# Creating a recursive control system

I am trying to design a digital control system of which output depends on the difference between 1 input signal 1 tick ago and the output 1 tick ago ,and 1 another input signal.

The output $$y(n)$$ = input $$i(n)$$ $$\cdot$$ $$b(n)$$ and $$b(n)$$ = input $$d(n-1)$$ $$-$$ $$y(n-1)$$ Here is what I have come up with so far:

I have made the time delay part.The inverse Z transform of $$z^{-1}$$ is $$\delta(n-1)$$ so I have created the time delay in both $$D(s)$$ and the feedback loop.However I need to somehow put I(s) in the overall system so that it does what I want but so far I have been unsuccesfull on doing so.How should I keep on?

Here is what I have come up with:

• Is this an IIR filter plus second signal? Commented Oct 14, 2023 at 23:30
• No it is not a IIR filter. Commented Oct 14, 2023 at 23:44
• Give more information. What is "the overall system". Why could you not "put I(s) in" it? Please also tell us how you "made the time delay part". Use edit option below the question to add the details. Do not insert the details as comments.
– AJN
Commented Oct 15, 2023 at 3:03
• Your math says that the output is an internal signal multiplied by input $i$. Your words say the output signal is an internal signal added to an input $i$. Could you edit your question for consistency? Commented Oct 15, 2023 at 17:02
• @TimWescott Ooops I forgot to add the n parameter in one of the equations.Thanks Commented Oct 15, 2023 at 18:45

First, recognize that your system, as given, is not linear time-invariant, because of the multiplication by $$i(n)$$. Depending on how you treat $$i(n)$$ and it's effects, your system is either nonlinear (if you take $$i(n)$$ as a regular old input) or time-varying (if you take $$i(n)$$ as a predefined signal, and make it part of the system's time dependence).
Either way, using $$z$$ notation on the signals is misleading. Even for non-linear systems, using $$\frac 1 z$$ inside a block to denote a unit delay is pretty common; that's what I do with the expectation that the reader will still understand it's not an LTI system. Here's what I would draw, using $$\Sigma$$ to denote a summing junction, and $$\Pi$$ to denote a product junction. It's pretty much the same except I've left out the $$A$$ term which you did not mention in your question.