Control-Lyapunov function

In control theory, a control-Lyapunov function $V(x,u)$ is a generalization of the notion of Lyapunov function $V(x)$ used in stability analysis. The ordinary Lyapunov function is used to test whether a dynamical system is stable, that is whether the system started in a state $x\neq 0$ will eventually return to $x=0$. The control-Lyapunov function is used to test whether a system is feedback stabilizable, that is whether for any state x there exists a control $u(x,t)$ such that the system can be brought to the zero state by applying the control u.

More formally, suppose we are given a dynamical system

${\dot {x}}(t)=f(x(t))+g(x(t))\,u(t),$

where the state x(t) and the control u(t) are vectors.

Definition. A control-Lyapunov function is a function $V(x,u)$ that is continuous, positive-definite (that is V(x,u) is positive except at $x=0$ where it is zero), proper (that is $V(x)\to \infty$ as $|x|\to \infty$), and such that

$\forall x\neq 0,\exists u\qquad {\dot {V}}(x,u)<0.$

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

$u^{*}(x)=\arg \min _{u}\nabla V(x,u)\cdot f(x,u)$

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.

See also

Artstein's theorem