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

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. 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 THROUGH IN THE TEXT, FOR A STEP CHANGE
TYPES OF DAMPING FACTORS
UNDERDAMPED
CRITICALLY DAMPED
OVERDAMPED

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

10. UNDERDAMPED CHARACTERISTICS
Effect of ζ (0.1 to 1.0) on Underdamped Response:

11. UNDERDAMPED CHARACTERISTICS
Effect of ζ (0.0 to -0.1) on Underdamped Response:

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

13. CRITICALLY DAMPED CHARACTERISTICS
FIGURE ANd 6.4.2
𝜁=1
nONPERIODIC BEHAVIOR
REPEATED REAL ROOTS
FOR THE STEP CHANGE, t > 0:

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