Parameter Estimation 1

Parameter Estimation 1#

\[min \sum ((X_{1}(t_{i}) - X_{1,meas}(t_i))^{2})\]
\[s.t \;\; X_{1,dot} = X_{2} \;\;\;\;\;\;\;\; X_{1}(0) = p_{1}\]
\[\;\; X_{2,dot} = 1-2*X_{2}-X_{1} \;\;\;\;\;\;\;\; X_{2}(0) = p_{2}\]
\[-1.5 \leq p_{1}, p_{2} \leq 1.5\]
\[t_{f} = 6\]
import pyomo.environ as pyo
from pyomo.dae import ContinuousSet, DerivativeVar

measurements = {1:0.264, 2:0.594, 3:0.801, 5:0.959}

model = pyo.ConcreteModel()
model.t = ContinuousSet(initialize=measurements.keys(),bounds=(0, 6))	

model.x1 = pyo.Var(model.t)
model.x2 = pyo.Var(model.t)

model.p1 = pyo.Var(bounds=(-1.5,1.5))
model.p2 = pyo.Var(bounds=(-1.5,1.5))

model.x1dot = DerivativeVar(model.x1,wrt=model.t)
model.x2dot = DerivativeVar(model.x2)

def _init_conditions(model):
	yield model.x1[0] == model.p1
	yield model.x2[0] == model.p2
model.init_conditions = pyo.ConstraintList(rule=_init_conditions)

# Alternate way to declare initial conditions
#def _initx1(model):
#	return model.x1[0] == model.p1		
#model.initx1 = pyo.Constraint(rule=_initx1)

#def _initx2(model):
#	return model.x2[0] == model.p2
#model.initx2 = pyo.Constraint(rule=_initx2)

def _x1dot(model,i):
	return model.x1dot[i] == model.x2[i]
model.x1dotcon = pyo.Constraint(model.t, rule=_x1dot)

def _x2dot(model,i):
	return model.x2dot[i] == 1-2*model.x2[i]-model.x1[i]
model.x2dotcon = pyo.Constraint(model.t, rule=_x2dot)

def _obj(model):
	return sum((model.x1[i]-measurements[i])**2 for i in measurements.keys())
model.obj = pyo.Objective(rule=_obj)

# Discretize model using Orthogonal Collocation
discretizer = pyo.TransformationFactory('dae.collocation')
discretizer.apply_to(model,nfe=8,ncp=5)

solver = pyo.SolverFactory('ipopt')

results = solver.solve(model,tee=True)

t_meas = sorted(list(measurements.keys()))
x1_meas = [pyo.value(measurements[i]) for i in sorted(measurements.keys())]

t = list(model.t)
x1 = [pyo.value(model.x1[i]) for i in model.t]
    
import matplotlib.pyplot as plt

plt.plot(t,x1)
plt.plot(t_meas,x1_meas,'o')
plt.xlabel('t')
plt.ylabel('x')
plt.title('Dynamic Parameter Estimation Using Collocation')
plt.show()
Ipopt 3.14.17: 

******************************************************************************
This program contains Ipopt, a library for large-scale nonlinear optimization.
 Ipopt is released as open source code under the Eclipse Public License (EPL).
         For more information visit https://github.com/coin-or/Ipopt
******************************************************************************

This is Ipopt version 3.14.17, running with linear solver MUMPS 5.7.3.

Number of nonzeros in equality constraint Jacobian...:      769
Number of nonzeros in inequality constraint Jacobian.:        0
Number of nonzeros in Lagrangian Hessian.............:        4

Total number of variables............................:      166
                     variables with only lower bounds:        0
                variables with lower and upper bounds:        2
                     variables with only upper bounds:        0
Total number of equality constraints.................:      164
Total number of inequality constraints...............:        0
        inequality constraints with only lower bounds:        0
   inequality constraints with lower and upper bounds:        0
        inequality constraints with only upper bounds:        0

iter    objective    inf_pr   inf_du lg(mu)  ||d||  lg(rg) alpha_du alpha_pr  ls
   0  1.9838140e+00 1.00e+00 7.15e-02  -1.0 0.00e+00    -  0.00e+00 0.00e+00   0
   1  3.7714229e-07 2.54e-15 7.19e-16  -1.0 1.00e+00    -  1.00e+00 1.00e+00h  1
   2  3.6867493e-07 2.57e-15 5.91e-17  -2.5 1.45e-04    -  1.00e+00 1.00e+00h  1
   3  3.5639520e-07 2.83e-15 6.61e-17  -3.8 1.72e-03    -  1.00e+00 1.00e+00h  1
   4  3.5566463e-07 5.86e-15 8.20e-17  -5.7 5.27e-04    -  1.00e+00 1.00e+00h  1
   5  3.5566442e-07 1.07e-14 8.02e-17  -8.6 9.04e-06    -  1.00e+00 1.00e+00h  1

Number of Iterations....: 5

                                   (scaled)                 (unscaled)
Objective...............:   3.5566442170075718e-07    3.5566442170075718e-07
Dual infeasibility......:   8.0230960763927328e-17    8.0230960763927328e-17
Constraint violation....:   1.0651202142497596e-14    1.0651202142497596e-14
Variable bound violation:   0.0000000000000000e+00    0.0000000000000000e+00
Complementarity.........:   2.5137424147587802e-09    2.5137424147587802e-09
Overall NLP error.......:   2.5137424147587802e-09    2.5137424147587802e-09


Number of objective function evaluations             = 6
Number of objective gradient evaluations             = 6
Number of equality constraint evaluations            = 6
Number of inequality constraint evaluations          = 0
Number of equality constraint Jacobian evaluations   = 6
Number of inequality constraint Jacobian evaluations = 0
Number of Lagrangian Hessian evaluations             = 5
Total seconds in IPOPT                               = 0.005

EXIT: Optimal Solution Found.

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