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

CHE 185 – PROCESS CONTROL AND DYNAMICS

PERFORMANCE OF P, PI & PID CONTROL LOOPS


P only control

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 k c is increased

Dynamic Changes as Kc is Increased

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


Fopdt components

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 components1

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

Root Locus Diagram

  • (Kc increases a to g) figure 8.2.1


Root locus derivation http lpsa swarthmore edu root locus rootlocuswhy html

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 html1

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 html2

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 html3

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 html4

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 html5

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 html6

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 html7

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 html8

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 html9

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 html10

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 html11

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 html12

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 html13

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 html14

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

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 rules1

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 rules2

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

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 k c on closed loop stability

Effect of Kc on Closed-Loop stability, ζ


Effect of k c on closed loop process time constant p

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


P only controller applied to first order process without deadtime

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

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

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 control1

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 control2

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 control3

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 k c

Effect of Variations in Kc

  • Reference figure 8.3.5

  • Low

  • Well-tuned

  • Too large


Effect of variations in i

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

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

Well tuned pi controller

  • Reference figure 8.3.7

  • Rapid convergence with minimal overshoot


Pi controller with too much proportional action

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

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

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 control1

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 control2

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

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 placement1

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 placement2

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

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 pid1

Control response Comparison between PI and PID

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


Pid controller response with high d

PID ControlLER response WITH HIGH τD

  • REFERENCEFIGURE 8.4.2


Pid controller damping response with d

PID ControlLER DAMPING RESPONSE ζ WITH τD

  • REFERENCEFIGURE 8.4.3 WITH ALL OTHER PARAMETERS FIXED


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