pid

1. CHE 185 – PROCESS
CONTROL AND DYNAMICS
SECOND AND HIGHER ORDER
PROCESSES

2. 2
SECOND ORDER
PROCESSES
CHARACTERIZATION
• CAN RESULT FROM TWO FIRST ORDER
OR ONE SECOND ORDER ODE
• GENERAL FORM OF THE SECOND
ORDER EQUATION AND THE
ASSOCIATED TRANSFER FUNCTION

3. CHARACTERISTIC EQUATION
• POLYNOMIAL FORMED FROM THE COEFFICIENTS OF
THE EQUATION IN TERMS OF y:
• THREE POSSIBLE SOLUTIONS FOR THE STEP
RESPONSE OF PROCESSES DESCRIBED BY THIS
EQUATION. USING THE NORMAL QUADRATIC SOLUTION
FORMULA:

4. ROOT OPTIONS 1 𝝵 > 1
• TWO REAL, DISTINCT ROOTS WHEN 𝝵 > 1,
OVERDAMPED. SOLUTION FOR A UNIT STEP (STEP SIZE
1) IS GIVEN BY:
• SEE FIGURE 6.4.1
• RESPONSE TAKES TIME TO BUILD UP TO ITS MAXIMUM
GRADIENT.
• THE MORE SLUGGISH THE RATE OF RESPONSE THE LARGER
THE DAMPING FACTOR
• FOR ALL DAMPING FACTORS, RESPONSES HEAD TOWARDS
THE SAME FINAL STEADY-STATE VALUE

5. ROOT OPTIONS 2 𝝵 = 1
• TWO REAL EQUAL ROOTS WHEN 𝝵 = 1, CRITICALLY
DAMPED. SOLUTION FOR A UNIT STEP (STEP SIZE 1) IS
GIVEN BY:
• SEE FIGURE 6.4.1
• RESULTS LOOK VERY SIMILAR TO THE OVERDAMPED
RESPONSES.
• THIS REPRESENTS THE LIMITING CASE - IT IS THE FASTEST
FORM OF THIS NON-OSCILLATORY RESPONSE

6. ROOT OPTIONS 3 𝝵 < 1
• TWO COMPLEX CONJUGATE ROOTS (a + ib, a- ib) WHEN
𝝵 < 1, UNDERDAMPED. SOLUTION FOR A UNIT STEP
(STEP SIZE 1) IS GIVEN BY:
• SEE FIGURE 6.4.2
• THE RESPONSE IS SLOW
TO BUILD UP SPEED.
• RESPONSE BECOMES FASTER
AND MORE OSCILLATORY AND
AMOUNT OF OVERSHOOT
INCREASES, AS FACTOR FALLS
FURTHER BELOW 1.
• REGARDLESS OF THE DAMPING FACTOR, ALL THE
RESPONSES SETTLE AT THE SAME FINAL STEADY-STATE
VALUE (DETERMINED BY THE STEADY-STATE GAIN OF THE
PROCESS)

7. 7
SECOND ORDER
PROCESSES
CHARACTERIZATION
• NOTE THAT THE GAIN, TIME CONSTANT,
AND THE DAMPING FACTOR DEFINE THE
DYNAMIC BEHAVIOR OF 2ND ORDER
PROCESS.

8. DAMPING FACTORS, ζ
• DAMPING FACTORS, ζ , ARE
REPRESENTED BY FIGURES 6.4.1
THROUGH 6.4.4 IN THE TEXT, FOR A
STEP CHANGE
• TYPES OF DAMPING FACTORS
– UNDERDAMPED
– CRITICALLY DAMPED
– OVERDAMPED
8

9. UNDERDAMPED
CHARACTERISTICS
• FIGURES 6.4.2 THROUGH 6.4.4
• 𝜁 < 1
• PERIODIC BEHAVIOR
• COMPLEX ROOTS
• FOR THE STEP CHANGE, t > 0:
9

10. UNDERDAMPED
CHARACTERISTICS
• EFFECT OF ζ (0.1 TO 1.0) ON
UNDERDAMPED RESPONSE:
10

11. UNDERDAMPED
CHARACTERISTICS
• EFFECT OF ζ (0.0 TO -0.1) ON
UNDERDAMPED RESPONSE:
11

12. OVERDAMPED
CHARACTERISTICS
• FIGURE 6.4.1
• 𝜁 > 1
• NONPERIODIC BEHAVIOR
• REAL ROOTS
• FOR THE STEP CHANGE, t > 0:
12

13. CRITICALLY DAMPED
CHARACTERISTICS
• FIGURE 6.4.1 AND 6.4.2
• 𝜁 = 1
• NONPERIODIC BEHAVIOR
• REPEATED REAL ROOTS
• FOR THE STEP CHANGE, t > 0:
13

14. CHARACTERISTICS OF AN
UNDERDAMPED RESPONSE
• RISE TIME
• OVERSHOOT
(B)
• DECAY RATIO
(C/B)
• SETTLING OR
RESPONSE
TIME
• PERIOD (T)
• FIGURE 6.4.4

15. EXAMPLES OF 2ND ORDER
SYSTEMS
• THE GRAVITY DRAINED TANKS AND THE
HEAT EXCHANGER IN THE SIMULATION
PROGRAM ARE EXAMPLES OF SECOND
ORDER SYSTEMS
• PROCESSES WITH INTEGRATING
FUNCTIONS ARE ALSO SECOND ORDER.

16. 2ND ORDER PROCESS
EXAMPLE
• THE CLOSED LOOP PERFORMANCE OF A PROCESS
WITH A PI CONTROLLER CAN BEHAVE AS A SECOND
ORDER PROCESS.
• WHEN THE AGGRESSIVENESS OF THE CONTROLLER IS
VERY LOW, THE RESPONSE WILL BE OVERDAMPED.
• AS THE AGGRESSIVENESS OF THE CONTROLLER IS
INCREASED, THE RESPONSE WILL BECOME
UNDERDAMPED.

17. DETERMINING THE
PARAMETERS OF A 2ND ORDER
SYSTEM
• SEE EXAMPLE 6.6 TO SEE METHOD FOR
OBTAINING VALUES FROM TRANSFER
FUNCTION
• SEE EXAMPLE 6.7 TO SEE METHOD FOR
OBTAINING VALUES FROM MEASURED
DATA

18. 2ND ORDER PROCESS RISE
TIME
• TIME REQUIRED FOR CONTROLLED VARIABLE
TO REACH NEW STEADY STATE VALUE AFTER
A STEP CHANGE
• NOTE THE EFFECT FOR VALUES OF ζ FOR
UNDER, OVER AND CRITICALLY DAMPED
SYSTEMS.
• SHORT RISE TIMES ARE PREFERRED

19. 2ND ORDER PROCESS
OVERSHOOT
• MAXIMUM AMOUNT THE CONTROLLED
VARIABLE EXCEEDS THE NEW STEADY STATE
VALUE
• THIS VALUE BECOMES IMPORTANT IF THE
OVERSHOOT RESULTS IN EITHER
DEGRADATION OF EQUIPMENT OR UNDUE
STRESS ON THE SYSTEM

20. 2ND ORDER PROCESS DECAY
RATIO
• RATIO OF THE MAGNITUDE OF
SUCCESSIVE PEAKS IN THE RESPONSE
• A SMALL DECAY RATIO IS PREFERRED

21. 2ND ORDER PROCESS
OSCILLATORY PERIOD
• THE OSCILLATORY PERIOD OF A CYCLE
• IMPORTANT CHARACTERISTIC OF A
CLOSED LOOP SYSTEM

22. 2ND ORDER PROCESS
RESPONSE OR SETTLING TIME
• TIME REQUIRED TO ACHIEVE 95% OR
MORE OF THE FINAL STEP VALUE
• RELATED TO RISE TIME AND DECAY
RATIO
• SHORT TIME IS NORMALLY THE TARGET

23. HIGHER ORDER PROCESSES
• MAY BE CONSIDERED AS FIRST ORDER
FUNCTIONS
• GENERAL FORM

24. HIGHER ORDER PROCESSES
• THE LARGER n, THE MORE SLUGGISH
THE PROCESS RESPONSE (I.E., THE
LARGER THE EFFECTIVE DEADTIME
• TRANSFER FUNCTION