Graphical Stabilization Approach for Unstable First Order Linear Systems with Time Delay

Abstract

In this paper, a graphical stabilization approach is proposed and analyzed for a class of unstable first order linear systems with time delay. We first show that the control designs based on time invariant models are unable to guarantee stability and asymptotic tracking for unstable first order linear systems in general case. So, the condition stability is analysed graphically by computing the first derivative and plotting the graph of a function with precision; the first derivative allows us to determine the critical points and several conditions of stability. Therefore, it’s important to note that the method can guarantee the existence of a proportional gain to ensure the stability of the closed-loop system such that the time delay is small relatively to the time constant. Finally, a numerical example illustrates the efficiency and performances of the proposed approach.