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.