PROPT Turbo Generator: Difference between revisions
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<math> J = \int_0^{t} (alpha_1*( (x_1-x_1^{s})^2 + (x_4-x_4^{s})^2) + </math> | <math> J = \int_0^{t} (alpha_1*( (x_1-x_1^{s})^2 + (x_4-x_4^{s})^2) + </math> | ||
<math> alpha_2*x_2^{2} + alpha_3*(x_3-x_3^{s})^{2} + </math> | <math> alpha_2*x_2^{2} + alpha_3*(x_3-x_3^{s})^{2} + </math> | ||
<math> beta_1*(u_1-u_1^{s})^{2} + beta_2*(u_2-u_2^{s})^{2} ) \mathrm{d}t </math> | <math> beta_1*(u_1-u_1^{s})^{2} + beta_2*(u_2-u_2^{s})^{2} ) \mathrm{d}t </math> | ||
subject to: | subject to: | ||
<math> \frac{dx_1}{dt} = x_2*x_4 </math> | <math> \frac{dx_1}{dt} = x_2*x_4 </math> | ||
<math> \frac{dx_2}{dt} = \frac{1}{M}*(u_1-s_4*x_1*x_4-s_5*x_1*x_3-kappa_d*x_2</math> | <math> \frac{dx_2}{dt} = \frac{1}{M}*(u_1-s_4*x_1*x_4-s_5*x_1*x_3-kappa_d*x_2</math> | ||
<math> \frac{dx_3}{dt} = u_2-A*x_3+c*x_4 </math> | <math> \frac{dx_3}{dt} = u_2-A*x_3+c*x_4 </math> | ||
<math> \frac{dx_4}{dt} = -x_1*x_2 </math> | <math> \frac{dx_4}{dt} = -x_1*x_2 </math> | ||
The initial condition are: | The initial condition are: | ||
<math> x(0) = [x_1^{s} \ x_2^{s} \ x_3^{s} \ x_4^{s}] </math> | <math> x(0) = [x_1^{s} \ x_2^{s} \ x_3^{s} \ x_4^{s}] </math> | ||
<math> x_{1:4}^{s} = [0.60295 \ 0.0 \ 1.87243 \ 0.79778] </math> | <math> x_{1:4}^{s} = [0.60295 \ 0.0 \ 1.87243 \ 0.79778] </math> | ||
<math> alpha = [2.5 \ 1.0 \ 0.1] </math> | <math> alpha = [2.5 \ 1.0 \ 0.1] </math> | ||
<math> beta = [1.0 \ 1.0] </math> | <math> beta = [1.0 \ 1.0] </math> | ||
<math> M = 0.04225 </math> | <math> M = 0.04225 </math> | ||
<math> s_{4:5} = [0.0 \ 0.0] </math> | <math> s_{4:5} = [0.0 \ 0.0] </math> | ||
<math> c = 0 </math> | <math> c = 0 </math> | ||
<math> A = 0.17 </math> | <math> A = 0.17 </math> | ||
<math> u_{1:2}^{s} = [0.80 \ 0.73962] </math> | <math> u_{1:2}^{s} = [0.80 \ 0.73962] </math> | ||
<math> kappa_d = 0.02535 </math> | <math> kappa_d = 0.02535 </math> | ||
<source lang="matlab"> | <source lang="matlab"> | ||
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[[File:turboGenerator_01.png]] | [[File:turboGenerator_01.png]] | ||
[[Category:PROPT Examples]] |
Latest revision as of 05:26, 14 February 2012
This page is part of the PROPT Manual. See PROPT Manual. |
OCCAL - A Mixed symbolic-numeric Optimal Control CALcluator
Section 4 Example 1
Problem Formulation
Find u over t in [0; t ] to minimize
subject to:
The initial condition are:
% Copyright (c) 2007-2008 by Tomlab Optimization Inc.
Problem setup
toms t
p = tomPhase('p', t, 0, 20, 30);
setPhase(p);
tomStates x1 x2 x3 x4
tomControls u1 u2
% Initial guess
x0i = [0.60295;0;1.87243;0.79778];
x0 = {icollocate({x1 == x0i(1); x2 == x0i(2)
x3 == x0i(3); x4 == x0i(4)})
collocate({u1 == 0; u2 == 0})};
% Boundary constraints
cbnd = initial({x1 == x0i(1); x2 == x0i(2)
x3 == x0i(3); x4 == x0i(4)});
% ODEs and path constraints
u1s = 0.80; u2s = 0.73962;
A = 0.17; c = 0;
s4 = 0; s5 = 0;
M = 0.04225; alpha1 = 2.5;
alpha2 = 1.0; alpha3 = 0.1;
beta1 = 1.0; beta2 = 1.0;
kappa_d = 0.02535;
ceq = collocate({dot(x1) == x2.*x4
dot(x2) == 1/M.*(u1-s4*x1.*x4-s5*x1.*x3-kappa_d*x2)
dot(x3) == u2-A*x3+c*x4; dot(x4) == -x1.*x2});
% Objective
objective = integrate(alpha1*( (x1-x0i(1)).^2 + ...
(x4-x0i(4)).^2) + alpha2*x2.^2 + alpha3*(x3-x0i(3)).^2 + ...
beta1*(u1-u1s).^2 + beta2*(u2-u2s).^2);
Solve the problem
options = struct;
options.name = 'Turbo Generator';
solution = ezsolve(objective, {cbnd, ceq}, x0, options);
t = subs(collocate(t),solution);
x1 = subs(collocate(x1),solution);
x2 = subs(collocate(x2),solution);
x3 = subs(collocate(x3),solution);
x4 = subs(collocate(x4),solution);
u1 = subs(collocate(u1),solution);
u2 = subs(collocate(u2),solution);
Problem type appears to be: qpcon Time for symbolic processing: 0.17864 seconds Starting numeric solver ===== * * * =================================================================== * * * TOMLAB - TOMLAB Development license 999007. Valid to 2011-12-31 ===================================================================================== Problem: --- 1: Turbo Generator f_k 15.019841547670794000 sum(|constr|) 0.000000000046608960 f(x_k) + sum(|constr|) 15.019841547717403000 f(x_0) -57.012069754799789000 Solver: snopt. EXIT=0. INFORM=1. SNOPT 7.2-5 NLP code Optimality conditions satisfied FuncEv 1 ConstrEv 35 ConJacEv 35 Iter 23 MinorIter 119 CPU time: 0.046800 sec. Elapsed time: 0.048000 sec.
Plot result
subplot(2,1,1)
plot(t,x1,'*-',t,x2,'*-',t,x3,'*-',t,x4,'*-');
legend('x1','x2','x3','x4');
title('Turbo Generator state variables');
subplot(2,1,2)
plot(t,u1,'+-',t,u2,'+-');
legend('u1','u2');
title('Turbo Generator control');