Lab 3 nust control

PAKISTAN NAVY ENGINEERING COLLEGE NUSTTALHA WAQAR EE-805

LAB # 3(A)

System Response

OBJECT:

To study the System response for different order systems, natural frequency and damping ratio, Peak response, settling time, Rise time, steady state, using MATLAB commands and LTI Viewer.

THEORY:

Generally we have two types of responses, Steady State Response and Transient response such as rise time, peak time, maximum overshoot, settling time etc.

n=[1 1];

d=[2 4 6];

S1=tf(n,d)

size(S1) %no. of inputs and outputs.

pole(S1) % no.of poles .

pzmap(S1) %pole/zero map.

K=dcgain(S1)

zpk(S1) % zero /pole/gain .

damp (S1) % damping coefficients.

[wn,z] = damp(S1) % naturalfrequency

step(S1) % assinging input for analysis

Transfer function:

s + 1

---------------

2 s^2 + 4 s + 6

Transfer function with 1 outputs and 1 inputs.

ans =

-1.0000 + 1.4142i

-1.0000 - 1.4142i

K=

0.1667

Zero/pole/gain:

0.5 (s+1)

--------------

(s^2 + 2s + 3)

Eigenvalue Damping Freq. (rad/s)

-1.00e+000 + 1.41e+000i 5.77e-001 1.73e+000

-1.00e+000 - 1.41e+000i 5.77e-001 1.73e+000

wn =

1.7321

1.7321

z =

0.5774

0.5774

CONTROL SYSTEM LAB SYSTEM RESPONSEDATED: 25-FEB-14


zeta= [0.3 0.6 0.9 1.5]; % zeta funtion is assigned with four different values.for k=1:4; % k is assigned from 1-4 so as to run the program four times in a loop. num=[0 1 2]den=[1 2*zeta(k) 1]; % den take out four different values of zeta . TF=tf(num,den) step(TF)hold on; % hold on restores the previous graphs.end; % end represent the completion

num =

0 1 2 Transfer function: s + 2---------------s^2 + 0.6 s + 1

num = 0 1 2

Transfer function: s + 2---------------s^2 + 1.2 s + 1 num =

0 1 2

Transfer function: s + 2---------------s^2 + 1.8 s + 1 num =

0 1 2

Transfer function: s + 2-------------s^2 + 3 s + 1

Exercise:

1. Given the transfer function, G(s) = a/(s+a), Evaluate settling time and rise time for the following values of a= 1, 2, 3, 4. Also, plot the poles.

for k=1:4; num=[k]den=[1 k]; TF=tf(num,den) step(TF)hold on; end



Lab task:

Task# 1:



Task# 2a:num=[25];>> den=[1 4 25];>> trans=tf(num,den);>> step(trans);>> zero(trans)

p=pole(t1)p =-2.0000 + 4.5826i-2.0000 - 4.5826i>> z=zero(t1)z =Empty matrix: 0-by-1>> y=pzmap (t1)



Task # 2b >> trans=tf(num,den)



G(s)= b/s^2+as+b

Coefficent of damping I will represent with C

poles= -C wn + -j wn sqrt 1-C^2

wn sqrt 1-C^2= 5*sqrt 1-a4^2=4.5826

C wn= 4 now wn=6.0828 & C=0.6575

Tp= .6949, Ts=1.0139, OS = .0645

a=7.89, b=36

num=[36];

den=[1 7.86 36];

Transfer function:

36

-----------------

s^2 + 7.86 s + 36

>> step(trans)



p=pole(t1)p =-4.0000 + 4.5826i-4.0000 - 4.5826i>> z=zero(t1)z =Empty matrix: 0-by-1>> y=pzmap(t1)y =-4.0000 + 4.5826i -4.0000 - 4.5826i



Task # 2c:Calculate the values of a and b so that the imaginary part of the poles remains the same, but the real part is decreased ½ time over that of (a), and repeat the 2(a). num=[22];>> den=[1 2 22];>> trans=tf(num,den);>> step(trans)>>zero(trans)ans =

Empty matrix: 0-by-1



p=pole(t1) p =-1.0000 + 4.5826i-1.0000 - 4.5826i>> z=zero(t1)z =Empty matrix: 0-by-1>> y=pzmap(t1)y =-1.0000 + 4.5826i-1.0000 - 4.5826i

Task # 3a:For the system of prelab 2(a) calculate the values of a and b so that the realpart of the poles remains



the same but the imaginary part is increased 2times ove that of prelab 2(a) and repeat prelab 2(a)A=4,b=88num=[88];>> den=[1 4 88];>> trans=tf(num,den);>> step(trans)



p=pole(t1)p =-2.0000 + 9.1652i-2.0000 - 9.1652iz=zero(t1)z = Empty matrix: 0-by-1>> y=pzmap(t1)y =-2.0000 + 9.1652i-2.0000 - 9.1652i

Task # 3bFor the system of prelab 2(a) calculate the values of a and b so that the realpart of the poles remains the same but the imaginary part is increased 4times over that of prelab 2(a) and repeat prelab 2(a)A=4,b=340num=[340];>> den=[1 4 340];>> trans=tf(num,den) Transfer function: 340---------------s^2 + 4 s + 340 >> step(trans)





p=pole(t1)p =-2.0000 +18.3303i-2.0000 -18.3303i>> z=zero(t1)z =Empty matrix: 0-by-1>> y=pzmap(t1)y =-2.0000 +18.3303i-2.0000 -18.3303i

Task # 4aFor the system of 2(a), calculate the values of a and b so that the damping ratio remains the same, but the natural frequency is increased 2 times over that of 2(a), and repeat 2(a). num=[100];>> den=[1 8 100];>> trans=tf(num,den)Transfer function: 100---------------s^2 + 8 s + 100 >> step(trans)





Task # 4b:For the system of 2(a), calculate the values of a and b so that the damping ratio remains the same, but the natural frequency is increased 4 times over that of 2(a), and repeat 2(a).eeta=0.4>> omega=20omega=20>> b=omega*omegab =400>> a=2*eeta*omegaa =16>> num=[b]num=400>> den=[ 1 a b]den =1 16 400>> t=tf([num],[den]) Transfer function:400s^2 + 16 s + 400





Exercise:Using Simulink, set up the systems of Q 2. Using the Simulink LTI Viewer, plot the step response of each of the 3 transfer functions on a single graph.



a=tf([25],[1 4 25]);

>> b=tf([37],[1 8 37]);

>> c=tf([22],[1 2 22]);

>> step(a,b,c)

task # 3:Using Simulink, set up the systems of Q2(a) and Q3. Using the Simulink LTI Viewer, plot the step response of each of the 3 transfer functions on a single graph.c=tf([25],[1 4 25]);

>> b=tf([88],[1 4 88]);



>> a=tf([340],[1 4 340]);

>> step(a,b,c)



Task # 4:Using Simulink, set up the systems of Q 2(a) and Q 4. Using the Simulink LTI Viewer, plot the step response of each of the 3 transfer functions on a single graph.a=tf([25],[1 4 25]);

>> b=tf([100],[1 8 100]);

>> c=tf([400],[1 16 400]);

>> step(a,b,c)




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Lab 3 nust control