4.1.3. Hypersensitive OCP#
2023 Devakumar Thammisetty
MPOPT is an open-source Multi-phase Optimal Control Problem (OCP) solver based on pseudo-spectral collocation with customized adaptive grid refinement techniques.
Download this notebook: hypersensitive.ipynb
Install mpopt from pypi using the following. Disable after first usage
Import mpopt (Contains main solver modules)
[1]:
#!pip install mpopt
from mpopt import mp
4.1.3.1. Defining OCP#
Hypersensitive OCP: https://www.gpops2.com/Examples/HyperSensitive.html
We first create an OCP object and then polulate the object with dynamics, path_constraints, terminal_constraints and objective (running_costs, terminal_costs)
[2]:
ocp = mp.OCP(n_states=1, n_controls=1, n_phases=1)
[3]:
ocp.dynamics[0] = lambda x, u, t: [-x[0] * x[0] * x[0] + u[0]]
ocp.running_costs[0] = lambda x, u, t: 0.5 * (x[0] * x[0] + u[0] * u[0])
ocp.terminal_constraints[0] = lambda xf, tf, x0, t0: [xf[0] - 1.0]
Initial state
[4]:
ocp.x00[0] = 1
Box constraints
[5]:
ocp.lbtf[0] = ocp.ubtf[0] = 1000.0
Scale the time variable
[6]:
ocp.scale_t = 1 / 1000.0
[7]:
ocp.validate()
4.1.3.2. Solve and plot the results in one line#
Lets solve the OCP using following pseudo-spectral approximation * Collocation using Legendre-Gauss-Radau roots * Let’s plot the position and velocity evolution with time starting from 0.
[8]:
mpo, post = mp.solve(ocp, n_segments=5, poly_orders=50, scheme="LGR", plot=True)
******************************************************************************
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 http://projects.coin-or.org/Ipopt
******************************************************************************
Total number of variables............................: 501
variables with only lower bounds: 0
variables with lower and upper bounds: 0
variables with only upper bounds: 0
Total number of equality constraints.................: 252
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
Number of Iterations....: 12
(scaled) (unscaled)
Objective...............: 1.1498050755273090e+00 1.1498050755273090e+00
Dual infeasibility......: 4.7184478546569153e-16 4.7184478546569153e-16
Constraint violation....: 1.4589005542615039e-14 1.1368683772161603e-13
Complementarity.........: 0.0000000000000000e+00 0.0000000000000000e+00
Overall NLP error.......: 1.4589005542615039e-14 1.1368683772161603e-13
Number of objective function evaluations = 30
Number of objective gradient evaluations = 13
Number of equality constraint evaluations = 42
Number of inequality constraint evaluations = 0
Number of equality constraint Jacobian evaluations = 13
Number of inequality constraint Jacobian evaluations = 0
Number of Lagrangian Hessian evaluations = 12
Total CPU secs in IPOPT (w/o function evaluations) = 0.066
Total CPU secs in NLP function evaluations = 0.010
EXIT: Optimal Solution Found.
solver : t_proc (avg) t_wall (avg) n_eval
nlp_f | 340.00us ( 11.33us) 340.41us ( 11.35us) 30
nlp_g | 4.85ms (115.38us) 4.78ms (113.76us) 42
nlp_grad | 266.00us (266.00us) 265.36us (265.36us) 1
nlp_grad_f | 322.00us ( 23.00us) 322.67us ( 23.05us) 14
nlp_hess_l | 592.00us ( 49.33us) 595.87us ( 49.66us) 12
nlp_jac_g | 2.76ms (197.43us) 2.75ms (196.68us) 14
total | 76.82ms ( 76.82ms) 76.14ms ( 76.14ms) 1
Retrive the solution
x: states, u: Controls, t:time, a:Algebraic variables
[9]:
x, u, t, a = post.get_data()
print(f"Terminal time, state : {t[-1][0]:.4f} vs 1000 (Exact), {x[-1]}")
Terminal time, state : 1000.0000 vs 1000 (Exact), [1.]
4.1.3.3. Solve again with Chebyshev-Gauss-Lobatto (CGL) roots#
[10]:
mpo, post = mp.solve(ocp, n_segments=5, poly_orders=50, scheme="CGL", plot=True)
Total number of variables............................: 501
variables with only lower bounds: 0
variables with lower and upper bounds: 0
variables with only upper bounds: 0
Total number of equality constraints.................: 252
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
Number of Iterations....: 12
(scaled) (unscaled)
Objective...............: 1.1406025536022588e+00 1.1406025536022588e+00
Dual infeasibility......: 5.5128124287762148e-13 5.5128124287762148e-13
Constraint violation....: 5.6878507039202523e-12 1.8587797967484221e-11
Complementarity.........: 0.0000000000000000e+00 0.0000000000000000e+00
Overall NLP error.......: 5.6878507039202523e-12 1.8587797967484221e-11
Number of objective function evaluations = 36
Number of objective gradient evaluations = 13
Number of equality constraint evaluations = 49
Number of inequality constraint evaluations = 0
Number of equality constraint Jacobian evaluations = 13
Number of inequality constraint Jacobian evaluations = 0
Number of Lagrangian Hessian evaluations = 12
Total CPU secs in IPOPT (w/o function evaluations) = 0.061
Total CPU secs in NLP function evaluations = 0.010
EXIT: Optimal Solution Found.
solver : t_proc (avg) t_wall (avg) n_eval
nlp_f | 425.00us ( 11.81us) 407.26us ( 11.31us) 36
nlp_g | 5.59ms (114.14us) 5.58ms (113.88us) 49
nlp_grad | 264.00us (264.00us) 263.05us (263.05us) 1
nlp_grad_f | 325.00us ( 23.21us) 325.10us ( 23.22us) 14
nlp_hess_l | 614.00us ( 51.17us) 617.11us ( 51.43us) 12
nlp_jac_g | 2.82ms (201.79us) 2.81ms (201.06us) 14
total | 78.41ms ( 78.41ms) 77.65ms ( 77.65ms) 1
[11]:
x, u, t, a = post.get_data()
print(f"Terminal time, state : {t[-1][0]:.4f} vs 1000s (Exact), {x[-1]}")
Terminal time, state : 1000.0000 vs 1000s (Exact), [1.]
4.1.3.4. Solve again with Legendre-Gauss-Lobatto (LGL) roots#
[12]:
mpo, post = mp.solve(ocp, n_segments=5, poly_orders=50, scheme="LGL", plot=True)
Total number of variables............................: 501
variables with only lower bounds: 0
variables with lower and upper bounds: 0
variables with only upper bounds: 0
Total number of equality constraints.................: 252
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
Number of Iterations....: 12
(scaled) (unscaled)
Objective...............: 1.1502075893651909e+00 1.1502075893651909e+00
Dual infeasibility......: 1.0234087882698972e-13 1.0234087882698972e-13
Constraint violation....: 4.7414571594509346e-13 1.3997691894473974e-12
Complementarity.........: 0.0000000000000000e+00 0.0000000000000000e+00
Overall NLP error.......: 4.7414571594509346e-13 1.3997691894473974e-12
Number of objective function evaluations = 36
Number of objective gradient evaluations = 13
Number of equality constraint evaluations = 49
Number of inequality constraint evaluations = 0
Number of equality constraint Jacobian evaluations = 13
Number of inequality constraint Jacobian evaluations = 0
Number of Lagrangian Hessian evaluations = 12
Total CPU secs in IPOPT (w/o function evaluations) = 0.064
Total CPU secs in NLP function evaluations = 0.011
EXIT: Optimal Solution Found.
solver : t_proc (avg) t_wall (avg) n_eval
nlp_f | 423.00us ( 11.75us) 431.21us ( 11.98us) 36
nlp_g | 6.12ms (124.82us) 6.12ms (124.86us) 49
nlp_grad | 264.00us (264.00us) 263.12us (263.12us) 1
nlp_grad_f | 326.00us ( 23.29us) 327.97us ( 23.43us) 14
nlp_hess_l | 611.00us ( 50.92us) 610.94us ( 50.91us) 12
nlp_jac_g | 2.82ms (201.07us) 2.82ms (201.54us) 14
total | 77.49ms ( 77.49ms) 76.93ms ( 76.93ms) 1
[13]:
x, u, t, a = post.get_data()
print(f"Terminal time, state : {t[-1][0]:.4f} vs 1000s (Exact), {x[-1]}")
Terminal time, state : 1000.0000 vs 1000s (Exact), [1.]
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