Analisi Stabilità tramite nuove disuguaglianze nonlineari di tipo Halanay di sistemi a tempo discreto con ritardi vincolati

The research presented in this thesis covers the stability analysis of discrete-time systems with delays where the delay signals are contained to follow a dynamic driven by a delay digraph. The interest in this class of systems has been motivated traditionally by sampled-data systems in which a process is sampled periodically and then controlled via a machine. This setting leads to relatively cheap control solutions but requires the discretization of signals which typically introduces time delays. Time-delays often lead to complex behaviors in the dynamics of a system and may cause the loss of stability. Therefore, the investigation of stability is a fundamental problem, which often turns to be highly challenging. More recently the interest in discrete-time systems with delay has been motivated by networked control systems in which the connection between the process and the controller is made through a shared communication network. This communication network increases the flexibility of the control architecture but also introduces effects such as packet dropouts, uncertain time-varying delays and timing jitter. Motivated by the fact that almost every system in practice is subject to constraints and Lyapunov theory is one of the few methods that can be easily adapted to deal with constraints, all results in the thesis are based on Lyapunov theory. The delay variation modeled via a delay digraph can have a beneficial effect on the stability of a system, and can guarantee stabilization even when instability occurs for one or more values of the delays, taken constant. The delay digraph can describe a large range of cases, including the case of bounded delay variation and the case of arbitrary delay-signals. Some tractable sufficient Lyapunov conditions exploiting a novel nonlinear discrete-time Halanay-type inequality are provided in the stability and the input-to-state stability analysis. The more information is provided by the delays digraph, the more is the reduction of the number of involved inequalities in the provided sufficient Lyapunov conditions. The key role of using the nonlinear Halanay-type inequality is to provide simpler conditions with respect to Lyapunov-Razumikhin and Lyapunov-Krasovskii techniques. Furthermore, the nonlinear Halany-inequality allows to chose simple Lyapunov function candidate in the analysis. In the linear case, by using simple quadratic Lyapunov candidates, the stability properties are obtained by using Linear Matrix Inequality techniques. For nonlinear discrete-time delay systems, the case where the delays are subjected to follow a Markov chain is discussed and some sufficient conditions for the exponential mean square stability are provided.