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**CHE 185 – PROCESS CONTROL AND DYNAMICS**

SECOND AND HIGHER ORDER PROCESSES

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**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

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**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:

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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

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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

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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)

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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.

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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

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**UNDERDAMPED CHARACTERISTICS**

FIGURES THROUGH 6.4.4 𝜁<1 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 𝜁>1 nONPERIODIC BEHAVIOR REAL ROOTS FOR THE STEP CHANGE, t > 0:

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**CRITICALLY DAMPED CHARACTERISTICS**

FIGURE ANd 6.4.2 𝜁=1 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

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**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.

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**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.

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**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

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**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

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**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

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**2ND ORDER PROCESS DECAY RATIO**

RATIO OF THE MAGNITUDE OF SUCCESSIVE PEAKS IN THE RESPONSE A SMALL DECAY RATIO IS PREFERRED

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**2ND ORDER PROCESS OSCILLATORY PERIOD**

THE oscillatory PERIOD OF A CYCLE IMPORTANT CHARACTERISTIC OF A CLOSED LOOP SYSTEM

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**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

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**HIGHER ORDER PROCESSES**

MAY BE CONSIDERED AS FIRST ORDER FUNCTIONS GENERAL FORM

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**HIGHER ORDER PROCESSES**

The larger n, the more sluggish the process response (i.e., the larger the effective deadtime Transfer function

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