Control-Lyapunov function: Difference between revisions

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In [[control theory]] a '''control-Lyapunov function''' <math>V(x,u)</math> is a generalization of the notion of [[Lyapunov function]] <math>V(x)</math> used in [[Lyapunov stability|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 <math>x \ne 0</math> will eventually return to <math>x = 0</math>. 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 <math> u(x,t)</math> 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

The theory and application of CLF's were developed by Z. Artstein and [[Eduardo D. Sontag|E. D. Sontag]] in the 1980's.

:<math>

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

</math>

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

:<math>

\forall x \ne 0, \exists u \qquad \dot{V}(x,u) < 0

</math>

(this 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 clf if and only if there exists a regular stabilizing feedback u(x).

It may not be easy to find a clf 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

:<math>

u^*(x) = \arg\min_u \nabla V(x,u) \cdot f(x,u)

</math>

for each state x.

The theory and application of CLF's were developed by Z. Artstein and [[Eduardo D. Sontag|E. D. Sontag]] in the 1980's. and 1990's.

[[Category:Control theory]]