I'm trying to show global asymptotic stability (GAS) for a system with a feedback controller. I managed to show GAS for the perfect feedback signal, that is
$$\gamma(x(t)) = K_p x_1(t) + K_d x_2(t).$$
Now, I'm interested in showing GAS if - like in a real system - I measure $x_1$ and $x_2$ thus introducing uncertainties and compose that to be my feedback signal like $$ \gamma(x(t),t) = K_p (x_1(t) + \Delta x_1(t)) + K_d (x_2(t) + \Delta x_2(t)).$$
When I try and do the proof as I've done it with the "perfect" signal, when calculation the derivative of the lyapunov function, I no longer have a lyapunov function which only depends on $x$ but also on $t$. This leads to derivatives of the uncertainties $\Delta x_1$ and $\Delta x_2$ which I do not know.
Question is: Are there some kind of standard ways to handle that or to argue that the system is still GAS?