In control theory, a control-Lyapunov function is a Lyapunov function
More formally, suppose we are given an autonomous dynamical system
where
Definition. A control-Lyapunov function is a function
The last condition is the key condition; in words it says that for each state x we can find a control u that will reduce the "energy" V. Intuitively, if in each state we can always find a way to reduce the energy, we should eventually be able to bring the energy to zero, that is to bring the system to a stop. This is made rigorous by the following result:
Artstein's theorem. The dynamical system has a differentiable control-Lyapunov function if and only if there exists a regular stabilizing feedback u(x).
It may not be easy to find a control-Lyapunov function for a given system, but if we can find one thanks to some ingenuity and luck, then the feedback stabilization problem simplifies considerably, in fact it reduces to solving a static non-linear programming problem
for each state x.
The theory and application of control-Lyapunov functions were developed by Z. Artstein and E. D. Sontag in the 1980s and 1990s.
Example
Here is a characteristic example of applying a Lyapunov candidate function to a control problem.
Consider the non-linear system, which is a mass-spring-damper system with spring hardening and position dependent mass described by
Now given the desired state,
A Control-Lyapunov candidate is then
which is positive definite for all
Now taking the time derivative of
The goal is to get the time derivative to be
which is globally exponentially stable if
Hence we want the rightmost bracket of
to fulfill the requirement
which upon substitution of the dynamics,
Solving for
with
This control law will guarantee global exponential stability since upon substitution into the time derivative yields, as expected
which is a linear first order differential equation which has solution
And hence the error and error rate, remembering that
If you wish to tune a particular response from this, it is necessary to substitute back into the solution we derived for
which can then be solved using any linear differential equation methods.