Figure 4.8:
Closed-loop performance of combined nonlinear MHE/MPC for varying disturbance size. The system is controlled between two steady states.
Code for Figure 4.8
Text of the GNU GPL.
main.py
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221 | # [makes] cstr_nonlinear_mpc_mhe.mat
#
# Control of a nonlinear CSTR using nonlinear MPC and MHE.
# Translated from cstr_nonlinear_mpc_mhe.m.
import numpy as np
import casadi
import scipy.io
import mpctools as mpc
def runcstr(wmax, vmax):
Delta = 0.5
Nx = 2; Nu = 1; Ny = 1; Nw = 1
Nsim = 200
Ncontroller = 10; Nmhe = 10
# CSTR parameters.
T0 = 350.0; c0 = 1.0; r_val = 0.219; k0 = 7.2e10; E = 8750.0
U = 54.94; rho = 1e3; Cp = 0.239; DeltaH = -5e4
A = np.pi * r_val**2; rhoCp = rho * Cp; h_val = 0.659; F = 0.1
def ode(x, u):
c, T, Tc = x[0], x[1], u[0]
k = k0 * np.exp(-E / T)
rate = k * c
dcdt = F * (c0 - c) / (A * h_val) - rate
dTdt = (F * (T0 - T) / (A * h_val) - DeltaH / rhoCp * rate
+ 2 * U / (r_val * rhoCp) * (Tc - T))
return np.array([dcdt, dTdt])
cstrsim = mpc.getCasadiIntegrator(ode, Delta, [Nx, Nu], ['x', 'u'],
funcname='cstrsim', verbosity=0)
# Steady state (iterate a few steps to ensure dx/dt = 0).
xs = np.array([0.878, 324.5])
us = np.array([300.0])
for _ in range(10):
xs = np.array(cstrsim(xs, us)).flatten()
C = np.array([[0.0, 1.0]]) # Measure temperature only.
G = np.array([[1.0], [0.0]]) # Process noise enters concentration only.
ys = C @ xs # shape (1,)
CVs = [0] # Concentration is controlled variable (0-based).
# Bounds.
xlb = np.array([0.1, 250.0])
xub = np.array([1.0, 350.0])
umax_val = 0.25 * us[0]
ulb = us - umax_val
uub = us + umax_val
# CasADi functions.
print(' Building controllers.')
flin = mpc.getCasadiFunc(ode, [Nx, Nu], ['x', 'u'], funcname='flin')
fmpc = mpc.getCasadiFunc(ode, [Nx, Nu], ['x', 'u'], funcname='fmpc',
rk4=True, Delta=Delta, M=2)
fsstarg = mpc.getCasadiFunc(ode, [Nx, Nu], ['x', 'u'], funcname='fsstarg')
x_sym = casadi.SX.sym('x', Nx)
u_sym = casadi.SX.sym('u', Nu)
w_sym = casadi.SX.sym('w', Nw)
fmhe = casadi.Function('fmhe', [x_sym, u_sym, w_sym],
[fmpc(x_sym, u_sym) + casadi.mtimes(G, w_sym)],
['x', 'u', 'w'], ['fmhe'])
def measurement(x):
return x[1]
h_func = mpc.getCasadiFunc(measurement, [Nx], ['x'], funcname='h')
# Linearize continuous model and compute LQR terminal cost matrix P.
ss = mpc.util.getLinearizedModel(flin, [xs, us], names=['A', 'B'], Delta=Delta)
A_lin, B_lin = ss['A'], ss['B']
Q_mpc = 0.01 * np.diag(xs**-2)
R_mpc = np.diag(us**-2)
[_, P_dare] = mpc.util.dlqr(A_lin, B_lin, Q_mpc, R_mpc)
S_mpc = 10 * R_mpc
# MHE noise covariance inverses.
Qinv_mhe = 1.0 / 0.05
Rinv_mhe = 1.0 / 50.0
# Stage costs (closures capture Q_mpc, R_mpc, S_mpc, P_dare, Qinv, Rinv).
def l_ctrl(x, u, Du, xsp, usp):
dx = x - xsp; du = u - usp
return (mpc.mtimes(dx.T, Q_mpc, dx) + mpc.mtimes(du.T, R_mpc, du)
+ mpc.mtimes(Du.T, S_mpc, Du))
l_ctrl_func = mpc.getCasadiFunc(l_ctrl, [Nx, Nu, Nu, Nx, Nu],
['x', 'u', 'Du', 'x_sp', 'u_sp'],
funcname='l')
def costtogo(x, xsp):
dx = x - xsp
return mpc.mtimes(dx.T, P_dare, dx)
Pf_func = mpc.getCasadiFunc(costtogo, [Nx, Nx], ['x', 'x_sp'], funcname='Pf')
def lmhe_func(w, v):
return Qinv_mhe * w[0]**2 + Rinv_mhe * v[0]**2
lmhe = mpc.getCasadiFunc(lmhe_func, [Nw, Ny], ['w', 'v'], funcname='l')
# Assemble controller.
N_ctrl = {'x': Nx, 'u': Nu, 't': Ncontroller}
controller = mpc.nmpc(f=fmpc, l=l_ctrl_func, N=N_ctrl, x0=xs,
Vf=Pf_func,
par={'xsp': xs, 'usp': us},
lb={'x': xlb, 'u': ulb}, ub={'x': xub, 'u': uub},
guess={'x': np.tile(xs, (Ncontroller + 1, 1)),
'u': np.tile(us, (Ncontroller, 1))},
uprev=us,
funcargs={'l': ['x', 'u', 'Du', 'x_sp', 'u_sp']},
verbosity=0)
# Steady-state target finder.
N_ss = {'x': Nx, 'u': Nu, 'y': Ny}
sstarg_solver = mpc.sstarg(fsstarg, h_func, N_ss,
lb={'x': xlb, 'u': ulb}, ub={'x': xub, 'u': uub},
guess={'x': xs, 'u': us, 'y': ys},
discretef=False, verbosity=0)
# Random disturbances (same seed each call to runcstr, matching Matlab behavior).
np.random.seed(0)
urand = lambda n, m: 2 * np.random.rand(n, m) - 1
v = vmax * urand(Ny, Nsim + 1) # (Ny, Nsim+1)
w_dist = wmax * urand(Nw, Nsim) # (Nw, Nsim)
y_init_perturb = vmax * urand(Nmhe + 1, Ny) # (Nmhe+1, Ny)
# MHE.
y_init = np.tile(ys, (Nmhe + 1, 1)) + y_init_perturb # (Nmhe+1, Ny)
u_init = np.tile(us, (Nmhe, 1)) # (Nmhe, Nu)
N_mhe = {'x': Nx, 'u': Nu, 'w': Nw, 'y': Ny, 't': Nmhe}
mhe = mpc.nmhe(fmhe, h_func, u_init, y_init, lmhe, N_mhe,
lb={'x': xlb}, ub={'x': xub},
guess={'x': np.tile(xs, (Nmhe + 1, 1))},
inferargs=True, verbosity=0)
# Closed-loop simulation.
x = np.full((Nx, Nsim + 1), np.nan)
x[:, 0] = xs
y_meas = np.full((Ny, Nsim + 1), np.nan)
u_ctrl = np.full((Nu, Nsim), np.nan)
xhat = np.full((Nx, Nsim + 1), np.nan)
xtarg = np.full((Nx, Nsim), np.nan)
utarg = np.full((Nu, Nsim), np.nan)
# Setpoint: concentration steps down 10% every Nchange/2 steps.
t_idx = np.arange(Nsim + 1)
Nchange = 100
xsp = np.full((Nx, Nsim + 1), np.nan)
xsp[CVs[0], :] = xs[CVs[0]] * (1 - 0.1 * (np.mod(t_idx, Nchange) >= Nchange / 2))
xsp[:, -1] = xsp[:, -2]
u_prev = us.copy()
for i in range(Nsim + 1):
print('(%3d) ' % (i + 1), end='')
# Steady-state target.
sstarg_solver.fixvar('x', 0, np.array([xsp[CVs[0], i]]), CVs)
sstarg_solver.solve()
print('Target: %s, ' % sstarg_solver.stats['status'], end='')
if sstarg_solver.stats['status'] != 'Solve_Succeeded':
print()
print('Warning: sstarg failed at step %d!' % i)
break
if i < Nsim:
xtarg[:, i] = np.array(sstarg_solver.var['x', 0]).flatten()
utarg[:, i] = np.array(sstarg_solver.var['u', 0]).flatten()
sstarg_solver.saveguess()
# Measurement and MHE.
y_meas[:, i] = (C @ x[:, i]) + v[:, i]
mhe.newmeasurement(y_meas[:, i], u_prev)
mhe.solve()
print('Estimator: %s, ' % mhe.stats['status'], end='')
if mhe.stats['status'] != 'Solve_Succeeded':
print()
print('Warning: MHE failed at step %d!' % i)
break
xhat[:, i] = np.array(mhe.var['x', Nmhe]).flatten()
mhe.saveguess()
if i == Nsim:
print('Done')
break
# Controller.
controller.fixvar('x', 0, xhat[:, i])
controller.par['x_sp'] = [xtarg[:, i]] * (Ncontroller + 1)
controller.par['u_sp'] = [utarg[:, i]] * Ncontroller
controller.par['u_prev'] = [u_prev]
controller.solve()
print('Controller: %s' % controller.stats['status'], end='')
if controller.stats['status'] != 'Solve_Succeeded':
print()
print('Warning: controller failed at step %d!' % i)
break
u_ctrl[:, i] = np.array(controller.var['u', 0]).flatten()
controller.saveguess()
u_prev = u_ctrl[:, i].copy()
print()
# Simulate plant.
x[:, i + 1] = (np.array(cstrsim(x[:, i], u_ctrl[:, i])).flatten()
+ G[:, 0] * w_dist[0, i])
t = Delta * np.arange(Nsim + 1)
return dict(x=x, u=u_ctrl, y=y_meas, xhat=xhat, xsp=xsp, t=t,
v=v, w=w_dist, vmax=float(vmax), wmax=float(wmax))
wmaxes = [0, 0.001, 0.001, 0.01, 0.01, 0.025]
vmaxes = [0, 0.1, 1, 1, 10, 10]
data = []
for (wmax_i, vmax_i) in zip(wmaxes, vmaxes):
print('Running wmax=%g, vmax=%g' % (wmax_i, vmax_i))
data.append(runcstr(wmax_i, vmax_i))
scipy.io.savemat('cstr_nonlinear_mpc_mhe.mat', {'data': data})
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