« Previous « Start » Next »

110  Room temperature control

110.1  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)

110.2  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
Starting numeric solver
===== * * * =================================================================== * * *
TOMLAB - Tomlab Optimization Inc. Development license  999001. Valid to 2011-02-05
=====================================================================================
Problem:  1: Temperature control at night       f_k   12892.776351482709000000
                                       sum(|constr|)      0.000000000002029896
                              f(x_k) + sum(|constr|)  12892.776351482711000000
                                              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
CPU time: 0.062500 sec. Elapsed time: 0.031000 sec.
Problem type appears to be: qp
Starting numeric solver
===== * * * =================================================================== * * *
TOMLAB - Tomlab Optimization Inc. Development license  999001. Valid to 2011-02-05
=====================================================================================
Problem:  1: Temperature control at night       f_k   12882.457401725025000000
                                       sum(|constr|)      0.000000006475484391
                              f(x_k) + sum(|constr|)  12882.457401731501000000
                                              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
CPU time: 0.062500 sec. Elapsed time: 0.047000 sec.

« Previous « Start » Next »