Optimal Control Problems

ocp. A object of class ode is a structure that contains a all information of a ordinary differential equation. This have a dynamic equation, initial condition, time span, etc. $$ \min_{u \in \mathcal{U}} \bigg[ \psi(x(T)) + \int_0^T L(t,x(t),u(t)) dt \bigg] \\ \text{subject to:} \\ \dot{x} = f(t,x,u)$$

Class:: ocp

A object of class ode is a structure that contains a all information of a ordinary differential equation. This have a dynamic equation, initial condition, time span, etc. $$ \min_{u \in \mathcal{U}} \bigg[ \psi(x(T)) + \int_0^T L(t,x(t),u(t)) dt \bigg] \\ \text{subject to:} \\ \dot{x} = f(t,x,u)$$

Method::ocp of Class::ocp

Index⤴

Syntax

iocp = ocp(idynamics,PathCostFcn,FinalCostFcn)

Interface

INPUTS
Name	Class	Description
idynamics	{ode,pde1d,pde2d,pdefem}	This is a object { `ode` ,`pde1d`,`pde2d`,`pdefem`} that represent the dynamic. Inside this object we have a time span, dynamics equation, initial condition, etc.
PathCostFcn	casadi function	it is a casadi function that $ L:\mathbb{R}^+ \times \mathcal{X} \times \mathcal{U} \rightarrow \mathbb{R}$
FinalCostFcn	casadi function	it is a casadi function that $ \psi:\mathcal{X} \rightarrow \mathbb{R}$

OUTPUTS
Name	Class	Description
iocp	ode	ocp object that contain all information of optimal control problem.

Example

clear all;close all
import casadi.*

Xs = SX.sym('x',2,1);
Us = SX.sym('u',2,1);
ts = SX.sym('t');
%
A = [ -2 +1;
      +1 -2];

B = [1 0;
     0 1];

EvolutionFcn = Function('f',{ts,Xs,Us},{ A*Xs + B*Us });
%
tspan = linspace(0,2,10);
iode = ode(EvolutionFcn,Xs,Us,tspan);
SetIntegrator(iode,'RK4')
iode.InitialCondition = [1;2];

epsilon = 1e4;
PathCost  = Function('L'  ,{ts,Xs,Us},{ Us'*Us           });
FinalCost = Function('Psi',{Xs}      ,{ epsilon*(Xs'*Xs) });

iocp = ocp(iode,PathCost,FinalCost)

iocp = 

  ocp with properties:

      DynamicSystem: [1x1 ode]
            CostFcn: [1x1 CostFcn]
       VariableTime: 0
        constraints: [1x1 constraints]
        TargetState: []
        Hamiltonian: [0x0 casadi.Function]
      AdjointStruct: [1x1 AdjointStruct]
    ControlGradient: [1x1 casadi.Function]

Method::ArmijoGradient of Class::ocp

Index⤴

Syntax

[uopt,xopt] = ArmijoGradient(iocp,uguess)
[uopt,xopt] = ArmijoGradient(iocp,uguess,'MaxIter',MaxIter)

Interface

INPUTS
Name	Class	Description
iocp	ocp	Optimal Control Problems object
uguess	double

OPTIONAL INPUTS
Name	Class	Description	Default
MaxIter			50

OUTPUTS
Name	Class	Description
uopt	double	Optimal Control
xopt	double	Optimal State

Example

import casadi.*

A = [-2  1;
      1 -2];
B = [1;0];
%
tspan = linspace(0,1,7);
idyn = linearode(A,B,tspan);
idyn.InitialCondition = [1;2];

ts = idyn.ts;
Xs = idyn.State.sym;
Us = idyn.Control.sym;
%
epsilon = 1e4;
PathCost  = Function('L'  ,{ts,Xs,Us},{ Us'*Us           });
FinalCost = Function('Psi',{Xs}      ,{ epsilon*(Xs'*Xs) });

iocp = ocp(idyn,PathCost,FinalCost);

uguess = tspan;
[uopt,xopt] = ArmijoGradient(iocp,uguess);

===================================================================================================
|   iter   |   norm(dJ/du)  |  abs(Jc-Ja)/Jc  |  LengthStep  |       J       |   Distance2Target  |
===================================================================================================
|   008    |    2.017e+03   |    1.780e-01    |   2.50e-04   |  1.13750e+03  |       NaN          |
|   012    |    1.651e+03   |    2.549e-01    |   2.50e-04   |  6.18609e+02  |       NaN          |
|   016    |    2.011e+03   |    1.540e-01    |   1.00e-03   |  4.12580e+02  |       NaN          |
|   020    |    3.068e+03   |    4.962e-02    |   1.00e-03   |  3.26251e+02  |       NaN          |
|   024    |    8.769e+02   |    8.368e-02    |   1.00e-03   |  2.07381e+02  |       NaN          |
|   028    |    8.176e+02   |    1.259e-01    |   1.00e-03   |  1.28015e+02  |       NaN          |
|   032    |    1.172e+03   |    7.778e-02    |   1.00e-03   |  1.02105e+02  |       NaN          |
|   036    |    3.997e+02   |    1.316e-01    |   2.50e-04   |  8.07760e+01  |       NaN          |
|   040    |    4.187e+02   |    2.306e-02    |   1.00e-03   |  6.74948e+01  |       NaN          |
===================================================================================================
|   iter   |   norm(dJ/du)  |  abs(Jc-Ja)/Jc  |  LengthStep  |       J       |   Distance2Target  |
===================================================================================================
|   048    |    1.864e+02   |    3.809e-02    |   2.50e-04   |  5.13924e+01  |       NaN          |
|   052    |    1.509e+02   |    3.159e-02    |   2.50e-04   |  4.71590e+01  |       NaN          |
|   056    |    1.803e+02   |    1.423e-02    |   1.00e-03   |  4.55183e+01  |       NaN          |
|   060    |    8.734e+01   |    8.959e-03    |   1.00e-03   |  4.47686e+01  |       NaN          |
|   064    |    7.475e+01   |    1.117e-02    |   2.50e-04   |  4.39292e+01  |       NaN          |
|   068    |    5.289e+01   |    3.286e-04    |   2.50e-04   |  4.35600e+01  |       NaN          |
|   072    |    9.023e+01   |    2.091e-03    |   1.00e-03   |  4.32197e+01  |       NaN          |
|   076    |    3.367e+01   |    1.155e-03    |   1.00e-03   |  4.31266e+01  |       NaN          |
|   080    |    2.741e+01   |    3.512e-05    |   2.50e-04   |  4.30802e+01  |       NaN          |
===================================================================================================
|   iter   |   norm(dJ/du)  |  abs(Jc-Ja)/Jc  |  LengthStep  |       J       |   Distance2Target  |
===================================================================================================

    Mininum Length Step have been achieve !! 

uopt

uopt =

   -7.9370   -8.1077   -7.8352   -6.7968   -4.4915   -0.1483    7.4126

xopt

xopt =

    1.0000   -0.0772   -0.9118   -1.4622   -1.6301   -1.2395   -0.0019
    2.0000    1.4904    1.0038    0.5701    0.2238    0.0129    0.0094

Method::IpoptSolver of Class::ocp

Index⤴

Syntax

[uopt,xopt] = IpoptSolver(iocp,uguess)
[uopt,xopt] = IpoptSolver(iocp,uguess,'Integrator',Integrator)

Interface

INPUTS
Name	Class	Description
iocp	ocp	Optimal Control Problems object
uguess	double
Integrator	String	Integrator define the numerical schema in time. This variable must be: `BackwardEuler` , `ForwardEuler` , `rk4` , `rk5`, `CrankNicolson` , `rk8` , `SemiLinearBackwardEuler`

OUTPUTS
Name	Class	Description
uopt	double	Optimal Control
xopt	double	Optimal State

Example

import casadi.*

A = [-2  1;
      1 -2];
B = [1;0];
%
tspan = linspace(0,1,7);
idyn = linearode(A,B,tspan);
idyn.InitialCondition = [1;2];

ts = idyn.ts;
Xs = idyn.State.sym;
Us = idyn.Control.sym;
%
epsilon = 1e4;
PathCost  = Function('L'  ,{ts,Xs,Us},{ Us'*Us           });
FinalCost = Function('Psi',{Xs}      ,{ epsilon*(Xs'*Xs) });

iocp = ocp(idyn,PathCost,FinalCost);

uguess = tspan;
[uopt,xopt] = IpoptSolver(iocp,uguess);

This is Ipopt version 3.12.3, running with linear solver mumps.
NOTE: Other linear solvers might be more efficient (see Ipopt documentation).

Number of nonzeros in equality constraint Jacobian...:       62
Number of nonzeros in inequality constraint Jacobian.:        0
Number of nonzeros in Lagrangian Hessian.............:        9

Total number of variables............................:       21
                     variables with only lower bounds:        0
                variables with lower and upper bounds:        0
                     variables with only upper bounds:        0
Total number of equality constraints.................:       14
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  2.0000338e+04 1.00e+00 1.33e+01  -1.0 0.00e+00    -  0.00e+00 0.00e+00   0
   1  2.1420978e+01 3.33e-16 1.66e-14  -1.0 5.90e+00    -  1.00e+00 1.00e+00f  1

Number of Iterations....: 1

                                   (scaled)                 (unscaled)
Objective...............:   1.0710489227567338e-01    2.1420978455134676e+01
Dual infeasibility......:   1.6577017536434369e-14    3.3154035072868737e-12
Constraint violation....:   3.3306690738754696e-16    3.3306690738754696e-16
Complementarity.........:   0.0000000000000000e+00    0.0000000000000000e+00
Overall NLP error.......:   1.6577017536434369e-14    3.3154035072868737e-12


Number of objective function evaluations             = 2
Number of objective gradient evaluations             = 2
Number of equality constraint evaluations            = 2
Number of inequality constraint evaluations          = 0
Number of equality constraint Jacobian evaluations   = 2
Number of inequality constraint Jacobian evaluations = 0
Number of Lagrangian Hessian evaluations             = 1
Total CPU secs in IPOPT (w/o function evaluations)   =      0.006
Total CPU secs in NLP function evaluations           =      0.001

EXIT: Optimal Solution Found.
               t_proc [s]   t_wall [s]    n_eval
       nlp_f      3.3e-05      3.4e-05         2
       nlp_g      5.8e-05      5.8e-05         2
  nlp_grad_f     0.000101       0.0001         3
  nlp_hess_l      0.00807      6.2e-05         1
   nlp_jac_g     0.000236      0.00024         3
      solver       0.0208      0.00761         1
Elapsed time is 0.022389 seconds.

uopt

uopt =

   -5.5768   -5.6623   -5.5679   -4.7070   -2.4409    2.3186    5.4928

xopt

xopt =

    1.0000    0.1623   -0.5028   -0.9714   -1.1358   -0.7915   -0.0010
    2.0000    1.5116    1.0554    0.6485    0.3127    0.0857    0.0046

Method::SteptestGradientDescent of Class::ocp

Index⤴

Syntax

[uopt,xopt] = SteptestGradientDescent(iocp,uguess)
[uopt,xopt] = SteptestGradientDescent(iocp,uguess,'MaxIter',MaxIter)
[uopt,xopt] = SteptestGradientDescent(iocp,uguess,'InitialLengthStep',InitialLengthStep)

Interface

INPUTS
Name	Class	Description
iocp	ocp	Optimal Control Problems object
uguess	double

OUTPUTS
Name	Class	Description
uopt	double	Optimal Control
xopt	double	Optimal State

Example

import casadi.*

A = [-2  1;
      1 -2];
B = [1;0];
%
tspan = linspace(0,1,7);
idyn = linearode(A,B,tspan);
idyn.InitialCondition = [1;2];

ts = idyn.ts;
Xs = idyn.State.sym;
Us = idyn.Control.sym;
%
epsilon = 1e4;
PathCost  = Function('L'  ,{ts,Xs,Us},{ Us'*Us           });
FinalCost = Function('Psi',{Xs}      ,{ epsilon*(Xs'*Xs) });

iocp = ocp(idyn,PathCost,FinalCost);

uguess = tspan;
[uopt,xopt] = SteptestGradientDescent(iocp,uguess);

iteration: 005 | error: 3.672e+03 | LengthStep: 8.705e-04 | J: 1.234e+03 | Distance2Target: NaN 
iteration: 010 | error: 2.642e+02 | LengthStep: 2.640e-04 | J: 5.972e+01 | Distance2Target: NaN 
iteration: 015 | error: 1.706e+02 | LengthStep: 1.358e-04 | J: 5.118e+01 | Distance2Target: NaN 
iteration: 020 | error: 3.506e+01 | LengthStep: 1.806e-04 | J: 4.304e+01 | Distance2Target: NaN 
iteration: 025 | error: 2.688e+01 | LengthStep: 2.964e-06 | J: 4.304e+01 | Distance2Target: NaN 
iteration: 030 | error: 2.658e+01 | LengthStep: 3.377e-07 | J: 4.304e+01 | Distance2Target: NaN 
iteration: 035 | error: 2.655e+01 | LengthStep: 4.453e-08 | J: 4.304e+01 | Distance2Target: NaN 

    Mininum Length Step have been achive !! 

uopt

uopt =

   -7.9333   -8.1012   -7.8269   -6.7884   -4.4855   -0.1486    7.3999

xopt

xopt =

    1.0000   -0.0763   -0.9101   -1.4598   -1.6275   -1.2375   -0.0019
    2.0000    1.4905    1.0041    0.5706    0.2245    0.0137    0.0100

Method::ClassicalGradient of Class::ocp

Index⤴

Syntax

[uopt,xopt] = ArmijoGradient(iocp,uguess)
[uopt,xopt] = ArmijoGradient(iocp,uguess,'MaxIter',MaxIter)

Interface

INPUTS
Name	Class	Description
iocp	ocp	Optimal Control Problems object
uguess	double

OUTPUTS
Name	Class	Description
uopt	double	Optimal Control
xopt	double	Optimal State

Example

import casadi.*

A = [-2  1;
      1 -2];
B = [1;0];
%
tspan = linspace(0,1,7);
idyn = linearode(A,B,tspan);
idyn.InitialCondition = [1;2];

ts = idyn.ts;
Xs = idyn.State.sym;
Us = idyn.Control.sym;
%
epsilon = 1e4;
PathCost  = Function('L'  ,{ts,Xs,Us},{ Us'*Us           });
FinalCost = Function('Psi',{Xs}      ,{ epsilon*(Xs'*Xs) });

iocp = ocp(idyn,PathCost,FinalCost);

uguess = tspan;
[uopt,xopt] = ClassicalGradient(iocp,uguess);

iteration: 005 | error: 2.525e+03 | LengthStep: 1.000e-04 | J: 1.981e+03 | Distance2Target: NaN 
iteration: 010 | error: 2.253e+03 | LengthStep: 1.000e-04 | J: 1.688e+03 | Distance2Target: NaN 
iteration: 015 | error: 2.086e+03 | LengthStep: 1.000e-04 | J: 1.453e+03 | Distance2Target: NaN 
iteration: 020 | error: 1.933e+03 | LengthStep: 1.000e-04 | J: 1.252e+03 | Distance2Target: NaN 
iteration: 025 | error: 1.790e+03 | LengthStep: 1.000e-04 | J: 1.079e+03 | Distance2Target: NaN 
iteration: 030 | error: 1.658e+03 | LengthStep: 1.000e-04 | J: 9.315e+02 | Distance2Target: NaN 
iteration: 035 | error: 1.536e+03 | LengthStep: 1.000e-04 | J: 8.046e+02 | Distance2Target: NaN 
iteration: 040 | error: 1.423e+03 | LengthStep: 1.000e-04 | J: 6.957e+02 | Distance2Target: NaN 
iteration: 045 | error: 1.318e+03 | LengthStep: 1.000e-04 | J: 6.024e+02 | Distance2Target: NaN 
iteration: 050 | error: 1.221e+03 | LengthStep: 1.000e-04 | J: 5.223e+02 | Distance2Target: NaN 

uopt

uopt = 

[[-4.66594, -4.75716, -4.63036, -4.11885, -2.96299, -0.762059, 3.09807]]

xopt

xopt = 


[[1, 0.354971, -0.164069, -0.534352, -0.705436, -0.587689, -0.0341006], 
 [2, 1.54437, 1.13777, 0.786533, 0.501721, 0.302829, 0.222859]]

Method::ConjugateGradient of Class::ocp

Index⤴

Syntax

[uopt,xopt] = ArmijoGradient(iocp,uguess)
[uopt,xopt] = ArmijoGradient(iocp,uguess,'MaxIter',MaxIter)

Interface

INPUTS
Name	Class	Description
iocp	ocp	Optimal Control Problems object
uguess	double

OUTPUTS
Name	Class	Description
uopt	double	Optimal Control
xopt	double	Optimal State

Example

import casadi.*

A = [-2  1;
      1 -2];
B = [1;0];
%
tspan = linspace(0,1,7);
idyn = linearode(A,B,tspan);
idyn.InitialCondition = [1;2];

ts = idyn.ts;
Xs = idyn.State.sym;
Us = idyn.Control.sym;
%
epsilon = 1e4;
PathCost  = Function('L'  ,{ts,Xs,Us},{ Us'*Us           });
FinalCost = Function('Psi',{Xs}      ,{ epsilon*(Xs'*Xs) });

iocp = ocp(idyn,PathCost,FinalCost);

uguess = tspan;
[uopt,xopt] = ConjugateGradient(iocp,uguess);

iteration: 005 | error: 2.473e+03 | norm(dJ): 2.473e+03 | LengthStep: +2.762e-07 | beta: 1.093e+00 | J: 1.9944e+03 | Distance2Target: NaN 
iteration: 010 | error: 2.997e+03 | norm(dJ): 2.997e+03 | LengthStep: +1.976e-07 | beta: 1.073e+00 | J: 1.9937e+03 | Distance2Target: NaN 
iteration: 015 | error: 3.521e+03 | norm(dJ): 3.521e+03 | LengthStep: +1.609e-07 | beta: 1.063e+00 | J: 1.9930e+03 | Distance2Target: NaN 
iteration: 020 | error: 4.065e+03 | norm(dJ): 4.065e+03 | LengthStep: +1.405e-07 | beta: 1.057e+00 | J: 1.9922e+03 | Distance2Target: NaN 
iteration: 025 | error: 4.638e+03 | norm(dJ): 4.638e+03 | LengthStep: +1.278e-07 | beta: 1.053e+00 | J: 1.9913e+03 | Distance2Target: NaN 
iteration: 030 | error: 5.248e+03 | norm(dJ): 5.248e+03 | LengthStep: +1.195e-07 | beta: 1.049e+00 | J: 1.9902e+03 | Distance2Target: NaN 
iteration: 035 | error: 5.897e+03 | norm(dJ): 5.897e+03 | LengthStep: +1.138e-07 | beta: 1.047e+00 | J: 1.9889e+03 | Distance2Target: NaN 
iteration: 040 | error: 6.586e+03 | norm(dJ): 6.586e+03 | LengthStep: +1.098e-07 | beta: 1.044e+00 | J: 1.9874e+03 | Distance2Target: NaN 
iteration: 045 | error: 7.312e+03 | norm(dJ): 7.312e+03 | LengthStep: +1.070e-07 | beta: 1.042e+00 | J: 1.9855e+03 | Distance2Target: NaN 
iteration: 050 | error: 8.066e+03 | norm(dJ): 8.066e+03 | LengthStep: +1.049e-07 | beta: 1.039e+00 | J: 1.9833e+03 | Distance2Target: NaN 

uopt

uopt =

   -2.6217   -2.6272   -2.5285   -2.2421   -1.6369   -0.5107    1.4475

xopt

xopt =

    1.0000    0.6188    0.3006    0.0606   -0.0738   -0.0560    0.1892
    2.0000    1.5773    1.2206    0.9230    0.6830    0.5053    0.4026

Optimal Control Problems

Table of Contents

Methods

Class:: ocp

Method::ocp of Class::ocp

Syntax

Interface

Example

Method::ArmijoGradient of Class::ocp

Syntax

Interface

Example

Method::IpoptSolver of Class::ocp

Syntax

Interface

Example

Method::SteptestGradientDescent of Class::ocp

Syntax

Interface

Example

Method::ClassicalGradient of Class::ocp

Syntax

Interface

Example

Method::ConjugateGradient of Class::ocp

Syntax

Interface

Example