Stability and Stabilization of Nonlinear Systems

Phenomenal progress in nonlinear systems theory has been made during the last
decades. It has been reflected in two aspects. On the one hand, internal and external
global stability notions have been studied intensely for uncertain nonlinear systems.
On the other hand, the applications of these advanced stability results to control
engineering systems have led to numerous novel methodologies for the design of
nonlinear feedback controllers. It is fair to say that input-to-state stability (ISS),
a notion invented by E.D. Sontag in the late 1980s, plays an influential role in the
work of many researchers including the authors of this book. ISS has bridged the gap
which previously existed between the input–output and the state-space methods, two
popular approaches within the control systems community. Roughly speaking, the
importance of ISS for the study of nonlinear systems is reflected by the intriguing
fact that it captures two main stability notions: Lyapunov stability (i.e., the behavior
of the zero-input response with respect to nonzero initial conditions) and input–
output stability (i.e., the behavior of the zero-state response with respect to nonzero
external inputs).