# 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 $P(D) = 1$. Feb 27, 2020 at 23:28
• TimWescott you are very right, I erred in saying it's unknown. It's certainly very known.
– EE18
Feb 28, 2020 at 0:37

You have discovered pole-zero cancellation, which is one of the reasons that state-space systems description was invented. Another reason is because when you go from a differential equation to a transfer function, you lose information about the structure of a system that you can choose to preserve in a state-space description.

In general, you should be very careful when you see a pole-zero cancellation, because even if the pole is stable, it can still affect the behavior of the system on startup, if there is a transient that pushes the system into nonlinear behavior and excites the extra pole, or if the pole and the zero don't actually match up perfectly in the real world.

And yes, if you have an unstable pole that's cancelled by a zero then Bad Things will happen to the system that simply aren't described by the transfer function.

• If I can, you mention that "you should be very careful when you see a pole-zero cancellation, because even if the pole is stable, it can still affect the behavior of the system on startup". I think that this answers the crux of my question, but I'd like to confirm - you're saying that the natural (I should be more rigorous, zero input) solution given above would be an incorrect description of the zero-input response. That is, it misses some characteristic modes...and so this procedure of recovering the ODE from the transfer function is wrong in general, and especially for cascaded systems.
– EE18
Feb 28, 2020 at 0:39
• I'm not sure that it's entirely wrong, or even much more wrong than starting with a linear state-space description. It really all depends on what you're doing with the differential equation. If you just want to know the system behavior in steady state when it's behaving in a linear fashion, then going back to a differential equation is probably unnecessary. If you have any linear description, and the nonlinear behavior matters, then you probably need to go all the way back to the physical description of the system (circuits & mechanics) and move forward from there. Feb 28, 2020 at 13:21
• The question was ultimately to obtain a zero-input response from this cascaded system. The solution was to recover the DE as mentioned above, and then extract the characteristic modes as is standard...I just figured that doing so would miss some characteristic modes destroyed when cancellation occurs.
– EE18
Feb 29, 2020 at 2:14