Systems with time delays

Time delays are commonly encountered in control systems and can occur in various components, including sensors, actuators, and communication channels. Time-delayed systems are typically represented by differential or difference equations that include delay terms. Time delays can result in instability or reduced performance. Since these delays are unavoidable, they need to be considered in the design and analysis of control systems.

Passivity in the face of time delays

In the context of passivity-based control, it is possible to perform an input-output transformation to make a communication channel with delays passive.

Observer-predictors for systems with time delays

One approach to designing control systems involves devising a state predictor-observer and then feeding the prediction into a control law originally designed for the delay-free system. This strategy was used for the social distancing policy proposed during the covid-19 pandemic. 

Infinite-dimensional observer-predictor

In the delay-free scenario, a commonly used observer for nonlinear systems is the high-gain observer.  The observer is popular for two main reasons:

•  Generality. The assumptions required for constructing it are uniform observability (necessary for any observer) and uniform Lypschitz continuity of the nonlinear vector field (a reasonable assumption).

• Ease of tuning. A single scalar gain is adjusted to dominate the system's nonlinearities. The gain can be set as high as desired without compromising the stability of the estimation error dynamics. 

For systems with time delays, the same architecture can still be used. However, the scalar gain can no longer be set as high as desired without destroying stability. The infinite-dimensional predictor presented in [BCM25] addresses this limitation.