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)

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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

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UNDERDAMPED CHARACTERISTICS

• FIGURES 6.4.2 THROUGH 6.4.4• • PERIODIC BEHAVIOR • COMPLEX ROOTS• FOR THE STEP CHANGE, t > 0:

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UNDERDAMPED CHARACTERISTICS

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

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UNDERDAMPED CHARACTERISTICS

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

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OVERDAMPED CHARACTERISTICS

• FIGURE 6.4.1 • • NONPERIODIC BEHAVIOR • REAL ROOTS• FOR THE STEP CHANGE, t > 0:

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CRITICALLY DAMPED CHARACTERISTICS

• FIGURE 6.4.1 AND 6.4.2• • NONPERIODIC BEHAVIOR • REPEATED REAL ROOTS• FOR THE STEP CHANGE, t > 0:

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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