On computing the worst-case norm of linear systems subject to inputs with magnitude bound and rate limit

摘要

In this paper, we propose a practical and effective approach to compute the worst-case norm of finite-dimensional convolution systems. System inputs are modelled to have bounded magnitude and rate limit. The computation of the worst-case norm is formulated as a fixed-terminal-time optimal control problem. Applying Pontryagin's maximum principle with the generalized Karush-Kuhn-Tucker theorem, we obtain necessary conditions which are subsequently exploited to characterize the worst-case input. Furthermore, we develop a novel algorithm called successive pang interval search (SPIS) to construct the worst-case input for general finite-dimensional convolution systems. The algorithm is guaranteed to converge and give an accurate solution within a prescribed error bound. To verify the accuracy of the algorithm, we derive bounds on computational errors including the truncation error and the discretization error. Then, the bounds on the errors yielded by our algorithm are compared with those of a comparative discrete-time method. This suggests that SPIS is deemed to be more accurate, analytically. Numerical results based on second-order linear systems show that both approaches give the worst-case norms with comparable errors, but SPIS requires much less computation time than the discrete-time method.