Che 185 process control and dynamics
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CHE 185 – PROCESS CONTROL AND DYNAMICS. PERFORMANCE OF P, PI & PID CONTROL LOOPS. P-only Control. For an open loop overdamped process, as K c is increased, the process dynamics goes through the following sequence of behavior overdamped critically damped oscillatory ringing

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CHE 185 – PROCESS CONTROL AND DYNAMICS

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CHE 185 – PROCESS CONTROL AND DYNAMICS

PERFORMANCE OF P, PI & PID CONTROL LOOPS


P-only Control

  • For an open loop overdampedprocess, as Kc is increased, the process dynamics goes through the following sequence of behavior

    • overdamped

    • critically damped

    • oscillatory

    • ringing

    • sustained oscillations

    • unstable oscillations


Dynamic Changes as Kc is Increased

  • for a FOPDT Process – closed loop - example 8.3, Figure 8.2.2


Fopdt components

  • Use padÉ approximation for process deadtime:

  • Transfer function for p-only:

  • Poles can Be determined for this function using quadratic equations 8.2.2 and 8.2.3


Fopdt components

  • Analysis YIELDs A CHARACTERISTIC FUNCTION OF THE FORM:

  • TWO ROOTS OF THE FUNCTION CAN BE OBTAINED

  • ASSUMING THESE ARE BASED ON ALLOWING ONLY Kc TO VARY, A ROOT-LOCUS DIAGRAM CAN BE DEVELOPED


Root Locus Diagram

  • (Kc increases a to g) figure 8.2.1


Root locus derivation(http://lpsa.swarthmore.edu/Root_Locus/RootLocusWhy.html)

  • Example for proportional only loop

  • System transfer function

  • And the controller gain is k


Root locus derivation(http://lpsa.swarthmore.edu/Root_Locus/RootLocusWhy.html)

  • closed loop gain is given by:

  • TRIAL & ERROR SOLUTION FOR k yields a range of response curves to a STEP FUNCTION


Root locus derivation(http://lpsa.swarthmore.edu/Root_Locus/RootLocusWhy.html)

  • For k = 1:

  • RESPONSE IS TOO SLOW


Root locus derivation(http://lpsa.swarthmore.edu/Root_Locus/RootLocusWhy.html)

  • For k = 10:

  • RESPONSE LOOKS GOOD


Root locus derivation(http://lpsa.swarthmore.edu/Root_Locus/RootLocusWhy.html)

  • For k = 100:

  • RESPONSE too oscillatory


Root locus derivation(http://lpsa.swarthmore.edu/Root_Locus/RootLocusWhy.html)

  • ALTERNATELY, analytical method TO CALCULATEthepoles of the closed loop transfer function

  • roots of TRANSFER FUNCTION denominator are

    • overdamped response for 9>4K

    • underdamped response for 9<4K

    • critically damped response for 9=4K

  • FOR EXAMPLE WITH an underdamped response with =1/√2

  • we must set the magnitudes of the real and imaginary parts of the roots equal to each other.


Root locus derivation(http://lpsa.swarthmore.edu/Root_Locus/RootLocusWhy.html)

  • starting WITH:

  • FOR real and imaginary parts TO have equal magnitude

  • SOLVING FOR K

  • YIELDS transfer function


Root locus derivation(http://lpsa.swarthmore.edu/Root_Locus/RootLocusWhy.html)

  • RESULTING step response

  • SAME TECHNIQUE WILL NOT BE USEFUL FOR SYSTEMS LIKE


Root locus derivation(http://lpsa.swarthmore.edu/Root_Locus/RootLocusWhy.html)

  • "root locus" method

  • Calculate and plot roots for example equation


Root locus derivation(http://lpsa.swarthmore.edu/Root_Locus/RootLocusWhy.html)

  • For the example, at k = 0 the initial roots for s are 0 and -3

  • As k is increased the roots move together with a limit at s = -1.5

  • Above s = -1.5, roots diverge vertically


Root locus derivation(http://lpsa.swarthmore.edu/Root_Locus/RootLocusWhy.html)

  • Use root locus results to evaluate k

  • At s = -1.5


Root locus derivation(http://lpsa.swarthmore.edu/Root_Locus/RootLocusWhy.html)

  • Plotting the root locus

  • From example - open loop transfer function

  • n=2 poles at s = 0, -3.  We have m=0 finite zeros.  So there exists q=2 zeros as s goes to infinity (q = n-m = 2-0 = 2)

  • rewrite the open loop transfer function as G(s)H(s)=N(s)/D(s)

    • where N(s) is the numerator polynomial

    • D(s) is the denominator polynomial.  N(s)= 1, and D(s)= s2 + 3 s.


Root locus derivation(http://lpsa.swarthmore.edu/Root_Locus/RootLocusWhy.html)

  • Characteristic Equation is 1+KG(s)H(s)=0, or 1+KN(s)/D(s)=0, or D(s)+KN(s) = s2 + 3 s+ K( 1 ) = 0

  • COMPLETED

    ROOT LOCUS


Root locus derivation(http://lpsa.swarthmore.edu/Root_Locus/RootLocusWhy.html)

  • FOR A MORE COMPLEX EQUATION

  • n=3 poles at s = 0, -3, -2.  m=0 finite zeros.  So there exists q=3 zeros as s goes to infinity (q = n-m = 3-0 = 3)

  • open loop transfer function G(s)H(s)=N(s)/D(s). N(s)= 1 and D(s)= s3 + 5 s2 + 6 s.


Root locus derivation(http://lpsa.swarthmore.edu/Root_Locus/RootLocusWhy.html)

  • Characteristic Equation is 1+KG(s)H(s)=0, or 1+KN(s)/D(s)=0,or D(s)+KN(s) = s3 + 5 s2 + 6 s+ K( 1 ) = 0

  • COMPLETED

    ROOT LOCUS


Root locus rules

  • Rule 1 – Symmetry of Root Locus. The Root locus is symmetric about the real axis.

  • Rule 2 – Number of Branches of Root Locus is equal to the order of characteristic equation.

  • Rule 3 – Starting and Ending Points of Root Locus. The locus starts (when K=0) at poles of the loop gain, and ends (when K→∞ ) at the zeros. Note: there are q zeros of the loop gain as s→∞ .

  • Rule 4 – Root Locus exists on Real Axis to the left of an odd number of poles and zeros on the axis.


Root locus rules

  • Rule 5 – Asymptotes of the Root Locus as |s|→∞ . If q>0 there are asymptotes of the root locus that intersect real axis at

    asymptotes radiate out with angles

    where r=1, 3, 5…

  • Rule 6 – Break-Away and Break-In Points on Real Axis occur wherever

  • Rule 7 – Angle of Departure from a Complex Pole pj, is 180 degrees + (sum of angles between pj and all zeros) - (sum of angles between pj and all other poles).


Root locus rules

  • Rule 8 – Angle of Arrival at a Complex polezj, is 180 degrees + (sum of angles between zj and all other zeros) - (sum of angles between zj and all poles)

  • Rule 9 – determine Where Locus Crosses Imaginary Axis Using Routh-Horwitz.

  • Rule 10 - Find Location of Closed Loop Poles from Value of "K“.

    • Rewrite characteristic equation as D(s)+KN(s)=0. Put value of K into equation, and find roots of characteristic equation.

  • Rule 11 - Find Value of "K" from Location of Closed Loop Pole

    • Rewrite characteristic equation as

    • replace "s" by the desired pole location and solve for K. 


Root locus summary

  • OVERDAMPED BEHAVIOR HAS REAL ROOTS ONLY

  • A SINGLE VALUE FOR THE ROOT OCCURS AT THE OVER-UNDERDAMPING BOUNDARY

  • IMAGINARY ROOTS INDICATE UNDERDAMPED AND OSCILLATIONS

  • Kc= 0 REPRESENTS THE LOCATION OF SUSTAINED OSCILLATIONS

  • FOR POSITIVE VALUES OF Kc, THE SYSTEM IS UNSTABLE

  • THERE TYPICALLY IS A SINGLE VALUE OF Kc WHERE THE SYSTEM OFFSET AND TIME TO STABILIZE ARE MINIMAL


Effect of Kc on Closed-Loop stability, ζ


Effect of Kc on Closed-Loop process time constant, τp


P-only Controller Applied to First-Order Process without Deadtime

  • Without deadtime, the system will not become unstable regardless of how large Kc is.

  • First-order process model does not consider combined actuator/process/sensor system.

  • Therefore, first-order process model without deadtime is not a realistic model of a process under feedback control.


P-ONLY CONTROLLER APPLIED TO A SECOND ORDER PROCESS

  • A CLOSED LOOP CHARACTERISTIC EQUATION CAN BE DEVELOPED (EXAMPLE 8.2)

  • VALUES OF ζ` CAN BE PLOTTED AGAINST Kc TO DETERMINE THE RANGES OF OVERDAMPING (ζ`>1), UNDERDAMPING (0 < ζ`< 1) AND UNSTABLE OPERATION ζ`< 0


PI Control

  • Effect of tuning parameters

  • As Kc is increased or τIis decreased (i.e., more aggressive control), the closed loop dynamics goes through the same sequence of changes as the P-only controller:

    • overdamped,

    • critically damped,

    • oscillatory,

    • ringing,

    • sustained oscillations,

    • and unstable oscillations.


PI Control

  • Effect of tuning parameters

  • PROPORTIONAL ACTION INCREASES THE SPEED OF THE RESPONSE

  • INTEGRAL ACTION REMOVES OFFSET

  • ADJUSTING THE CONTROLLER GAIN ONLY:

  • A ROOT LOCUS DIAGRAM CAN BE GENERATED (SEE FIGURE 8.3.1 FOR A FOPDT SYSTEM)

  • A MONTONIC RELATIONSHIP BETWEEN DAMPING AND Kc CAN BE DEVELOPED, AS SHOWN IN FIGURE 8.3.2


PI Control

  • PLOTTING THE INTEGRAL ABSOLUTE ERROR (IAE) WITH Kc, WILL SHOW A UNIMODAL RELATIONSHIP (WITH A SINGLE MINIMUM)

    • FOR THIS SYSTEM, THE Kc VALUES BELOW THE MINIMUM ARE TOO SLOW

    • VALUES ABOVE THE MINIMUM ARE TOO AGGRESSIVE AND CAN LEAD TO OSCILLATIONS


PI Control

  • A THREE DIMENSIONAL ANALYSIS CAN ALSO BE USED WHERE IAE IS PLOTTED AS A FUNCTION OF Kc AND THE INTEGRAL TIME, τi

    • THIS ANALYSIS LEADS TO A SURFACE

    • THE ELLIPSES SHOWN ON THE SURFACE REPRESENT CONSTANT IAE VALUES

    • THE LIMIT OF STABILITY IS A COMBINATION OF BOTH VARIABLES


Effect of Variations in Kc

  • Reference figure 8.3.5

  • Low

  • Well-tuned

  • Too large


Effect of Variations in τI

  • Reference figure 8.3.6

  • Large

  • Well-tuned

  • Too small


Analysis of the Effect of proportional and integral action on fopdt processes

  • When there is too little proportional action or too little integral action, it is easy to identify.

  • it is difficult to differentiate between too much proportional action and too much integral action because both lead to ringing.


Well tuned pi controller

  • Reference figure 8.3.7

  • Rapid convergence with minimal overshoot


PI Controller with Too Much Proportional Action

  • Reference figure 8.3.8

  • Reduces lag but more overshoot and slower convergence


Response of a PI Controller with Too Much Integral Action

  • Reference figure 8.3.9

  • increases lag with less overshoot but slower convergence


PID Control

  • Kc and τIhave the same general effect as observed for PI control.

  • Derivative action tends to reduce the oscillatory nature of the response and results in faster settling for systems with larger deadtime to time constant ratios, θp/τp.


PID Control

  • IT IS NOT PRACTICAL TO GRAPHICALLY SHOW THE RELATIONSHIP BETWEEN 4 VARIABLES: Kc, τI, τD, AND IAE

    • THE RESULT WOULD BE A SOLID SURFACE

    • THE ANALYSIS CAN BE DONE USING SEARCH ROUTINES


PID Control

  • ONE OF THE GOALS OF PID IS MINIMUM VARIATION OF MANIPULATED VARIABLE BEHAVIOR

  • EXCESSIVE DERIVATIVE CONTROL CAN CAUSE STALLING, AS SHOWN IN FIGURE 8.4.2

  • THE LIMIT OF VARIATION IN THE MANIPULATED VARIABLE SHOULD BE SPECIFIED FOR THE CONTROLLER TUNING ANALYSIS WITH PID.


TUNING WITH POLE PLACEMENT

  • PI EXAMPLE

  • THIS METHOD IS REFERRED TO AS DIRECT SYNTHESIS OR POLE-ASSIGNMENT DESIGN METHOD

  • THE POLES OF THE CLOSE LOOP ARE SPECIFIED IN THE FORM OF PROCESS TIME CONSTANT AND DAMPING FACTOR

  • THEN THE APPROPRIATE VALUE OF Kc AND τIARE CALCULATED

  • SPECIFY τ`pAND ζ` FOR THE CLOSED LOOP EQUATION: (7.2.7)


TUNING WITH POLE PLACEMENT

  • USING THESE VALUES:

    PROVIDES A VALUE RELATED TO THE AGGRESSIVENESS OF THE CONTROLLER

  • VALUES FOR KcAND τI ARE THEN CALCULATED (eqn. 9.4.4)

  • WITH: (eqn. 9.4.3)


TUNING WITH POLE PLACEMENT

  • THIS METHOD IS USEFUL FOR DISCUSSING THE RESPONSE OF THEORETICAL SYSTEMS

  • IT IS NOT APPLIED IN INDUSTRY BECAUSE DEVELOPING THE RIGOROUS PROCESS MODEL IS NOT PRACTICAL


Control response Comparison between PI and PID

  • for a Low θp/τp Ratio (small θp)FIGURE 8.4.1


Control response Comparison between PI and PID

  • for a HIGH θp/τp Ratio (LARGE θp) FIGURE 8.4.1


PID ControlLER response WITH HIGH τD

  • REFERENCEFIGURE 8.4.2


PID ControlLER DAMPING RESPONSE ζ WITH τD

  • REFERENCEFIGURE 8.4.3 WITH ALL OTHER PARAMETERS FIXED


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