Giving a rectangular pulse in ode45
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Kashish Pilyal
le 15 Août 2022
Modifié(e) : Kashish Pilyal
le 15 Août 2022
I am trying to use a rectangular pulse in my system through ode 45, so I decided to split my code and run it 3 times. Once at 0, then at 1 and at 0 again. I am not getting results as expected. As you can see, I have a system code as:
clear all
%%
%paramters of leader vehicle
q=0; %position
v=0; %velocity
a=0; %acceleration
x_0=[q;
v;
a]; %initial conditions
t=linspace(0,5,100);
%parameters of second vehicle
q1=0; %position
v1=0; %velocity
a1=0; %acceleration
x_1=[q1;
v1;
a1]; %initial conditions
%parameters of third vehicle
q2=0; %position
v2=0; %velocity
a2=0; %acceleration
x_2=[q2;
v2;
a2]; %initial conditions
%initial conditions of the platoon
x=[x_0; x_1; x_2];
[t,y1]=ode45(@code,t,x);
%% second step
%paramters of leader vehicle
q=1; %position
v=0; %velocity
a=0; %acceleration
x_0=[q;
v;
a]; %initial conditions
t=linspace(5,50,100);
%parameters of second vehicle
q1=0; %position
v1=0; %velocity
a1=0; %acceleration
x_1=[q1;
v1;
a1]; %initial conditions
%parameters of third vehicle
q2=0; %position
v2=0; %velocity
a2=0; %acceleration
x_2=[q2;
v2;
a2]; %initial conditions
%initial conditions of the platoon
x=[x_0; x_1; x_2];
[t,y2]=ode45(@code,t,x);
%% third step
%paramters of leader vehicle
q=0; %position
v=0; %velocity
a=0; %acceleration
x_0=[q;
v;
a]; %initial conditions
t=linspace(50,60,100);
%parameters of second vehicle
q1=y2(end,4); %position
v1=0; %velocity
a1=0; %acceleration
x_1=[q1;
v1;
a1]; %initial conditions
%parameters of third vehicle
q2=y2(end,7); %position
v2=0; %velocity
a2=0; %acceleration
x_2=[q2;
v2;
a2]; %initial conditions
%initial conditions of the platoon
x=[x_0; x_1; x_2];
[t,y3]=ode45(@code,t,x);
%% concating
time=linspace(0,20,300);
%parameters of 1st
q_1=[y1(:,1);y2(:,1);y3(:,1)];
%parameters of 2nd
q_2=[y1(:,4);y2(:,4);y3(:,4)];
%parameters of 3rd
q_3=[y1(:,7);y2(:,7);y3(:,7)];
plot(time',q_1,time',q_2,time',q_3)
The system response is not as expected from ode45. I found out that the reason it does not match is due to the vehicles depending upon the predecessor's acceleration which does not happen as I shift to ode 45 mid code. I tried giving initial condition as some acceleration but it just messed up the rectangular pulse input that I am giving. Is there any other way to feed the ode45 solver a rectangular pulse?
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Torsten
le 15 Août 2022
Modifié(e) : Torsten
le 15 Août 2022
I can't evaluate the equations and initial conditions you use. This is what the solver returns - slight over-and undershooting.
options = odeset('RelTol',1e-12,'AbsTol',1e-12);
%paramters of leader vehicle
q=0; %position
v=0; %velocity
a=0; %acceleration
x_0=[q;
v;
a]; %initial conditions
t=linspace(0,5,100);
%parameters of second vehicle
q1=0; %position
v1=0; %velocity
a1=0; %acceleration
x_1=[q1;
v1;
a1]; %initial conditions
%parameters of third vehicle
q2=0; %position
v2=0; %velocity
a2=0; %acceleration
x_2=[q2;
v2;
a2]; %initial conditions
%initial conditions of the platoon
x=[x_0; x_1; x_2];
[t1,y1]=ode45(@code,t,x,options);
%% second step
%paramters of leader vehicle
q=1; %position
v=0; %velocity
a=0; %acceleration
x_0=[q;
v;
a]; %initial conditions
t=linspace(5,50,100);
%parameters of second vehicle
q1=0; %position
v1=0; %velocity
a1=0; %acceleration
x_1=[q1;
v1;
a1]; %initial conditions
%parameters of third vehicle
q2=0; %position
v2=0; %velocity
a2=0; %acceleration
x_2=[q2;
v2;
a2]; %initial conditions
%initial conditions of the platoon
x=[x_0; x_1; x_2];
[t2,y2]=ode45(@code,t,x,options);
%% third step
%paramters of leader vehicle
q=0; %position
v=0; %velocity
a=0; %acceleration
x_0=[q;
v;
a]; %initial conditions
t=linspace(50,70,100);
%parameters of second vehicle
q1=y2(end,4); %position
v1=0; %velocity
a1=0; %acceleration
x_1=[q1;
v1;
a1]; %initial conditions
%parameters of third vehicle
q2=y2(end,7); %position
v2=0; %velocity
a2=0; %acceleration
x_2=[q2;
v2;
a2]; %initial conditions
%initial conditions of the platoon
x=[x_0; x_1; x_2];
[t3,y3]=ode45(@code,t,x,options);
%% concating
time=[t1;t2;t3];
%parameters of 1st
q_1=[y1(:,1);y2(:,1);y3(:,1)];
%parameters of 2nd
q_2=[y1(:,4);y2(:,4);y3(:,4)];
%parameters of 3rd
q_3=[y1(:,7);y2(:,7);y3(:,7)];
plot(time,[q_1,q_2,q_3])
function y=code(t,x)
%-----------legend------
% x(1),x(2),x(3) [position, velocity, acceleration] 1st
% x(4),x(5),x(6) [position, velocity, acceleration] 2nd
% x(7),x(8),x(9) [position, velocity, acceleration] 3rd
tau_i=0.1;
h_i=0.5;
k_d=0.68;
k_p=0.2;
T=tau_i/h_i;
%% leader
u_i=( (1-T)*x(3) );
%x(2)=0.5*t;
% CACC equations
x_1dot=0; %q(position)dot
x_1ddot=x(3); %v(velocity)dot
x_1dddot=(-(1/tau_i)*x(3))+((1/tau_i)*u_i); %a(accelration)dot
%% second vehicle
tau_i=0.2;
h_i=0.5;
T=tau_i/h_i;
e_i1_1=x(1)-( x(4)+(h_i*x(5)) );
e_i2_1=x(2)-( x(5)+(h_i*x(6)) );
%e_i0_1=int(e_i1_1,t);
u_i=T*[k_p k_d]*[e_i1_1;e_i2_1]+ ( (1-T)*x(6) ) + ( T*x(3) );
% CACC equations
x_2dot=x(5); %q(position)dot
x_2ddot=x(6); %v(velocity)dot
x_2dddot=(-(1/tau_i)*x(6))+((1/tau_i)*u_i); %a(accelration)dot
%% third vehicle
tau_i=0.1;
h_i=0.5;
T=tau_i/h_i;
e_i1_2=x(4)-( x(7)+(h_i*x(8)) );
e_i2_2=x(5)-( x(8)+(h_i*x(9)) );
u_i=T*[k_p k_d]*[e_i1_2;e_i2_2]+ ( (1-T)*x(9) )+ ( T*x(6) );
% CACC equations
x_3dot=x(8); %q(position)dot
x_3ddot=x(9); %v(velocity)dot
x_3dddot=(-(1/tau_i)*x(9))+((1/tau_i)*u_i); %a(accelration)dot
y=[x_1dot; %position of 1st
x_1ddot; %velocity of 1st
x_1dddot; %acceleration of 1st
x_2dot; %position of 2nd
x_2ddot; %velocity of 2nd
x_2dddot; %acceleration of 2nd
x_3dot; %position of 3rd
x_3ddot; %velocity of 3rd
x_3dddot]; %acceleration of 3rd
end
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