1. ERT 210 DYNAMICS AND PROCESS CONTROL CHAPTER 11 – MATLAB TUTORIAL
Prepared by:
Miss Hairul Nazirah Abdul Halim

2. Transfer function
To model a transfer function in MATLAB, define the numerator and denominator polynomials as 1-row matrices first.
Then use the ‘ tf ’ command in Matlab.

3. Example 1 Transfer function is given by:
Model the transfer function in MATLAB.
Solution
Using MATLAB, define the denominator and numerator as matrices:
>> num = 1;
>> den = [1 2 3];
>> G = tf(num,den)
OR >> G = tf(1, [1 2 3])

4. Example 2 Model the following transfer function in MATLAB. Solution
Using MATLAB, define the denominator and numerator as matrices:
>> num = 1;
>> den = [1e-9 1e-6 1];
>> G = tf(num,den)
OR >> G = tf(1, [1e-9 1e-6 1])

5. Exercise 1 1) 2)

6. Model interconnections
Multiple models can be manipulated using normal mathematical functions (addition and multiplication)
For models in series, multiply them
For models in parallel, add them

7. Model interconnections

8. Model interconnections

9. Model interconnections
Example 3
Given that
Find the total transfer function if both of them are:
a) Connected in series (multiply)
b) Connected in parallel (add)

10. Model interconnections
Solution
>> G1 = tf([1 4], [1 3 2])
>> G2 = tf(1, [1 5])
>> G_series = G1*G2
>> G_parallel = G1+G2

11. Model interconnections
Example 4
For feedback loop above, overall transfer
function is

12. >> gl = tf ([1 4] , [1 3 2])
>> g2 = tf (1, [1 5]);
>> g_closed = feedback(gl,g2)

13. Example 5 In the diagram above, it is given that
Find the overall transfer function using MATLAB

14. Solution
>> g1=tf(1, [1 2]);
>> g2=tf([1 3] , [1 4 5]);
>> g3=tf(4, [1 0]);
>> g4=tf(10, [1 10]);
>> g12=g1+g2;
>> g123=g12*g3;
>> g_overall=feedback(g123,g4);

15. Analyzing Dynamic System
Poles and Zeros
Characteristic equation
System stability
System Performance

16. Effects of Poles and Zeros
Poles and zeros determine the performance of a system, especially poles
For a transfer function, zeros are the roots of the numerator and poles are the roots of the denominator
To find zeros, make the numerator equal to zero, and solve for s.
To find poles, make the denominator equal to zero and solve for s.

17. Poles and zeros To find zeros, equate numerator to zero
To find poles, equate denominator to zero

18. Poles and zeros Example 6
Poles and zeros can be automatically found in MATLAB using the ‘pole’ and ‘zero’ command
>> g=tf ([1 1] , [ ])
>> z=zero(g)
>> p=pole(g)

19. Poles and zeros Exercise 3 Find the poles and zeros of the following
transfer function using MATLAB

20. Poles and zeros Complicated polynomials
Multiplying and solving complicated polynomials can be very difficult
The operation above can be done using
The ‘conv’ command
>> conv([1 3 1], [ ])

21. Complicated polynomials
The solution of the equation above can be calculated using the ‘roots’ command
>> roots([ ])

22. Characteristic equation
‘denominator = 0’ has a special name:
THE CHARACTERISTIC EQUATION
It determines the main characteristics of the system

23. Characteristic equation
The characteristic equation of the system above is
The poles of the system are the solutions of the characteristic equation.

24. System Stability
If all the poles are negative then the output will go towards D when t goes towards infinity.
If any of the poles are positive then the output will go towards infinity when t goes towards infinity (unstable).
Therefore, for a stable response, the poles must all lie in the negative left half plane.

25. System Stability
Stability of system by pole location

26. System Stability Effect of poles and zeros
Poles of a system can be classified into 2 categories:
• Real s = -1, -1.5, -5
• Complex conjugate (always appear in pairs with imaginary part)
s = -1 ± j3, -2 ± j(2)-1/2

27. System Stability
Real and complex poles

28. Effect of real poles

29. Effect of real poles
• Will get an ‘aperiodic’ (no oscillation) response
• No overshoot
• Will converge exponentially towards a final value
• If the pole has a bigger ‘magnitude’, the convergence will be faster

30. Effect of complex poles
• Will have overshoot
• Converges with oscillation (decaying sine wave) towards a final value for stable systems

31. Effect of complex poles

32. Zeros
Zeros have not been studied in great details in Control.
Rule of thumb - if a zero is near a pole, it will ‘cancel’ out the effect of the pole, depending on how near it is. The nearer it is, the less is the effect of the pole.

33. Poles and zeros Exercise 4
Check the stability of the following systems and verify it with the ‘step’ response

34. Example 7
Perform the step response of the following system:

35. >> g1 = tf(1, [1 1])
>> step(g1)

36. >> g2 = tf(4, [1 4 8])
>> step(g2)

37. Root Locus
Used to graphically show the movement of the closed loop poles when the system gain is changed from 0 to infinity

38. Root Locus
For the system above:
The closed loop transfer function is

39. Root Locus To find the poles of the closed loop system
The closed loop characteristic equation depends on K.
So the closed loop poles will also depend on K.
The root locus technique shows the locus of the closed-loop poles as K is changed.

40. Root Locus
Example 8
Draw the root locus of the following system:

41. Solution
>> num=10;
>> den1=[1 0];
>> den2=[1 3];
>> den3=[1 5];
>> denl2=conv(den1,den2);
>> den=conv(denl2,den3);
>> g=tf(num,den);
>>r1ocus(g)

42. Root Locus

43. Root Locus Interpretation of the root locus plot
• When the gain K = 0, the closed loop poles are the open loop poles at s = 0, -3, -5.
• As the gain K is increased, the poles at 0 and -3 move towards each other, while the pole at -5 moves further into the LHP.
• When K = 0.821, the poles from 0 and -3 meet at -1.21, and move upwards.
• When K = 12, the upward moving poles reach the imaginary axis at If the gain K is increased any further, the closed loop system goes unstable.
• To get those points, click on the root locus plot and drag along. The information is given.