1. Lec 14. PID Controller Design
Some commonly used controllers
Proportional Controller
Integration Controller
Derivative Controller
PID controller
Ziegler-Nichols tuning rules
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2. Design of Unity-Feedback Systems
Objectives in designing controller C(s)
Stability (first priority)
Steady state error (static)
Time specifications (dynamic)
controller
plant +

3. Some Commonly Used Controllers
Proportional (P) controller:
Integration (I) controller:
Derivative (D) controller:
Or combination of them, such as
Proportional plus Integration (PI) controller:
Controllers using root locus & frequency-response design
Phase-lead compensator
Phase-lag compensator
Lag-lead compensator

4. Proportional Controller
plant
𝐺(𝑠) +
Can change closed-loop pole locations along root locus
Increasing 𝐾 can decrease steady state tracking error, but cannot change system type
Stability may become an issue for large 𝐾

5. Example
1 𝑠+1 +

6. Integration Controller
plant
1 𝑠+1 +
A integration controller can increase the system type by 1
For a type 0 system, the compensated system is of type 1
Hence, steady state error for step input is completely eliminated
Compared with proportional controller, instability in general a bigger issue for large K

7. Step Responses

8. PI Controller 1 𝑠+1 Proportional controller: increase system response
Integration controller: increase system type (reduce tracking error)
Proportional plus Integration (PI) controller:
controller
plant
1 𝑠+1 +

9. Step Responses (T=1)

10. Derivative Controller
plant
𝐾 𝑑 𝑠 +
Derivative controller C(s)=Kds
Anticipate and correct error before it becomes too large
Add damping to the system
Highly sensitive to the error e(t)
Tend to increase the stability of the system
Almost always used along with P or PI controller

11. Example 1 𝐽 𝑠 2 𝐶(𝑠) Plant is a double integrator
controller
plant
1 𝐽 𝑠 2
𝐶(𝑠) +
Plant is a double integrator
Two choices for the controller C(s)
A proportional controller 𝐶 𝑠 = 𝐾 𝑝
A proportional plus derivative (PD) controller 𝐶 𝑠 = 𝐾 𝑝 + 𝐾 𝑑 𝑠

12. Proportional Controller
Step Response (J=10, Kp=1):
Changing Kp can only change the frequency of the undamped oscillation

13. PD Controller Proportional plus derivative controller
Closed-loop transfer function is a (non-standard) second order system
Step Response
(J=10, Kp=1):
Increasing the derivative component will lead to more damping in the step response, and improve the stability of the system

14. Step Response (J=10, Kd=10)
The improved stability caused by the derivative component allows us to choose larger gain for the proportional component, thus reducing steady state error (indirectly)

15. PID Controller
A combination of proportional, integration and derivative controllers:
Three parameters to adjust (more flexibility)
Widely used for controlling practical systems

16. PID Controller PID (Proportional-Integral-Derivative) Controller:
Controller output:
Three adjustable parameters:
Often used for plant with unknown model
controller
plant +

17. PID Tuning

18. First Method Obtain plant’s unit step response experimentally
For many plants, unit step response is S-shaped
Delay time L
Time constant T

19. Determining L and T

20. Ziegler-Nichols Turning Rule (First Method)
Type of Controller P PI
PID
(provided educated guesses for the parameter values)

21. Second Method (If no oscillation occurs for all values of Kp,
(If no oscillation occurs for all values of Kp,
this method is not applicable)
controller
plant +

22. Ziegler-Nichols Turning Rule (Second Method)
Type of Controller P PI
PID
(provided educated guesses for the parameter values)

23. Example
PID controller (second method):
controller
plant +

24. Step Responses
compensated by
compensated by
uncompensated

25. Example
PID controller (first method):