Anesthesia System
Anesthesia machines are classic examples of feedback control. One significant difficulty in designing them is modeling the human "plant", which can vary wildly from individual to individual, and even in the same individual person depending on physiological state. As shown in the figure below, a desired anesthetic level, A_i, is provided to an anesthesia machine, which compares that value to the actual patient anesthetic level, A_o, and releases an output stream of anesthetic based on the supply gain, S.
The human plant can be modeled as a basic accumulator similar to what you've already encountered, but with several modifications. The human body's metabolism of the anesthetic agents in modeled by the variable D, where D \lt 1. Because it takes a relatively long time for the body to respond to changes in an anesthetic concentration, there is a delay in the accumulator (\mathcal{R}^P, where P \gt 0).
The sample time, T, of the anesthesia machine is T=0.1~\text{seconds}. In parts 1 through 4 below, the supply gain is S=2.
Part 1
Determine the pole(s) of the anethesia feedback system when attached to a
patient named Benedict. Benedict's healthy body can be modeled as having a
D=0.96, and delay, P=2.
Enter the poles as a python list of (possibly complex) numbers, e.g.,
[1.2, 2.3+0.5j, 2.3-0.5j].
Don't forget the commas or the brackets.
Part 2
After a long time running, what is the actual value of anesthetic in Benedict's system, $A_o$, if the desired level, $A_i$ is $0.5$? Enter your answer as a decimal number, accurate to 2 decimal places. _Hint: Both this part and Part 4 below can be solved either through simulations with the python libraries or mathematically using the system functions and difference equation you derived. Secondary Hint: How is the output at $n$ different from the output at $n-1$ for large $n$ in a converging system?_
Part 3
While under anesthesia, Benedict's body reacts poorly to a drug, and his body's metabolism changes such that now $D=0.5$. Determine the new poles for the system.
Part 4
_Retrograde Hint: The anesthetic levels were oscillating in the previous part _.After the oscillations die down, what is the new actual level, A_o, given the desired level, A_i, remains at 0.5?
Part 5
Oscillation is obviously not a good thing to have happen on an anesthesia machine. If the level swings too low, the patient will wake up under the knife. If the level swings too high, the patient will never wake up again. With Benedict in his dangerous physiological state of D=0.5, determine the maximum value for S that will result in non-oscillatory converging behavior of the system.