Fixed-time and finite-time control and estimation

Lhe theory of control is used to design the regulators for dynamical systems, to estimate the internal state variables or parameters of the plants, to monitor and detect the faults, in many domains of engineering, technical or natural sciences. Important quality requirements, which are associated with these designs, deal with the time of convergence of the estimation or control errors to zero, and their robustness with respect to unmodeled disturbances and measurement noises, or delays in actuation or sensing. To this end, the concepts of convergence in a finite or in a fixed time show their importance, since imposing these acceleration constraints on the closed-loop dynamics also implies remarkable robustness features of these systems. This tutorial provides the basic motivations, definitions of properties finite-time and fixed-time convergent systems, with corresponding design of controls and state observers. A special attention is paid to discrete-time implementation of this kind of algorithms, which is particularly challenging. The exposition is accompanied by illustrative simple examples.