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

$u=\varphi (t,x)+v$

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

Further reading

• Khalil, H.K. (2002). Nonlinear Systems (3rd ed.). Upper Saddle River, NJ: Prentice Hall. ISBN 0-13-067389-7. http://www.egr.msu.edu/~khalil/NonlinearSystems/. 
• .Freeman, R. A.; Kokotović, P. (1996). Robust Nonlinear Control Design: State-Space and Lyapunov Techniques. Boston, MA: Birkhäuser. ISBN 978-0-8176-4758-2.