Greenhouse Climate Control

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This problem is taken from the book: Optimal Control of Greenhouse Cultivation G. van Straten, R.J.C. van Ooteghem, L.G. van Willigenburg, E. van Henten.

The problem models the growth of crop in a greenhouse depending on the temperature and light. The problem is modeled with external input for the outside temperature and light coming in, and the control is a heating element inside the greenhouse. The goal is to maximize the profit of the greenhouse which comes from the weight of crops minus the cost of heating.

Problem formulation

The problem formulation is the following:

$J = - p_{5} x_1 + \int_{t_0}^{t_f} p_{4} u$ $\dot{x}_1 = p_1 I x_2$ $\dot{x}_2 = p_2 (T_0 - x_2)+p_3 u$ $I = max(0, 800 \sin( 4 \pi \frac{t}{t_f} - 0.65 \pi))$ $T_0 = 15+10 \sin(4\pi \frac{t}{t_f} - 0.65\pi)$ $p_1 = 7.5 \cdot 10^{-8}, \quad p_2 = 1, \quad p_3 = 0.1$ $p_4 = 4.55 \cdot 10^{-4}, \quad p_{5} = 136.4$ $t_0 = 0$ $t_f = 48$ $x_1(0) = 0$ $x_2(0) = 10$ $0 \leq u \leq 10$

Variable descriptions:

variable	Description	Unit
$x_1$	Dry weight of the crop	$\textrm{kgm}^{-2}$
$x_2$	Temperature in the greenhouse	°C
$I$	Intensity of the light entering the greenhouse	$\textrm{Wm}^{-2}$
$T_0$	Temperature outside the greenhouse	°C
$u$	Heat input from the heating system	$\textrm{Wm}^{-2}$

Yop implementation

This example shows how to model external input with the help of YopInterpolant.

Warning: YopInterpolant assumes that the input is row vectors and might cause Matlab to crash if they’re not.

%% Greenhouse Climate Control, with external input
% Author: Dennis Edblom
sys = YopSystem('states', 2, 'controls', 1, ...
    'model', @greenhouseModel);
% Symbolic variables
t = sys.t;
x = sys.x;
u = sys.u;

%% Formulate optimal control problem
p4 = 4.55e-4;
p5 = 136.4;
tf = 48;

ocp = YopOcp();
ocp.min({ t_f(-p5*x(1)) '+' timeIntegral(p4*u) });
ocp.st(...
     'systems', sys, ...
     ... % Initial conditions
    { 0  '==' t_0(x(1)) }, ...
    { 10 '==' t_0(x(2)) }, ...
    ... % Constraints
    { 0  '<=' u '<=' 10 }, ...
    ... % Final conditions
    { tf '==' t_f(t)    } ...
    );
sol = ocp.solve('controlIntervals', 100);

%% Plot
% States, time and control
x1 = sol.signal(sys.x(1))';
x2 = sol.signal(sys.x(2))';
u = sol.signal(sys.u)';
t = sol.signal(sys.t)';

% External input for plots, comes from greenhouseModel
te = sys.y.te;
I  = sys.y.I;
T0 = sys.y.T0;

% Plot external inputs and control
figure(1);
plot(te,I./40,te,T0,t,u); axis([0 tf -1 30]);
xlabel('Time [h]');
ylabel('Heat input, temperatures & light');
legend('Light [W]','Outside temp. [oC]','Heat input [W]');
title('Optimal heating, outside temperature and light');

% Plot the optimal states
figure(2)
sf1=1200; sf3=60;
x3 = cumtrapz(t,p4*u); % Integral(pHc*u)
plot(t,[sf1*x1 x2 sf3*x3]); axis([0 tf -5 30]);
xlabel('Time [h]'); ylabel('states');
legend('1200*Dry weight [kg]','Greenhouse temp. [oC]','60*Integral(pHc*u dt) [J]');
title('Optimal system behavior and the running costs');

%% Model
function [dx, y] = greenhouseModel(t, x, u)
% Constants
p1 = 7.5e-8;
p2 = 1;
p3 = 0.1;
tf = 48;
% External inputs: [time, sunlight, outside temperature]
te  = (-1:0.2:49);
I = max(0, 800*sin(4*pi*te/tf-0.65*pi));
T0 = 15+10*sin(4*pi*te/tf-0.65*pi);

% Extract external inputs from table tue through interpolation
d1 = YopInterpolant(te, I);
d2 = YopInterpolant(te,T0);

dx1 = p1*d1(t)*x(2);
dx2 = p2*(d2(t)-x(2))+p3*u;
dx = [dx1; dx2];
y.te = te;
y.I = I;
y.T0 = T0;
end

Plots

greenhouseControl — Greenhouse control and external input