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Given the following text description, write Python code to implement the functionality described below step by step
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Description:
 Batch Bioreactor
In this Jupyter Notebook we illustrate the example batch_reactor.
Open an interactive online Jupyter Notebook with this content on Binder
Step1:  Model
In the following we will present the configuration, setup and connection between these blocks, starting with the model. 
The considered model of the batch bioreactor is continuous and has 4 states and 1 control input, which are depicted below
Step2:  States and control inputs
The four states are concentration of the biomass $X_{\text{s}}$, the concentration of the substrate $S_{\text{s}}$, the concentration of the product $P_{\text{s}}$ and the volume $V_{\text{s}}$
Step3:  The control input is the feed flow rate $u_{\text{inp}}$ of $S_{\text{s}}$
Step4:  ODE and parameters
The system model is described by the ordinary differential equation
Step5:  In the next step, the ODE for each state is set
Step6:  Finally, the model setup is completed
Step7:  Controller
Next, the controller is configured.
First, one member of the mpc class is generated with the prediction model defined above
Step8:  We choose the prediction horizon n_horizon, set the robust horizon n_robust to 3. The time step t_step is set to one second and parameters of the applied discretization scheme orthogonal collocation are as seen below
Step9:  Objective
The batch bioreactor is used to produce penicillin.
Hence, the objective of the controller is to maximize the concentration of the product $P_{\text{s}}$.
Additionally, we add a penalty on input changes, to obtain a smooth control performance.
Step10:  Constraints
In the next step, the constraints of the control problem are set.
In this case, there are only upper and lower bounds for each state and the input
Step11:  Uncertain values
The explicit values of the two uncertain parameters $Y_{\text{x}}$ and $S_{\text{in}}$, which are considered in the scenario tree, are given by
Step12:  This means with n_robust=1, that 9 different scenarios are considered.
The setup of the MPC controller is concluded by
Step13:  Estimator
We assume, that all states can be directly measured (state-feedback)
Step14:  Simulator
To create a simulator in order to run the MPC in a closed-loop, we create an instance of the do-mpc simulator which is based on the same model
Step15:  For the simulation, we use the time step t_step as for the optimizer
Step16:  Realizations of uncertain parameters
For the simulatiom, it is necessary to define the numerical realizations of the uncertain parameters in p_num.
First, we get the structure of the uncertain parameters
Step17:  We define a function which is called in each simulation step, which gives the current realization of the uncertain parameters, with respect to defined inputs (in this case t_now)
Step18:  By defining p_fun as above, the function will always return the same values.
To finish the configuration of the simulator, call
Step19:  Closed-loop simulation
For the simulation of the MPC configured for the batch bioreactor, we inspect the file main.py.
We define the initial state of the system and set for all parts of the closed-loop configuration
Step20:  Prepare visualization
For the visualization of the control performance, we first import matplotlib and change some basic settings
Step21:  We use the plotting capabilities, which are included in do-mpc.
The mpc_graphics contain information like the current estimated state and the predicted trajectory of the states and inputs based on the solution of the control problem.
The sim_graphics contain the information about the simulated evaluation of the system.
Step22:  A figure containing the 4 states and the control input are created
Step23:  Run closed-loop
The closed-loop system is now simulated for 50 steps (and the ouput of the optimizer is suppressed)
Step24:  Results
The next cell converts the results of the closed-loop MPC simulation into a gif (might take a few minutes)
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Python Code:
import numpy as np
import sys
from casadi import *
# Add do_mpc to path. This is not necessary if it was installed via pip
sys.path.append('../../../')
# Import do_mpc package:
import do_mpc
Explanation: Batch Bioreactor
In this Jupyter Notebook we illustrate the example batch_reactor.
Open an interactive online Jupyter Notebook with this content on Binder:
The example consists of the three modules template_model.py, which describes the system model, template_mpc.py, which defines the settings for the control and template_simulator.py, which sets the parameters for the simulator.
The modules are used in main.py for the closed-loop execution of the controller.
In the following the different parts are presented. But first, we start by importing basic modules and do-mpc.
End of explanation
model_type = 'continuous' # either 'discrete' or 'continuous'
model = do_mpc.model.Model(model_type)
Explanation: Model
In the following we will present the configuration, setup and connection between these blocks, starting with the model. 
The considered model of the batch bioreactor is continuous and has 4 states and 1 control input, which are depicted below:
The model is initiated by:
End of explanation
# States struct (optimization variables):
X_s = model.set_variable('_x',  'X_s')
S_s = model.set_variable('_x',  'S_s')
P_s = model.set_variable('_x',  'P_s')
V_s = model.set_variable('_x',  'V_s')
Explanation: States and control inputs
The four states are concentration of the biomass $X_{\text{s}}$, the concentration of the substrate $S_{\text{s}}$, the concentration of the product $P_{\text{s}}$ and the volume $V_{\text{s}}$:
End of explanation
# Input struct (optimization variables):
inp = model.set_variable('_u',  'inp')
Explanation: The control input is the feed flow rate $u_{\text{inp}}$ of $S_{\text{s}}$:
End of explanation
# Certain parameters
mu_m  = 0.02
K_m	  = 0.05
K_i	  = 5.0
v_par = 0.004
Y_p	  = 1.2
# Uncertain parameters:
Y_x  = model.set_variable('_p',  'Y_x')
S_in = model.set_variable('_p', 'S_in')
Explanation: ODE and parameters
The system model is described by the ordinary differential equation:
\begin{align}
\dot{X}{\text{s}} &=\mu(S{\text{s}})X_{\text{s}}-\frac{u_{\text{inp}}}{V_{\text{s}}}X_{\text{s}},\
\dot{S}{\text{s}} &=-\frac{\mu(S{\text{s}})X_{\text{s}}}{Y_{\text{x}}}-\frac{vX_{\text{s}}}{Y_{\text{p}}}+\frac{u_{\text{inp}}}{V_{\text{s}}}(S_{\text{in}}-S_{\text{s}}),\
\dot{P}{\text{s}} &=vX{\text{s}}-\frac{u_{\text{inp}}}{V_{\text{s}}}P_{\text{s}},\
\dot{V}{\text{s}} &=u{\text{inp}},\
\end{align}
where:
\begin{align}
\mu(S_{\text{s}})&=\frac{\mu_{\text{m}} S_{\text{s}}}{K_{\text{m}}+S_{\text{s}}+(S_{\text{s}}^2/K_{\text{i}})},
\end{align}
$S_{\text{in}}$ is the inlet substrate concentration, $\mu_{\text{m}}$, $K_{\text{m}}$, $K_{\text{i}}$ and $v$ are kinetic parameters $Y_{\text{x}}$ and $Y_{\text{p}}$ are yield coefficients.
The inlet substrate concentration $S_{\text{in}}$ and the $Y_{\text{x}}$ are uncertain while the rest of the parameters is considered certain:
End of explanation
# Auxiliary term
mu_S = mu_m*S_s/(K_m+S_s+(S_s**2/K_i))
# Differential equations
model.set_rhs('X_s', mu_S*X_s - inp/V_s*X_s)
model.set_rhs('S_s', -mu_S*X_s/Y_x - v_par*X_s/Y_p + inp/V_s*(S_in-S_s))
model.set_rhs('P_s', v_par*X_s - inp/V_s*P_s)
model.set_rhs('V_s', inp)
Explanation: In the next step, the ODE for each state is set:
End of explanation
# Build the model
model.setup()
Explanation: Finally, the model setup is completed:
End of explanation
mpc = do_mpc.controller.MPC(model)
Explanation: Controller
Next, the controller is configured.
First, one member of the mpc class is generated with the prediction model defined above:
End of explanation
setup_mpc = {
    'n_horizon': 20,
    'n_robust': 1,
    'open_loop': 0,
    't_step': 1.0,
    'state_discretization': 'collocation',
    'collocation_type': 'radau',
    'collocation_deg': 2,
    'collocation_ni': 2,
    'store_full_solution': True,
    # Use MA27 linear solver in ipopt for faster calculations:
    #'nlpsol_opts': {'ipopt.linear_solver': 'MA27'}
}
mpc.set_param(**setup_mpc)
Explanation: We choose the prediction horizon n_horizon, set the robust horizon n_robust to 3. The time step t_step is set to one second and parameters of the applied discretization scheme orthogonal collocation are as seen below:
End of explanation
mterm = -model.x['P_s'] # terminal cost
lterm = -model.x['P_s'] # stage cost
mpc.set_objective(mterm=mterm, lterm=lterm)
mpc.set_rterm(inp=1.0) # penalty on input changes
Explanation: Objective
The batch bioreactor is used to produce penicillin.
Hence, the objective of the controller is to maximize the concentration of the product $P_{\text{s}}$.
Additionally, we add a penalty on input changes, to obtain a smooth control performance.
End of explanation
# lower bounds of the states
mpc.bounds['lower', '_x', 'X_s'] = 0.0
mpc.bounds['lower', '_x', 'S_s'] = -0.01
mpc.bounds['lower', '_x', 'P_s'] = 0.0
mpc.bounds['lower', '_x', 'V_s'] = 0.0
# upper bounds of the states
mpc.bounds['upper', '_x','X_s'] = 3.7
mpc.bounds['upper', '_x','P_s'] = 3.0
# upper and lower bounds of the control input
mpc.bounds['lower','_u','inp'] = 0.0
mpc.bounds['upper','_u','inp'] = 0.2
Explanation: Constraints
In the next step, the constraints of the control problem are set.
In this case, there are only upper and lower bounds for each state and the input:
End of explanation
Y_x_values = np.array([0.5, 0.4, 0.3])
S_in_values = np.array([200.0, 220.0, 180.0])
mpc.set_uncertainty_values(Y_x = Y_x_values, S_in = S_in_values)
Explanation: Uncertain values
The explicit values of the two uncertain parameters $Y_{\text{x}}$ and $S_{\text{in}}$, which are considered in the scenario tree, are given by:
End of explanation
mpc.setup()
Explanation: This means with n_robust=1, that 9 different scenarios are considered.
The setup of the MPC controller is concluded by:
End of explanation
estimator = do_mpc.estimator.StateFeedback(model)
Explanation: Estimator
We assume, that all states can be directly measured (state-feedback):
End of explanation
simulator = do_mpc.simulator.Simulator(model)
Explanation: Simulator
To create a simulator in order to run the MPC in a closed-loop, we create an instance of the do-mpc simulator which is based on the same model:
End of explanation
params_simulator = {
    'integration_tool': 'cvodes',
    'abstol': 1e-10,
    'reltol': 1e-10,
    't_step': 1.0
}
simulator.set_param(**params_simulator)
Explanation: For the simulation, we use the time step t_step as for the optimizer:
End of explanation
p_num = simulator.get_p_template()
Explanation: Realizations of uncertain parameters
For the simulatiom, it is necessary to define the numerical realizations of the uncertain parameters in p_num.
First, we get the structure of the uncertain parameters:
End of explanation
p_num['Y_x'] = 0.4
p_num['S_in'] = 200.0
# function definition
def p_fun(t_now):
    return p_num
# Set the user-defined function above as the function for the realization of the uncertain parameters
simulator.set_p_fun(p_fun)
Explanation: We define a function which is called in each simulation step, which gives the current realization of the uncertain parameters, with respect to defined inputs (in this case t_now):
End of explanation
simulator.setup()
Explanation: By defining p_fun as above, the function will always return the same values.
To finish the configuration of the simulator, call:
End of explanation
# Initial state
X_s_0 = 1.0 # Concentration biomass [mol/l]
S_s_0 = 0.5 # Concentration substrate [mol/l]
P_s_0 = 0.0 # Concentration product [mol/l]
V_s_0 = 120.0 # Volume inside tank [m^3]
x0 = np.array([X_s_0, S_s_0, P_s_0, V_s_0])
# Set for controller, simulator and estimator
mpc.x0 = x0
simulator.x0 = x0
estimator.x0 = x0
mpc.set_initial_guess()
Explanation: Closed-loop simulation
For the simulation of the MPC configured for the batch bioreactor, we inspect the file main.py.
We define the initial state of the system and set for all parts of the closed-loop configuration:
End of explanation
import matplotlib.pyplot as plt
plt.ion()
from matplotlib import rcParams
rcParams['text.usetex'] = True
rcParams['text.latex.preamble'] = [r'\usepackage{amsmath}',r'\usepackage{siunitx}']
rcParams['axes.grid'] = True
rcParams['lines.linewidth'] = 2.0
rcParams['axes.labelsize'] = 'xx-large'
rcParams['xtick.labelsize'] = 'xx-large'
rcParams['ytick.labelsize'] = 'xx-large'
Explanation: Prepare visualization
For the visualization of the control performance, we first import matplotlib and change some basic settings:
End of explanation
mpc_graphics = do_mpc.graphics.Graphics(mpc.data)
sim_graphics = do_mpc.graphics.Graphics(simulator.data)
Explanation: We use the plotting capabilities, which are included in do-mpc.
The mpc_graphics contain information like the current estimated state and the predicted trajectory of the states and inputs based on the solution of the control problem.
The sim_graphics contain the information about the simulated evaluation of the system.
End of explanation
%%capture
fig, ax = plt.subplots(5, sharex=True, figsize=(16,9))
fig.align_ylabels()
for g in [sim_graphics,mpc_graphics]:
    # Plot the state on axis 1 to 4:
    g.add_line(var_type='_x', var_name='X_s', axis=ax[0], color='#1f77b4')
    g.add_line(var_type='_x', var_name='S_s', axis=ax[1], color='#1f77b4')
    g.add_line(var_type='_x', var_name='P_s', axis=ax[2], color='#1f77b4')
    g.add_line(var_type='_x', var_name='V_s', axis=ax[3], color='#1f77b4')
    # Plot the control input on axis 5:
    g.add_line(var_type='_u', var_name='inp', axis=ax[4], color='#1f77b4')
    
ax[0].set_ylabel(r'$X_s~[\si[per-mode=fraction]{\mole\per\litre}]$')
ax[1].set_ylabel(r'$S_s~[\si[per-mode=fraction]{\mole\per\litre}]$')
ax[2].set_ylabel(r'$P_s~[\si[per-mode=fraction]{\mole\per\litre}]$')
ax[3].set_ylabel(r'$V_s~[\si[per-mode=fraction]{\mole\per\litre}]$')
ax[4].set_ylabel(r'$u_{\text{inp}}~[\si[per-mode=fraction]{\cubic\metre\per\minute}]$')
ax[4].set_xlabel(r'$t~[\si[per-mode=fraction]{\minute}]$')
Explanation: A figure containing the 4 states and the control input are created:
End of explanation
%%capture
n_steps = 100
for k in range(n_steps):
    u0 = mpc.make_step(x0)
    y_next = simulator.make_step(u0)
    x0 = estimator.make_step(y_next)
Explanation: Run closed-loop
The closed-loop system is now simulated for 50 steps (and the ouput of the optimizer is suppressed):
End of explanation
from matplotlib.animation import FuncAnimation, FFMpegWriter, ImageMagickWriter
# The function describing the gif:
def update(t_ind):
    sim_graphics.plot_results(t_ind)
    mpc_graphics.plot_predictions(t_ind)
    mpc_graphics.reset_axes()
    
if False:
    anim = FuncAnimation(fig, update, frames=n_steps, repeat=False)
    gif_writer = ImageMagickWriter(fps=10)
    anim.save('anim_batch_reactor_final.gif', writer=gif_writer)
Explanation: Results
The next cell converts the results of the closed-loop MPC simulation into a gif (might take a few minutes):
End of explanation
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