1. Response of Higher Order Systems
Steady-State
In terms of the deviation variables
deviation variables

2. Response of Higher Order Systems
Reduce the parameters from 4 to 3

3. Response of Higher Order Systems
Laplace Transform
Characteristic roots

4. Response of Higher Order Systems

5. Response of Higher Order Systems

6. Response of Higher Order Systems

7. Response of Higher Order Systems

8. Response of Higher Order Systems

9. Response of Higher Order Systems
The two time constants are equal

10. Response of Higher Order Systems

11. Response of Higher Order Systems

12. Response of Higher Order Systems

13. Response of Higher Order Systems

14. Response of Higher Order Systems

15. Response of Higher Order Systems

16. Response of Higher Order Systems

17. Response of Higher Order Systems

18. Rules of block diagrams

19. Rules of block diagrams

20. The Output Responses Output Response:
If all the roots of the denominator of the
transfer function are real, then
1- The response is monotonic (nonoscillatory).
2- It is stable only if all the roots are negative.

21. The Output Responses Output Response: Monotonic stable
Monotonic unstable

22. The Output Responses Output Response:
The time it takes for the transients to die out, we can see that each exponential term starts at unity (e0 = 1) and, if the root is negative, decays to zero with time. We define the threshold for each term of the response to become less than 1% of its initial value, as the threshold of each term to die out. To use a good round number, let erf = e-5 = , or 0.67%, which is less than 1%. Then the time required for the kth exponential term to reach 0.67% of its initial value is
tk = -5/rk
Thus the root with the smallest absolute value (least negative) will take the longest to die out. Such a root is called the dominant root of the response.

23. The Output Responses
Output Response:
Pair of Complex Conjugate Roots

24. The Output Responses Output Response: 1- The response is oscillatory.
Pair of Complex Conjugate Roots
1- The response is oscillatory.
2- The oscillations grow with time (unstable) if any of the pairs of complex roots has a positive real part.

25. The Output Responses Output Response: Pair of Complex Conjugate Roots
Oscillatory Unstable
Oscillatory Stable

26. The Output Responses
Output Response:
Final Steady-State Value
SUMMARY

27. The Output Responses Output Response: RESPONSE OF FIRST-ORDER SYSTEMS
Step Response

28. The Output Responses Output Response: RESPONSE OF FIRST-ORDER SYSTEMS
Ramp Response

29. The Output Responses Output Response: RESPONSE OF FIRST-ORDER SYSTEMS
Sinusoidal Response

30. The Output Responses Output Response: RESPONSE OF FIRST-ORDER SYSTEMS
Response with Time Delay

31. The Output Responses Output Response: RESPONSE OF FIRST-ORDER SYSTEMS
Response of a Lead-Lag Unit

32. The Output Responses
Output Response:
RESPONSE OF SECOND-ORDER SYSTEMS

33. The Output Responses
Output Response:
RESPONSE OF SECOND-ORDER SYSTEMS

34. The Output Responses Output Response: RESPONSE OF SECOND-ORDER SYSTEMS
Step Response
a- overdamped
b- critically damped

35. The Output Responses Output Response: RESPONSE OF SECOND-ORDER SYSTEMS
Ramp Response

36. The Output Responses Output Response: RESPONSE OF SECOND-ORDER SYSTEMS
Sinusoidal Response

37. The Output Responses Output Response: RESPONSE OF SECOND-ORDER SYSTEMS
Sinusoidal Response

38. The Output Responses Output Response: RESPONSE OF SECOND-ORDER SYSTEMS
Underdamped step response

39. The Output Responses Output Response: RESPONSE OF SECOND-ORDER SYSTEMS
Rise Time. This is the time it takes for the response to first reach its final steady-state
value, tR
Settling Time. This is the time it takes for the response to come within some prescribed
band of the final steady-state value and remain in this band. Typical band limits
are +/- 5%, +/- 3%, and +/-1% of the total change.