1. Control System Toolbox

2. Outline Transfer Function Models Pole-Zero maps
Simplification of Block Diagrams
Time Domain Analysis of 1st and 2nd Order Systems
Frequency Domain Analysis of Control Systems
Root Locus of LTI models
Excercises

3. Transfer Function Model Using Numerator & Denominator Coefficients
This transfer function can be stored into the MATLAB
num = 100;
den = [ ];
sys=tf(num,den)
To check your entry you can use the command printsys as shown below:
printsys(num,den);
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4. Transfer Function Model Using Zeros, Poles and Gain (ZPK model)
This transfer function can be stored into the MATLAB
Zeros=-3;
Poles= [ ];
K=100;
sys=zpk(Zeros,Poles,K)
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5. Poles & Zeros
To plot the poles and zeros of any transfer function there is a built in function pzmap in the MATLAB
pzmap(num,den)
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6. Series Blocks
Blocks in series can be simplified by using series command S
9S + 17
9(S+3)
2S2 + 9s + 27
num1 = [1 0];
den1 = [9 17];
num2 = 9*[1 3];
den2 = [ ];
[num12, den12] = series (num1,den1,num2,den2);
printsys(num12,den12);
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7. Contd… Series Blocks
Blocks in series can also be simplified by using conv command S
9S + 17
9(S+3)
2S2 + 9s + 27
num1 = [1 0];
den1 = [9 17];
num2 = 9*[1 3];
den2 = [ ];
num12 =conv(num1,num2);
den12 = conv(,den1,den2);
printsys(num12,den12);
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8. Parallel Block
Blocks in parallel can be simplified by using parallel command
num1 = [1 2];
den1 = [ ];
num2 = [1 3];
den2 = [ ];
[num, den]=parallel(num1,den1,num2,den2);
printsys(num,den);
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9. Closed-loop transfer function
The block in feedback can be reduced using feedback command.
C(S)
R(S) - 1
S + 1 2 S
num1 = 1;
den1 = [1 1];
num2 = 2;
den2 = [1 0];
[numcl,dencl] = feedback(num1,den1,num2,den2,-1);
printsys(numcl,dencl)
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10. Time Response of 1st order Systems
The first order system has only one pole.
Where K is the D.C gain and T is the time constant of the system.
Time constant is a measure of how quickly a 1st order system responds to a unit step input.
D.C Gain of the system is ratio between the input signal and the steady state value of output.

11. Step Response To obtain the step response of the system num = 10;
den = [ ];
step(num,den) or
num = 10;
den = [ ];
sys=tf(num, den)
step(sys)

12. Impulse Response To find the step response of the system num = 10;
den = [ ];
impulse(num,den)
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13. Ramp Response To find the ramp response of the system t = 0:0.01:10
r = t;
num = 10;
den = [2 1];
lsim(num,den,r,t)
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14. Time Response of 2nd Order Systems
A general second-order system is characterized by the following transfer function.

15. Undamped Natural Frequency & Damping ratio
Determine the un-damped natural frequency and damping ratio of the following second order system.
To find the undamped natural frequency and damping ratio in matlab
num=4
den=[ ]
sys=tf(num, den)
damp(sys)

16. Bode Plots To obtain the bode plot use following MATLAB code
num =10*[ ];
den1= [ ];
den2=[ ];
den3=[ ];
den12=conv(den1,den2);
den=conv(den12,den3);
bode(num,den)
grid on
K =10;
zero=-10;
pole=[ ];
sys=zpk(k, zero, pole);
bode(sys)
grid on
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17. Bode Plots

18. Bode Plots (for specific Frequency range)
To obtain the bode plot for specific frequency range
w=0.1:0.1:100;
num =10*[ ];
den1= [ ];
den2=[ ];
den3=[ ];
den12=conv(den1,den2);
den=conv(den12,den3);
bode(num,den,w)
grid on
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19. Bode Plots (for specific Frequency range)

20. [mag, pha, w] = bode(num, den, w)
Bode Plots (with left hand arguments)
When invoked with left-hand arguments, such as
[mag, pha, w] = bode(num, den, w)
bode returns the magnitude (mag), phase (pha) and frequency (w) of the system in matrices.

21. Polar plots or Nyquist plot
To obtain the Nyquist plot use the following MATLAB code
num =2500;
den1= [ ];
den2=[ ];
den3=[ ];
den12=conv(den1,den2);
den=conv(den12,den3);
nyquist(num,den)
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22. Polar plots or Nyquist plot-w w
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23. Polar plots or Nyquist plot
To adjust the default axes of Nyquist plot use axis command
num =2500;
den1= [ ];
den2=[ ];
den3=[ ];
den12=conv(den1,den2);
den=conv(den12,den3);
nyquist(num,den)
axis([ ])
grid on
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24. Polar plots or Nyquist plot
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25. Nyquist plot (Open-loop & Closed Loop Frequency Response )

26. Magnitude-Phase plot or Nichols Chart
To obtain the Nichols plot use following MATLAB code
num =2500;
den1= [ ];
den2=[ ];
den3=[ ];
den12=conv(den1,den2);
den=conv(den12,den3);
nichols(num,den)
ngrid
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27. Magnitude-Phase plot or Nichols Chart

28. Magnitude-Phase plot or Nichols Chart
To obtain the Nichols plot use following MATLAB code
num =2500;
den1= [ ];
den2=[ ];
den3=[ ];
den12=conv(den1,den2);
den=conv(den12,den3);
nichols(num,den)
ngrid; axis([ ])
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29. Magnitude-Phase plot or Nichols Chart

30. Magnitude-Phase plot or Nichols Chart

31. Phase & Gain Margins
To obtain the gain margin and phase use the following mat lab code.
num =2500;
den1= [ ];
den2=[ ];
den3=[ ];
den12=conv(den1,den2);
den=conv(den12,den3)
[GM, PM, wp, wg]=margin(num,den)

32. Phase & Gain Margins GM = 5.5000 PM = 31.7124 num =2500; den1= [1 0];
To obtain the gain margin and phase use the following mat lab code.
GM =
5.5000
PM =
wp =
wg =
6.2184
num =2500;
den1= [ ];
den2=[ ];
den3=[ ];
den12=conv(den1,den2);
den=conv(den12,den3)
[GM, PM, wp, wg]=margin(num,den)

33. Phase & Gain Margins
To obtain the gain margin and phase use the following mat lab code.
num =2500;
den1= [ ];
den2=[ ];
den3=[ ];
den12=conv(den1,den2);
den=conv(den12,den3)
margin(num,den)

34. Phase & Gain Margins

35. System Characteristics (Bode Plot)

36. System Characteristics (Nyquist Plot)

37. System Characteristics (Nichol’s Chart)

38. Root Locus of 1st Order System
Consider the following unity feedback system
Matlab Code
num=1;
den=[1 0];
G=tf(num,den);
rlocus(G)
sgrid

39. Root Locus of 1st Order System
Consider the following unity feedback system
num=[1 0];
den=[1 1];
G=tf(num,den);
rlocus(G)
sgrid
Continued…..

40. Root Locus 2nd order systems
Consider the following unity feedback system
num=1;
den1=[1 0];
den2=[1 3];
den=conv(den1,den2);
G=tf(num,den);
rlocus(G)
sgrid
Determine the location of closed loop poles that will modify the damping ratio to 0.8 and natural undapmed frequency to 1.7 r/sec. Also determine the gain K at that point.

41. Root Locus of Higher Order Systems
Consider the following unity feedback system
num=1;
den1=[1 0];
den2=[1 1];
den3=[1 2];
den12=conv(den1,den2);
den=conv(den12,den3);
G=tf(num,den);
rlocus(G)
sgrid
Determine the closed loop gain that would make the system marginally stable.

42. Root Locus of a Zero-Pole-Gain Model
k=2;
z=-5;
p=[ ];
G=zpk(z,p,k);
rlocus(G)
sgrid

43. Root Locus of a State-Space Model
B=[1;0];
C=[1 0];
D=0;
sys=ss(A,B,C,D);
rlocus(sys)
sgrid

44. Choosing Desired Gain num=[2 3]; den=[3 4 1]; G=tf(num,den);
[kd,poles]=rlocfind(G)
sgrid

45. Please attach the hard copy of following exercises under the title solution to lab-6 in your practical book.
Exercises

46. Exercise#1
Simplify the following block diagram and determine the following (Assume K=10).
Closed loop transfer function (C/R)
Poles
Zeros
Pole-zero-map

47. Exercise#2
Simplify the following block diagram and determine the following.
Closed loop transfer function (C/R)
Poles
Zeros
Pole-Zero-map + - R C

48. Exercise-3: Obtain the step and ramp responses of the following 1st order system for T=1, 2, 3 5, 10 seconds.
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49. Exercise-4: Obtain the step and ramp responses of the following 1st order system for K=1, 5, 10 and 15.
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50. Exercise-5: Obtain the step response of the following 1st order system with zero for
K=1, α=4 and T=3 sec
K=1, α=3 and T=4 sec
K=1, α=3 and T=3 sec
And compare the results with system w/o zero
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51. Exercise#6
Describe the nature of the second-order system response via the value of the damping ratio for the systems with transfer function

52. Exercise-7: Obtain the pole zero map and step response of the 2nd order system and determine the mode of damping in the system. If the system is underdamped obtain the time domain specifications. On the pole zero map show that corresponding damping ratio and natural undamped frequency of the poles.
ωn=3 r/s and ζ=1
ωn=3 r/s and ζ=2
ωn=3 r/s and ζ=0.1
ωn=3 r/s and ζ=0.5
ωn=3 r/s and ζ=0

53. Exercise-8: Obtain the step response of the 2nd order system if
ωn=0.1 r/s and ζ=0.5
ωn=0.3 r/s and ζ=0.5
ωn=0.6 r/s and ζ=0.5
ωn=1 r/s and ζ=0.5
ωn=1.5 r/s and ζ=0.5

54. Exercise#9
Consider the system shown in following figure, where damping ratio is 0.6 and natural undamped frequency is 5 rad/sec. Obtain the rise time tr, peak time tp, maximum overshoot Mp, and settling time 2% and 5% criterion ts when the system is subjected to a unit-step input.

55. Bode Plots
Exercise-10: - Obtain the bode plot of the second order system
ωn = 0.1 rad / sec
ζ = 0.1, 0.5, 1, 1.5
G(S) =
(S S+0.01)
0.01
G(S) =
(S S+0.01)
0.01
G(S) =
(S S+0.01)
0.01
G(S) =
(S S+0.01)
0.01
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56. Bode Plots
Exercise-11: - Calculate the magnitude and phase of the following system at w (rad/sec)=0.1, 0.5, 1, 10, 100.

57. Polar plots or Nyquist plot
Exercise-12: - consider following transfer function
Obtain the Nyquist plot of the following system (when w>0).
Determine the open-loop & closed-loop magnitude responses when w=2.5 rad/sec
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58. Exercise-13: - For the following Transfer Function
(i) Obtain the Nichols Chart
(ii) Determine the open-loop as well as closed-loop magnitude and phase when w=1.39 rad/sec.
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59. Exercise#14
Obtain the phase and gain margins of the system shown in following figure for the two cases where K=10 and K=100.

60. Exercise#15
Consider the system shown in following figure. Obtain Bode diagram for the closed-loop transfer function. Obtain also the resonant peak, resonant frequency, and bandwidth.

61. Plot the root locus of following first order systems.
Exercise#16
Plot the root locus of following first order systems.

62. Plot the root locus of following 2nd order systems.
Exercise#17
Plot the root locus of following 2nd order systems.
(1)
(2)

63. Exercise#18
Plot the root locus of following systems.
(1)
(2)

64. Exercise#19
Plot the root Loci for the above ZPK model and find out the location of closed loop poles for =0.505 and n=8.04 r/sec.
b=0.505;
wn=8.04;
sgrid(b, wn)
axis equal
Open File  New  M-File
Save File as H:\ECE4115\Module1\Mod1_1.m

65. Exercise#20: Consider the following unity feedback system
Plot the root Loci for the above transfer function
Find the gain when both the roots are equal
Also find the roots at that point
Determine the settling of the system when two roots are equal.
Open File  New  M-File
Save File as H:\ECE4115\Module1\Mod1_1.m

66. Exercise#21 Consider the following velocity feedback system
Plot the root Loci for the above system
Determine the gain K at which the system produces sustained oscillations with frequency 8 rad/sec.
Open File  New  M-File
Save File as H:\ECE4115\Module1\Mod1_1.m

67. End of Tutorial You can Download this tutorial from
End of Tutorial