On this basis, discontinuous feedback control strategies appeared in the
middle of the 20th century under the name of theory of variable-structure
systems. Within this viewpoint, the control inputs typically take values
from a discrete set, such as the extreme limits of a relay, or from a limited
collection of prespecified feedback control functions. The switching logic is
designed in such a way that a contracting property dominates the closedloop
dynamics of the system thus leading to a stabilization on a switching
manifold, which induces desirable trajectories. Based on these principles,
one of the most popular techniques was created, developed since the 1950s
and popularized by the seminal paper by Utkin (see [30] in chapter 7): the
sliding mode control. The essential feature of this technique is the choice
of a switching surface of the state space according to the desired dynamical
specifications of the closed-loop system. The switching logic, and thus the
control law, are designed so that the state trajectories reach the surface
and remain on it.