Lyapunov redesign
# 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.

