Time-Delay Systems

Systems theory is a foundational discipline that abstracts real-world processes into “systems”, a perspective now ubiquitous in fields like physics, chemistry, biology, engineering, and economics. A system comprises a state defining its current configuration, inputs representing external influences, and outputs demonstrating its effects on the environment. This framework is supported by efficient methods for the representation, the simulation, the analysis, and the control of such systems.

When it comes to representation, differential equations have long been prized for their ability to accurately describing physical phenomena, showcasing simple behaviors, such as convergence to an equilibrium point, to more complex ones, such as multistability, oscillations, and even chaos. As a result, a multitude number of methods for their simulation and their qualitative analysis have been developed and are now well-established – Lyapunov theory of stability being one of them. In spite of that, this field is still an active field of research, again due to its richness and broad applicability.

The control of systems described by differential equations – consisting of finding suitable system inputs in order to enforce a desired behavior for the controlled system – is also well-established and benefits from a rich array of tools ranging from frequency-domain techniques to time-domain methodologies.

Delay-differential equations, a more recent development, incorporate past values alongside current states and inputs, often the result of propagation or memory effects within the system and its interaction with its environment. Systems described by such delay-differential equations are referred to as time-“delay systems”. The presence of delays adds complexity to their behavior, demanding more sophisticated tools for their analysis compared to those described by more standard differential equations.

As it can be expected, controlling such systems may also pose significant challenges as the history of the process now plays a major role in its future evolution. In fact, longer delays often pose larger risks than small ones: it is certainly more difficult to drive a vehicle with a longer actuation delay than with a short one and one may easily sense here the existence of a tradeoff between speed and actuation delay that would guarantee a desired safety profile for the controlled vehicle. In this regard, it is natural expected that some properties of the system, such as its stability, are now properties which may be depending on the value of the delays.

The objective of this article is to give to the readers a glimpse into the realm of time-delay systems through the lens of systems and control theory, and provide...