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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
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
• TWO REAL, DISTINCT ROOTS WHEN 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
ROOT OPTIONS 2
• TWO REAL EQUAL ROOTS WHEN 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 • TWO COMPLEX CONJUGATE ROOTS (a + ib, a- ib) WHEN
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.
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• • 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 • • NONPERIODIC BEHAVIOR • REAL ROOTS• FOR THE STEP CHANGE, t > 0:
12
CRITICALLY DAMPED CHARACTERISTICS
• FIGURE 6.4.1 AND 6.4.2• • 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 DEGRADATION OF EQUIPMENT OR UNDUE 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 LARGER THE EFFECTIVE DEADTIME
• TRANSFER FUNCTION