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:
$$\ln(y) = b\cdot x + \ln(a).$$
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 $K_p$, $T_i$ and $T_d$. 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.
I failed to understand why you think that by taking the $ln$ of your model you would obtain a linearized expression. Because your output is now $ln(y)$ which is definitely not linear. Could you cite a reference/source that you are using?
Usually, a linearization of a single function involves taking the first order Maclaurin's series and keeping the linear terms
In your example $y = ae^{bx}$ would equal to $y = 1 + abx$, $y$ is your output, $x$ is your state variable
You are using Ziegler Nichols which means you are tuning a PID controller. You need to simulate this system somewhere, do you have a model of the system? What software/hardware are you using to simulate your system? You need to increase your $Kp$ value until your output becomes sinusoidal with a constant amplitude, that is your ultimate gain $Ku$.
You need to find $Tu$, which is the period of oscillation at $Ku$.
After that follow the heuristic below to find your PID gain.
https://en.wikipedia.org/wiki/Ziegler%E2%80%93Nichols_method
Please attach a model of your system (plant + controller) and let me know if you encounter any other question