Fig. $6$ Fuzzy Logic PID Controller

The bioreactor model under consideration can be described as a tank containing nutrients, biological

cells, and water. The state of this process is characterized by the cell concentration $c_1(t)$ and the

nutrient amount $c_2(t)$ per unit volume. The volume in the tank remains constant by matching the rate

of content removal from the tank with the incoming rate $u(t).$ This flow rate is the control input of the

bioreactor. The primary objective of bioreactor control is to maintain the cell concentration $c_1(t)$ at

the desired level using the only control input of externally supplied pure water.

$c_1^{}(t)=-c_1(t)u(t)+c_1(t)(1-c_2(t))e^{c_2\frac{t}{}}$

$c_2^{}(t)=-c_2(t)u(t)+c_1(t)(1-c_2(t))e^{c_2\frac{t}{}}\frac{1+}{1+-c_2(t)}$

Note that the state variables $c1(t)$ and $c2(t)$ can only take values between zero and one since they are

concentration rates. The initial conditions are $c1(t)=0\mathrm{.}1207,c2(t)=0\mathrm{.}8801,$ and $u(t)=$

$0\mathrm{.}7500.$ Be aware that the initial values of these state variables may fluctuate within a range of $+-10\%$

from their stable values. The parameters for the growth rate and nutrient inhibition are specified as

$=0\mathrm{.}02$ and $=0\mathrm{.}48$ respectively.

Simulate the bioreactor system using Matlab. $($You can use Runge $-$ Kutta algorithm you used

for question $1.)$

Develop a Mamdani type fuzzy logic PID controller $($Fig $6-8)$ for the described bioreactor

system and determine the scaling factors according to the bioreactor system inputs and given

membership functions. Use Matlab and Simulink. Please note "AND" operation is min, and

the centroid method should be employed for defuzzification.

Perform system simulations for unit step and sinusoidal $($with $0\mathrm{.}3$ magnitude, $0\mathrm{.}3$ bias, and $1\mathrm{.}5$

$ra\frac{d}{sec}$ frequency$)$ inputs. Graph both the output and control signals.

Introduce a unit step input disturbance at $t=3$ to the system during simulations for unit step

input. Again, provide the resulting output and control signals. Discuss your observations and

interpretations of the system's response.

Add white noise to your system during simulations for unit step input, while keeping the

disturbance active. Display the resulting output and control signals. Provide commentary on

the outcomes and what they suggest about the system's behavior under these conditions.

$......$

mbership Functions for A$)$ error signal, B$)$ derivative of error signal and C$)$ output signal