# Recovering a Differential Equation From the Transfer Function of a Cascaded System

With respect to the below discussion, consider that we are talking about LTIC systems characterized by constant coefficient ODEs.

Consider a cascaded system whose transfer function H(s) is given by

$$H(s) = \frac {s+2}{(s+2)(s+3)(s+1)} =\frac 1 {(s+1)(s+3)}$$

"Recovering" the defining differential equation from this transfer function yields

$$(D^2 + 4D + 3)y = P(D)x$$

where $$P(D)$$ is unknown but irrelevant for the purposes of this discussion. What are the characteristic modes of the system?

Is it acceptable to simply say that the natural response $$y_n$$ is given as below?

$$y_n = Ae^{-t} + Be^{-3t}$$

It would seem to me that the answer is no. Why can we recover an ODE from a transfer function in general. For instance, what if the cascaded system had've had $$(s-2)$$ factors rather than $$(s+2)$$ factors that cancelled (ie. one of the subsystems had a pole at $$s=2$$). In such a case, would it not be wildly incorrect to recover the ODE above since then we are ignoring an internal characteristic mode that blows up $$(e^{2t})$$?

• If you are defining $P(D)$ as some polynomial in $D$, then $P(D) = 1$. – TimWescott Feb 27 at 23:28
• TimWescott you are very right, I erred in saying it's unknown. It's certainly very known. – 1729_SR Feb 28 at 0:37