Lyapunov Methods for Time-Invariant Delay Difference Inclusions

Motivated by the fact that delay difference inclusions (DDIs) form a rich modeling class that includes, for example, uncertain time-delay systems and certain types of networked control systems, this paper provides a comprehensive collection of Lyapunov methods for DDIs. Firstly, the Lyapunov-Krasovskii approach, which is an extension of the classical Lyapunov theory to time-delay systems, is considered. It is shown that a DDI is KL-stable if and only if it admits a Lyapunov-Krasovskii function (LKF). Secondly, the Lyapunov-Razumikhin method, which is a type of small-gain approach for time-delay systems, is studied. It is proven that a DDI is KL-stable if it admits a Lyapunov-Razumikhin function (LRF). Moreover, an example of a linear delay difference equation which is globally exponentially stable but does not admit a LRF is provided. Thus, it is established that the existence of a LRF is not a necessary condition for KL-stability of a DDI. Then, it is shown that the existence of a LRF is a sufficient condition for the existence of a LKF and that only under certain additional assumptions the converse is true. Furthermore, it is shown that a LRF induces a family of sets with certain contraction properties that are particular to time-delay systems. On the other hand, a LKF is shown to induce a type of contractive set similar to those induced by a classical Lyapunov function. The class of quadratic candidate functions is used to illustrate the results derived in this paper in terms of both LKFs and LRFs, respectively. Both stability analysis and stabilizing controller synthesis methods for linear DDIs are proposed. 1. Introduction. Systems affected by time-delay can be found within many applications in the control field, see, e.g., [25] for an extensive list of examples. Delay difference inclusions (DDIs) form a rich modeling class that includes, for example, uncertain systems, time-delay systems and certain types of networked control systems [14, 45]. However, while stability analysis of delay-free systems is often based on the existence of a Lyapunov function (LF), see, e.g., [2], for systems affected by delays the classical Lyapunov theory does not apply straightforwardly. This is due to the fact that the influence of the delayed states can cause a violation of the monotonic decrease condition that a standard LF obeys. To solve this issue, two types of functions were proposed: the Lyapunov-Krasovskii function (LKF) [27], which is an extension of the classical LF to time-delay systems, …