Stability analysis of time‐delay systems via Bessel inequality: A quadratic separation approach

Abstract

In this paper, new sufficient stability conditions for the asymptotic stability of time-delay systems are presented using the quadratic separation approach. The time-delay system is modeled as an interconnected closed-loop system involving a linear transformation and delay-dependent functions, representing the uncertainties brought by the delay. Those complex-valued functions are then embedded into adequate norm-bounded uncertainties, which lead to several stability results. The novelty of this approach relies on the introduction of new dedicated functions that are built in accordance to the Bessel inequality. They allow us to model the system as an uncertain feedback system and to control the accuracy of the inequality. Then, a sequence of linear matrix inequality conditions is proposed, which tends to the analytical bounds for both delay-dependent stability and delay range stability, at least on examples.

Input/output stability of a damped string equation coupled with ordinary differential system

“…To this end, a complementary approach is to use robust stability analysis tools, as for example in the work of Espitia et al, which considers an event‐trigger control. In the same direction, the small‐gain theorem and quadratic separation (QS) provide, as we show, efficient frameworks to reveal the stability conditions for the interconnected ODE/PDE system in a parameter region of interest.…”

“…The following system is borrowed from the work of Ariba et al, where …”

Robust stability analysis of fractional‐order interval systems with multiple time delays

“…It has been illustrated that time delay frequently cause instability in control systems and generates oscillations . Therefore, the problem of the analysis and control of systems whose models have delays is of both theoretical and practical importance …”

“…3 Therefore, the problem of the analysis and control of systems whose models have delays is of both theoretical and practical importance. [4][5][6] Recently, using fractional-order calculus for modeling physical systems widely has been spread because it can describe the behavior of many systems more accurately than its integer-order counterpart, [7][8][9] including delay systems. 10,11 Papers were published to introduce some criteria for the robust stability of fractional-order systems based on the concepts of value set, Young and Jensen inequalities, and principal characteristic equation.…”

Parabolic Set Simulation for Reachability Analysis of Linear Time Invariant Systems with Integral Quadratic Constraint

“…Let n = (E −Ẽ)x * − (f −f ). Using(1,12,(8)(9)(10), n satisfies the linear time varying differential equation:ṅ = (−A + M xw M -1 w B +ẼBM -1 w B )n. Sincet belongs to I(P ) and by time-continuity ofẼ(·) over [0, t], there is a scalar K > 0 that bounds Ẽ (·) over [0, t]. Then, there exists L > 0 upper bound of −A +M xw M -1 w B +Ẽ(·)BM -1 w B over [0, t].…”