This paper presents a stability analysis of the iterative learning control (ILC) problem for discrete-time systems when the plant Markov parameters are subject to interval uncertainty. Using the so-called super-vector approach to ILC, vertex impulse response matrices are employed to develop sufficient conditions for both asymptotic stability and monotonic convergence of the ILC process. It is shown that the stability of such interval ILC systems can be determined by checking the stability of the system using only the vertex points of the interval Markov parameters.     

Robust design of feedback feed-forward iterative learning control based on 2D system theory for linear uncertain systems

For a class of linear discrete-time uncertain systems, a feedback feed-forward iterative learning control (ILC) scheme is proposed, which is comprised of an iterative learning controller and two current iteration feedback controllers. The iterative learning controller is used to improve the performance along the iteration direction and the feedback controllers are used to improve the performance along the time direction. First of all, the uncertain feedback feed-forward ILC system is presented by an uncertain two-dimensional Roesser model system. Then, two robust control schemes are proposed. One can ensure that the feedback feed-forward ILC system is bounded-input bounded-output stable along time direction, and the other can ensure that the feedback feed-forward ILC system is asymptotically stable along time direction. Both schemes can guarantee the system is robust monotonically convergent along the iteration direction. Third, the robust convergent sufficient conditions are given, which contains a linear matrix inequality (LMI). Moreover, the LMI can be used to determine the gain matrix of the feedback feed-forward iterative learning controller. Finally, the simulation results are presented to demonstrate the effectiveness of the proposed schemes.     

Robust Optimization-Based Iterative Learning Control for Nonlinear Systems With Nonrepetitive Uncertainties

This paper aims to solve the robust iterative learning control(ILC)problems for nonlinear time-varying systems in the presence of nonrepetitive uncertainties.A new optimization-based method is proposed to design and analyze adaptive ILC,for which robust convergence analysis via a contraction mapping approach is realized by leveraging properties of substochastic matrices.It is shown that robust tracking tasks can be realized for optimization-based adaptive ILC,where the boundedness of system trajectories and estimated parameters can be ensured,regardless of unknown time-varying nonlinearities and nonrepetitive uncertainties.Two simulation tests,especially implemented for an injection molding process,demonstrate the effectiveness of our robust optimization-based ILC results.     

Non-linear iterative learning by an adaptive Lyapunov technique

We consider the iterative learning control problem from an adaptive control viewpoint. It is shown that some standard Lyapunov adaptive designs can be modified in a straightforward manner to give a solution to either the feedback or feedforward ILC problem. Some of the common assumptions of non-linear iterative learning control are relaxed: e.g. we relax the common linear growth asssumption on the non-linearities and handle systems of arbitrary relative degree. It is shown that generally a linear rate of convergence of the MSE can be achieved, and a simple robustness analysis is given. For linear plants we show that a linear rate of MSE convergence can be achieved for non-minimum phase plants.     

A guaranteed monotonically convergent iterative learning control

This paper presents a new iterative learning control (ILC) scheme for linear discrete time systems. In this scheme, the input of the controlled system is modified by applying a semi‐sliding window algorithm, with a maximum length of n + 1, on the tracking errors obtained from the previous iteration (n is the order of the controlled system). The convergence of the presented ILC is analyzed. It is shown that, if its learning gains are chosen proportional to the denominator coefficients of the system transfer function, then its monotonic convergence condition is independent of the time duration of the iterations and depends only on the numerator coefficients of the system transfer function. The application of the presented ILC to control second‐order systems is described in detail. Numerical examples are added to illustrate the results. Copyright © 2011 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society     

A numerical method for determining monotonicity and convergence rate in iterative learning control

In iterative learning control (ILC), a lifted system representation is often used for design and analysis to determine the convergence rate of the learning algorithm. Computation of the convergence rate in the lifted setting requires construction of large N×N matrices, where N is the number of data points in an iteration. The convergence rate computation is O(N²) and is typically limited to short iteration lengths because of computational memory constraints. As an alternative approach, the implicitly restarted Arnoldi/Lanczos method (IRLM) can be used to calculate the ILC convergence rate with calculations of O(N). In this article, we show that the convergence rate calculation using IRLM can be performed using dynamic simulations rather than matrices, thereby eliminating the need for large matrix construction. In addition to faster computation, IRLM enables the calculation of the ILC convergence rate for long iteration lengths. To illustrate generality, this method is presented for multi-input multi-output, linear time-varying discrete-time systems.     

Monotonically convergent iterative learning control for linear discrete-time systems

In iterative learning control schemes for linear discrete time systems, conditions to guarantee the monotonic convergence of the tracking error norms are derived. By using the Markov parameters, it is shown in the time-domain that there exists a non-increasing function such that when the properly chosen constant learning gain is multiplied by this function, the convergence of the tracking error norms is monotonic, without resort to high-gain feedback.     

On the P-type and Newton-type ILC schemes for dynamic systems with non-affine-in-input factors

In this paper, P-type learning scheme and Newton-type learning scheme are proposed for quite general nonlinear dynamic systems with non-affine-in-input factors. Using the contraction mapping method, it is shown that both schemes can achieve asymptotic convergence along learning repetition horizon. In order to quantify and evaluate the learning performance, new indices—Q-factor and Q-order—are introduced in particular to evaluate the learning convergence speed. It is shown that the P-type iterative learning scheme has a linear convergence order with limited learning convergence speed under system uncertainties. On the other hand, if more of system information such as the input Jacobian is available, Newton-type iterative learning scheme, which is originated from numerical analysis, can greatly speed up the learning convergence speed. The effectiveness of the two learning control methods are demonstrated through a switched reluctance motor system.     

Iterative Learning Control for a Class of Fractional Order Distributed Parameter Systems

This paper concerns a second‐order P‐type iterative learning control (ILC) scheme for a class of fractional order linear distributed parameter systems. First, by analyzing of the control and learning processes, a discrete system for P‐type ILC is established and the ILC design problem is then converted to a stability problem for such a discrete system. Next, a sufficient condition for the convergence of the control input and the tracking errors is obtained by using generalized Gronwall inequality, which is less conservative than the existing one. By incorporating the convergent condition obtained into the original system, the ILC scheme is derived. Finally, the validity of the proposed method is verified by a numerical example.     

H∞ approach to monotonically convergent ILC for uncertain time-varying delay systems

This paper deals with iterative learning control (ILC) design for uncertain time-delay systems. Monotonic convergence of the resulting ILC process is studied, and a sufficient condition within an H_∞-based framework is developed. It is shown that under this framework, delay-dependent conditions can be obtained in terms of linear matrix inequalities (LMIs), together with formulas for gain matrices design. A numerical example is provided to illustrate the effectiveness of the robust H_∞-based approach to ILC designed via LMIs.     

An iterative learning control theory for a class of nonlinear dynamic systems

An iterative learning control scheme is presented for a class of nonlinear dynamic systems which includes holonomic systems as its subset. The control scheme is composed of two types of control methodology: a linear feedback mechanism and a feedforward learning strategy. At each iteration, the linear feedback provides stability of the system and keeps its state errors within uniform bounds. The iterative learning rule, on the other hand, tracks the entire span of a reference input over a sequence of iterations. The proposed learning control scheme takes into account the dominant system dynamics in its update algorithm in the form of scaled feedback errors. In contrast to many other learning control techniques, the proposed learning algorithm neither uses derivative terms of feedback errors nor assumes external input perturbations as a prerequisite. The convergence proof of the proposed learning scheme is given under minor conditions on the system parameters.     

Sampled‐data iterative learning control for continuous‐time nonlinear systems with iteration‐varying lengths

In this work, sampled‐data iterative learning control (ILC) method is extended to a class of continuous‐time nonlinear systems with iteration‐varying trial lengths. In order to propose a unified ILC algorithm, the tracking errors will be redefined when the trial length is shorter or longer than the desired one. Based on the modified tracking errors, 2 sampled‐data ILC schemes are proposed to handle the randomly varying trial lengths. Sufficient conditions are derived rigorously to guarantee the convergence of the nonlinear system at each sampling instant. To verify the effectiveness of the proposed ILC laws, simulations for a nonlinear system are performed. The simulation results show that if the sampling period is set to be small enough, the convergence of the learning algorithms can be achieved as the iteration number increases.     

随机分布控制系统的自适应迭代学习控制

介绍输出概率密度函数(PDF)常规的迭代学习控制(ILC)的收敛条件,并利用此条件设计相应的迭代学习律.主要讨论如何解决输出PDF迭代学习控制(ILC)中的过迭代,收敛速度等问题.以离散输出概率密度函教(PDF)控制模型为基础,介绍了直接迭代学习控制算法收敛的必要条件,提出自适应的迭代学习参数调节方法和避免过迭代的迭代结束条件,这些措施能够保证输出PDF的迭代控制收敛且具有较快的收敛速度.仿真结果表明,输出PDF的自适应迭代学习控制具有较快的收敛速度,而学习终止条件能很好地避免过迭代.