I have the following question from an exercise set of the course "Control of Non Linear Mechanical Systems."
It involves so called integrator backstepping. And I've got a vague idea from a website how it must be done but I'm looking for a simple step by step guide for dummies.
The question is as follows: Consider the third order system:
$\dot{x}_1 = x_2+x_1^2-x_1^3$
$\dot{x}_2 = x_{3}$
$\dot{x}_3 = u$
Check that after one step of backstepping we can globally stabilize the second order system:
$\dot{x}_1 = x_2+x_1^2-x_1^3$
$\dot{x}_2 = x_3$
With $x_3$ as an input by the control $x_3 = -2x_1(x_2+x_1^2-x_1^3)-x_2-x_1^2-x_1$ Hint: Use the Lyapunov function $V(x_1,x_2)=\frac{1}{2}x_1^2+\frac{1}{2}(x_2+x_1^2)^2$. Using this result, design a controller that stabilizes the third order system.
What I do know:
I know that $x_2$ must cancel the non-linear quadratic term in $\dot{x}_1$, so $x_2 = -x_1^2$ And $x_3$ = $\phi(x_1,x_2)$ and $V(x_1,x_2)=\frac{1}{2}x_1^2+\frac{1}{2}x_2^2$
The more I try it, the more confused I get. There also has to be a transformation to $z$ coordinates for some reason, but that's not entirely clear to me.
Any help will be greatly appreciated.
Anybody looking for a solution to this problem can check out example 14.9 on page 593 of Non Linear Systems by Khalil. ($3^{rd}$edition).
Which is exactly the same problem as posted here.