The closed-form solution for the worst-case norm of linear systems subject to inputs with magnitude bound and rate limit

There have been several numerical methods to approximately compute the worst-case norm of finite-dimensional linear systems subject to inputs with magnitude bound and rate limit. Since the closed-form solution has not been obtained in general, it is difficult to decide which method gives the best numerical results. This paper presents explicit formulas for the worst-case norm of the second-order linear systems. To obtain the closed-form expressions, we derive a sharp upper bound of the worst-case norm and its associated input. The formulas are given in terms of system parameters, i.e., natural frequency and damping ratio. In particular, we derive the worst-case norm for three cases, namely, lightly-damped, underdamped, and overdamped, which also includes the critically-damped case. For general higher order systems, the derivation of the formulas becomes very sophisticated so that we need to incorporate analytical and numerical techniques to compute the worst-case norm. For such reason, the formulas given in this work are useful for examining the accuracy of computational methods.