Control-Lyapunov function

Control-Lyapunov function

In control theory, a control-Lyapunov function V(x,u) [1]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 (more restrictively, asymptotically stable). That is, whether the system starting in a state x \ne 0 in some domain D will remain in D, or for asymptotic stability 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 \ne 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.



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, qd, and actual state, q, with error, e = qdq, define a function r as

r=\dot{e}+\alpha e

A Control-Lyapunov candidate is then


which is positive definite for all  q \ne 0, \dot{q} \ne 0.

Now taking the time derivative of V


\dot{V}=(\dot{e}+\alpha e)(\ddot{e}+\alpha \dot{e})

The goal is to get the time derivative to be

\dot{V}=-\kappa V

which is globally exponentially stable if V is globally positive definite (which it is).

Hence we want the rightmost bracket of \dot{V},

(\ddot{e}+\alpha \dot{e})=(\ddot{q}_d-\ddot{q}+\alpha \dot{e})

to fulfill the requirement

(\ddot{q}_d-\ddot{q}+\alpha \dot{e}) = -\frac{\kappa}{2}(\dot{e}+\alpha e)

which upon substitution of the dynamics, \ddot{q}, gives

(\ddot{q}_d-\frac{u-K_0q-K_1q^3-b\dot{q}}{m(1+q^2)}+\alpha \dot{e}) = -\frac{\kappa}{2}(\dot{e}+\alpha e)

Solving for u yields the control law

u= m(1+q^2)(\ddot{q}_d + \alpha \dot{e}+\frac{\kappa}{2}r )+K_0q+K_1q^3+b\dot{q}

with κ and α, both greater than zero, as tunable parameters

This control law will guarantee global exponential stability since upon substitution into the time derivative yields, as expected

\dot{V}=-\kappa V

which is a linear first order differential equation which has solution

V = V(0)e − κt

And hence the error and error rate, remembering that V=\frac{1}{2}(\dot{e}+\alpha e)^2, exponentially decay to zero.

If you wish to tune a particular response from this, it is necessary to substitute back into the solution we derived for V and solve for e. This is left as an exercise for the reader but the first few steps at the solution are:



r=r(0)e^{-\frac{\kappa}{2} t}

\dot{e}+\alpha e= (\dot{e}(0)+\alpha e(0))e^{-\frac{\kappa}{2} t}

which can then be solved using any linear differential equation methods.


  1. ^ Freeman (46)


See also

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