International Journal of Control 2009, 1–8, iFirst A numerical method for determining monotonicity and convergence rate in iterative learning control Kira L. Barton a , Douglas A. Bristow b and Andrew G. Alleyne a * a Department of Mechanical Science and Engineering, University of Illinois at Urbana-Champaign, 1206 W. Green Street, Urbana, IL 61801, USA; b Department of Mechanical and Aerospace Engineering, Missouri University of Science and Technology, 210 Toomey Hall, Rolla, MO 65409, USA (Received 7 December 2008; final version received 18 June 2009) 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(N 2 ) 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. Keywords: iterative learning control; monotonic convergence; convergence rate; implicitly restarting Lanczos method 1. Introduction Iterative learning control (ILC) is a feedforward control method which focuses on improving the tracking performance of manufacturing systems that perform the same task repetitively. This control technique is based on the idea that the repetitive nature of these systems allows a controller to learn from previous iterations and modify a feedforward control input for improved tracking performance (Moore 1993; Longman 2000; Bristow, Tharayil, and Alleyne 2006). This approach has been shown to be successful in a variety of applications (Kim and Kim 1996; Havlicsek and Alleyne 1999; de Roover and Bosgra 2000; Norrlof 2002; Zheng and Alleyne 2003; Bristow and Alleyne 2006). One important aspect of ILC algorithm design is to ensure rapid convergence of the system. In Section 2 we define the ILC norm as a concise term that captures the convergence rate behaviour for multi-input multi- output (MIMO), linear time-varying (LTV) discrete- time systems. A small ILC norm means that the convergence occurs very quickly, while an ILC norm larger than one means that convergence is not monotonic. Monotonic convergence is often desirable because, in the absence of monotonicity, many stable ILC systems will exhibit initial convergence properties followed by temporarily divergent properties due to large transients (Huang and Longman 1996). When considering different design approaches, including iteration-varying learning controllers, the ILC norm provides a good parameter for comparison. Calculation of the ILC norm involves the con- struction and maximum singular value calculation of large matrices (Bristow et al. 2006), where matrix size is NN and N is the number of discrete time steps in an iteration. For long iterations, especially when sample rates are high as in robotic applications (Kim and Kim 1996; de Roover and Bosgra 2000; Norrlof 2002), calculation of the ILC norm is very slow, or not possible because of computational memory limitations. An alternative method for calculating the maxi- mum singular value of structured matrices, such as the lifted structure used in the ILC-norm computation, is well known in some areas of mathematics (Boyd 2008; Saad 1992). This method does not require explicit construction of the matrix, but rather uses subspace calculations whereby functional descriptions of the matrix are sufficient. For the ILC norm, the subspace calculation can be replaced by a dynamic simulation. This approach is most useful for designs that yield a filter-description for the learning algorithm since this can be used directly in the simulation. *Corresponding author. Email: alleyne@uiuc.edu ISSN 0020–7179 print/ISSN 1366–5820 online ß 2009 Taylor & Francis DOI: 10.1080/00207170903131177 http://www.informaworld.com Downloaded By: [University of Illinois] At: 16:42 8 January 2010