# Lyapunov function

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## Lyapunov function

[lē′ap·ə‚nȯf ‚fəŋk·shən]
(mathematics)
A function of a vector and of time which is positive-definite and has a negative-definite derivative with respect to time for nonzero vectors, is identically zero for the zero vector, and approaches infinity as the norm of the vector approaches infinity; used in determining the stability of control systems. Also spelled Liapunov function.
McGraw-Hill Dictionary of Scientific & Technical Terms, 6E, Copyright © 2003 by The McGraw-Hill Companies, Inc.
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Also Lyapunov function is necessary for the process of designing adaptive controllers.
The Lyapunov function will determine the gains of the PI-fuzzy controller to ensure the convergence of the actual speed of the DSIM to its reference value.
Taking time derivative of the Lyapunov function, (29) can be written as
If one can find a Lyapunov function V(t, x(t)) as well as three class-K functions [g.sub.i], i = 1, 2, 3 such that
In the first part, the Lyapunov function will be constructed and the stability of [??] subsystem will be proved.
Akagi, "Modified Model Predictive Control of a Bidirectional AC-DC Converter Based on Lyapunov Function for Energy Storage Systems," IEEE Transactions on Industrial Electronics, vol.
firstly, we give two results about the existence and uniqueness of the periodic solution of (2) by using the fixed point theory; then, we use Lyapunov function method and discuss the stability of the periodic solution; further, we discuss (1) and get the existence of the periodic solution of (1); some new results are obtained.
The method generalizes the idea which shows that the system is stable if there are some Lyapunov function candidates for the system.
In  the authors propose a control law based on backstepping  and Lyapunov function so as to stabilize the global system.
We firstly consider the poultry subsystem in farms and define a Lyapunov function
Selecting the Lyapunov function using sliding mode is a natural and reasonable approach to get to the desired control goals that is tracking desired trajectory.

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