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

2. Outline of Today’s Lecture
Review
Ideal PID Controller
Proportional Control
Proportional-Integral Control
Proportional-Integral Derivative Control
Ziegler Nichols Tuning
PID Theory
Integrator Windup
Noise Improvement

3. 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

4. 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 for the problems
Can be unstable unless tuned properly
Not dependent on the process
Hunting (oscillation about an operating point)
Derivative noise amplification

5. 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 + -1

6. Proportional Control + -1
Y(s)
R(s)

7. Proportional – Integral Controller
Most controllers using this technology are of this form:
This reacts to the system error and reduces it + -1
R(s)
Y(s)

8. 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

9. 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.

10. 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
You also have to adjust for the gain K of the system by multiplying compensator by 1/K
Lag L
Rise
Time T
Type kp Ti Td P PI
PID

11. 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

12. PID: A Little Theory
Consider a 1st order function where the 1st method of Ziegler Nichols applies
The general transfer function for this system is
The term is the transport lag and delays the action for t0 seconds. Therefore
The term Ta is the time constant for the system. T measured on the graph is an estimate of this.

13. PID: A Little Theory
The method 1 PI controller applied to the loop equation is

14. PID: A Little Theory
In Method 2, the gain was increased until the system was nearly a perfect oscillatory system.
Since the gain changes the oscillatory patterns, the lowest order system that this could represent would by a 3rd order system.
For this system to oscillate, there must be a solution of the characteristic function for K real and positive where s=±wi

15. PID: A Little Theory
Applying the PI Controller:

16. Integrator Windup
We have tacitly assumed that the controlled devices could meet the demands of the controls that we designed.
However real devices have limitations that may prevent the system from responding adequately to the control signal
When this occurs with an integrating controller, the error which is used to amplify the control signal may build up and saturate the controller.
We refer to this as “integrator windup”:
the system can’t respond and the integrator signal is extremely large (often maxed out on a real controller)
the result is an uncontrolled system that can not return to normal operating conditions until the controller is reset

17. Integrator Windup
To avoid windup, a possible solution is to provide a correcting error from the actuator by adding another loop: (the actuator has to be extracted from the plant) + -1

18. Derivative Noise Improvement
A major problem with using the derivative part of the PID controller that the derivative has the effect of amplifying the high frequency components which, for most systems, is likely to be noise.
Without PID
With PID

19. Derivative Noise Improvement
One way to improve the noise rejection at higher frequencies is to apply a second order filter that passes low frequency and rejects high frequency
The natural frequency of the filter should be chosen as with N chosen to give the controller the bandwidth necessary, usually in the range of 2 to 20
The controller then has the design

20. Summary PID Theory Integrator Windup Noise Improvement
Next Class Loop Shaping