Controller design for an exponential plant of the form y=a*exp(bx)

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

ln(y) = b*x + 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