Robust Finite-Time Control of Linear System with Non-Differentiable Time-Varying Delay

Abstract

Practical systems such as hybrid power systems are currently implemented around the world. In order to get the system to work properly, the systems usually require their behavior to be maintained or state values to stay within a certain threshold. However, it is difficult to form a perfect mathematical model for describing behavior of the practical systems since there may be some information (uncertainties) that is not observed. Thus, in this article, we studied the stability of an uncertain linear system with a non-differentiable time-varying delay. We constructed Lyapunov-Krasovskii functionals (LKFs) containing several symmetric positive definite matrices to obtain robust finite-time stability (RFTS) and stabilization (RFTU) of the uncertain linear system. With the controller and uncertainties in the considered system, there exist nonlinear terms occurring in the formulation process. Past research handled these nonlinear terms as new variables but this led to some difficulty from a computation point of view. Instead, we applied a novel approach via Cauchy-like matrix inequalities to handle these difficulties. In the end, we present three numerical simulations to show the effectiveness of our proposed theory.

Robust Finite-Time Control of Discrete-Time Switched Positive Time-Varying Delay Systems with Exogenous Disturbance and Their Application

“…Furthermore, stability theory and optimal control have been studied and there is interest in them [31,32]. Different from the Lyapunov stability concept, finite-time stability (FTS) requires that the state trajectories of the considered systems do not exceed a certain bound during a fixed finite-time interval under some given constraints of the initial condition [33][34][35][36]. Besides, the concept of FTS is extended to a finite-time boundedness (FTB) if exogenous disturbances or the influence of perturbing forces are taken into account together [37].…”

Finite-Time Boundedness of Linear Uncertain Switched Positive Time-Varying Delay Systems with Finite-Time Unbounded Subsystems and Exogenous Disturbance

“…Nonetheless, the dynamical systems, including the slight uncertainties, may lead to the instability of those systems. Consequently, many researchers have devoted themselves to studying the FTS, FTB, and stability of the systems with uncertainties during the last decades [45][46][47][48][49][50][51][52]. However, to the best of our knowledge, there is no result on the FTB for a class of SPSs, including time-varying delay, interval uncertainties, exogenous disturbance, and finitetime unbounded subsystems in the literature.…”

An LMI Approach to Nonlinear State-Feedback Stability of Uncertain Time-Delay Systems in the Presence of Lipschitzian Nonlinearities

“…[1][2][3][4]. Because states of the system depend on the present time and a time period in the past, time delays occur in dynamical systems, and ignoring their effect yields severe deterioration in system performance or even system instability [5,6]. Thus, control of time-delay systems is known to have practical significance [7][8][9].…”