Corentin Briat

Time-delay systems

Delays are ubiquitous in nature as they arise a wide range of fields such as biology, ecology, economics, particle physics and communication networks. Time-delay systems (or hereditary systems) are a class of dynamical systems whose state evolution depends on past state values. For instance, discrete-delay systems with single delay can be generally expressed as

\[ \begin{array}{rcl} \dot{x}(t)&=&f(x(t),x(t-h))\\ x(s)&=&\phi(s), s\in[-h,0] \end{array} \]

where \(h>0\) is the delay, \(x(t)\) is the state value of the system at time \(t\), and \(\phi\) is the functional initial condition. They can be seen as a generalization of usual dynamical systems whose future evolution only depends on the current value of the state; e.g. \(\dot{x}(t)=f(x(t))\). In general, delays are detrimental to stability and will more likely destabilize systems and deteriorate performance. In some other problems, however, delays can improve stability properties or induce desirable oscillations (e.g. limit cycles in biological systems).

The analysis of delay-systems is slightly more complicated than for usual dynamical systems, mainly due to their infinite dimensional nature. Many techniques have been developed both in frequency- and time-domains. When delays are time-varying, time-domains approaches are preferable through the use of the Lyapunov-Krasovskii and Lyapunov-Razumikhin Theorems, and robust analysis methods (such as IQCs, well-posedness, etc.). When delays are state-dependent (as in communication networks), the problem is much more complicated and still open since very few general results have been obtained.

Controlling time-delay systems is also a challenging problem and many approaches have been developed over the past years. For input-delay systems, state-predictors can be considered to predict future values of the state in order to use them in the control law. When the delay acts on the state of the system, controllers involving “delay components” can be considered, either by using delayed signals (controllers involving memory) or by having the delay scheduling the controller (delay-scheduled controllers) in an LPV-way (see section above). In any case, however, the knowledge of the delay is critical for implementation. When the delay is time-varying or uncertain, it is difficult to know the exact value of the delay in real time and controllers must be made resilient with respect to delay uncertainty (memory-resilient controllers).

I am personally interested in the development of new theoretical results for the analysis and control of time-delay systems. The application of the theory of time-delay systems to real world processes is part of my research interests, notably the application to communication and biological networks.

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