PROPT LQR Problem

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

Problem: LQR: RIOTS 95 Manual

Problem Description

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

$J=\int _{0}^{1}(0.625*x^{2}+0.5*x*u+0.5*u^{2})\mathrm {d} t$

subject to:

${\frac {dx}{dt}}={\frac {1}{2}}*x+u$

$x(0)=1$

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

Problem setup

toms t
p = tomPhase('p', t, 0, 1, 20);
setPhase(p);

tomStates x
tomControls u

% Initial guess
x0 = icollocate(x == 1-t);

% ODEs and constraints
ceq = {collocate(dot(x) == 0.5*x+u)
    initial(x == 1)};

% Objective
objective = integrate(0.625*x.^2+0.5*x.*u+0.5*u.^2);

Solve the problem

options = struct;
options.name = 'LQR Problem';
solution = ezsolve(objective, ceq, x0, options);
t = subs(collocate(t),solution);
x = subs(collocate(x),solution);
u = subs(collocate(u),solution);

Problem type appears to be: qp
Time for symbolic processing: 0.053812 seconds
Starting numeric solver
===== * * * =================================================================== * * *
TOMLAB - TOMLAB Development license  999007. Valid to 2011-12-31
=====================================================================================
Problem:  1: LQR Problem                        f_k       0.380797077977355130
                                       sum(|constr|)      0.000000000050276185
                              f(x_k) + sum(|constr|)      0.380797078027631300
                                              f(x_0)      0.000000000000000000

Solver: CPLEX.  EXIT=0.  INFORM=1.
CPLEX Barrier QP solver
Optimal solution found

FuncEv    3 GradEv    3 ConstrEv    3 Iter    3 
Elapsed time: 0.003000 sec. 

Plot result

subplot(2,1,1)
plot(t,x,'*-');
legend('x');
title('LQR Problem state variable');

subplot(2,1,2)
plot(t,u,'+-');
legend('u');
title('LQR Problem control');