Complete quadratic Lyapunov functionals for distributed delay systems

Abstract

International audienceThis paper is concerned with the stability analysis of distributed delay systems using complete-Lyapunov functionals. Numerous articles aim at approximating their parameters thanks to a discretization method or polynomial modeling. The interest of such approximations is the design of tractable sufficient stability conditions. In the present article, we provide an alternative method based on polynomial approximation which takes advantages of the Legendre polynomials and their properties. The resulting stability conditions are scalable with respect to the maximum degree of the polynomials and are expressed in terms of tractable linear matrix inequalities. Several examples of delayed systems are tested to show the effectiveness of the method

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

“…They are also based on Lyapunov functionals with extended state variables() at the cost of increased computational complexity. At the heart of all these techniques is the use of integral inequalities as Jensen's, Wirtinger's ones,() or Bessel inequalities (see the works of Seuret et al() for the general case and the works of Zeng et al and Park et al for some particular refined cases). These inequalities are essential and have been developed to take advantage of information on delayed signals.…”

“…Compared with previous results on quadratic separation for time‐delay systems with a constant delay, even if the methodology studied here is the same, the embedding in the work of Gouaisbaut and Peaucelle depends on Jensen's inequality, which makes it more conservative and can be viewed as a particular case of the proposed results. Notice that the use of Bessel inequality has been also studied successfully in the works of Seuret et al() in the Lyapunov‐Krasovskii functional framework. In this approach, the Bessel inequality is used to reduce the pessimism of some integral inequalities.…”

Bessel summation inequalities for stability analysis of discrete‐time systems with time‐varying delays

“…Time delay is a common phenomenon in many dynamic systems such as chemical systems, biological systems, mechanical engineering systems, and networked control systems . Since such phenomenon often causes control performance degradation or even system instability, there have been considerable efforts to solve stability analysis problems for time‐delay systems before implementing control strategies . Stability analysis for delayed discrete‐time systems can be classified into two types of problems depending on delay properties: constant time delays and time‐varying delays.…”

“…Furthermore, the Jensen integral inequality was successfully extended to general ones called the Bessel‐Legendre (B‐L) inequalities, which are constructed with the Bessel inequality and arbitrary‐order Legendre polynomials . At the same time, it was shown that high‐order orthogonal polynomials provide more precise bounds of integral quadratic functions and contribute to the developing less conservative stability conditions than the low‐order ones. Similarly, for stability analysis of delayed discrete‐time systems, there have been various efforts to reduce a bounding gap between the Jensen summation inequality and a summation quadratic function by using zero‐, first‐, second‐, and third‐order discrete orthogonal polynomials obtained from Gram‐Schmidt orthogonalization process .…”

Multiple integral inequalities and stability analysis of time delay systems

“…In[14], a delay bounding interval [0.200, 1.877] was obtained with 16 decision variables, while the authors of[2] derived the delay bounding interval [0.2000, 1.9504] from Theorem 1 with 59 decision variables. In[16], the lower bound of the delay was found to be 0.2001, while…”