Iterative Learning Control for Discrete-time Stochastic Systems with Quantized Information

An iterative learning control (ILC) algorithm using quantized error information is given in this paper for both linear and nonlinear discrete-time systems with stochastic noises. A logarithmic quantizer is used to guarantee an adaptive improvement in tracking performance. A decreasing learning gain is introduced into the algorithm to suppress the effects of stochastic noises and quantization errors. The input sequence is proved to converge strictly to the optimal input under the given index. Illustrative simulations are given to verify the theoretical analysis.      

Almost sure convergence of iterative learning control for stochastic systems

This paper proposes an iterative learning control (ILC) algorithm with the purpose of controling the output of a linear stochastic system presented in state space form to track a desired realizable trajectory. It is proved that the algorithm converges to the optimal one a.s. under the condition that the product input-output coupling matrices are full-column rank in addition to some assumptions on noises. No other knowledge about system matrices and covariance matrices is required.     

Iterative learning control of molten steel level in a continuous casting process

In this paper, an iterative learning control (ILC) method is introduced to control molten steel level in a continuous casting process, in the presence of disturbance, noise and initial errors. The general ILC method was originally developed for processes that perform tasks repetitively but it can also be applied to periodic time-domain signals. To propose a more realistic algorithm, an ILC algorithm that consists of a P-type learning rule with a forgetting factor and a switching mechanism is introduced. Then it is proved that the input signal error, the state error and the output error are ultimately bounded in the presence of model uncertainties, periodic bulging disturbances, measurement noises and initial state errors. Computer simulation and experimental results establish the validity of the proposed control method.     

Stochastic Iterative Learning Control With Faded Signals

Stochastic iterative learning control (ILC) is designed for solving the tracking problem of stochastic linear systems through fading channels. Consequently, the signals used in learning control algorithms are faded in the sense that a random variable is multiplied by the original signal. To achieve the tracking objective, a two-dimensional Kalman filtering method is used in this study to derive a learning gain matrix varying along both time and iteration axes. The learning gain matrix minimizes the trace of input error covariance. The asymptotic convergence of the generated input sequence to the desired input value is strictly proved in the mean-square sense. Both output and input fading are accounted for separately in turn, followed by a general formulation that both input and output fading coexists. Illustrative examples are provided to verify the effectiveness of the proposed schemes.      

测量数据丢失的一类非线性系统迭代学习控制

迭代学习控制方法应用于网络控制系统时,由于通信网络的约束导致数据包丢失现象经常发生.针对存在输出测量数据丢失的一类非线性系统,研究P型迭代学习控制算法的收敛性问题.将数据丢失描述为一个概率已知的随机伯努利过程,在此基础上给出P型迭代学习控制算法的收敛条件,理论上证明了算法的收敛性,并通过仿真验证理论结果.研究表明,当非线性系统存在输出测量数据丢失时,迭代学习控制算法仍然可以保证跟踪误差的收敛性.     

A Kiefer‐Wolfowitz Algorithm Based Iterative Learning Control for Hammerstein‐Wiener Systems

The iterative learning control (ILC) is considered for the Hammerstein‐Wiener (HW) system, which is a cascading system consisting of a static nonlinearity followed by a linear stochastic system and then a static nonlinearity. Except the structure, the system is unknown, but the system output is observed with additive noise. Both the linear and nonlinear parts of the system may be time‐varying. The optimal control sequence under the tracking performance is first characterized, which, is however, unavailable since the system is unknown. By using the observations on system output the ILC is generated by a Kiefer‐Wolfowitz (KW) algorithm with randomized differences, which aims at minimizing the tracking error. It is proved that ILC converges to the optimal one with probability one and the resulting tracking error tends to its minimal value. Copyright © 2011 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society     

Output tracking for nonlinear stochastic systems by iterative learning control

An iterative learning control (ILC) algorithm, which in essence is a stochastic approximation algorithm, is proposed for output tracking for nonlinear stochastic systems with unknown dynamics and unknown noise statistics. The nonlinear function of the system dynamics is allowed to grow up as fast as a polynomial of any degree, but the system is linear with respect to control. It is proved that the ILC generated by the algorithm a.s. converges to the optimal one at each time t/spl isin/[0,1,...,N] and the output tracking error is asymptotically minimized in the mean square sense as the number of iterates tends to infinity, although the convergence rate is rather slow. The only information used in the algorithm is the noisy observation of the system output and the reference signal y/sub d/(t). When the system state equation is free of noise and the system output is realizable, then the exact state tracking is asymptotically achieved and the tracking error is purely due to the observation noise.     

Improved D‐Type Anticipatory Iterative Learning Control for a Class of Inhomogeneous Heat Equations

This paper aims at providing a practical iterative learning control (ILC) scheme for a wide class of heat transfer systems in the sense that it avoids high‐gain learning of ILC, thus a potential non‐monotonic convergence issue, and the risk of violating the hardware limitation of input profile in implementation. Meanwhile, the ILC scheme guarantees the identical initial condition of heat process. As a result, the output tracking precision may be improved while not reducing the anticipatory step size as in 1 . All the benefits of the proposed ILC scheme are achieved by applying a heuristic selection algorithm for the anticipatory step size and rectifying the output reference simultaneously.     

Fixed point theorem-based iterative learning control for LTV systems with input singularity

In this paper, we address a challenging and open problem: how to design a suitable iterative learning control (ILC) system in the presence of input singularity, which is incurred by the singularities of the system direct feed-through term. Considering two typical types of input singularities, we first revise the ILC operators accordingly by adding a forgetting factor and incorporating a time-varying learning gain, in the sequel guarantee ILC operators to be contractible. Next, using the Banach fixed-point theorem, we demonstrate that the output sequence can either enter and remains ultimately in a designated neighborhood of the target trajectory, or is bounded by a class K function. Finally, an illustrative example is presented.     

Assessment of gradient-based iterative learning controllers using a multivariable test facility with varying interaction

A multiple input, multiple output (MIMO) experimental test facility has been developed for the evaluation, benchmarking and comparison of iterative learning control (ILC) strategies. The system addresses the distinct lack of experimental studies for the multivariable case and enables controller performance and robustness to be rigorously investigated over a broad range of operating conditions. The electromechanical facility is multi-configurable with up to 3 inputs and permits both exogenous disturbance injection and a variable level of coupling to be applied between input and output pairs. To confirm its suitability for evaluation and comparison of ILC, theoretical results are derived for two popular forms of gradient-type ILC algorithm, linking interaction with fundamental performance limitations. The test facility is then used to establish how well theoretical predictions match experimental results. The analysis is then extended to provide solutions to address this performance degradation, and these are again confirmed using the test facility.     

Parameter estimation of stochastic linear systems with noisy input

This paper addresses the problem of parameter estimation of stochastic liner systems with noisy input–output measurements. A new and simple estimation scheme for the variances of the white input and output measurement noises is presented, which is only based on expanding the denominator polynomial of the system transfer function and makes no use of the average least-squares errors. The attractive feature of the iterative least-square based parametric algorithm thus developed is its improved convergence property. The effectiveness of the developed identification algorithm is demonstrated through numerical illustrations.     

A convex optimization approach to robust iterative learning control for linear systems with time-varying parametric uncertainties

In this paper, we present a new robust iterative learning control (ILC) design for a class of linear systems in the presence of time-varying parametric uncertainties and additive input/output disturbances. The system model is described by the Markov matrix as an affine function of parametric uncertainties. The robust ILC design is formulated as a min–max problem using a quadratic performance criterion subject to constraints of the control input update. Then, we propose a novel methodology to find a suboptimal solution of the min–max optimization problem. First, we derive an upper bound of the worst-case performance. As a result, the min–max problem is relaxed to become a minimization problem in the form of a quadratic program. Next, the robust ILC design is cast into a convex optimization over linear matrix inequalities (LMIs) which can be easily solved using off-the-shelf optimization solvers. The convergences of the control input and the error are proved. Finally, the robust ILC algorithm is applied to a physical model of a flexible link. The simulation results reveal the effectiveness of the proposed algorithm.     

A statistical analysis of certain iterative learning control algorithms

Iterative learning control (ILC) is a technique used to improve the tracking performance of systems carrying out repetitive tasks, which are affected by deterministic disturbances. The achievable performance is greatly degraded, however, when non-repeating, stochastic disturbances are present. This paper aims to compare a number of different ILC algorithms, proposed to be more robust to the presence of these disturbances, first by a statistical analysis and then by simulation results and their application to a linear motor. New expressions for the expected value and variance of the controlled error are developed for each algorithm. The different algorithms are then tested in simulation and finally applied to the linear motor system to test their performance in practice. A filtered ILC algorithm is proposed when the noise and desired output spectra are separated. Otherwise an algorithm with a decreasing gain gives good robustness to noise and achievable precision but at a slower convergence rate.     

Optimal selection of the forgetting matrix into an iterative learning control algorithm

A recursive optimal algorithm, based on minimizing the input error covariance matrix, is derived to generate the optimal forgetting matrix and the learning gain matrix of a P-type iterative learning control (ILC) for linear discrete-time varying systems with arbitrary relative degree. This note shows that a forgetting matrix is neither needed for boundedness of trajectories nor for output tracking. In particular, it is shown that, in the presence of random disturbances, the optimal forgetting matrix is zero for all learning iterations. In addition, the resultant optimal learning gain guarantees boundedness of trajectories as well as uniform output tracking in presence of measurement noise for arbitrary relative degree.     

Approximate adjoint-based iterative learning control

This paper characterises stochastic convergence properties of adjoint-based (gradient-based) iterative learning control (ILC) applied to systems with load disturbances, when provided only with approximate gradient information and noisy measurements. Specifically, conditions are discussed under which the approximations will result in a scheme which converges to an optimal control input. Both the cases of time-invariant step sizes and cases of decreasing step sizes (as in stochastic approximation) are discussed. These theoretical results are supplemented with an application on a sequencing batch reactor for wastewater treatment plants, where approximate gradient information is available. It is found that for such case adjoint-based ILC outperforms inverse-based ILC and model-free P-type ILC, both in terms of convergence rate and measurement noise tolerance.     

An ILC-Based Adaptive Control for General Stochastic Systems With Strictly Decreasing Entropy

In this paper, a new method for adaptive control of general nonlinear and non-Gaussian unknown stochastic systems has been proposed. The method applies the minimum entropy control scheme to decrease the closed-loop randomness of the output under an iterative learning control (ILC) basis. Both modeling and control of the plant are performed using dynamic neural networks. For this purpose, the whole control horizon is divided into a certain number of time domain subintervals called batches and a pseudo-D-type ILC law is employed to train the plant model and controller parameters so that the entropy of the closed-loop tracking error is made to decrease batch by batch. The method has the advantage of decreasing the output uncertainty versus the advances of batches along the time horizon. The analysis on the proposed ILC convergence is made and a set of demonstrable experiment results is also provided to show the effectiveness of the obtained control algorithm, where encouraging results have been obtained.      

基于向量图分析的一种迭代学习控制算法 及其鲁棒性

为了增强迭代学习控制的鲁棒性,加快学习过程的收敛速度,而又不过多地依赖于系统内部信息,本文基于向量图分析思路,利用输入空间的向量构造三角形修正结构,得到了一种新的迭代学习控制算法．该算法根据跟踪误差的大小,调节输入控制量在三角形的一条边上滑动,在跟踪误差较大时,算法能找到控制期望的大致位置并加速收敛,在跟踪误差较小时,能将控制量稳定在其期望的很小邻域内,理论上证明了该邻域直径大小为跟踪误差的二阶无穷小．数值仿真结果说明了它的有效性和优越性．     

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.     

Adaptive fuzzy ILC of nonlinear discrete-time systems with unknown dead zones and control directions

This paper presents an adaptive fuzzy iterative learning control (ILC) design for non-parametrized nonlinear discrete-time systems with unknown input dead zones and control directions. In the proposed adaptive fuzzy ILC algorithm, a fuzzy logic system (FLS) is used to approximate the desired control signal, and an additional adaptive mechanism is designed to compensate for the unknown input dead zone. In dealing with the unknown control direction of the nonlinear discrete-time system, a discrete Nussbaum gain technique is exploited along the iteration axis and applied to the adaptive fuzzy ILC algorithm. As a result, it is proved that the proposed adaptive fuzzy ILC scheme can drive the ILC tracking errors beyond the initial time instants into a tunable residual set as iteration number goes to infinity, and keep all the system signals bounded in the adaptive ILC process. Finally, a simulation example is used to demonstrate the feasibility and effectiveness of the adaptive fuzzy ILC scheme.