Exponential Stability Criteria for Positive Systems with Time-Varying Delay: A Delay Decomposition Technique

Abstract

This paper is concerned with exponential stability analysis for linear continuous positive systems with bounded time-varying delay. Our approach is based on Lyapunov-Krasovskii functional with delay decomposition. The main idea is to divide uniformly the discrete delay interval into multiple segments and then to choose proper functionals with different weighted matrices corresponding to different segments. The synthesis of our delay-dependent sufficient stability criteria is formulated in terms of linear matrix inequalities. The proposed results are shown to be less conservative compared to other results from the literature. Some illustrative examples are given to show the effectiveness of the obtained results.

Robust Controlled Positive Delayed Systems with Interval Parameter Uncertainties: A Delay Uniform Decomposition Approach

“…Lemma 4. (Elloumi et al, 2015) For some given scalars h M , α > 0 and h a (h a = h M /2), if there exist positive definite matrices Q 3 , P , Z i , (i = 1, 2, 3) and Q 1 = Q 1 ,Q 12 , Q 2 = Q 2 with appropriate dimensions, system (6) is simultaneously positive and exponentially stable, where α is defined as its decay rate for h(t) satisfying (2), such that…”

“…This is due to the fact that we use the bi-decomposition of the interval [0, h M ]. Thus, as demonstrated by Elloumi et al (2015), conservatism can be reduced by increasing the decomposition of [0, h M ] although the number of LMIs is then greater.…”

“…The result proposed in this paper can be extended to the case of a delay N -decomposition. In fact, an α-exponential stability analysis was made by Elloumi et al (2015) by using a uniform delay N -decomposition technique in order to demonstrate that increasing the number of divisions N allows us to us reduce conservatism. This fact can consequently be used to improve our result.…”

“…In this work, we are interested in the exponential stability and stabilization known for its effect on the faster convergence of states. In the other hand, we are motivated by the result presented by Elloumi et al (2015) demonstrating that increasing the delay decomposition provides less conservative results. Unfortunately, this leads to solving a large-scale system of LMIs when faced with a large number of delay decomposition.…”

“…The main contribution of this paper concerns the robust α-exponential stabilization for continuous positive systems with time-varying delay when the control design might include parametric uncertainties. The proposed approach is based on the exponential stability conditions proposed by Elloumi et al (2015), especially those founded by an equidistant bi-decomposition of the delay. By using Lyapunov-Krasovskii functions (LKFs), the elaborated conditions can easily be solved by using LMIs.…”