Lyapunov stability, mathematics vs reality

[Liferider]

Homework Statement

I have not been doing Lyapunov for a while and when doing an ordinary Lyapunov problem the other day, I ran into a funny situation.

The correct way of doing it:
\begin{align}
\dot{e} &= \frac{1}{L}(u - R(e + x_{ref})) \\
V(e) &= \frac{1}{2}Le^2 \\
\dot{V} &= Le\dot{e} = Le \left( \frac{1}{L} \left[ u - R(e + x_{ref}) \right] \right) \\
&= - Re^2 + e(u - Rx_{ref})
\end{align}
The system is a modified system, with e defined in terms of the original state x
\begin{equation}
e = x - x_{ref} \ \Rightarrow \ x = e + x_{ref}
\end{equation}
To stabilize the system, we define the feedback control law u to be
\begin{align}
u &= Rx_{ref} - K_pe \\
\Rightarrow \ \dot{V} &= -Re^2 - K_pe^2 < 0
\end{align}

However, mathematically, one could define $u$ to be
\begin{align}
u &= Rx_{ref} \\
\Rightarrow \ \dot{V} &= -Re^2 < 0
\end{align}

I know this will not work, u is not a constant, it is a variable... but still, the mathematics checks out, kind of. What is the best way of explaining why this does not work?

 

[Liferider]

Hmm, if I think correctly, then
\begin{equation}
u = Rx_{ref}
\end{equation}
is the steady state value when
\begin{equation}
x=x_{ref}
\end{equation}
Soooo setting u to this value at all times should drive the system state to
\begin{equation}
x_{ref}
\end{equation}
It's just not a very good controller?

 

[Liferider]

You just add the extra Kp*e to make the solution converge to the equilibrium faster...