Finite-time control of perturbed dynamical systems based on a generalized time transformation approach

Abstract

We study finite-time control of perturbed dynamical systems based on the time transformation approach. For addressing time-critical applications, where the execution of a control algorithm over a prescribed time interval [0,τ) is necessary with τ being a user-defined convergence time, we introduce a new class of scalar, time-varying gain functions entitled as “generalized finite-time gain functions” that have the capability to convert an original baseline control algorithm into a time-varying one. Based on these generalized finite-time gain functions, in particular, the corresponding “generalized time transformation functions” are obtained and used to transform a resulting algorithm over the prescribed time interval [0,τ) to an equivalent algorithm over the stretched infinite-time interval [0,∞) for stability analysis, where the connection between the generalized finite-time gain functions and their corresponding generalized time transformation functions are investigated in detail. A procedure for designing finite-time control algorithms is further proposed and illustrated by numerical examples showing that the method is applicable to, but not limited to, a class of nonlinear systems as well as multiagent systems. In addition, we show all the conditions on the proposed generalized finite-time gain functions that guarantee the boundedness and convergence of the state and control signals. An application of our theoretical findings to the distributed control of networked multiagent systems problem over a prescribed time interval is also presented.