Similar outputs from different transfer functions

Question

I have two rational transfer functions with the same denominator: $$ H_{0}(s) = N_{0}(s)/D(s),\,\,H_{1}(s) = N_{1}(s)/D(s)$$ I would like for the two outputs from the system, $Y_0(t)=\mathcal{L}^{-1}(H_0(s)X(s))$ and $Y_1(t)=\mathcal{L}^{-1}(H_1(s)X(s))$ to be close in some sense (either having a large inner product, or forcing the difference between the outputs to have a small $L_2$ norm). The transfer functions are fixed, but I can alter $X$. Is this a common problem in control theory? Should I be using some representation of the system other than the transfer function?

Answer 1

We can think $X(s)$ as the input signal. Let us select $X(s)=0$, then $Y_0(t)=Y_1(t)=0$, the responses are exactly same. We can select the input as a sinus at a certain frequency and measure the output signal by the amplitude of it. If $H_0(s)$ and $H_1(s)$ happen to have the same gain $A$ for some frequency $\omega$, then $Y_0(t) = A \sin (\omega t + \phi_0)$ and $Y_1(t) = A \sin (\omega t + \phi_1)$ where $\phi_i$ is the phase of the output signal. The two systems have the same "measures".

Both of the examples can be used but none of them are meaningful, as they do not actually "make" systems closer, but only "reveals" the closer "components". For an arbitrary $X(s)$ the systems may behave very different. A meaningful closeness measure for systems should consider all the inputs.

You may however match two different systems using feedback, which is a different problem.