Lyapunov redesign

In nonlinear control, the technique of Lyapunov redesign refers to the design where a stabilizing state feedback controller can be constructed with knowledge of the Lyapunov function V. Consider the system

:\dot{x} = f(t,x)+G(t,x)[u+\delta(t, x, u)]

where x \in R^n is the state vector and u \in R^p is the vector of inputs. The functions f, G, and \delta are defined for (t, x, u) \in [0, \inf) \times D \times R^p, where D \subset R^n is a domain that contains the origin. A nominal model for this system can be written as

:\dot{x} = f(t,x)+G(t,x)u

and the control law

:u = \phi(t, x)+v

stabilizes the system. The design of v is called Lyapunov redesign.