Robust stability, control, and observation for nonlinear delay systems – Robusta

Time delays are ubiquitous in engineering, physics, and biology, with frequently degrading effects on the stability and robustness of control systems.
From a theoretical perspective, time-delay systems (TDS) occupy an intermediate position between ordinary differential equations and general infinite-dimensional systems. They benefit from a vast body of literature, particularly in the linear case. Despite this extensive research, several central results are missing in the robust stability theory of nonlinear TDS. In this project, we develop them by exploiting recent advances in the treatment of nonlinear infinite-dimensional systems and their robustness theory, in combination with results on TDS obtained by the applicants.

In ROBUSTA, we develop the input-to-state stability (ISS) framework, investigate contraction properties of various orders, and apply our results in neuroscience and observer-based control. ISS is a cornerstone of nonlinear control theory, essential for both system analysis and design. The versatile use of this tool requires solution-based and Lyapunov characterizations with minimal assumptions. None of these requirements are yet met for TDS. This is particularly true in the context of large-scale networks, in which existing results do not take into account a priori bounds on the delays involved, or use overly conservative assumptions on interconnection structures. Beyond ISS, k-contraction theory offers powerful tools for analyzing systems dynamics. Depending on the order k, it can determine the uniqueness of fixed points or rule out limit cycles or higher-order chaotic attractors. In nonlinear TDS applications, such qualitative analysis tools are sorely missing. It is ROBUSTA's ambition to address this deficiency by pioneering the development for k-contraction theory for TDS.

The practical significance of the developed tools will be demonstrated through observer design for finite-dimensional and time-delay systems, as well as applications in neuroscience. In particular, the development of k-contraction theory for TDS could prove transformative in disrupting Parkinsonian brain oscillations through closed-loop deep brain stimulation without compromising healthy neural dynamics.