Control-Lyapunov function: Difference between revisions

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In [[control theory]], a '''control-Lyapunov function (cLf)'''<ref>Isidori</ref><ref>Freeman (46)</ref><ref>Khalil</ref><ref>Sontag</ref> is an extension of the idea of [[Lyapunov function]] <math>V(x)</math> to systems with control inputs. The ordinary Lyapunov function is used to test whether a [[dynamical system]] is ''stable'' (more restrictively, ''asymptotically stable''). That is, whether the system starting in a state <math>x \ne 0</math> in some ___domain ''D'' will remain in ''D'', or for ''asymptotic stability'' will eventually return to <math>x = 0</math>. The control-Lyapunov function is used to test whether a system is ''asymptotically controllablestabilizable'', that is whether for any state ''x'' there exists a control <math> u(x,t)</math> such that the system can be brought to the zero state asymptotically by applying the control ''u''.

More formally, suppose we are given an autonomous dynamical system with inputs

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for each state ''x''.

The theory and application of control-Lyapunov functions were developed by Z. Artstein and [[Eduardo D. Sontag|E. D. Sontag]] in the 1980s and 1990s.

==Example==