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CTRL_TEST.m
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CTRL_TEST.m
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% =================================================
% CONTROLADORES PARA BITBOT
% José Eduardo Molina Calzia
% Carné 16063
% Agosto 2020
% =================================================
% Borrar estado anterior
close all;
clear all;
% ================================================
% Selector de trayectoria o meta fija --> fg, traj
sel = 'fg';
%Opciones: pid, pidexp, nonlin, lqr, lqi
ctrl = 'nonlin';
graf = 1;
%% Definición del sistema dinámico
% xi = [x; y; theta] u = [v;omega]
% Parámetros
ell =0.01; %m
% Campo vectorial del sisrema dinámico
f = @(xi,u)[u(1)*cos(xi(3)); ...
u(1)*sin(xi(3)); ...
u(2)];
% Difeomorfismo
finv = @(xi,mu) [1,0; 0,1/ell] * [cos(xi(3)), -sin(xi(3)); sin(xi(3)), cos(xi(3))]' * [mu(1); mu(2)];
% Matrices del sistema linealizado
A = zeros(2);
B = eye(2);
%% Parámetros de simulación
dt = 0.001;
t0 = 0;
tf = 15;
N = (tf-t0)/dt;
%% Condiciones iniciales
%xi0 = zeros(3,1);
xi0 = [-4;-5;0];
u0 = zeros(2,1);
% Vector de estado
xi = xi0;
% Vector de entradas
u = u0;
% Array de históticos
XI = zeros(numel(xi),N+1);
U = zeros(numel(u),N+1);
XI(:,1) = xi;
U(:,1) = u;
%% Meta fija
xg = 5;
yg = 5;
thetag = pi;
%% Trayectoria
t = 0:dt:tf;
traj = 2*[cos(0.1*2*pi*t); sin(0.1*2*pi*t)];
%% Controladores
% PID
% Posición
kpP = 1;
kiP = 0.001;
kdP = 0.1;
EP = 0;
eP_1 = 0;
% Orientación
kpO = 1;
kiO = 0.001;
kdO = 0.1;
EO = 0;
eO_1 = 0;
% Acercamiento exp
v0 = 2;
alpha = 0.5;
% Controlador no lineal
k_rho = 10;
k_alpha = 20;
k_beta = -10;
% LQR
Q = eye(2);
Q(1,1) = 0.0001;
Q(2,2) = 0.0001;
R = eye(2);
R(1,1) = 1;
R(2,2) = 1;
% Calculo de Klqr
Klqr = lqr(A,B,Q,R);
% LQI
Cr = eye(2);
Dr = zeros(2);
AA = [A, zeros(size(Cr')); Cr, zeros(size(Cr,1))];
BB = [B; Dr];
QQ = eye(size(A,1) + size(Cr,1));
QQ(3,3) = 1;
QQ(4,4) = 1;
RR = eye(2);
RR(1,1) = 1;
RR(2,2) = 1;
%Calculo de Klqr
Klqi = lqr(AA,BB,QQ,RR)
ref = [xg;yg];
sigma = 0;
%% Solución del sistema dinámico
for n = 0:N
if (strcmp(sel,'traj'))
xg = traj(1,n+1);
yg = traj(2,n+1);
end
if(strcmp(ctrl, 'pid'))
fprintf("PID sin exp\n\n");
x = xi(1);
y = xi(2);
theta = xi(3);
% Error
e = [xg-x; yg-y];
thetag = atan2(e(2),e(1));
eP = norm(e);
eO = thetag-theta;
eO = atan2(sin(eO), cos(eO));
% Pos --> Velocidad lineal
eP_D = eP-eP_1;
EP = EP + eP;
v = kpP*eP + kiP*EP + kdP*eP_D;
eP_1 = eP;
% Ori --> Velocidad angular
eO_D = eO - eO_1;
EO = EO + eO;
w = kpO*eO + kiO*EO + kdO*eO_D;
eO_1 = eO;
%Vector de entradas
u = [v; w];
elseif(strcmp(ctrl, 'pidexp'))
fprintf("PID con exp\n\n");
x = xi(1);
y = xi(2);
theta = xi(3);
% Error
e = [xg-x; yg-y];
thetag = atan2(e(2),e(1));
eP = norm(e);
eO = thetag - theta;
eO = atan2(sin(eO), cos(eO));
% Pos --> Velocidad lineal
constP = v0 * (1-exp(-alpha*eP^2)) / eP;
v = constP*eP;
% Ori --> Velocidad angular
eO_D = eO - eO_1;
EO = EO + eO;
w = kpO*eO + kiO*EO + kdO*eO_D;
eO_1 = eO;
% Vector de entradas
u = [v; w];
elseif(strcmp(ctrl, 'nonlin'))
fprintf("No linear\n\n");
x = xi(1);
y = xi(2);
theta = xi(3);
e = [xg - x; yg - y];
rho = norm(e);
alpha = -theta + atan2(e(2), e(1));
beta = -theta - alpha;
v = k_rho*rho;
w = k_alpha*alpha + k_beta*beta;
% Vector de entradas
u = [v;w];
elseif(strcmp(ctrl, 'lqr'))
fprintf("LQR\n\n");
x = xi(1);
y = xi(2);
e = [x - xg; y - yg];
mu = -Klqr*e;
% Vector de entradas
u = finv(xi, mu);
elseif(strcmp(ctrl, 'lqi'))
fprintf("LQI\n\n");
x = xi(1:2);
ref = [xg;yg];
sigma = sigma + (Cr*x - ref)*dt;
mu = -Klqi*[x; sigma];
% Vector de entradas
u = finv(xi, mu);
else %default pid
fprintf("DEF\n\n");
x = xi(1);
y = xi(2);
theta = xi(3);
% Error
e = [xg-x; yg-y];
thetag = atan2(e(2),e(1));
eP = norm(e);
eO = thetag-theta;
eO = atan2(sin(eO), cos(eO));
% Pos --> Velocidad lineal
eP_D = eP-eP_1;
EP = EP + eP;
v = kpP*eP + kiP*EP + kdP*eP_D;
eP_1 = eP;
% Ori --> Velocidad angular
eO_D = eO - eO_1;
EO = EO + eO;
w = kpO*eO + kiO*EO + kdO*eO_D;
eO_1 = eO;
%Vector de entradas
u = [v; w];
end
% RK4 --> actualización del sistema dinámico
k1 = f(xi, u);
k2 = f(xi+(dt/2)*k1, u);
k3 = f(xi+(dt/2)*k2, u);
k4 = f(xi+dt*k3, u);
xi = xi + (dt/6)*(k1+2*k2+2*k3+k4);
% Se guardan las trayectorias del estado y las entradas
XI(:,n+1) = xi;
U(:,n+1) = u;
end
if(graf == 1)
%% Gráficas
figure('units','normalized','outerposition',[0,0,1,1]);
t = t0:dt:tf;
plot(t,XI');
xlabel('$t[s]$', 'Interpreter', 'latex', 'Fontsize', 15);
ylabel('$x(t)$', 'Interpreter', 'latex', 'Fontsize', 15);
l = legend('$x(t)$', '$y(t)$', '$\theta(t)$');
set(l, 'Interpreter', 'latex', 'FontSize', 12);
grid minor
if(strcmp(ctrl, 'pid'))
title('PID','Interpreter', 'latex', 'Fontsize', 25 )
elseif(strcmp(ctrl, 'pidexp'))
title('PID acercamiento exponencial','Interpreter', 'latex', 'Fontsize', 25 )
elseif(strcmp(ctrl, 'lqi'))
title('LQI','Interpreter', 'latex', 'Fontsize', 25 )
elseif(strcmp(ctrl, 'lqr'))
title('LQR','Interpreter', 'latex', 'Fontsize', 25 )
elseif(strcmp(ctrl, 'nonlin'))
title('No linear','Interpreter', 'latex', 'Fontsize', 25 )
end
figure('units','normalized','outerposition',[0,0,1,1]);
t = t0:dt:tf;
plot(t,U');
xlabel('$t[s]$', 'Interpreter', 'latex', 'Fontsize', 15);
ylabel('$u(t)$', 'Interpreter', 'latex', 'Fontsize', 15);
l = legend('$v(t)$', '$w(t)$');
set(l, 'Interpreter', 'latex', 'FontSize', 12);
grid minor
if(strcmp(ctrl, 'pid'))
title('PID','Interpreter', 'latex', 'Fontsize', 25 )
elseif(strcmp(ctrl, 'pidexp'))
title('PID acercamiento exponencial','Interpreter', 'latex', 'Fontsize', 25 )
elseif(strcmp(ctrl, 'lqi'))
title('LQI','Interpreter', 'latex', 'Fontsize', 25 )
elseif(strcmp(ctrl, 'lqr'))
title('LQR','Interpreter', 'latex', 'Fontsize', 25 )
elseif(strcmp(ctrl, 'nonlin'))
title('No linear','Interpreter', 'latex', 'Fontsize', 25 )
end
%% Simulación visual
figure('units','normalized','outerposition',[0,0,1,1]);
s = max(max(abs(XI(1:2,:))));
xlim(s*[-1, 1]+[-2, 2]);
ylim(s*[-1, 1]+[-2, 2]);
grid minor;
hold on;
q = XI(:,1);
x = q(1);
y = q(2);
theta = q(3);
if(strcmp(sel,'traj'))
plot(traj(1,:), traj(2,:),'k');
elseif (strcmp(sel,'fg'))
plot(xg,yg,'o');
end
trajplot = plot(x, y, '--k', 'LineWidth', 1, 'Color', [0,1,0]);
p1 = [0-3.5;4];p2 = [3-3.5;4];p3 = [3-3.5;2];p4 = [7-3.5;2];
p5 = [7-3.5;-2];p6 = [3-3.5;-2];p7 = [3-3.5;-4];p8 = [0-3.5;-4];
BBt =[p1,p2,p3,p4,p5,p6,p7,p8]/20;
bodyplot = fill(BBt(1,:)+x,BBt(2,:)+y,[1,0,0]);
xlabel('$x$', 'Interpreter', 'latex', 'Fontsize', 16);
ylabel('$y$', 'Interpreter', 'latex', 'Fontsize', 16);
if(strcmp(ctrl, 'pid'))
title('PID','Interpreter', 'latex', 'Fontsize', 25 )
elseif(strcmp(ctrl, 'pidexp'))
title('PID acercamiento exponencial','Interpreter', 'latex', 'Fontsize', 25 )
elseif(strcmp(ctrl, 'lqi'))
title('LQI','Interpreter', 'latex', 'Fontsize', 25 )
elseif(strcmp(ctrl, 'lqr'))
title('LQR','Interpreter', 'latex', 'Fontsize', 25 )
elseif(strcmp(ctrl, 'nonlin'))
title('No linear','Interpreter', 'latex', 'Fontsize', 25 )
end
hold off;
for n = 2:N+1
q = XI(:,n);
x = q(1); y = q(2); theta = q(3);
trajplot.XData = [trajplot.XData, x];
trajplot.YData = [trajplot.YData, y];
rot = [cos(theta-0), -sin(theta-0); sin(theta-0), cos(theta-0)] * BBt;
bodyplot.XData = rot(1,:) + x;
bodyplot.YData = rot(2,:) + y;
pause(dt/2);
end
end