Input-Output Finite-Time Stability of Linear Systems: Necessary and Sufficient Conditions

In the recent paper "Input-output finite-time stabilization of linear systems," (F. Amato et al.) a sufficient condition for input-output finite-time stability (IO-FTS), when the inputs of the system are signals, has been provided; such condition requires the existence of a feasible solution to an optimization problem involving a certain differential linear matrix inequality (DLMI). Roughly speaking, a system is said to be input-output finite-time stable if, given a class of norm bounded input signals over a specified time interval of length, the outputs of the system do not exceed an assigned threshold during such time interval. IO-FTS constraints permit to specify quantitative bounds on the controlled variables to be fulfilled during the transient response. In this context, this paper presents several novel contributions. First, by using an approach based on the reachability Gramian theory, we show that the main theorem of F. Amato et al. is actually also a necessary condition for IO-FTS; at the same time we provide an alternative-still necessary and sufficient-condition for IO-FTS, in this case based on the existence of a suitable solution to a differential Lyapunov equality (DLE). We show that the last condition is computationally more efficient; however, the formulation via DLMI allows to solve the problem of the IO finite-time stabilization via output feedback. The effectiveness and computational issues of the two approaches for the analysis and the synthesis, respectively, are discussed in two examples; in particular, our methodology is used in the second example to minimize the maximum displacement and velocity of a building subject to an earthquake of given magnitude.