Section: New Results

Set invariance for discrete-time delay systems

We studied the existence of positively invariant sets for linear delay-difference equations. In particular, we regarded two strong stability notions: robust (with respect to delay parameter) asymptotic stability for the discrete-time case and delay-independent stability for the continuous-time case. The correlation between these stability concepts is also considered. Furthermore, for the delay-difference equations with two delay parameters, we provided a computationally efficient numerical routine which is necessary to guarantee the existence of contractive sets of Lyapunov–Razumikhin type. This condition also appears to be necessary and sufficient for the delay-independent stability and sufficient for the robust asymptotic stability. The results are published in [25] .

We proposed a new perspective on the structural properties of invaraint sets for time delay systems via set factorization. This novel perspective describes, in a unified framework, different existing notions of invariant sets [60] . Additionally, it is shown that the (possible non-minimal) state space representation is a key element in the description of low complexity invariant sets.