Challenge

Project - Position control of mass spring damper system

HHariram Senthilkumaran20 Jul 2023

Our Model needs to have a transfer function and PID controller to adjust the force on mass so that its position follows a reference signal.

Things to do

for transfer function,as per formula, we need m,k,c values which are already defined in the problem

For PID controller, We need Kp,Kd,Ki which we need to determine using pole placement design and can validate it through pzmap. so lets proceed with MATLAB code to determine the needed paramters.

MATLAB code in MATLAB file

clc;

clear all;

%inputs given

m=3.6; % mass in kg

k=400; %

c=100;

%matrices

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

B=[0;1/m];

%pole placement design values

fn=1; %unit frequency

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

zeta = 0.5; %under damping behaviour

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

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

P=[p1;p2];

K=place(A,B,P);

Kp=K(1);

Kd=K(2);

%pzmap function

s=tf('s');

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

pzmap(G);

grid on;

[p,z]=pzmap(G);

%End

Validation of values in Command Window

>> who

Your variables are:

A G Kd P fn m p1 s z
B K Kp c k p p2 wn zeta

>> A

A =

0 1.0000
-111.1111 -27.7778

>> B

B =

0
0.2778

>> P

P =

-3.1416 + 5.4414i
-3.1416 - 5.4414i

>> K

K =

-257.8777 -77.3805

>> Kp

Kp =

-257.8777

>> Kd

Kd =

-77.3805

>> m

m =

3.6000

>> k

k =

400

>> c

c =

100

>> zeta

zeta =

0.5000

>> wn

wn =

6.2832

Model

1)As usual we will start with clock for time

2) input can be given through multiple input signal types such as constant,step,ramp but here we choose pulse generator for signal wave input. its paramters used for model(in order to match the given reference signal vs time) are

amplitude =1;

period = 10 secs;

pulse width = 50%

3)a Sum block needed to add the error signal which we will input from the transfer function later in this model.

4) all thre gain blocks (Kp,Ki,Kd) are added where Kd,Kp will take values from the MATLAB code and Ki will be an manual input. Also, Ki,the intergral gain needs an integral block before it.

But for Kd, the derivative gain, we can input it in two ways. one way is to derivate the input signal value or to use the output of the model and latter would be suggested and used here.

5)The output of the PID controller design is sent to the Transfer function block which has m,k,c as paramters (rest all are default paramters).

6)Scope is used to plot the input signal and output of the model.and another scope is added in order to analyse the behaviour of the PID values based on the manual input of Ki values.

Plots inference from attached snips

1. Final solution snip is taken when the Ki value was initialized as 2500 and we got our needed output with respect to our input signal.

Note: Multiple inputs were tested for Ki and for every positive value above 100,the needed output started to show and after optimizing the value at 2500, we got the optimzed result.

2. another snip taken at Ki=0, to see other gains(Kp,Kd) and their plot.

Conclusion

Position control of mass spring damper system is modelled and output is verfied

Project - Position control of mass spring damper system

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in your own course.

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Ship work like thisin your own course.

Talk to a career counsellor

Ship work like this
in your own course.