Challenge

Project - Position control of mass spring damper system

PPrasann Jaiswal13 May 2023

Solution: The TF is obtained from the given Spring mass damper system and modelling is done:

Note: The Modelling parameters are

Fixed step size of 0.01 with solver as ODE4(Runga-Kutta) and Input as Pulse generator of amplitude 1 and period 10s and width 50%.

% PID design for Spring mass system

m=3.6; k=400; c=100; % Spring Mass system parameters

wn=sqrt(k/m);

zeta=c/(2*sqrt(k*m));

% Obtaining TF from the Spring mass Dynamics

% x1_dot=x2;

% x2_dot= -k/m*x1 -c/m*x2 + (1/m)*u

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

B=[0;1/m];

C=[1 0];

D=[0];

[N D]=ss2tf(A,B,C,D,1) % Finding TF from State Space eqn

% Controller Gain design

zeta1=0.5; %desired zeta

wn1=2*pi*1; % desired wn for f=1Hz

Ki=0.0;

p1= -zeta1*wn1+wn1*sqrt(zeta1^2-1);

p2= -zeta1*wn1-wn1*sqrt(zeta1^2-1);

P=[p1;p2];

K=place(A,B,P);

>> PID_desgn

N =

0 0 0.2778

D =

1.0000 27.7778 111.1111

>> P

P =

-3.1416 + 5.4414i
-3.1416 - 5.4414i

>> K

K =

-257.8777 -77.3805

Note: The Kp and Kd obtained from the above pole placement design is not appropriate and hence we will assume the following Kp and Kd to start with:

>> Kp=wn^2

Kp =

111.1111

>> Kd=2*zeta*wn

Kd =

27.7778

>> Ki=0.0

Case1:

Result: The response Obtained is fast but has huge steady state error.

Case 2: Keeping Kp and Kd same, Ki is increased from 0 to 500.

Result: The Steady state error is reduced but response is sluggish and underdamped.

Case 3: The Ki is increased to 1000.

Result: The Steady state error is reduced to zero but response is faster and overdamped.

Case 4: The Ki is increased to 1500 to improve the response further.

Result: The response is more faster but overshoot is near to 10%.

Case 5: The Ki is reduced to 1250 to reduce the overshoot<10%.

Conclusion: The optimum response is obtained with the controller gains as Kp=111.11, Kd=27.778 and Ki=1250. All the three important parameters like rise time, peak overshoot and steady error is best obtained at the above set PID controller gains.

Project - Position control of mass spring damper system

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Ship work like this
in your own course.