The goal is to design the control U as an anesthetic drug infusion rate such that the level of hypnosis (in our case Bis index) converges to a desired value. After researching I saw a pharmackinetic plus a pharmacodynamic model can be used to model the dynamics.
But I have some problems trying to derive the closed loop transfer function.
So I am making use of a pharmacokinetic model to describe the distribution of the anesthetic drug in the body. I use a three compartmental model illustrated as follows, where the constants 
represent the frequency at which the drug is redistributed from compartment i to compartment j. The input of this model is U which is the drug's infusion rate while its output is the drug concetration of the primary compartment C1.
I then use a pharmacodynamic model to model the drugs effect, mapping the primary compartment concentration C1 to a level of hypnosis through 2 steps.
Step 1 : Map 
to 
represents the drug concentration of the site where the drug actually causes its effect on the body.Step 2 : Map to Bis index The last step is to then map to an actual level of hypnosis to measure the patients consciousness. I used a hill sigmoid function for this. 
where a andγrepresents given parameters of the patient.This last step introduces non linearity to the model which I attempted solving with feedback linarization in the following way.
In the laplace domain we have the following 
, rewriting Bis as 
hence the feedback linearization can be applied as 
The problem is simulating this model and trying to make the Bis(t) converge to a certain value setting v as a pid control.
The goal isto find the control such that the bis index from an initial value of 1 decreases to a desired bis index in the range of 0.5.
My questions are :
1) Given the feedback linearization I proposed, How do I calculate the closed loop transfer function of this system (the problem is the non linear hill block), I tried feedback(bis*u,1) in matlab but I am not sure.
2) given the error as e(t)= (Bisdesired - Bis), which functions or commands in matlab can I apply to I make e(t) converge to zero with a pid controller through feedback linearization.
