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 ${\displaystyle V}$. Consider the system

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

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

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

and the control law

${\displaystyle u=\phi (t,x)+v}$

stabilizes the system. The design of ${\displaystyle v}$ is called Lyapunov redesign.