1. CHE 185 – PROCESS CONTROL AND DYNAMICS
PID CONTROLLER FUNDAMENTALS

2. CLOSED LOOP COMPONENTS
GENERAL DEFINITIONS
OPEN LOOPS ARE MANUAL CONTROL
FEEDBACK LOOPS ARE CLOSED
EXAMPLE P&ID FOR FEEDBACK CONTROL LOOP

3. CLOSED LOOP COMPONENTS
GENERAL BLOCK DIAGRAM FOR FEEDBACK CONTROL LOOP (FIGURE FROM TEXT)

4. CLOSED LOOP COMPONENTS
OVERALL TRANSFER FUNCTION

5. TYPICAL TRANSFER FUNCTIONS FOR A FEEDBACK LOOP
CONSIDER THE RESPONSE TO A DISTURBANCE
WITH CONSTANT S/P (Ysp(s) = 0)
REGULATORY CONTROL OR DISTURBANCE REJECTION
THIS REPRESENTS A PROCESS AT
STEADY STATE RESPONDING TO
BACKGROUND DISTURBANCES

6. TYPICAL TRANSFER FUNCTIONS FOR A FEEDBACK LOOP
CONSIDER THE SETPOINT RESPONSE
WITH NO DISTURBANCE (D(s) = 0)
SETPOINT TRACKING OR SERVO CONTROL
THIS MODEL REPRESENTS THE SYSTEM RESPONSE TO A S/P ADJUSTMENT

7. TYPICAL TRANSFER FUNCTIONS FOR A FEEDBACK LOOP
GENERALIZATIONS REGARDING THE FORM OF THE TRANSFER FUNCTIONS
THE NUMERATOR IS THE PRODUCT OF ALL TRANSFER FUNCTIONS BETWEEN THE INPUT AND THE OUTPUT
THE DENOMINATOR IS EQUAL TO THE NUMERATOR + 1

8. TYPICAL TRANSFER FUNCTIONS FOR A FEEDBACK LOOP
CHARACTERISTIC EQUATION
oBTAINED BY SETTING THE DENOMINATOR = 0
ROOTS FOR THIS EQUATION WILL BE:
OVERDAMPED LOOP
COMPLEX ROOTS, FOR AN OSCILLATORY LOOP
AT LEAST ONE REAL POSITIVE ROOT FOR AN UNSTABLE LOOP

9. Feedback Control Analysis
The loop gain (KcKaKpKs) should be positive for stable feedback control.
An open-loop unstable process can be made stable by applying the proper level of feedback control.

10. Characteristic Equation Example
Consider the dynamic behavior of a P-only controller applied to a CST thermal mixer (Kp=1; τp=60 sec) where the temperature sensor has a τs=20 sec and τa is assumed small. Note that Gc(s)=Kc.

11. Characteristic Equation Example- closed loop poles
When Kc =0, poles are and which correspond to the inverse of τp and τs.
As Kc is increased from zero, the values of the poles begin to approach one another.
Critically damped behavior occurs when the poles are equal.
Underdamped behavior results when Kc is increased further due to the imaginary components in the poles.

12. PID ALGORITHM - POSITION FORM
ISA POSITION FORM FOR PID:
FOR PROPORTIONAL ONLY

13. Definition of Terms e(t) - the error from setpoint [e(t)=ysp-ys].
Kc - the controller gain is a tuning parameter and largely determines the controller aggressiveness.
τI - the reset time is a tuning parameter and determines the amount of integral action.
τD - the derivative time is a tuning parameter and determines the amount of derivative action.

14. PID Controller Transfer Function

15. PID ALGORITHM - POSITION FORM
FOR PROPORTIONAL/INTEGRAL:
FOR PROPORTIONAL/DERIVATIVE

16. PID ALGORITHM - POSITION FORM
TRANSFER FUNCTION FOR PID CONTROLLER:

17. PID ALGORITHM - POSITION FORM
DERIVATIVE KICK:
RESULTS FROM AN ERROR SPIKE (INCREASE IN 𝑑𝑒(𝑡) 𝑑𝑡 ) WHEN A SETPOINT CHANGE IS INITIATED
CAN BE ELIMINATED BY REPLACING THE CHANGE IN ERROR WITH A CHANGE IN THE CONTROLLED VARIABLE −𝑑 𝑦 𝑠 (𝑡) 𝑑𝑡 IN THE PID ALGORITHM
RESULTING EQUATION IS CALLED THE DERIVATIVE-ON-MEASUREMENT FORM OF THE PID ALGORITHM

18. DIGITAL VERSIONS OF THE PID ALGORITHM
DIGITAL CONTROL SYSTEMS REQUIRE CONVERSION OF ANALOG SIGNALS TO DIGITAL SIGNALS FOR PROCESSING.
DIGITAL VERSION OF THE PREVIOUS EQUATION IN DIGITAL FORMAT BASED ON A SINGLE TIME INTERVAL, Δt: YIELDS THE VELOCITY FORM OF THE PID ALGORITHM

19. DIGITAL VERSIONS OF THE PID ALGORITHM
FOR INTEGRATION OVER A TIME PERIOD, t, WHERE n = t/Δt:

20. DIGITAL VERSIONS OF THE PID ALGORITHM
PROPORTIONAL KICK
RESULTS FROM THE INITIAL RESPONSE TO A SETPOINT CHANGE
CAN BE ELIMINATED IN THE VELOCITY EQUATION BY REPLACING THE ERROR TERM IN THE ALGORITHM WITH THE SENSOR TERM

21. First Order Process with a PI Controller Example

22. PI Controller Applied to a Second Order Process Example

23. PROPORTIONAL ACTION
USES A MULTIPLE OF THE ERROR AS A SIGNAL TO THE CONTROLLER, CONTROLLER GAIN,
HAS INVERSE UNITS TO PROCESS GAIN

24. Proportional Action Properties
Closed loop transfer function base on a P-only controller applied to a first order process.
Properties of P control
Does not change order of process
Closed loop time constant is smaller than open loop τp
Does not eliminate offset.

25. P-only Control Offset

26. Proportional Response Action with a PI Controller

27. PROPORTIONAL CONTROL RESPONSE OF FIRST ORDER PROCESS TO STEP FUNCTION
OPEN LOOP - NO CONTROL
CLOSED LOOP - PROPORTIONAL CONTROL

28. PROPORTIONAL CONTROL
PROPORTIONAL CONTROL MEANS THE CLOSED SYSTEM RESPONDS QUICKER THAN THE OPEN SYSTEM TO A CHANGE.
OFFSET IS A RESULT OF PROPORTIONAL CONTROL. AS t INCREASES, THE RESULT IS:

29. Integral Action
The primary benefit of integral action is that it removes offset from setpoint.
In addition, for a PI controller all the steady-state change in the controller output results from integral action.

30. INTEGRAL ACTION
WHERE PROPORTIONAL MODE GOES TO A NEW STEADY-STATE VALUE WITH OFFSET, INTEGRAL DOES NOT HAVE A LIMIT IN TIME, AND PERSISTS AS LONG AS THERE IS A DIFFERENCE.
INTEGRAL WORKS ON THE CONTROLLER GAIN
INTEGRAL SLOWS DOWN THE RESPONSE OF THE CONTROLLER WHEN PRESENT WITH PROPORTIONAL

31. INTEGRAL ACTION
INTEGRAL ADDS AN ORDER TO THE CONTROL FUNCTION FOR A CLOSED LOOP
FOR THE FIRST ORDER PROCESS WITH PI CONTROL, THE TRANSFER FUNCTION IS:
WHERE
AND

32. Derivative Action Properties
THE DERIVATIVE MODE RESPONDS TO THE SLOPE
THIS MODE AMPLIFIES SUDDEN CHANGES IN THE CONTROLLER INPUT SIGNAL - INCREASES CONTROLLER SENSITIVITY

33. Derivative Action Properties
DERIVATIVE MODE CAN COUNTERACT INTEGRAL MODE TO SPEED UP THE RESPONSE OF THE CONTROLLER.
DERIVATIVE DOES NOT REMOVE OFFSET
IMPROPER TUNING CAN RESULT IN HIGH-FREQUENCY VARIATION IN THE MANIPULATED VARIABLE
7.6 DOES NOT WORK WELL WITH NOISY SYSTEMS

34. Derivative Action Properties
Properties of derivative control:
Does not change the order of the process
Does not eliminate offset
Reduces the oscillatory nature of the feedback response
Closed loop transfer function for derivative-only control applied to a second order process.

35. Derivative Action Response for a PID Controller

36. Derivative Action
The primary benefit of derivative action is that it reduces the oscillatory nature of the closed-loop response.