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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. 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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  1. CHE 185 – PROCESS CONTROL AND DYNAMICS PERFORMANCE OF P, PI & PID CONTROL LOOPS

  2. 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

  3. Dynamic Changes as Kc is Increased • for a FOPDT Process – closed loop - example 8.3, Figure 8.2.2

  4. 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

  5. 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

  6. Root Locus Diagram • (Kc increases a to g) figure 8.2.1

  7. 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

  8. 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

  9. Root locus derivation(http://lpsa.swarthmore.edu/Root_Locus/RootLocusWhy.html) • For k = 1: • RESPONSE IS TOO SLOW

  10. Root locus derivation(http://lpsa.swarthmore.edu/Root_Locus/RootLocusWhy.html) • For k = 10: • RESPONSE LOOKS GOOD

  11. Root locus derivation(http://lpsa.swarthmore.edu/Root_Locus/RootLocusWhy.html) • For k = 100: • RESPONSE too oscillatory

  12. 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.

  13. 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

  14. Root locus derivation(http://lpsa.swarthmore.edu/Root_Locus/RootLocusWhy.html) • RESULTING step response • SAME TECHNIQUE WILL NOT BE USEFUL FOR SYSTEMS LIKE

  15. Root locus derivation(http://lpsa.swarthmore.edu/Root_Locus/RootLocusWhy.html) • "root locus" method • Calculate and plot roots for example equation

  16. 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

  17. Root locus derivation(http://lpsa.swarthmore.edu/Root_Locus/RootLocusWhy.html) • Use root locus results to evaluate k • At s = -1.5

  18. 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.

  19. 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

  20. 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.

  21. 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

  22. 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.

  23. 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).

  24. 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. 

  25. 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

  26. Effect of Kc on Closed-Loop stability, ζ

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

  28. 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.

  29. 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

  30. 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.

  31. 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

  32. 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

  33. 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

  34. Effect of Variations in Kc • Reference figure 8.3.5 • Low • Well-tuned • Too large

  35. Effect of Variations in τI • Reference figure 8.3.6 • Large • Well-tuned • Too small

  36. 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.

  37. Well tuned pi controller • Reference figure 8.3.7 • Rapid convergence with minimal overshoot

  38. PI Controller with Too Much Proportional Action • Reference figure 8.3.8 • Reduces lag but more overshoot and slower convergence

  39. Response of a PI Controller with Too Much Integral Action • Reference figure 8.3.9 • increases lag with less overshoot but slower convergence

  40. 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.

  41. 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

  42. 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.

  43. 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)

  44. 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)

  45. 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

  46. Control response Comparison between PI and PID • for a Low θp/τp Ratio (small θp)FIGURE 8.4.1

  47. Control response Comparison between PI and PID • for a HIGH θp/τp Ratio (LARGE θp) FIGURE 8.4.1

  48. PID ControlLER response WITH HIGH τD • REFERENCEFIGURE 8.4.2

  49. PID ControlLER DAMPING RESPONSE ζ WITH τD • REFERENCEFIGURE 8.4.3 WITH ALL OTHER PARAMETERS FIXED

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