Project - Position control of mass spring damper system

Project - Speed Control of a Direct Current (DC) motor

Obtain the transfer function of a mass spring damper system and use it in the model. Add a PID controller to adjust the force on mass so that its position follows a reference signal.

Take m = 3.6; %kg k = 400; %N/m c = 100; %Ns/m

The reference signal is shown in the following figure.

Answer

Spring mass damper system

X2dot = [F(t) - c*x_dot - kx]/m

F(t) = Kp*(x_ref-x)+Ki*∫(x_ref-x)*dt+Kd*x_dot

x = position ( or displacement)

x_dot = velocity = dx/dt

x2dot = acceleration = d2x/dt2

matlab script

clc;

clear;

m = 3.6; % mass of spring mass damper system in Kg

c = 100; % damping cofficient of spring mass damper system in Ns/m

k = 400; % spring stifness on spring mass damper system in N/m

wn=sqrt(k/m); % natural frequency in Hz

zeta=c/(2*m*wn); % damping ratio

Kp=wn^2; % proportaional gain

Kd=2*zeta*wn; % derivative gain

Ki = 900; % integral gain

%% Transfer function

s=tf('s');

G = wn^2/(s^2+2*zeta*wn*s+wn^2);

simulink model

pulse parameter

time parameter to workspace

model setting

plot on matlab script

%% plot

figure(1);

subplot(511);plot(out.time,out.xref,'m-');grid;title('reference');

subplot(512);plot(out.time,out.u);grid;title('control');

subplot(513);plot(out.time,out.a);grid;title('acceleration');

subplot(514);plot(out.time,out.v);grid;title('velocity');

subplot(515);plot(out.time,out.x);grid;title('position');

figure(2);

plot(out.time,out.xref,'r--');grid;

plot(out.time,out.x,'b-');hold;grid;title('position with reference');

these figure 2 shows the position follows a reference signal.