# Does the plant transfer function of a system change with variations in the reference variable?

I am designing a PID controller for a system that requires several step changes in the reference variable (concentration of solute in a tank) over a set period of time. I can find the plant transfer function experimentally, by applying a step change in the input (flow rate) and measuring the resulting changes in concentration over time.

What is not clear to me is: does the plant transfer function change each time I apply a different step change in the input variable (e.g. from 0.1 L/hr to 0.08 L/hr then after a while from 0.08 L/hr to 0.05 L/hr), or is the plant transfer function fixed for a given system (regardless of changes in the reference/input variable)? Thanks in advance.

Not enough detail, but in your case very possibly not.

A transfer function is a description of a linear system, using the Laplace transform (or the z transform if you're sticking to sampled time).

The basic problem with the above is that most real-world systems are nonlinear. We get around that by holding one eye closed and pretending that our system is linear (if we're measuring its response), or by finding its linearization around an operating point (if we're designing from a mathematical model).

I'm not up on the chemical engineering details, but I suspect that there are all sorts of nonlinearities in your system.

So, if the apparent transfer function that you extract from your measurements at one flow rate is considerably different than the one you extract at a significantly different flow rate, then yes, your linearized model of your nonlinear system's behavior at that different flow rate is, indeed, probably different.

If you've got the math chops for it, write out the differential equations that describe the reaction -- if they're nonlinear, then you can (A) expect that the transfer function is only an approximation, and (B) you can linearize them yourself around various operating points and see if the gains are different.

• Thanks for your reply, it was very helpful. In my case, there is a fixed quantity of salt in my tank (there are no reactions taking place). However, I am adding a flow rate of water to dilute the salt concentration over time. Eventually, the water flow rates will be very small (<0.01 L/hr), so I am thinking I may have to assume linearity to be able to estimate the transfer function at some point. Jul 22 '19 at 5:35