1. Chapter 5 Dynamics and Regulation of Low-order Systems
§ General Effects of Feedback
§ Dynamics and Regulation of 1st-order System
§ Dynamics and Regulation of 2nd-order System
§ Time-domain Specifications of System Performance

2. § 5.1 General Effects of Feedback (1)
Closed-loop System:
Negative Feedback Control:

3. § 5.1 General Effects of Feedback (2)
Proportional Regulator:
Under Regulation
Out of Regulation

4. § 5.1 General Effects of Feedback (3)
Proportional Controller:
Mechanical and Hydraulic Controller
Electronic Controller
Governing of mechanical plant is liberated by using OP to realize electronic controller.

5. § 5.1 General Effects of Feedback (4)
Effectiveness of Feedback:
(1) Set point regulation
Error Response
Following the first law of Newtonian Mechanics:
At rest Zero error in position regulation
At constant velocity Zero error in speed regulation

6. § 5.1 General Effects of Feedback (5)
Dynamic Response
Poles of Closed-loop Transfer Function:
The dynamic response is governed by new poles as D1(s)D2(s)+N1(s)N2(s)=0
For unity feedback system, feedback has no effect on system zeros.

7. § 5.1 General Effects of Feedback (6)
(2) Parameter Variations
I/O transmission is dominated by sensor for high loop gain.
System variations
The effect of large variation is investigated by using robust analysis.

8. § 5.1 General Effects of Feedback (7)
Sensitivity analysis (small variation)
Variation of G
Loop gain
The Sensitivity of I/O variation is reduced by employing high loop gain.
Variation of H
Feedback sensor directly and constantly affect the I/O transmission.
In general, is higher than High loop gain reduces the
I/O variation due to system parameters.

9. § 5.1 General Effects of Feedback (8)
(3) Disturbance Rejection
Disturbance Transmission

10. § 5.1 General Effects of Feedback (9)
Disturbance Error
Note: steady state = set point - |offset error| - |disturbance error|
|parameter uncertain error|
Overall Effects of Feedback
Advantages : Command following
Disturbance rejection
Improve system robustness
Disadvantages : Reduce gain
Increase system complexity
Introduce instability

11. § 5.2 Dynamics and Regulation of 1st-order Systems (1)
Definition – A system that can store energy in only one form and location Physical Examples:
System pole-zero diagram
t=0+ , ON
Input signal
Output signal
Input signal
Output signal
t=0+
Open gate
Input signal

12. § 5.2 Dynamics and Regulation of 1st-order Systems (2)
System and Input Model
Differential Eq. Model:
Transfer Function Model:
Standard Form of Pure Dynamics

13. § 5.2 Dynamics and Regulation of 1st-order Systems (3)
System Dynamics:
(1) Step response from differential equation

14. § 5.2 Dynamics and Regulation of 1st-order Systems (4)
Response of initial relaxed system

15. § 5.2 Dynamics and Regulation of 1st-order Systems (5)
(2) Step response from pole-zero diagram
Steady state (Pure static gain)
Transient (Pure dynamics)

16. § 5.2 Dynamics and Regulation of 1st-order Systems (6)
Overall response

17. § 5.2 Dynamics and Regulation of 1st-order Systems (7)
Closed-loop Regulation
Ex: Liquid-level regulation

18. § 5.2 Dynamics and Regulation of 1st-order Systems (8)
Unity Feedback Regulator
Set Point Regulation
Equivalent I/O transmission
Proportional regulator (Kp) changes the static amplification (K’) and response
speed ( ).

19. § 5.2 Dynamics and Regulation of 1st-order Systems (9)
(1) Effect of Kp on Pure Dynamics
Pole-zero diagram
Response is faster when the proportional gain is increased.

20. § 5.2 Dynamics and Regulation of 1st-order Systems (10)
(2) Effect of Kp on Steady State Response
Steady state error (offset) is reduced when the proportional gain is increased.

21. § 5.2 Dynamics and Regulation of 1st-order Systems (11)
Set Point and Response Regulation
Response speed is a pure dynamic behavior, therefore the comparisons of response speeds are referred to the steady state responses at different Kp.

22. § 5.2 Dynamics and Regulation of 1st-order Systems (12)
Disturbance Rejection

23. § 5.2 Dynamics and Regulation of 1st-order Systems (13)
High Gain Regulation
For infinite high gain Kp, the unity feedback regulator approaches ideal static system.
Finite gain regulation will improve the speed of dynamic response but it has offset
error in steady state.

24. § 5.3 Dynamics and Regulation of 2nd-order Systems (1)
Definition – A system having two separate energy-storage elements.
Physical Examples:
System pole-zero distribution

25. § 5.3 Dynamics and Regulation of 2nd-order Systems (2)
System and Input Model
Differential Eq. Model:
Transfer Function Model:
Standard Form of Pure Dynamics

26. § 5.3 Dynamics and Regulation of 2nd-order Systems (3)
Step Response From Differential Equation
For underdamping and initial relaxation system

27. § 5.3 Dynamics and Regulation of 2nd-order Systems (4)

28. § 5.3 Dynamics and Regulation of 2nd-order Systems (5)
Step Response From Pole-zero distribution
Pole-zero distribution
Poles distribution
Response:
Steady state
Transient

29. § 5.3 Dynamics and Regulation of 2nd-order Systems (6)
Overall response

30. § 5.3 Dynamics and Regulation of 2nd-order Systems (7)
Closed-loop Regulation
Ex: DC Servomechanism
Servo motor
Motion and power transmission by reduction gear
(Gear Ratio; N)
Position sensor by potentiometer

31. § 5.3 Dynamics and Regulation of 2nd-order Systems (8)
Command response
Disturbance response

32. § 5.3 Dynamics and Regulation of 2nd-order Systems (9)
Unity Feedback Regulator
Set Point Regulation
Equivalent I/O transmission
Proportional regulator (Kp) changes the static amplification (K’) and dynamic
response( ) through increasing the stiffness of a system.

33. § 5.3 Dynamics and Regulation of 2nd-order Systems (10)
(1) Regulation of Pure Dynamics
Pole-zero distribution
(2) Steady-state response
The same result as that of 1st-order system with offset error

34. § 5.3 Dynamics and Regulation of 2nd-order Systems (11)
Disturbance Rejection

35. § 5.4 Time-domain Specifications of System Performance (1)
Step Testing of Black-Box System
(2) Dynamic test (Within linear range)
(1) Static test

36. § 5.4 Time-domain Specifications of System Performance (2)
Step Testing and Identification of Low-order Systems
(1) Performance testing

37. § 5.4 Time-domain Specifications of System Performance (3)
(2) System Identification
1st-order system:
2nd-order system:

38. § 5.4 Time-domain Specifications of System Performance (4)
Effects of Additional Poles an Zeros
Effect of real pole
The effect of the real pole is to make the response more sluggish.
Effect of real LHP zero
The effect of the real zero is to make the response more oscillatory.

39. § 5.4 Time-domain Specifications of System Performance (5)
Effect of real RHP zero
RHP zero: Nonminimum-phase zero
The effect of nonminimum-phase zero is to cause initial reversal motion in step response.

40. § 5.4 Time-domain Specifications of System Performance (6)
Dynamic Model Simplification
Order reduction: High order system Low order model
Ignore real pole (zero) in oscillatory system modes:
Pole-zero cancellation:
Ignorance of far away poles and zeros:
Poles and zeros only have s.s. effects