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 [math]\displaystyle{ V }[/math]. Consider the system

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

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

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

and the control law

[math]\displaystyle{ u = \phi(t, x)+v }[/math]

stabilizes the system. The design of [math]\displaystyle{ v }[/math] 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/.