A Lyapunov theorem establishes the existence of a Lyapunov function, which is a real-valued function that uniquely describes the long-term behavior of a control...
A Lyapunov theorem establishes the existence of a Lyapunov function, which is a real-valued function that uniquely describes the long-term behavior of a controlled system near an equilibrium point. The Lyapunov function provides information about the stability of the equilibrium point, indicating whether it is asymptotically stable, stable, or unstable.
The Lyapunov function typically involves the system's state variables and a parameter related to the system's dynamics. The Lyapunov function must satisfy specific properties, such as being radially unbounded and having a negative definite derivative with respect to the state variables.
The Lyapunov stability theorem guarantees that, under certain conditions, the equilibrium point of the controlled system is Lyapunov stable. This means that the system converges asymptotically to the equilibrium point as time goes to infinity.
To illustrate the Lyapunov stability theorem, consider the simple linear system:
where u is the control input. A Lyapunov function for this system is:
Taking the derivative of V(x), we get:
Setting V'(x) = 0, we get:
Substituting this value of u into the Lyapunov function, we get:
The Lyapunov function is radially unbounded, negative definite, and has a global minimum at the equilibrium point. Therefore, the equilibrium point is asymptotically stable under the Lyapunov stability theorem