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

• Published in: IEEE transactions on automatic control Vol. 57; no. 12; pp. 3051 - 3063
• Main Authors: Amato, F., Carannante, G., De Tommasi, G., Pironti, A.
• Format: Journal Article
• Language: English
• Published: New York, NY IEEE 01.12.2012; Institute of Electrical and Electronics Engineers; The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
• Subjects: Applied sciences; Computer science; control theory; systems; Control system analysis; Control system synthesis; Control theory. Systems; Differential linear matrix inequality (DLMI); differential Lyapunov Equation (DLE); Earthquakes; Equations; Exact sciences and technology; input-output finite-time stability (IO-FTS); Intervals; Linear matrix inequalities; Linear systems; LMI; Mathematical analysis; Mathematical models; Modelling and identification; Numerical stability; Output feedback; reachability Gramian; Stability; Stability analysis; Stabilization; Studies; System theory; Theorems; time-varying systems; time-varying systems; Input-output stability; Input signal; differential Lyapunov Equation (DLE); Necessary and sufficient condition; Differential equation; Feedback regulation; L2 approximation; reachability Gramian; Input output controller; Linear control; Optimization; Differential inequality; Time varying system; Maximum velocity; Earthquakes; Linear matrix inequality; Time interval; Transient response; Existence theorem; input-output finite-time stability (IO-FTS); Differential linear matrix inequality (DLMI); LMI; linear systems; Lyapunov equation; Reachability; Output feedback; Lyapunov function
• ISSN: 0018-9286; 1558-2523
• DOI: 10.1109/TAC.2012.2199151

In the recent paper "Input-output finite-time stabilization of linear systems," (F. Amato ) a sufficient condition for input-output finite-time stability (IO-FTS), when the inputs of the system are L 2 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 T , 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 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.