the relation between input and output of linear systems

Question

I have a question regarding to a simple LTI system \begin{array}{l} \dot x = Ax + Bu\\ y = Cx \end{array} where $A$ is a $n\times n$ matrix; $B$ is of size $n\times p$; $C$ is of size $q\times n$. $y$ is the output; $u$ is the input; $x$ is the state.

I am wondering if the output $y$ and input $u$ have a positive correlation. That is, the bigger the magnitude of $u$ is, the bigger the magnitude of $y$ is. If so, how to prove it.

Answer 1

Let $x_0=0$, then $u(t) \to y(t)$ implies $\alpha u(t) \to \alpha y(t)$ for any $\alpha \in \mathbb{R}$. This is clear from the solution:

$$y(t) = \int_0^t C e^{A (t-\tau)} B u(\tau) d\tau$$

The system is called "linear" because of this property (plus additive property). We are generally interested in the "gain" of the system for specific inputs, i.e. $\lVert y(t) \rVert / \lVert u(t) \rVert$. The maximum gain of a system for all possible inputs is called the $H_\infty$ norm of the system.

Answer 2

You question is slightly ambiguous to me.

Why $u$ should depend on $y$?

If you mean you set input as $$u=Ky$$ for example through a feedback.

Then, you can write

$$\dot x=Ax+BK(Cx)=(A+BKC)x$$

For the system to be stable, all the eigenvalues of matrix $(A+BKC)$ must have negative real parts.