Elsevier

Automatica

Volume 63, January 2016, Pages 359-365
Automatica

Brief paper
On almost sure and mean square convergence of P-type ILC under randomly varying iteration lengths

https://doi.org/10.1016/j.automatica.2015.10.050Get rights and content

Abstract

This note proposes convergence analysis of iterative learning control (ILC) for discrete-time linear systems with randomly varying iteration lengths. No prior information is required on the probability distribution of randomly varying iteration lengths. The conventional P-type update law is adopted with Arimoto-like gain and/or causal gain. The convergence both in almost sure and mean square senses is proved by direct math calculating. Numerical simulations verifies the theoretical analysis.

Introduction

Repeatability may be a basic requirement for learning, because a learning algorithm usually works by trying and correcting. Meanwhile, many practical systems perform the same task again and again and thus satisfy the requirement of learning. For example, a chemical process is operated repeatedly in a plant to obtain batches of product. For this kind of systems, the information generated in the previous iterations could be used to improve the quality of products. This is the starting point of iterative learning control (ILC), aiming to improve the system performance by learning (Ahn et al., 2007, Bristow et al., 2006, Shen and Wang, 2014, Wang et al., 2009). As a matter of fact, ILC updates its control signal mainly using information of previous iterations rather than only the information of previous time instances in current iteration. Besides, it has simple control structure but effective capacity, thus has been applied to a lot of practical systems such as high-speed rail train (Sun, Hou, & Li, 2013), permanent magnet step motors (Bifaretti, Tomei, & Verrelli, 2011), robotic-assisted biomedical system (Xu, Chu, & Rogers, 2014), etc.

In order to gain a good performance, it is usually required that the tracking task of ILC repeats in a fixed time interval. However, in many practical applications, the trail length may vary among different iterations. Two biomedical systems, which are functional electrical stimulation for upper limb movement and for gait assistance, respectively, were given in Seel, Schauer, and Raisch (2011) to show this case. Subjected to complex factors and unknown dynamics, the learning process may end before the whole stimulation profile is completed. Another example was provided in Longman and Mombaur (2006), where humanoid robot was considered to follow some gait by ILC and repetitive control. One of the most important issues is that durations of the phases are not the same from iteration to iteration during the learning process. Therefore, the repeatability of iteration length is no longer valid in these cases. This motivates us to make efforts on the ILC under randomly varying iteration lengths.

Some studies have been made on this topic. In Seel et al. (2011), the maximum of iteration length was defined as full-length, and then the tracking error trajectory of all the non-full-length learning iterations were extended to be of full-length by appending zero elements at the lost positions. ILC algorithm was designed based on the modified tracking error trajectory to fulfill the requirements of standard ILC. It is shown that the algorithm could improve the first input samples sequentially, while maintaining the performance of last samples until long enough iteration occurs. In Li, Xu, and Huang (2014), the iteration length was assumed to be randomly varying and the iteration average operator technique was used to design ILC algorithms. It was proved that the mathematical expectation of tracking error converges to zero with the help of λ-norm technique. The varying iteration model there is that the actual iteration length could be either longer or shorter than the desired iteration length. If the actual iteration length exceeds the desired one, then the redundant points of tracking error are cut off, while on the contrary the absent points of tracking error are set as zero. The nonlinear and continuous-time case of Li et al. (2014) was given in Li, Xu, and Huang (2015) following similar techniques. In short, more efforts on the analysis and performance in probability sense should be made when iteration length varies randomly.

In this note, we build strong convergence property of ILC algorithms for linear system with randomly varying iteration lengths. It should be emphasized that the important significance is not to propose an alternative algorithm of previous results, but to reveal that the conventional P-type ILC algorithm retains robustness and strong convergence against randomly varying iteration lengths.

There are several major differences between this note and (Li et al., 2014, Seel et al., 2011) from the perspectives of convergence analysis technique and convergence property. In the first place, the analysis of Seel et al. (2011) was directly made according to tracking error of adjacent iterations following a determinate way. In Li et al. (2014), the random variable describing the varying iteration length was turned into a deterministic value by taking mathematical expectation and then no randomness had to be considered in the following derivations. Thus both Seel et al. (2011) and Li et al. (2014) showed the convergence following the determinate way. In contrast, this note proposes a novel analysis technique from the perspective of probability theory. Moreover, the convergence property of Li et al. (2014) and Seel et al. (2011) are much weaker than this note. In Seel et al. (2011), it proved that the tracking error was monotonically decreasing in 1-norm and -norm senses but the limitation of the tracking error might not be zero. In Li et al. (2014), the expectation of tracking error was shown convergent to zero. However, this does not imply that the actual tracking error was small enough since the variance could be very large. Unlike Li et al. (2014) and Seel et al. (2011), the almost sure and mean square convergence results proposed in this note are the strongest convergence in probability theory. Last but by no means least, both Seel et al. (2011) and this note require no prior information on the probability distribution of random varying lengths, while the latter information was used in Li et al. (2014) for the design of learning matrix.

Specifically speaking, the contributions of this note are listed as follows: (a) the conventional P-type ILC update law is used and shown effective and robust under randomly varying iteration lengths; (b) both almost sure and mean square convergence of the input sequence is proved by direct calculations; (c) no prior information on the probability distribution of randomly varying iteration lengths is required, thus is more suitable for practical applications.

The rest of the note is arranged as follows. Section  2 gives problem formulation, where the randomly varying iteration lengths is defined. In Section  3, the conventional ILC algorithm is used with minor modifications on tracking error. Strict convergence analysis both in almost sure sense and mean square sense is proposed in Section  4. Section  5 shows an illustration simulation and Section  6 concludes this note.

Notations

E denotes the mathematical expectation. λ(A) is the eigenvalue of a matrix A, while ρ(A) is the spectral radius. A is an induced norm of a matrix A. The subscript T denotes the transpose, and denotes the Kronecker product. P() denotes the probability of an event. EX denotes the mathematical expectation of a random variable X. The abbreviation “i.o.” is short for “infinitely often”.

Section snippets

Problem formulation

Consider the following linear time-varying system x(t+1,k)=Atx(t,k)+Btu(t,k)y(t,k)=Ctx(t,k) where x(t,k)Rn, u(t,k)Rp, and y(t,k)Rq denote state, input, and output, respectively. k=0,1, and t=0,1,,N denote the iteration index and discrete time index, respectively, and N is the maximum of iteration lengths. At, Bt, and Ct are system matrices with appropriate dimensions. It is assumed that Ct+1Bt is of full-column-rank, which means the relative degree is 1.

If the operation length of all

ILC design

Notice that the iteration length cannot exceed the maximum length, thus only two cases of tracking error need to be considered. If the iteration length equals the maximum length, then the tracking error is a normal one with dimension Nq; while if the iteration length is shorter than the maximum length, then the tracking error at the absent time instances are missing, which therefore could not be used for input update. For the latter case, we could append zeros to the absent time instances so

Strong convergence properties

Based on Lemma 1, the following theorem establishes the convergence in mathematical expectation sense.

Theorem 1

Consider system   (2)   with randomly varying iteration length and use control update law   (8). The mathematical expectation of tracking error Ek, i.e., EEk, converges to zero if the learning gain L satisfies   (19).

Proof

By (14) one has EΔUk+1=KkEΔU0. Then by recurrence of the mean Kk, i.e.,  (15), it is obvious to have EΔUk=(i=1mpiΓi)kEΔU0. Thus it is sufficient to show that ρ(i=1mpiΓi)<1. By

Illustrative simulations

In order to show the effectiveness and robustness of the conventional P-type ILC algorithm, a time-varying system is given as follows x(t+1,k)=(0.2exp(t/100)0.6000.50sin(t)000.7)x(t,k)+(00.3sin(t)1.00)u(t,k)y(t,k)=(00.11.00+0.1cos(t))x(t,k). The initial state is set as x(0,k)=[000]T. Let the desired trajectory be y(t,d)=sin(2πt/50)+sin(2πt/5). The maximum of iteration length is N=50. Without loss of generality, the input of the initial iteration is simply set to zero, i.e., u(t,0)=0,0tN.

The

Conclusion

The tracking performance of traditional P-type ILC for linear system with randomly varying iteration lengths is discussed in this note. The probability properties along sample path are first calculated for further analysis. Then the zero-convergence both in almost sure sense and mean square sense is established, as long as the probability of full-length iteration is not zero. The sufficient condition on the design of learning gain is also clarified. For further study, the nonlinear system case

Dong Shen received the B.S. degree in mathematics from Shandong University, Jinan, China, in 2005. He received the Ph.D. degree in mathematics from the Academy of Mathematics and System Science, Chinese Academy of Sciences (CAS), Beijing, China, in 2010. From 2010 to 2012, he was a Post-Doctoral Fellow with the Institute of Automation, CAS. Since 2012, he has been an associate professor with College of Information Science and Technology, Beijing University of Chemical Technology (BUCT),

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Dong Shen received the B.S. degree in mathematics from Shandong University, Jinan, China, in 2005. He received the Ph.D. degree in mathematics from the Academy of Mathematics and System Science, Chinese Academy of Sciences (CAS), Beijing, China, in 2010. From 2010 to 2012, he was a Post-Doctoral Fellow with the Institute of Automation, CAS. Since 2012, he has been an associate professor with College of Information Science and Technology, Beijing University of Chemical Technology (BUCT), Beijing, China. His current research interests include iterative learning controls, stochastic control and optimization. He has published more than 30 refereed journals and conference papers. Dr. Shen received IEEE CSS Beijing Chapter Young Author Prize in 2014 and Wentsun Wu Artificial Intelligence Science and Technology Progress Award in 2012.

Wei Zhang received the B.Eng. degree in automation from Beijing University of Chemical Technology, China, in 2012. He is currently working towards the Master degree at Beijing University of Chemical Technology, China. His research interest is in field of iterative learning control.

Youqing Wang received his B.S. degree from Shandong University, Jinan, Shandong, China, in 2003, and received his Ph.D. degree in control science and engineering from Tsinghua University, Beijing, China, in 2008. He worked as a Research Assistant in the Department of Chemical Engineering, Hong Kong University of Science and Technology, from February 2006 to August 2007. From February 2008 to February 2010, he worked as a Senior Investigator in the Department of Chemical Engineering, University of California, Santa Barbara, CA, USA. From March 2008 to December 2011, he was an Adjunct Senior Investigator in Sansum Diabetes Research Institute. From August 2015 to November 2015, he was a Visiting Professor in University of Alberta. Currently, he is a Full Professor in Beijing University of Chemical Technology. His research interests include fault-tolerant control, state monitoring, modeling and control of biomedical processes (e.g., artificial pancreas system), and iterative learning control. Prof. Wang is serving as an Associate Editor of Multidimensional Systems and Signal Processing and has served as a Guest Editor of Journal of Process Control. He is a member of IFAC Technical Committee on Chemical Process Control and IFAC Technical Committee on Biological and Medical Systems. He is the recipient of several research awards (including Journal of Process Control Survey Paper Prize, ADCHEM2015 Young Author Prize).

Chiang-Ju Chien received his Ph.D. degree in electrical engineering from National Taiwan University, Taipei, Taiwan, in 1992. Since 1993, he has been with the Department of Electronic Engineering, Huafan University, Taipei, Taiwan, where he is currently a Professor. He served as the Chair of the same Department from 2000 to 2005, the Secretary General from 2005 to 2011 and the Dean of College of Engineering and Management from 2012 to 2015 in Huafan University. He is now the Secretary General again in Huafan. He has published more than 100 peer reviewed international journals and conference papers in the area of control theory and applications. His current research interests are mainly in iterative learning control, adaptive control, fuzzy-neuro systems, and control circuit design. Dr. Chien is a member of IEEE and the Chinese Automatic Control Society.

This work is supported by National Natural Science Foundation of China (61304085 and 61374099), Beijing Natural Science Foundation (4152040). The material in this paper was partially presented at the 27th Chinese Control and Decision Conference, May 23–25, 2015, Qingdao, China. This paper was recommended for publication in revised form by Associate Editor Abdelhamid Tayebi under the direction of Editor Toshiharu Sugie.

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