Control-Lyapunov Function - Example

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

m(1+q^2)\ddot{q}+b\dot{q}+K_0q+K_1q^3=u

Now given the desired state, and actual state, with error, define a function as

r=\dot{e}+\alpha e

A Control-Lyapunov candidate is then

V=\frac{1}{2}r^2

which is positive definite for all, .

Now taking the time derivative of

\dot{V}=r\dot{r}
\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 is globally positive definite (which it is).

Hence we want the rightmost bracket of ,

(\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, 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 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 and, 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^{-\kappa t}

And hence the error and error rate, remembering that, 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 and solve for . This is left as an exercise for the reader but the first few steps at the solution are:

r\dot{r}=-\frac{\kappa}{2}r^2
\dot{r}=-\frac{\kappa}{2}r
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.

Read more about this topic:  Control-Lyapunov Function

Famous quotes containing the word example:

    “Our intellect is not the most subtle, the most powerful, the most appropriate, instrument for revealing the truth. It is life that, little by little, example by example, permits us to see that what is most important to our heart, or to our mind, is learned not by reasoning but through other agencies. Then it is that the intellect, observing their superiority, abdicates its control to them upon reasoned grounds and agrees to become their collaborator and lackey.”
    —Marcel Proust (1871–1922)