def simulate_pendulum_MPC(sim_options):
use_NN_model = get_parameter(sim_options, 'use_NN_model')
if use_NN_model:
Ts_fit = 10e-3 # model was fitted with this sampling time
ss_model = CartPoleStateSpaceModel(Ts=Ts_fit)
nn_solution = NeuralStateSpaceSimulator(ss_model, Ts=Ts_fit)
model_name = "model_SS_1step_nonoise.pkl"
nn_solution.ss_model.load_state_dict(
torch.load(os.path.join("models", model_name)))
f_ODE = nn_solution.f_ODE
else:
f_ODE = f_ODE_wrapped
seed_val = get_parameter(sim_options, 'seed_val')
if seed_val is not None:
np.random.seed(seed_val)
Ac = get_parameter(sim_options, 'Ac')
Bc = get_parameter(sim_options, 'Bc')
Cc = np.array([[1., 0., 0., 0.], [0., 0., 1., 0.]])
Dc = np.zeros((2, 1))
[nx, nu] = Bc.shape # number of states and number or inputs
ny = np.shape(Cc)[0]
Ts_MPC = get_parameter(sim_options, 'Ts_MPC')
ratio_Ts = int(Ts_MPC // Ts_fast)
Ts_MPC = (
(Ts_MPC //
Ts_fast)) * Ts_fast # make Ts_MPC an integer multiple of Ts_fast
# Brutal forward euler discretization
Ad = np.eye(nx) + Ac * Ts_MPC
Bd = Bc * Ts_MPC
Cd = Cc
Dd = Dc
# Standard deviation of the measurement noise on position and angle
std_npos = get_parameter(sim_options, 'std_npos')
std_nphi = get_parameter(sim_options, 'std_nphi')
# Force disturbance
std_dF = get_parameter(sim_options, 'std_dF')
# disturbance power spectrum
w_F = get_parameter(sim_options,
'w_F') # bandwidth of the force disturbance
tau_F = 1 / w_F
Hu = control.TransferFunction([1], [1 / w_F, 1])
Hu = Hu * Hu
Hud = control.matlab.c2d(Hu, Ts_MPC)
N_sim_imp = tau_F / Ts_MPC * 20
t_imp = np.arange(N_sim_imp) * Ts_MPC
t, y = control.impulse_response(Hud, t_imp)
y = y[0]
std_tmp = np.sqrt(np.sum(y**2)) # np.sqrt(trapz(y**2,t))
Hu = Hu / (std_tmp) * std_dF
N_skip = 1 #int(20 * tau_F // Ts_MPC) # skip initial samples to get a regime sample of d
t_sim_d = get_parameter(sim_options, 'len_sim') # simulation length (s)
N_sim_d = int(t_sim_d // Ts_MPC)
N_sim_d = N_sim_d + N_skip + 1
e = np.random.randn(N_sim_d)
te = np.arange(N_sim_d) * Ts_MPC
_, d, _ = control.forced_response(Hu, te, e)
d = d.ravel()
p_ref = d[N_skip:]
#td = np.arange(len(d)) * Ts_fast
Np = get_parameter(sim_options, 'Np')
Nc = get_parameter(sim_options, 'Nc')
Qx = get_parameter(sim_options, 'Qx')
QxN = get_parameter(sim_options, 'QxN')
Qu = get_parameter(sim_options, 'Qu')
QDu = get_parameter(sim_options, 'QDu')
# Reference input and states
xref_fun = get_parameter(sim_options, 'xref_fun') # reference state
xref_fun_v = np.vectorize(xref_fun, signature='()->(n)')
t0 = 0
xref_MPC = xref_fun(t0)
uref = get_parameter(sim_options, 'uref')
uminus1 = np.array(
[0.0]
) # input at time step negative one - used to penalize the first delta u at time instant 0. Could be the same as uref.
# Constraints
xmin = np.array([-1000, -1000, -1000, -1000])
xmax = np.array([1000, 1000.0, 1000, 1000])
umin = np.array([-10])
umax = np.array([10])
Dumin = np.array([-100 * Ts_MPC])
Dumax = np.array([100 * Ts_MPC])
# Initialize simulation system
phi0 = 0.0 #10*2*np.pi/360
x0 = np.array([0, 0, phi0, 0]) # initial state
# system_dyn = ode(f_ODE_wrapped).set_integrator('vode', method='bdf') # dopri5
system_dyn = ode(f_ODE).set_integrator('dopri5')
system_dyn.set_initial_value(x0, t0)
system_dyn.set_f_params(0.0)
QP_eps_rel = get_parameter(sim_options, 'QP_eps_rel')
QP_eps_abs = get_parameter(sim_options, 'QP_eps_abs')
# Emergency exit conditions
EMERGENCY_STOP = False
EMERGENCY_POS = 30.0
EMERGENCY_ANGLE = 90 * DEG_TO_RAD
K = MPCController(Ad,
Bd,
Np=Np,
Nc=Nc,
x0=x0,
xref=xref_MPC,
uminus1=uminus1,
Qx=Qx,
QxN=QxN,
Qu=Qu,
QDu=QDu,
xmin=xmin,
xmax=xmax,
umin=umin,
umax=umax,
Dumin=Dumin,
Dumax=Dumax,
eps_feas=1e3,
eps_rel=QP_eps_rel,
eps_abs=QP_eps_abs)
try:
K.setup(solve=True) # setup initial problem and also solve it
except:
EMERGENCY_STOP = True
if not EMERGENCY_STOP:
if K.res.info.status != 'solved':
EMERGENCY_STOP = True
# Basic Kalman filter design
Q_kal = get_parameter(sim_options, 'Q_kal')
R_kal = get_parameter(sim_options, 'R_kal')
L, P, W = kalman_design_simple(Ad,
Bd,
Cd,
Dd,
Q_kal,
R_kal,
type='predictor')
x0_est = x0
KF = LinearStateEstimator(x0_est, Ad, Bd, Cd, Dd, L)
# Simulate in closed loop
len_sim = get_parameter(sim_options, 'len_sim') # simulation length (s)
nsim = int(
np.ceil(len_sim / Ts_MPC)
) # simulation length(timesteps) # watch out! +1 added, is it correct?
t_vec = np.zeros((nsim, 1))
t_calc_vec = np.zeros(
(nsim, 1)) # computational time to get MPC solution (+ estimator)
status_vec = np.zeros((nsim, 1))
x_vec = np.zeros((nsim, nx))
x_ref_vec = np.zeros((nsim, nx))
y_vec = np.zeros((nsim, ny))
y_meas_vec = np.zeros((nsim, ny))
y_est_vec = np.zeros((nsim, ny))
x_est_vec = np.zeros((nsim, nx))
u_vec = np.zeros((nsim, nu))
x_MPC_pred = np.zeros(
(nsim, Np + 1, nx)) # on-line predictions from the Kalman Filter
nsim_fast = int(len_sim // Ts_fast)
t_vec_fast = np.zeros((nsim_fast, 1))
x_vec_fast = np.zeros(
(nsim_fast, nx)) # finer integration grid for performance evaluation
x_ref_vec_fast = np.zeros(
(nsim_fast, nx)) # finer integration grid for performance evaluatio
u_vec_fast = np.zeros(
(nsim_fast, nu)) # finer integration grid for performance evaluatio
Fd_vec_fast = np.zeros((nsim_fast, nu)) #
t_int_vec_fast = np.zeros((nsim_fast, 1))
emergency_vec_fast = np.zeros((nsim_fast, 1)) #
t_pred_all = t0 + np.arange(nsim + Np + 1) * Ts_MPC
Xref_MPC_all = xref_fun_v(t_pred_all)
t_step = t0
x_step = x0
u_MPC = None
for idx_fast in range(nsim_fast):
## Determine step type: fast simulation only or MPC step
idx_MPC = idx_fast // ratio_Ts
run_MPC = (idx_fast % ratio_Ts) == 0
# Output for step i
# Ts_MPC outputs
if run_MPC: # it is also a step of the simulation at rate Ts_MPC
t_vec[idx_MPC, :] = t_step
x_vec[idx_MPC, :] = x_step #system_dyn.y
xref_MPC = np.array([p_ref[idx_MPC], 0.0, 0.0, 0.0])
#xref_MPC = xref_fun(t_step) # reference state
t_pred = t_step + np.arange(Np + 1) * Ts_MPC
x_ref_vec[idx_MPC, :] = xref_MPC
if not EMERGENCY_STOP:
# u_MPC, info_MPC = K.output(return_x_seq=True, return_status=True) # u[i] = k(\hat x[i]) possibly computed at time instant -1
u_MPC, info_MPC = K.output(
return_status=True, return_x_seq=True
) # u[i] = k(\hat x[i]) possibly computed at time instant -1
else:
u_MPC = np.zeros(nu)
if not EMERGENCY_STOP:
if info_MPC['status'] != 'solved':
EMERGENCY_STOP = True
if not EMERGENCY_STOP:
#pass
x_MPC_pred[idx_MPC, :, :] = info_MPC[
'x_seq'] # x_MPC_pred[i,i+1,...| possibly computed at time instant -1]
u_vec[idx_MPC, :] = u_MPC
y_step = Cd.dot(x_step) # y[i] from the system
ymeas_step = np.copy(y_step)
ymeas_step[0] += std_npos * np.random.randn()
ymeas_step[1] += std_nphi * np.random.randn()
y_vec[idx_MPC, :] = y_step
y_meas_vec[idx_MPC, :] = ymeas_step
if not EMERGENCY_STOP:
status_vec[idx_MPC, :] = (info_MPC['status'] != 'solved')
if np.abs(ymeas_step[0]) > EMERGENCY_POS or np.abs(
ymeas_step[1]) > EMERGENCY_ANGLE:
EMERGENCY_STOP = True
# Ts_fast outputs
t_vec_fast[idx_fast, :] = t_step
x_vec_fast[idx_fast, :] = x_step #system_dyn.y
x_ref_vec_fast[idx_fast, :] = xref_MPC
u_fast = u_MPC
u_vec_fast[idx_fast, :] = u_fast
Fd_vec_fast[idx_fast, :] = 0.0
emergency_vec_fast[idx_fast, :] = EMERGENCY_STOP
## Update to step i+1
# Controller simulation step at rate Ts_MPC
if run_MPC:
time_calc_start = time.perf_counter()
# Kalman filter: update and predict
#KF.update(ymeas_step) # \hat x[i|i]
#KF.predict(u_MPC) # \hat x[i+1|i]
KF.predict_update(u_MPC, ymeas_step)
# MPC update
if not EMERGENCY_STOP:
#Xref_MPC = Xref_MPC_all[idx_MPC:idx_MPC + Np + 1]
K.update(
KF.x, u_MPC,
xref=xref_MPC) # update with measurement and reference
#K.update(KF.x, u_MPC, xref=xref_MPC) # update with measurement and reference
t_calc_vec[idx_MPC, :] = time.perf_counter() - time_calc_start
if t_calc_vec[idx_MPC, :] > 2 * Ts_MPC:
EMERGENCY_STOP = True
# System simulation step at rate Ts_fast
time_integrate_start = time.perf_counter()
system_dyn.set_f_params(u_fast)
system_dyn.integrate(t_step + Ts_fast)
x_step = system_dyn.y
#x_step = x_step + f_ODE(t_step, x_step, u_fast)*Ts_fast
t_int_vec_fast[
idx_fast, :] = time.perf_counter() - time_integrate_start
# Time update
t_step += Ts_fast
simout = {
't': t_vec,
'x': x_vec,
'u': u_vec,
'y': y_vec,
'y_meas': y_meas_vec,
'x_ref': x_ref_vec,
'x_MPC_pred': x_MPC_pred,
'status': status_vec,
'Fd_fast': Fd_vec_fast,
't_fast': t_vec_fast,
'x_fast': x_vec_fast,
'x_ref_fast': x_ref_vec_fast,
'u_fast': u_vec_fast,
'emergency_fast': emergency_vec_fast,
'KF': KF,
'K': K,
'nsim': nsim,
'Ts_MPC': Ts_MPC,
't_calc': t_calc_vec,
't_int_fast': t_int_vec_fast,
'use_NN_model': use_NN_model
}
return simout
Np=Np,
x0=x0,
xref=xref,
uminus1=uminus1,
Qx=Qx,
QxN=QxN,
Qu=Qu,
QDu=QDu,
xmin=xmin,
xmax=xmax,
umin=umin,
umax=umax,
Dumin=Dumin,
Dumax=Dumax,
eps_feas=1e3)
K.setup()
# Simulate in closed loop
[nx, nu] = Bd.shape # number of states and number or inputs
len_sim = 40 # simulation length (s)
nsim = int(len_sim / Ts) # simulation length(timesteps)
xsim = np.zeros((nsim, nx))
usim = np.zeros((nsim, nu))
tsim = np.arange(0, nsim) * Ts
time_start = time.time()
t_step = t0
uMPC = uminus1
for i in range(nsim):
xsim[i, :] = system_dyn.y
def simulate_pendulum_MPC(sim_options):
seed_val = get_parameter(sim_options,'seed_val')
if seed_val is not None:
np.random.seed(seed_val)
# In[Sample times]
Ts_MPC = get_parameter(sim_options, 'Ts_MPC')
ratio_Ts = int(Ts_MPC // Ts_PID)
# In[Real System]
Cc = np.array([[1., 0., 0., 0.],
[0., 0., 1., 0.]])
Cd = np.copy(Cc)
nx, nu = 4,1
ny = 2
# In[initialize simulation system]
t0 = 0
phi0 = -0.0 * 2 * np.pi / 360 # initial angle
x0 = np.array([0, 0, phi0, 0]) # initial state
#system_dyn = ode(f_ODE_wrapped).set_integrator('vode', method='bdf') # dopri5
system_dyn = ode(f_ODE_wrapped).set_integrator('dopri5') # dopri5
# system_dyn = ode(f_ODE_wrapped).set_integrator('dopri5')
system_dyn.set_initial_value(x0, t0)
system_dyn.set_f_params(0.0)
#In[MPC params --model]
Acl_c = get_parameter(sim_options, 'Acl_c')
Bcl_c = get_parameter(sim_options, 'Bcl_c')
Ccl_c = np.array([[1., 0., 0],
[0., 0., 1]])
Dcl_c = np.zeros((2, 1))
ncl_x, ncl_u = Bcl_c.shape # number of states and number or inputs
#ncl_y = np.shape(Ccl_c)[0]
#In[MPC matrices discretization]
Acl_d = np.eye(ncl_x) + Acl_c*Ts_MPC
Bcl_d = Bcl_c*Ts_MPC
Ccl_d = Ccl_c
Dcl_d = Dcl_c
x0_cl = np.array([0,0,phi0])
M_cl = LinearStateSpaceSystem(A=Acl_d, B=Bcl_d, C=Ccl_d, D=Dcl_d, x0=x0_cl)
# MPC parameters
Np = get_parameter(sim_options, 'Np')
Nc = get_parameter(sim_options, 'Nc')
Qx = get_parameter(sim_options, 'Qx')
QxN = get_parameter(sim_options, 'QxN')
Qr = get_parameter(sim_options, 'Qr')
QDr = get_parameter(sim_options, 'QDr')
# Constraints
#xmin = np.array([-1.5, -100, -100])
#xmax = np.array([1.5, 100.0, 100])
#umin = np.array([-10])
#umax = np.array([10])
#Dumin = np.array([-100 * Ts_MPC_def])
#Dumax = np.array([100 * Ts_MPC_def])
QP_eps_rel = get_parameter(sim_options, 'QP_eps_rel')
QP_eps_abs = get_parameter(sim_options, 'QP_eps_abs')
# Emergency exit conditions
EMERGENCY_STOP = False
EMERGENCY_POS = 2.0
EMERGENCY_ANGLE = 30 * DEG_TO_RAD
# Reference input and states
xref_cl_fun = get_parameter(sim_options, 'xref_cl_fun') # reference state
xref_cl_fun_v = np.vectorize(xref_cl_fun, signature='()->(n)')
t0 = 0
xref_MPC = xref_cl_fun(t0)
uref = get_parameter(sim_options, 'uref')
uminus1 = np.array([0.0]) # input at time step negative one - used to penalize the first delta u at time instant 0. Could be the same as uref.
kMPC = MPCController(Acl_d, Bcl_d, Np=Np, Nc=Nc, x0=x0_cl, xref=xref_MPC, uminus1=uminus1,
Qx=Qx, QxN=QxN, Qu=Qr, QDu=QDr,
eps_feas=1e3, eps_rel=QP_eps_rel, eps_abs=QP_eps_abs)
try:
kMPC.setup(solve=True) # setup initial problem and also solve it
except:
EMERGENCY_STOP = True
if not EMERGENCY_STOP:
if kMPC.res.info.status != 'solved':
EMERGENCY_STOP = True
# In[initialize PID]
# Default controller parameters -
P = -100.0
I = -1
D = -20
N = 100.0
kP = control.tf(P,1, Ts_PID)
kI = I*Ts_PID*control.tf([0, 1], [1,-1], Ts_PID)
kD = D*control.tf([N, -N], [1.0, Ts_PID*N - 1], Ts_PID)
PID_tf = kP + kD + kI
PID_ss = control.ss(PID_tf)
k_PID = LinearStateSpaceSystem(A=PID_ss.A, B=PID_ss.B, C=PID_ss.C, D=PID_ss.D)
# In[initialize noise]
# Standard deviation of the measurement noise on position and angle
std_npos = get_parameter(sim_options, 'std_npos')
std_nphi = get_parameter(sim_options, 'std_nphi')
# Force disturbance
std_dF = get_parameter(sim_options, 'std_dF')
# Disturbance power spectrum
w_F = get_parameter(sim_options, 'w_F') # bandwidth of the force disturbance
tau_F = 1 / w_F
Hu = control.TransferFunction([1], [1 / w_F, 1])
Hu = Hu * Hu
Hud = control.matlab.c2d(Hu, Ts_PID)
N_sim_imp = tau_F / Ts_PID * 20
t_imp = np.arange(N_sim_imp) * Ts_PID
t, y = control.impulse_response(Hud, t_imp)
y = y[0]
std_tmp = np.sqrt(np.sum(y ** 2)) # np.sqrt(trapz(y**2,t))
Hu = Hu / (std_tmp) * std_dF
N_skip = int(20 * tau_F // Ts_PID) # skip initial samples to get a regime sample of d
t_sim_d = get_parameter(sim_options, 'len_sim') # simulation length (s)
N_sim_d = int(t_sim_d // Ts_PID)
N_sim_d = N_sim_d + N_skip + 1
e = np.random.randn(N_sim_d)
te = np.arange(N_sim_d) * Ts_PID
_, d, _ = control.forced_response(Hu, te, e)
d = d.ravel()
# Simulate in closed loop
len_sim = get_parameter(sim_options, 'len_sim') # simulation length (s)
nsim = int(len_sim // Ts_MPC) #int(np.ceil(len_sim / Ts_MPC)) # simulation length(timesteps) # watch out! +1 added, is it correct?
t_vec = np.zeros((nsim, 1))
status_vec = np.zeros((nsim,1))
x_vec = np.zeros((nsim, nx))
x_ref_vec = np.zeros((nsim, ncl_x))
y_vec = np.zeros((nsim, ny))
y_meas_vec = np.zeros((nsim, ny))
u_vec = np.zeros((nsim, nu))
x_model_vec = np.zeros((nsim,3))
nsim_fast = int(len_sim // Ts_PID)
t_vec_fast = np.zeros((nsim_fast, 1))
x_vec_fast = np.zeros((nsim_fast, nx)) # finer integration grid for performance evaluation
ref_phi_vec_fast = np.zeros((nsim_fast, 1))
y_meas_vec_fast = np.zeros((nsim_fast, ny))
x_ref_vec_fast = np.zeros((nsim_fast, nx)) # finer integration grid for performance evaluatio
u_vec_fast = np.zeros((nsim_fast, nu)) # finer integration grid for performance evaluatio
Fd_vec_fast = np.zeros((nsim_fast, nu)) #
t_int_vec_fast = np.zeros((nsim_fast, 1))
emergency_vec_fast = np.zeros((nsim_fast, 1)) #
t_step = t0
x_step = x0
u_PID = None
t_pred_all = t0 + np.arange(nsim + Np + 1) * Ts_MPC
Xref_MPC_all = xref_cl_fun_v(t_pred_all)
for idx_fast in range(nsim_fast):
## Determine step type: fast simulation only or MPC step
idx_MPC = idx_fast // ratio_Ts
run_MPC_controller = (idx_fast % ratio_Ts) == 0
y_step = Cd.dot(x_step) # y[i] from the system
ymeas_step = np.copy(y_step)
ymeas_step[0] += std_npos * np.random.randn()
ymeas_step[1] += std_nphi * np.random.randn()
y_meas_vec_fast[idx_fast,:] = ymeas_step
# Output for step i
# Ts_MPC outputs
if run_MPC_controller: # it is also a step of the simulation at rate Ts_MPC
if idx_MPC < nsim:
t_vec[idx_MPC, :] = t_step
y_vec[idx_MPC,:] = y_step
y_meas_vec[idx_MPC,:] = ymeas_step
u_vec[idx_MPC, :] = u_PID
x_model_vec[idx_MPC, :] = M_cl.x.ravel()
xref_MPC = xref_cl_fun(t_step)
x_ref_vec[idx_MPC,:] = xref_MPC.ravel()
if not EMERGENCY_STOP:
phi_ref_MPC, info_MPC = kMPC.output(return_status=True) # u[i] = k(\hat x[i]) possibly computed at time instant -1
else:
phi_ref_MPC = np.zeros(nu)
# PID angle CONTROLLER
ref_phi = phi_ref_MPC.ravel()
error_phi = ref_phi - ymeas_step[1]
u_PID = k_PID.output(error_phi)
u_PID[u_PID > 10.0] = 10.0
u_PID[u_PID < -10.0] = -10.0
u_TOT = u_PID
# Ts_fast outputs
t_vec_fast[idx_fast,:] = t_step
x_vec_fast[idx_fast, :] = x_step #system_dyn.y
u_vec_fast[idx_fast,:] = u_TOT
Fd_vec_fast[idx_fast,:] = 0.0
ref_phi_vec_fast[idx_fast,:] = ref_phi
## Update to step i+1
k_PID.update(error_phi)
# Controller simulation step at rate Ts_MPC
if run_MPC_controller:
M_cl.update(ref_phi)
if not EMERGENCY_STOP:
x_cl = np.array([x_step[0], x_step[1], x_step[2]])
Xref_MPC = Xref_MPC_all[idx_MPC:idx_MPC + Np + 1]
xref_MPC = Xref_MPC_all[idx_MPC]
kMPC.update(x_cl, phi_ref_MPC, xref=xref_MPC) # update with measurement and reference
# System simulation step at rate Ts_fast
time_integrate_start = time.perf_counter()
system_dyn.set_f_params(u_TOT)
system_dyn.integrate(t_step + Ts_PID)
x_step = system_dyn.y
t_int_vec_fast[idx_fast,:] = time.perf_counter() - time_integrate_start
# Time update
t_step += Ts_PID
simout = {'t': t_vec, 'x': x_vec, 'u': u_vec, 'y': y_vec, 'y_meas': y_meas_vec, 'x_ref': x_ref_vec, 'status': status_vec, 'Fd_fast': Fd_vec_fast,
't_fast': t_vec_fast, 'x_fast': x_vec_fast, 'x_ref_fast': x_ref_vec_fast, 'u_fast': u_vec_fast, 'y_meas_fast': y_meas_vec_fast, 'emergency_fast': emergency_vec_fast,
'PID_tf': PID_tf, 'Ts_MPC': Ts_MPC, 'ref_phi_fast': ref_phi_vec_fast, 'x_model': x_model_vec,
't_int_fast': t_int_vec_fast
}
return simout