based on Haimo manuscript the below equation is finite time:
$$ \dot x(t) = - {\rm sign}(x(t)) \tag1 $$ and based on the manuscript many other equation are proposed, such as terminal sliding mode, fast terminal sliding mode, ... controllers in control systems theory which is not main problem for me, here. infact the -sign(x(t)) term is as control input. I want to have finite time control input which its integral is zero.
in an application like cascade control, integral of the right side of the equation (1), ${\rm sign}(x(t))$, is used which should reaches zero after time.
As you know and based on Haimo, finite time variables $x(t)$ do not spin around the origin, meaning its sign often is not changed and the variable reaches origin in finite time with constant slope.
How can I trigger the equation variable to force the integral of ${\rm sign}(x(t))$ function to zero value?
in other word we should trigger the variable, $x(t)$, to spin around origin and make $\int {\rm sign}(x(t))\,dt$ over time, zero.