Project - Position control of mass spring damper system

To obtain the transfer function of the mass-spring-damper system, we'll start by writing down the equations of motion:

��¨+��˙+��=�mx¨+cx˙+kx=F

Where:

• �m is the mass (3.6 kg)
• �c is the damping coefficient (100 Ns/m)
• �k is the spring constant (400 N/m)
• �x is the displacement of the mass from its equilibrium position
• �F is the force applied to the mass

Taking the Laplace transform, assuming zero initial conditions, we get:

��2�(�)+���(�)+��(�)=�(�)ms2X(s)+csX(s)+kX(s)=F(s)

Rearranging and solving for �(�)/�(�)X(s)/F(s), we obtain the transfer function:

�(�)�(�)=1��2+��+�F(s)X(s)​=ms2+cs+k1​

Substituting the given values �=3.6m=3.6, �=100c=100, and �=400k=400, we get:

�(�)�(�)=13.6�2+100�+400F(s)X(s)​=3.6s2+100s+4001​

Next, let's design a PID controller to adjust the force on the mass. The general form of a PID controller is:

�(�)=��+���+���C(s)=KP​+sKI​​+KD​s

Where:

• ��KP​ is the proportional gain
• ��KI​ is the integral gain
• ��KD​ is the derivative gain

The overall transfer function with the PID controller becomes:

�(�)�(�)=�(�)1+�(�)�(�)R(s)X(s)​=1+C(s)G(s)C(s)​

Substituting �(�)=13.6�2+100�+400G(s)=3.6s2+100s+4001​ and �(�)=��+���+���C(s)=KP​+sKI​​+KD​s, we get:

�(�)�(�)=���2+���+��3.6�3+(100+��)�2+(400+��)�+��R(s)X(s)​=3.6s3+(100+KD​)s2+(400+KP​)s+KI​KP​s2+KI​s+KD​​

This is the transfer function of the closed-loop system with the PID controller.

Matlab program