I have been given the following definition for uniform stability: the equilibrium state $x_e$ is uniformly stable, if for any $\epsilon > 0$ there is a $\delta > 0$ such that
$$\|x(0)-x_e\|<\delta~~\Rightarrow ~~\|x(t)-x_e\|<\epsilon, ~~\forall t\geq 0$$
In my opinion this definition does not have anything to do with stability. Imagine a system with $x(t)$ going to infinity and $x(t) \geq 10^{100} ~~\forall t \geq 0$ and $x_e = 0$.
Then the system would be uniformly stable following the above definition, as for any $\epsilon$ I can use $\delta = 10^{100}$ and that would fulfill the implication. The reason is that $x(0) – x_e$ is never smaller than $10^{100}$ so the left side of the implication is always false and therefore the implication always true.
What am I missing here?
Best Answer
Your definition of uniform stability is strange.
Usual definition:
Uniform stability: for each $\varepsilon>0$, there is $\delta=\delta(\mathbf{\epsilon})>0$, $\textbf{independent of}$ $\mathbf{t_0}$, such that $$\|x(t_0)-x_0(t_0)\|<\delta~~\Rightarrow ~~\|x(t)-x_0(t)\|<\epsilon, ~~\forall t\geq t_0\geq0$$