# Repetitive learning control of linear periodic plants

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- Purdue University
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## Abstract

The need to increase production and quality drives the need for better control algorithms with high performance specifications. Manufacturing and industrial applications often have plants that perform repetitive tasks. In these situations, exploiting the periodic properties of the design problem is an important part in maximizing performance. In a control application with a periodic task, we can model the behavior of the plant as a linear periodic (LP) system. We propose an architecture called repetitive learning control (RLC) that uses a linear learning law to control linear periodic plants and to guarantee zero asymptotic tracking error.^ A repetitive learning controller uses the tracking error and control input in the previous period to compute the control input for the present period. A key measure of the performance of these controllers is the convergence ratio, with smaller ratios translating to faster convergence. We propose several methods of designing a repetitive learning controller, based on eigenvalue assignment and matrix inequality techniques. The latter includes linear matrix inequality, P-K iterations, and C-C iterations.^ Controller complexity is an important issue in RLC design and we propose two approaches: gain structure and basis functions. The gain structure approach takes advantage of the redundancy in the design problem to reduce the number of free variables in the controller. The use of basis functions reduces the controller complexity by reducing the number of inputs and outputs that must be processed, at the cost of a larger non-zero error. One class of basis functions that are ideal in this application is the class of wavelet basis functions, due to its multiresolution properties and the availability of fast algorithms for performing this basis translation compared to other basis functions such as sinusoids. ^

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