PROPT Singular Control 5: Difference between revisions

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<math> J = x_4(t_F) </math>
<math> J = x_4(t_F) </math>


subject to:
subject to:


<math> \frac{dx_1}{dt} = x_2 </math>
<math> \frac{dx_1}{dt} = x_2 </math>
<math> \frac{dx_2}{dt} = -x_3*u + 16*x_5 - 8 </math>
<math> \frac{dx_2}{dt} = -x_3*u + 16*x_5 - 8 </math>
<math> \frac{dx_3}{dt} = u </math>
<math> \frac{dx_3}{dt} = u </math>
<math> \frac{dx_4}{dt} = x_1^2 + x_2^2 + 0.0005*(x_2+16*x_5-8-0.1*x_3*u^2)^2 </math>
<math> \frac{dx_4}{dt} = x_1^2 + x_2^2 + 0.0005*(x_2+16*x_5-8-0.1*x_3*u^2)^2 </math>
<math> \frac{dx_5}{dt} = 1 </math>
<math> \frac{dx_5}{dt} = 1 </math>


The initial condition are:
The initial condition are:


<math> x(0) = [0 \ -1 \ -sqrt(5) \ 0 \ 0] </math>
<math> x(0) = [0 \ -1 \ -sqrt(5) \ 0 \ 0] </math>
<math> -4 <= u <= 10 </math>
<math> -4 <= u <= 10 </math>


The state x4 is implemented as a cost directly. x4 in the implementation is x5. u has a low limit of 9 in the code.
The state x4 is implemented as a cost directly. x4 in the implementation is x5. u has a low limit of 9 in the code.

Revision as of 08:12, 9 November 2011

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This page is part of the PROPT Manual. See PROPT Manual.

ITERATIVE DYNAMIC PROGRAMMING, REIN LUUS

10.3 Yeo's singular control problem

CHAPMAN & HALL/CRC Monographs and Surveys in Pure and Applied Mathematics

Problem Formulation

Find u over t in [0; 1 ] to minimize:


subject to:


The initial condition are:


The state x4 is implemented as a cost directly. x4 in the implementation is x5. u has a low limit of 9 in the code.

% Copyright (c) 2007-2008 by Tomlab Optimization Inc.

Problem setup

toms t
p = tomPhase('p', t, 0, 1, 80);
setPhase(p)

tomStates x1 x2 x3 x4
tomControls u

% Initial guess
x0 = {icollocate({x1 == 0; x2 == -1
    x3 == -sqrt(5); x4 == 0})
    collocate(u == 3)};

% Box constraints
cbox = {0 <= collocate(u) <= 10};

% Boundary constraints
cbnd = initial({x1 == 0; x2 == -1
    x3 == -sqrt(5); x4 == 0});

% ODEs and path constraints
ceq = collocate({dot(x1) == x2
    dot(x2) == -x3.*u + 16*x4 - 8
    dot(x3) == u; dot(x4) == 1});

% Objective
objective = integrate(x1.^2 + x2.^2 + ...
    0.0005*(x2+16*x4-8-0.1*x3.*u.^2).^2);

Solve the problem

options = struct;
options.name = 'Singular Control 5';
solution = ezsolve(objective, {cbox, cbnd, ceq}, x0, options);
t = subs(collocate(t),solution);
u = subs(collocate(u),solution);
Problem type appears to be: con
Time for symbolic processing: 0.16034 seconds
Starting numeric solver
===== * * * =================================================================== * * *
TOMLAB - TOMLAB Development license  999007. Valid to 2011-12-31
=====================================================================================
Problem: ---  1: Singular Control 5             f_k       0.119318613152343460
                                       sum(|constr|)      0.000000483936371958
                              f(x_k) + sum(|constr|)      0.119319097088715420
                                              f(x_0)      1.024412849382206200

Solver: snopt.  EXIT=0.  INFORM=1.
SNOPT 7.2-5 NLP code
Optimality conditions satisfied

FuncEv  374 GradEv  372 ConstrEv  372 ConJacEv  372 Iter  345 MinorIter  938
CPU time: 4.180827 sec. Elapsed time: 4.185000 sec. 

Plot result

figure(1)
plot(t,u,'+-');
legend('u');
title('Singular Control 5 control');

SingularControl5 01.png