Lyapunov-Krasovskii Approach to Robust Stability of Time Delay Systems

Stabilization of time-delay systems through linear differential equations using a descriptor representation

“…It allows having a better understanding of the terms of the LKF and to have a better idea of where the conservatism is introduced [2] and especially obtaining less conservative results. In [15], a complete LKF (ie. which corresponds to necessary and sufficient conditions of stability) is constructed by solving a functional differential equation.…”

Necessary stability conditions for linear delay systems

“…In Kharitonov and Zhabko (2003), the functionals which admit a quadratic lower bound when the linear system is exponentially stable were presented.…”

Some necessary conditions for the exponential stability of one delay systems

“…Moreover, examples showing that the functional (2) does not admit a quadratic lower bound (9) were given in [10], leading to the presentation of a functional named complete that satisfies one.…”

“…The early proposal of the functional in the work of Repin in 1965 [23] was followed by contributions of Datko [3], Infante and Castelan [1], Huang [7] and Louisell [16], where substantial advances were presented. During the past decade, in the papers of Kharitonov [10], [11] fundamental concepts revealing the extend of the analogy with the delay free case were clarified: the Lyapunov matrix function is obtained as the solution of the dynamic, symmetric and algebraic properties that play the role of the Lyapunov equation; the existence and uniqueness of the solution of the Lyapunov matrix equation provided a spectrum Lyapunov like condition is satisfied is established in [12]. This lead to the presentation of a functional, named of complete type, having a quadratic lower bound when the system is exponentially stable, a property demanded by the Lyapunov-Krasovskii stability Theorem [14].…”

“…The results were extended to broader classes of systems such as neutral type and distributed delays systems and applications such as robust stability conditions, exponential estimates of the response [10], stability/instability critical values characterization [22], among others.…”