PROPT Room temperature control

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

Problem Description

Temperature control from 0.00 to 7.00 hours at night. Finds best heating policy at night that brings the temperature back to 20 [oC] in the morning irrespective of night temperatures.

Programmers: Gerard Van Willigenburg (Wageningen University)

% Copyright (c) 2007-2010 by Tomlab Optimization Inc.

Problem setup

Define tomSym variable t (time) and t_f (final time) if the final time is free

toms t; t_f=7; % Fixed final time

for n=[20 40]
    % Define & set time axis
    p = tomPhase('p', t, 0, t_f, n);
    setPhase(p);

    % Define the state and control variables
    tomStates x
    tomControls u

    % Initial state
    xi=20;

    % Initial guess
    if n==20
        x0 = {icollocate({x == xi(1)})
            collocate({u == 0})};
    else
        x0 = {icollocate({x == xopt})
            collocate({u == uopt})};
    end

    % Boundary conditions
    cbnd = {initial({x == xi}); final({x == xi})};

    % Equality constraints: state-space differential equations
    tau=2; pH=0.002; % Parameters

    % External input d1
    d1=15-10*sin(pi*t/t_f);

    %Differential equation
    ceq = collocate({dot(x) == 1/tau*(d1-x) + pH*u});

    % Inequality constraints
    cbox = {0 <= collocate(u) <= 3600; icollocate(x) >= 15};

    % Cost function to be minimized
    objective = integrate(u+1e-6*dot(u)^2);

    % Solve the problem after specifying its name
    options = struct;
    options.name = 'Temperature control at night';
    solution = ezsolve(objective, {cbox, cbnd, ceq}, x0, options);

    % Plot intermediate solution
    figure; subplot(2,1,1);
    ezplot(x); legend('x');
    title('State');

    subplot(2,1,2);
    ezplot(u); legend('u');
    title('Optimal control'); drawnow;

    % Obtain intermediate solution to initialize the next
    xopt = subs(x,solution);
    uopt = subs(u,solution);
end

% Obtain final solution t,x,...,u,..
% that overwrite the associated tomSym variables
t = subs(collocate(t),solution);
x = subs(collocate(x),solution);
u = subs(collocate(u),solution);

%Plot results
figure; plot(t,x,t,u/100); axis([0 t_f -1 50]);
xlabel('Time [h]'); ylabel('Heat input, Inside & Outside temperature');
title('Optimal heating, outside temperature');
legend('Inside temp. [oC]','Outside temp. [oC]');

Problem type appears to be: qp
Time for symbolic processing: 0.048438 seconds
Starting numeric solver
===== * * * =================================================================== * * *
TOMLAB - TOMLAB Development license  999007. Valid to 2011-12-31
=====================================================================================
Problem:  1: Temperature control at night       f_k   12892.776351482817000000
                                       sum(|constr|)      0.000000000005229143
                              f(x_k) + sum(|constr|)  12892.776351482822000000
                                              f(x_0)      0.000000000000000000

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

FuncEv   11 GradEv   11 ConstrEv   11 Iter   11 
Elapsed time: 0.004000 sec. 
Problem type appears to be: qp
Time for symbolic processing: 0.048291 seconds
Starting numeric solver
===== * * * =================================================================== * * *
TOMLAB - TOMLAB Development license  999007. Valid to 2011-12-31
=====================================================================================
Problem:  1: Temperature control at night       f_k   12882.457402361473000000
                                       sum(|constr|)      0.000000014487930886
                              f(x_k) + sum(|constr|)  12882.457402375961000000
                                              f(x_0)      0.000000000000000000

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

FuncEv   13 GradEv   13 ConstrEv   13 Iter   13 
Elapsed time: 0.006000 sec.