# Averaging and Lifting Techniques for Linear Discrete-Time Systems

## Abstract

Similar to Chap. 2, this chapter also considers a class of discrete-time linear systems with randomly varying trial lengths. However, in contrast to Chap. 2, this chapter aims to avoid using the traditional \(\lambda \)-norm in convergence analysis which may lead to a non-monotonic convergence. Compared to Chap. 2, the main contributions of the chapter can be summarized as follows: (i) A new formulation is presented for ILC of discrete-time systems with randomly varying trial lengths by defining a stochastic matrix. Comparing with Chap. 2, the introduction of the stochastic matrix is more straightforward, and the calculation of its probability distribution is less complex. (ii) Different from Chap. 2, we investigate ILC for systems with nonuniform trial lengths under the framework of lifted system and the utilization of \(\lambda \)-norm is avoided.

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