Please check my Python code and edit it$.$

Q$.$ Create a graphical output of a mathematical model using arbitrary variables and Python code, but you must use a PID controller.

import numpy as np

import matplotlib.pyplot as plt

from scipy.integrate import odeint

# Constants

R $=287\mathrm{.}1$ # N$($m$)/($k$)$gK

T $=293$ # K

V $=0\mathrm{.}8$ # L

k $=1\mathrm{.}4$

p$\_$atm $=101\mathrm{.}325$ # kPa

S$\_$v $=15\mathrm{.}1$ # $($m$^3)/($h$)$

p$\_$v $=0\mathrm{.}7$ # Pa

V$\_$t $=1\mathrm{.}3$ # L

# Variables

D $=1$ # Arbitrary diameter

A$\_$cu $=1$ # Arbitrary area

A$\_$du $=1$ # Arbitrary area

alpha $=1$ # Correction coefficient of the flow area of the valve port

sigma $=0\mathrm{.}5$ # Critical pressure ratio

d $=1$

# PID Controller Parameters

Kp $=0\mathrm{.}1$

Ki $=0\mathrm{.}01$

Kd $=0\mathrm{.}01$

# Initial conditions

p$0=$ p$\_$atm

p$\_$t$0=$ p$\_$atm

Q$\_$m$1=0$

Q$\_$m$2=0$

Q$\_$mv $=0$

x$\_$vm $=0$

# Time points

t $=$ np$.$linspace$(0,100,1000)$

# Pressure Differential Equation

def model$($y$,$ t$)$:

p$,$ p$\_$t$,$ Q$\_$m$1,$ Q$\_$m$2,$ Q$\_$mv$,$ x$\_$vm $=$ y

f$1=($k $*$ Q$\_$m$1*$ R $*$ T$)/$ V

f$2=($k $*$ R $*$ T$)/$ V$\_$t

Q$\_$m $=$ Q$\_$m$1-$ Q$\_$m$2$

f$3=($p$\_$t $*$ S$\_$v$)/($R $*$ T$)*(1-$ p$\_$v $/$ p$\_$t$)$

dpdt $=$ f$1-$ Kp $*($p $-$ p$0)-$ Ki $*($p $-$ p$0)*$ t $-$ Kd $*($p $-$ p$0)/$ t

dp$\_$tdt $=$ f$2*($Q$\_$m$2-$ Q$\_$mv$)$

dQ$\_$m$1$dt $=0\mathrm{.}0404*$ A$\_$cu $*$ p$\_$atm $/$ np$.$sqrt$($T$)*$ f$($p$,$ p$\_$atm$)*$ d$\_$cu$($t$)$

dQ$\_$m$2$dt $=0\mathrm{.}0404*$ A$\_$du $*$ p $/$ np$.$sqrt$($T$)*$ f$($p$,$ p$\_$t$)*$ d$\_$du$($t$)$

dQ$\_$mvdt $=$ f$3$

dx$\_$vm$\_$dt $=0\mathrm{.}0404*$ np$.$pi $*$ alpha $*$ D $*$ x$\_$vm $*$ p $/$ np$.$sqrt$($T$)*$ f$($p$,$ p$\_$t$)*$ d

return $[$dpdt$,$ dp$\_$tdt$,$ dQ$\_$m$1$dt$,$ dQ$\_$m$2$dt$,$ dQ$\_$mvdt$,$ dx$\_$vm$\_$dt$]$

# Function f$($p$1,$ p$2)$

def f$($p$\_1,$ p$\_2)$:

if p$\_2/($p$\_1*(1-$ sigma sigma:

return $1$

else:

return np$.$sqrt$(1-($p$\_2/(1-$ sigma$))**2)$

# Duty cycles

def d$\_$cu$($t$)$:

return $0\mathrm{.}5$ # Duty cycles for the process of charging

def d$\_$du$($t$)$:

return $0\mathrm{.}5$ # Duty cycles for the process of discharging

# Solve ODE

y$0=[$p$0,$ p$\_$t$0,$ Q$\_$m$1,$ Q$\_$m$2,$ Q$\_$mv$,$ x$\_$vm$]$

y $=$ odeint$($model$,$ y$0,$ t$)$

# Plot results

plt$.$figure$()$

plt$.$plot$($t$,$ y$[$:$,0],\text{'}$r$\text{'},$ label$=\text{'}$p$($t$)\text{'})$

plt$.$plot$($t$,$ y$[$:$,1],\text{'}$b$\text{'},$ label$=\text{'}$p$\_$t$($t$)\text{'})$

plt$.$legend$($loc$=$'best'$)$

plt$.$xlabel$(\text{'}$time$\text{'})$

plt$.$ylabel$(\text{'}$pressure$\text{'})$

plt$.$title$(\text{'}$Pressure Control with PID Controller'$)$

plt$.$grid$()$

plt$.$show$()$