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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

- 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

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

- 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

- 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

- (Kc increases a to g) figure 8.2.1

- Example for proportional only loop
- System transfer function
- And the controller gain is k

- closed loop gain is given by:
- TRIAL & ERROR SOLUTION FOR k yields a range of response curves to a STEP FUNCTION

- For k = 1:
- RESPONSE IS TOO SLOW

- For k = 10:
- RESPONSE LOOKS GOOD

- For k = 100:
- RESPONSE too oscillatory

- 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.

- starting WITH:
- FOR real and imaginary parts TO have equal magnitude
- SOLVING FOR K
- YIELDS transfer function

- RESULTING step response
- SAME TECHNIQUE WILL NOT BE USEFUL FOR SYSTEMS LIKE

- "root locus" method
- Calculate and plot roots for example equation

- 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

- Use root locus results to evaluate k
- At s = -1.5

- 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.

- 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

- 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.

- 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

- 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.

- 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).

- 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.

- 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

- 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.

- 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

- 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.

- 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

- 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

- 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

- Reference figure 8.3.5
- Low
- Well-tuned
- Too large

- Reference figure 8.3.6
- Large
- Well-tuned
- Too small

- 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.

- Reference figure 8.3.7
- Rapid convergence with minimal overshoot

- Reference figure 8.3.8
- Reduces lag but more overshoot and slower convergence

- Reference figure 8.3.9
- increases lag with less overshoot but slower convergence

- 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.

- 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

- 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.

- 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)

- 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)

- 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

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

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

- REFERENCEFIGURE 8.4.2

- REFERENCEFIGURE 8.4.3 WITH ALL OTHER PARAMETERS FIXED