Repetitive Process based Iterative Learning Control designed by LMIs and Experimentally Verified on a Gantry Robot Lukasz Hladowski, Zhonglun Cai, Krzysztof Galkowski, Eric Rogers, Chris T. Freeman, Paul L. Lewin, Wojciech Paszke Abstract— In this paper we use a 2D systems setting to develop new results on iterative learning control for linear single-input single-output (SISO) plants, where it is well known in the subject area that a trade-off exists between speed of convergence and the response along the trials. Here we give new results by designing the control scheme using a strong form of stability for repetitive processes/2D linear systems known as stability along the pass (or trial). The design computations are in terms of Linear Matrix Inequalities (LMIs) and results from experimental verification on a gantry robot are also given. I. INTRODUCTION Iterative learning control (ILC) is a technique for con- trolling systems operating in a repetitive (or pass-to-pass) mode with the requirement that a reference trajectory y ref (t) defined over a finite interval 0 ≤ t ≤ α is followed to a high precision. Examples of such systems include robotic manipulators that are required to repeat a given task, chemical batch processes or, more generally, the class of tracking systems. Since the original work [1] in the mid 1980s, the general area of ILC has been the subject of intense research effort. Initial sources for the literature here are the survey papers [2] and [3]. The analysis of ILC schemes is firmly outside standard, or 1D, control theory, although it still has a significant role to play in certain cases of practical interest. In this paper we deal with ILC schemes that can be represented as a repetitive process [4]. In ILC, a major objective is to achieve convergence of the trial-to-trial error and often this has been treated as the only one that needs to be considered. It is, however, possible that enforcing fast convergence could lead to unsatisfactory performance along the trial, and here we address this problem by first showing that ILC schemes can be designed for a class of discrete linear systems by extending techniques developed for linear repetitive processes. This allows us to use the strong concept of stability along the pass (or trial) for these processes, in an ILC setting, as a possible means of dealing with poor/unacceptable transients in the along the trial dynamics. The results developed give control law design algorithms that can be implemented via LMIs, and results L.Hladowski and K.Galkowski are with the Institute of Control and Computation Engineering, University of Zielona Gora, Poland L.Hladowski@issi.uz.zgora.pl, K.Galkowski@issi.uz.zgora.pl Z. Cai, E. Rogers, C. T. Freeman and P. L. Lewin are with the School of Electronics and Computer Science, University of Southampton, Southampton SO17 1BJ, UK W Paszke is with the Control Systems Technology Group, Eindhoven University of Technology, The Netherlands from their experimental implementation on a gantry robot executing a pick and place operation are also given. The remainder of this paper begins with a simulation study which demonstrates that it is possible for trial-to-trial error convergence to occur where the along the trial response is very poor. This is followed by analysis which shows how the design of a class of ILC laws can be formulated in a repetitive process setting and designed via LMIs to ensure stability along the trial, with the possibility of tuning to give desired along the trial performance. Finally, the experimental results are given. In this paper, the null and identity matrices with the re- quired dimensions are denoted by 0 and I respectively. Also Γ ≻ 0 and Γ ≺ 0 respectively are used to denote symmetric matrices which are positive definite and negative definite respectively. The symbol r(·) is used to denote the spectral radius of a given matrix. In particular if M is a p × p matrix with eigenvalues λ i , 1 ≤ i ≤ p, then r(M ) = max i≤i≤p |λ i |. II. BACKGROUND Consider the case when the plant to be controlled can be modeled as a single-input, single-output differential linear time-invariant system with state-space model defined by {A c ,B c ,C c }. In an ILC setting this is written as ˙ x k (t) = A c x k (t)+ B c u k (t), 0 ≤ t ≤ α, y k (t) = C c x k (t), (1) where on trial k, x k (t) ∈ R n is the state vector, y k (t) ∈ R m is the output vector, u k (t) ∈ R r is the vector of control inputs, and α< ∞ is the trial length. If the signal to be tracked is denoted by y ref (t) then e k (t)= y ref (t) − y k (t) is the error on trial k, and the most basic requirement is to force the error to converge in k. It is, however, possible that trial-to-trial convergence will occur but produce along the trial performance which is far from satisfactory for many practical applications. Consider, for example, a gantry robot executing the following set of operations: collect an object from a location and place it on a moving conveyor, ii) return to the original location and collect the next one and place it on the conveyor, and iii) repeat i) and ii) for the next one and so on. Then if the object has an open top and is filled with liquid, and/or is fragile in nature, unwanted vibrations during the transfer time could have very detrimental effects. Hence in such cases there is also a need to control the along the trial dynamics and in this paper the method used is a strong form of stability theory for linear repetitive processes. 2009 American Control Conference Hyatt Regency Riverfront, St. Louis, MO, USA June 10-12, 2009 WeB08.5 978-1-4244-4524-0/09/$25.00 ©2009 AACC 949