- #1

Liferider

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## 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?

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