PROPT Nagurka Problem

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

ITERATIVE DYNAMIC PROGRAMMING, REIN LUUS

6.4 Further example

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

n'th-order linear time-invariant system.

Problem description

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

$J=\int _{0}^{1}x'*x+u'*u\mathrm {d} t+10*x_{1}(t_{F})^{2}$

subject to:

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

A = [0 1 0 ... 0
     0 0 1 ... 0
     ... ... ...
     0 0 0 ... 1
     1 -2 3 ... (-1)^(n+1)*n]

The initial condition are:

$x(0)=[1\ 2\ ...\ n],$

$-\infty <=u(1:n)<=\infty .$

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

Problem setup

toms t
n  = 6;
t_F = 1;

p = tomPhase('p', t, 0, t_F, 25);
setPhase(p);

x = tomState('x', n, 1);
u = tomState('u', n, 1);

nvec = (1:n);
A = [sparse(n-1,1), speye(n-1); ...
    sparse(nvec.*(-1).^(nvec+1))];

% Initial guess
guess = icollocate(x == nvec');

% Initial conditions
cinit = (initial(x) == nvec');

% ODEs and path constraints
ceq = collocate(dot(x) == A*x+u);

% Objective
objective = 10*final(x(1))^2 + integrate(x'*x + u'*u);

Solve the problem

options = struct;
options.name = 'Nagurka Problem';
solution = ezsolve(objective, {ceq, cinit}, guess, 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: 0.067164 seconds
Starting numeric solver
===== * * * =================================================================== * * *
TOMLAB - TOMLAB Development license  999007. Valid to 2011-12-31
=====================================================================================
Problem:  1: Nagurka Problem                    f_k     109.074347751905190000
                                       sum(|constr|)      0.000000000001514694
                              f(x_k) + sum(|constr|)    109.074347751906710000
                                              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 
CPU time: 0.015600 sec. Elapsed time: 0.007000 sec. 

Plot result

subplot(2,1,1)
x1 = x(:,1);
plot(t,x1,'*-');
legend('x1');
title('Nagurka Problem - First state variable');

subplot(2,1,2)
u1 = u(:,1);
plot(t,u1,'+-');
legend('u1');
title('Nagurka Problem - First control variable');

figure(2)
surf(t, 1:n, x')
xlabel('t'); ylabel('i'); zlabel('x');
title('Nagurka Problem - All state variables');

figure(3)
surf(t, 1:n, u')
xlabel('t'); ylabel('i'); zlabel('u');
title('Nagurka Problem - All control variables');