1. PID Control Professor Walter W. Olson
Department of Mechanical, Industrial and Manufacturing Engineering
University of Toledo
PID Control

2. Outline of Today’s Lecture
Review
Margins from Nyquist Plots
Margins from Bode Plot
Non Minimum Phase Systems
Ideal PID Controller
Proportional Control
Proportional-Integral Control
Proportional-Integral Derivative Control
Ziegler Nichols Tuning

3. Margins
Margins are the range from the current system design to the edge of instability. We will determine
Gain Margin
How much can gain be increased?
Formally: the smallest multiple amount the gain can be increased before the closed loop response is unstable.
Phase Margin
How much further can the phase be shifted?
Formally: the smallest amount the phase can be increased before the closed loop response is unstable.
Stability Margin
How far is the the system from the critical point?

4. Gain and Phase Margin Definition Nyquist Plot-1

5. Gain and Phase Margin Definition Bode Plots
Magnitude, dB
Positive Gain Margin w
Phase, deg
-180
Phase Margin w
Phase Crossover Frequency

6. Stability Margin
It is possible for a system to have relatively large gain and phase margins, yet be relatively unstable.
Stability
margin, sm

7. Non-Minimum Phase Systems
Non minimum phase systems are those systems which have poles on the right hand side of the plane: they have positive real parts.
This terminology comes from a phase shift with sinusoidal inputs
Consider the transfer functions
The magnitude plots of a Bode diagram are exactly the same but the phase has a major difference:

8. Another Non Minimum Phase System A Delay
Delays are modeled by the function which multiplies the T.F.

9. Proportional-Integral-Derivative Controller
Based on a survey of over eleven thousand controllers in the refining, chemicals and pulp and paper industries, 97% of regulatory controllers utilize PID feedback.
L. Desborough and R. Miller, 2002 [DM02].
PID Control, originally developed in 1890’s in the form of motor governors, which were manually adjusted
The first theory of PID Control was published by a Russian (Minorsky) who was working for the US Navy in 1922
The first papers regarding tuning appeared in the early 1940’s
Today. there are several hundred different rules for tuning PID controllers (See Dwyer, 2003, who has cataloged the major methods)
While most of the discussion is about the “ideal” PID controller, there are many forms of the PID controller

10. PID Control Advantages Disadvantages Process independent
The best controller where the specifics of the process can not be modeled
Leads to a “reasonable” solution when tuned for most situations
Inexpensive: Most of the modern controllers are PID
Can be tuned without a great amount of experience required
Disadvantages
Not optimal
Can be unstable unless tuned properly
Not dependent on the process
Hunting (oscillation about an operating point)
Derivative noise amplification

11. The Ideal PID Controller
The input/output realtionship for the PID Controller is the Integral-Differential Equation
The ideal PID controller has the transfer function
Structurally it would look like + -

12. The Ideal PID Controller
The system transfer function is + -

13. Proportional Control + -
Y(s)
R(s)

14. Proportional Control

15. Proportional Controller
kp=7.2

16. Proportional Controller
With kp=7.2,
We could reduce the error with a prefilter: }
Error
Also: the response time is poor + -1
Y(s)=a(s)
R(s)
0.139

17. Proportional – Integral Controller
Most controllers using this technology are of this form:
This reacts to the system error and reduces it + -
R(s)
Y(s)

18. Proportion-Integral Control
Applying PI control to the F-16 Elevator,
Response time
improved with
no error

19. Proportional- Integral-Derivative Control
The derivative component is rarely used.
Reduces overshoot
May slows the response time depending on the system
Sensitive to noise
For the F-16 Elevator

20. PID Tuning
Tuning is the choosing of the parameters kd, ki, and kp, for a PID Controller
The oldest and most used method of tuning are the Ziegler- Nichols (ZN) methods developed in the 1940’s.
The first method is based on the assumption that the process without its feedback loop performs with a 1st order transfer function, perhaps with a transport delay
The second method assumes that a higher order system has dominant poles which can be excited by gain to the point of steady oscillation
In order to establish the constants for computing the parameters simple tests are performed of the process

21. Ziegler-Nichols PID Tuning Method 1 for First Order Systems
A system with a transfer function of the form has the time response to a unit step input:
This response might also be generated from a higher order system that is has high damping.

22. Ziegler-Nichols PID Tuning Method 1 for First Order Systems
The advice given is to draw a line tangent to the response curve through the inflection point of the curve.
However, a mathematical first order response doesn’t have a point of inflection as it is of the form (at no place does the 2nd derivative change sign.) My advice: place the line tangent to the initial curve slope
Adjust the gain K of the system by multiplying compensator by 1/K
Lag L
Rise
Time T
Type kp Ti Td P PI
PID

23. Ziegler-Nichols PID Tuning Method 1 for First Order Systems
For this example,
Type kp Ti Td P PI
PID
Lag L
Rise
Time T

24. Ziegler-Nichols PID Tuning Method 2 for Unknown Oscillatory System
The form of the transfer function unknown but the system can be put in steady oscillation by increasing the gain:
Increase gain, K, on closed loop system until the gain at steady oscillation, Kcr, is found
Then measure the critical period, Pcr
Apply table for controller constants and multiply by system gain 1/K 1
Cycle
Type kp Ti Td P PI
PID
Pcr

25. Ziegler-Nichols PID Tuning Method 2 Example
k=1000
k=2000
k=1500
k=1750
Type kp Ti Td P PI
PID
1.0 1
Cycle
17.8
Pcr
Kcr=1875

26. Ziegler-Nichols PID Tuning Method 2 Example
1.0 1
Cycle
17.8
Pcr
Kcr=1875

27. Summary Ideal PID Controller Proportional Control
Proportional-Integral Control
Proportional-Integral Derivative Control
Ziegler-Nichols Tuning
Next Class: PID Controls Continued