Lyapunov stability theory for linear repetitive processes — The 2D equation approach

This paper develops the basis of a so-called 2D Lyapunov equation based approach to the stability analysis of discrete linear repetitive processes. The key feature of this equation is that, in contrast to the so-called ID Lyapunov equation approach, it is defined in terms of matrices with constant entries and has a `similar´ structure to the Lyapunov equation for standard, or ID, discrete linear systems. Here it is shown that the 2D Lyapunov equation gives, in general, a characterization of stability which is sufficient but not necessary. Despite this deficiency, it is also shown how the 2D Lyapunov equation can be used to characterize stability margins and robustness to uncertainty in the model description - very important topics for which few results are currently available. Some areas for short to medium term further research are also briefly noted.