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91 Rayleigh Unconstrained
Lecture Notes for ECE/MAE 7360, Robust and Optimal Control (part 2) Fall 2003, Jinsong Liang, Nov. 20, 2003 Utah State University at Logan
91.1 Problem Formulation
Find u over t in [0; t_F ] to minimize
subject to:
| = −x1+(1.4−0.14*x22)*x2+4*u |
Reference: [22]
91.2 Problem setup
toms t
p = tomPhase('p', t, 0, 2.5, 50);
setPhase(p);
tomStates x1 x2
tomControls u
% Initial guess
x0 = {icollocate({x1 == -5; x2 == -5})
collocate(u == 0)};
% Box constraints
cbox = {-100 <= icollocate(x1) <= 100
-100 <= icollocate(x2) <= 100
-100 <= collocate(u) <= 100};
% Boundary constraints
cbnd = initial({x1 == -5; x2 == -5});
% ODEs and path constraints
ceq = collocate({dot(x1) == x2
dot(x2) == -x1+(1.4-0.14*x2.^2).*x2+4*u});
% Objective
objective = integrate(x1.^2+u.^2);
91.3 Solve the problem
options = struct;
options.name = 'Rayleigh Unconstrained';
solution = ezsolve(objective, {cbox, cbnd, ceq}, x0, options);
t = subs(collocate(t),solution);
x1 = subs(collocate(x1),solution);
x2 = subs(collocate(x2),solution);
u = subs(collocate(u),solution);
Problem type appears to be: qpcon
Starting numeric solver
===== * * * =================================================================== * * *
TOMLAB - Tomlab Optimization Inc. Development license 999001. Valid to 2011-02-05
=====================================================================================
Problem: --- 1: Rayleigh Unconstrained f_k 29.376079656023496000
sum(|constr|) 0.000000003740997838
f(x_k) + sum(|constr|) 29.376079659764493000
f(x_0) 62.499999999999986000
Solver: snopt. EXIT=0. INFORM=1.
SNOPT 7.2-5 NLP code
Optimality conditions satisfied
FuncEv 1 ConstrEv 47 ConJacEv 47 Iter 36 MinorIter 144
CPU time: 0.250000 sec. Elapsed time: 0.266000 sec.
91.4 Plot result
subplot(2,1,1)
plot(t,x1,'*-',t,x2,'*-');
legend('x1','x2');
title('Rayleigh Unconstrained state variables');
subplot(2,1,2)
plot(t,u,'+-');
legend('u');
title('Rayleigh Unconstrained System control');
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