def run_demo(with_plots=True):
"""
This example is based on the Hicks-Ray Continuously Stirred Tank Reactors
(CSTR) system. The system has two states, the concentration and the
temperature. The control input to the system is the temperature of the
cooling flow in the reactor jacket. The chemical reaction in the reactor is
exothermic, and also temperature dependent; high temperature results in
high reaction rate.
The problem is solved using the CasADi-based collocation algorithm through
the MPC-class. FMI is used for initialization and simulation purposes.
The following steps are demonstrated in this example:
1. How to generate an initial guess for a direct collocation method by
means of simulation with a constant input. The trajectories resulting
from the simulation are used to initialize the variables in the
transcribed NLP, in the first sample(optimization).
2. An optimal control problem is defined where the objective is to
transfer the state of the system from stationary point A to point B.
An MPC object for the optimization problem is created. After each
sample the NLP is updated with an estimate of the states in the next
sample. The estimate is done by simulating the model for one sample
period with the optimal input calculated in the optimization as input.
To each estimate a normally distributed noise, with the mean 0
and standard deviation 0.5% of the nominal value of each state, is
added. The MPC object uses the result from the previous optimization as
initial guess for the next optimization (for all but the first
optimization, where the simulation result from #1 is used instead).
(3.) If with_plots is True we compile the same optimization problem again
and define the options so that the op has the same options and
resolution as the op we solved through the MPC-class. By same
resolution we mean that both op should have the same mesh and blocking
factors. This allows us to compare the MPC-results to an open loop
optimization. Note that the MPC-results contains noise while the open
loop optimization does not.
"""
### 1. Compute initial guess trajectories by means of simulation
# Locate the Modelica and Optimica code
file_path = os.path.join(get_files_path(), "CSTR.mop")
# Compile and load the model used for simulation
sim_fmu = compile_fmu("CSTR.CSTR_MPC_Model", file_path,
compiler_options={"state_initial_equations":True})
sim_model = load_fmu(sim_fmu)
# Define stationary point A and set initial values and inputs
c_0_A = 956.271352
T_0_A = 250.051971
sim_model.set('_start_c', c_0_A)
sim_model.set('_start_T', T_0_A)
sim_model.set('Tc', 280)
opts = sim_model.simulate_options()
opts["CVode_options"]["maxh"] = 0.0
opts["ncp"] = 0
init_res = sim_model.simulate(start_time=0., final_time=150, options=opts)
### 2. Define the optimal control problem and solve it using the MPC class
# Compile and load optimization problem
op = transfer_optimization_problem("CSTR.CSTR_MPC", file_path,
compiler_options={"state_initial_equations":True})
# Define MPC options
sample_period = 3 # s
horizon = 33 # Samples on the horizon
n_e_per_sample = 1 # Collocation elements / sample
n_e = n_e_per_sample*horizon # Total collocation elements
finalTime = 150 # s
number_samp_tot = int(finalTime/sample_period) # Total number of samples to do
# Create blocking factors with quadratic penalty and bound on 'Tc'
bf_list = [n_e_per_sample]*(horizon/n_e_per_sample)
factors = {'Tc': bf_list}
du_quad_pen = {'Tc': 500}
du_bounds = {'Tc': 30}
bf = BlockingFactors(factors, du_bounds, du_quad_pen)
# Set collocation options
opt_opts = op.optimize_options()
opt_opts['n_e'] = n_e
opt_opts['n_cp'] = 2
opt_opts['init_traj'] = init_res
opt_opts['blocking_factors'] = bf
if with_plots:
# Compile and load a new instance of the op to compare the MPC results
# with an open loop optimization
op_open_loop = transfer_optimization_problem(
"CSTR.CSTR_MPC", file_path,
compiler_options={"state_initial_equations":True})
op_open_loop.set('_start_c', float(c_0_A))
op_open_loop.set('_start_T', float(T_0_A))
# Copy options from MPC optimization
open_loop_opts = copy.deepcopy(opt_opts)
# Change n_e and blocking_factors so op_open_loop gets the same
# resolution as op
open_loop_opts['n_e'] = number_samp_tot
bf_list_ol = [n_e_per_sample]*(number_samp_tot/n_e_per_sample)
factors_ol = {'Tc': bf_list_ol}
bf_ol = BlockingFactors(factors_ol, du_bounds, du_quad_pen)
open_loop_opts['blocking_factors'] = bf_ol
open_loop_opts['IPOPT_options']['print_level'] = 0
constr_viol_costs = {'T': 1e6}
# Create the MPC object
MPC_object = MPC(op, opt_opts, sample_period, horizon,
constr_viol_costs=constr_viol_costs, noise_seed=1)
# Set initial state
x_k = {'_start_c': c_0_A, '_start_T': T_0_A }
# Update the state and optimize number_samp_tot times
for k in range(number_samp_tot):
# Update the state and compute the optimal input for next sample period
MPC_object.update_state(x_k)
u_k = MPC_object.sample()
# Reset the model and set the new initial states before simulating
# the next sample period with the optimal input u_k
sim_model.reset()
sim_model.set(list(x_k.keys()), list(x_k.values()))
sim_res = sim_model.simulate(start_time=k*sample_period,
final_time=(k+1)*sample_period,
input=u_k, options=opts)
# Extract state at end of sample_period from sim_res and add Gaussian
# noise with mean 0 and standard deviation 0.005*(state_current_value)
x_k = MPC_object.extract_states(sim_res, mean=0, st_dev=0.005)
# Extract variable profiles
MPC_object.print_solver_stats()
complete_result = MPC_object.get_complete_results()
c_res_comp = complete_result['c']
T_res_comp = complete_result['T']
Tc_res_comp = complete_result['Tc']
time_res_comp = complete_result['time']
# Verify solution for testing purposes
try:
import casadi
except:
pass
else:
Tc_norm = N.linalg.norm(Tc_res_comp) / N.sqrt(len(Tc_res_comp))
assert(N.abs(Tc_norm - 311.7362) < 1e-3)
c_norm = N.linalg.norm(c_res_comp) / N.sqrt(len(c_res_comp))
assert(N.abs(c_norm - 653.5369) < 1e-3)
T_norm = N.linalg.norm(T_res_comp) / N.sqrt(len(T_res_comp))
assert(N.abs(T_norm - 328.0852) < 1e-3)
# Plot the results
if with_plots:
### 3. Solve the original optimal control problem without MPC
res = op_open_loop.optimize(options=open_loop_opts)
c_res = res['c']
T_res = res['T']
Tc_res = res['Tc']
time_res = res['time']
# Get reference values
Tc_ref = op.get('Tc_ref')
T_ref = op.get('T_ref')
c_ref = op.get('c_ref')
# Plot
plt.close('MPC')
plt.figure('MPC')
plt.subplot(3, 1, 1)
plt.plot(time_res_comp, c_res_comp)
plt.plot(time_res, c_res )
plt.plot([time_res[0],time_res[-1]],[c_ref,c_ref],'--')
plt.legend(('MPC with noise', 'Open-loop without noise', 'Reference value'))
plt.grid()
plt.ylabel('Concentration')
plt.title('Simulated trajectories')
plt.subplot(3, 1, 2)
plt.plot(time_res_comp, T_res_comp)
plt.plot(time_res, T_res)
plt.plot([time_res[0],time_res[-1]],[T_ref,T_ref], '--')
plt.grid()
plt.ylabel('Temperature [C]')
plt.subplot(3, 1, 3)
plt.step(time_res_comp, Tc_res_comp)
plt.step(time_res, Tc_res)
plt.plot([time_res[0],time_res[-1]],[Tc_ref,Tc_ref], '--')
plt.grid()
plt.ylabel('Cooling temperature [C]')
plt.xlabel('time')
plt.show()
def test_eliminated_variables(self):
"""
Test that the results when using eliminated variables are the same as when not using them.
"""
# Compile and load the model used for simulation
sim_fmu = compile_fmu(
"CSTR.CSTR_MPC_Model",
self.cstr_file_path,
compiler_options={"state_initial_equations": True})
sim_model = load_fmu(sim_fmu)
# Compile and load the model with eliminated variables used for simulation
sim_fmu_elim = compile_fmu("CSTR.CSTR_elim_vars_MPC_Model",
self.cstr_file_path,
compiler_options={
"state_initial_equations": True,
'equation_sorting': True,
'automatic_tearing': False
})
sim_model_elim = load_fmu(sim_fmu_elim)
# Define stationary point A and set initial values and inputs
c_0_A = 956.271352
T_0_A = 250.051971
sim_model.set('_start_c', c_0_A)
sim_model.set('_start_T', T_0_A)
sim_model.set('Tc', 280)
init_res = sim_model.simulate(start_time=0., final_time=150)
# Compile and load optimization problems
op = transfer_to_casadi_interface("CSTR.CSTR_MPC",
self.cstr_file_path,
compiler_options={
"state_initial_equations": True,
"common_subexp_elim": False
})
op_elim = transfer_to_casadi_interface("CSTR.CSTR_elim_vars_MPC",
self.cstr_file_path,
compiler_options={
"state_initial_equations":
True,
'equation_sorting': True,
'automatic_tearing': False,
"common_subexp_elim": False
})
# Define MPC options
sample_period = 5 # s
horizon = 10 # Samples on the horizon
n_e_per_sample = 1 # Collocation elements / sample
n_e = n_e_per_sample * horizon # Total collocation elements
finalTime = 50 # s
number_samp_tot = 5 # Total number of samples to do
# Create blocking factors with quadratic penalty and bound on 'Tc'
bf_list = [n_e_per_sample] * (horizon / n_e_per_sample)
factors = {'Tc': bf_list}
du_quad_pen = {'Tc': 50}
du_bounds = {'Tc': 30}
bf = BlockingFactors(factors, du_bounds, du_quad_pen)
# Set collocation options
opt_opts = op.optimize_options()
opt_opts['n_e'] = n_e
opt_opts['n_cp'] = 2
opt_opts['init_traj'] = init_res
constr_viol_costs = {'T': 1e6}
# Create the MPC object
MPC_object = MPC(op,
opt_opts,
sample_period,
horizon,
constr_viol_costs=constr_viol_costs,
noise_seed=1)
# Set initial state
x_k = {'_start_c': c_0_A, '_start_T': T_0_A}
# Update the state and optimize number_samp_tot times
for k in range(number_samp_tot):
# Update the state and compute the optimal input for next sample period
MPC_object.update_state(x_k)
u_k = MPC_object.sample()
# Reset the model and set the new initial states before simulating
# the next sample period with the optimal input u_k
sim_model.reset()
sim_model.set(list(x_k.keys()), list(x_k.values()))
sim_res = sim_model.simulate(start_time=k * sample_period,
final_time=(k + 1) * sample_period,
input=u_k)
# Extract state at end of sample_period from sim_res and add Gaussian
# noise with mean 0 and standard deviation 0.005*(state_current_value)
x_k = MPC_object.extract_states(sim_res, mean=0, st_dev=0.005)
# Extract variable profiles
complete_result = MPC_object.get_complete_results()
op_elim.eliminateAlgebraics()
assert (len(op_elim.getEliminatedVariables()) == 2)
opt_opts_elim = op_elim.optimize_options()
opt_opts_elim['n_e'] = n_e
opt_opts_elim['n_cp'] = 2
opt_opts_elim['init_traj'] = init_res
# Create the MPC object with eliminated variables
MPC_object_elim = MPC(op_elim,
opt_opts_elim,
sample_period,
horizon,
constr_viol_costs=constr_viol_costs,
noise_seed=1)
# Set initial state
x_k = {'_start_c': c_0_A, '_start_T': T_0_A}
# Update the state and optimize number_samp_tot times
for k in range(number_samp_tot):
# Update the state and compute the optimal input for next sample period
MPC_object_elim.update_state(x_k)
u_k = MPC_object_elim.sample()
# Reset the model and set the new initial states before simulating
# the next sample period with the optimal input u_k
sim_model_elim.reset()
sim_model_elim.set(list(x_k.keys()), list(x_k.values()))
sim_res = sim_model_elim.simulate(start_time=k * sample_period,
final_time=(k + 1) *
sample_period,
input=u_k)
# Extract state at end of sample_period from sim_res and add Gaussian
# noise with mean 0 and standard deviation 0.005*(state_current_value)
x_k = MPC_object_elim.extract_states(sim_res, mean=0, st_dev=0.005)
# Extract variable profiles
complete_result_elim = MPC_object_elim.get_complete_results()
N.testing.assert_array_almost_equal(complete_result['c'],
complete_result_elim['c'])
N.testing.assert_array_almost_equal(complete_result['T'],
complete_result_elim['T'])
N.testing.assert_array_almost_equal(complete_result['Tc'],
complete_result_elim['Tc'])