from gekko import GEKKO
import numpy as np

# True system: dx/dt = -k_true * x, with k_true = 0.4
k_true = 0.4
t_points = np.linspace(0, 5, 51)
x_true = np.exp(-k_true * t_points)
x_meas = x_true[-1]  # measurement at t=5

# GEKKO model for parameter estimation (MHE)
m = GEKKO(remote=False)
m.time = np.linspace(0, 5, 51)
k = m.FV(value=1.0, lb=0, ub=2)
k.STATUS = 1  # allow optimization to adjust k
x = m.Var(value=1.0)
m.Equation(x.dt() == -k * x)
# Define controlled variable for measurements
y = m.CV(value=np.nan * np.ones(len(m.time)))
y.FSTATUS = 0
y.MEAS = np.nan * np.ones(len(m.time))
y.FSTATUS[-1] = 1  # provide measurement feedback at final time
y.MEAS[-1] = x_meas
m.Equation(y == x)
m.options.IMODE = 5  # MHE mode
m.options.EV_TYPE = 1  # L2 norm in objective
m.solve(disp=False)
print("Estimated k:", k.value[0])
print("Predicted x(5):", x.value[-1], "Measured x(5):", x_meas)