### Lect. 15 CHE 185 – 2nd AND HIGHER ORDER PROCESSES

```CHE 185 – PROCESS
CONTROL AND DYNAMICS
SECOND AND HIGHER ORDER
PROCESSES
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
2
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:
ROOT OPTIONS 1  > 1
• TWO REAL, DISTINCT ROOTS WHEN  > 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
THE MORE SLUGGISH THE RATE OF RESPONSE THE LARGER
THE DAMPING FACTOR
FOR ALL DAMPING FACTORS, RESPONSES HEAD TOWARDS
ROOT OPTIONS 2  = 1
• TWO REAL EQUAL ROOTS WHEN  = 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
ROOT OPTIONS 3  < 1
• TWO COMPLEX CONJUGATE ROOTS (a + ib, a- ib) WHEN
< 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)
SECOND ORDER PROCESSES
CHARACTERIZATION
• NOTE THAT THE GAIN, TIME CONSTANT,
AND THE DAMPING FACTOR DEFINE THE
DYNAMIC BEHAVIOR OF 2ND ORDER
PROCESS.
7
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
UNDERDAMPED
CHARACTERISTICS
•
•
•
•
•
FIGURES 6.4.2 THROUGH 6.4.4
<1
PERIODIC BEHAVIOR
COMPLEX ROOTS
FOR THE STEP CHANGE, t > 0:
9
UNDERDAMPED
CHARACTERISTICS
• EFFECT OF ζ (0.1 TO 1.0) ON
UNDERDAMPED RESPONSE:
10
UNDERDAMPED
CHARACTERISTICS
• EFFECT OF ζ (0.0 TO -0.1) ON
UNDERDAMPED RESPONSE:
11
OVERDAMPED
CHARACTERISTICS
•
•
•
•
•
FIGURE 6.4.1
>1
NONPERIODIC BEHAVIOR
REAL ROOTS
FOR THE STEP CHANGE, t > 0:
12
CRITICALLY DAMPED
CHARACTERISTICS
•
•
•
•
•
FIGURE 6.4.1 AND 6.4.2
=1
NONPERIODIC BEHAVIOR
REPEATED REAL ROOTS
FOR THE STEP CHANGE, t > 0:
13
CHARACTERISTICS OF AN
UNDERDAMPED RESPONSE
• RISE TIME
• OVERSHOOT
(B)
• DECAY RATIO
(C/B)
• SETTLING OR
RESPONSE
TIME
• PERIOD (T)
• FIGURE 6.4.4
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.
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.
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
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
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
STRESS ON THE SYSTEM
2ND ORDER PROCESS DECAY
RATIO
• RATIO OF THE MAGNITUDE OF
SUCCESSIVE PEAKS IN THE RESPONSE
• A SMALL DECAY RATIO IS PREFERRED
2ND ORDER PROCESS
OSCILLATORY PERIOD
• THE OSCILLATORY PERIOD OF A CYCLE
• IMPORTANT CHARACTERISTIC OF A
CLOSED LOOP SYSTEM
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
HIGHER ORDER PROCESSES
• MAY BE CONSIDERED AS FIRST ORDER
FUNCTIONS
• GENERAL FORM
HIGHER ORDER PROCESSES
• THE LARGER n, THE MORE SLUGGISH
THE PROCESS RESPONSE (I.E., THE