I have a stationary model for a plant(a valve) given by y=a*exp(bx). I linearised this by taking log on both sides.

ln=bx+ln(a) <---- linear

Then, I estimated the plant transfer function(1st order with lag) from the step response. Then I used Ziegler-Nicols tuning rules to get a first guess for Kp, Ti and Td. My dilemma begins after this.

How do I modify my resulting controller output obtained from the above parameters to feed it to my plant. Note, the error input to the controller is the difference between the log of set point(SP) and the log of the process variable(PV).

error, e=ln(SP)- ln(PV)

Thus, the controller output is calculated for that modified error. The real error, obviously, is SP- PV.

I thought of just taking the exponential of the controller output, but could not justify the Maths involved.

I would be grateful for some feedback. Thanks

 

It's unclear what you are attempting to achieve - maybe prove theoretical mathematics is not always useful in the real world of
engineering.

I few years ago I had a conundrum.

A constantly filling tank was drained using a variable speed drive and one switching probe to detect liquor level.

Two probes would have been easy - start pumping on the high probe and continue to lower probe detecting low level. The resultant speed profile of the variable speed pump turned out to be logarithmic.

Start / stop pump control was logical - No liquor splashing the probe after tx: pump stop. Probe being splashed or submerged for tx: start pump. Speed control was non-linear between the above extremes.

The control software proved to be so successful (OK the pump did hunt a little due to not having a deadband), alarms were created for overfull and pump running dry using the same single probe.