Project - Position control of mass spring damper system

Q.1 - 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.

Sol-

Parameters used for position control of spring mass damper system

% Initial conditions

m = 3.6; % kg

k = 400; % N/m

c = 100; % Ns/m

% A and B matrices

A = [0 1; -k/m -c/m];

B = [0; 1/m];

% Placement of poles

fn = 1.0; % 1Hz natural frequency design

wn = 2*pi*fn; % Natural frequency in rad/s

% zeta = 1.0; % for a critical damped design

zeta = 0.99; % for more damped design

% Poles

p1 = -zeta*wn+wn*sqrt(zeta^2-1);

p2 = -zeta*wn-wn*sqrt(zeta^2-1);

P= [p1;p2];

% Poles placement design

K = place(A,B,P);

Kp = K(1);

Kd = K(2);

Ki = 0;

zeta = 1.0 % for an critical damped design

Both poles p1 and p2 are same -6.2832

when both poles are the place command cannot place poles multiplicity greater than rank(B)

Now, taking zeta = 0.99

Command window:

Simulation:

Block used:

Clock

To workspace

Pulse Generator (Ref_signal)

Sum block

Gain block

Derivative

Integrator

Transfer function block

Scope

As a reference signal using a pulse generator with PID design, we are using it to control the position of the spring mass damper system.

Pulse Generator parameters using as given:

Transfer Function Parameters

Num = 1

Den = [m k c]

num(s)/den(s) = 1/(m * s ^ 2 + c * s + k)

Model Settings Parameters

Scope:

Kp = -257.8777

Kd = -55.2135

Ki = 0

there is an error between the reference signal and the simulated signal when ki = 0

Kp = -257.8777

Kd = -55.2135

Ki = 1000

Kp = -257.8777

Kd = -55.2135

Ki = 5000

Ki value changes the position of the spring-mass damper system.

ki = 5000, the graph more accurate and stable system.