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PID Tuning using the SIMC rules Sigurd Skogestad NTNU, Trondheim, Norway September 2008 April PID Tuning using the SIMC rules Sigurd Skogestad NTNU, Trondheim, Norway September 2008 April 4 -8, 2004 1. Model 2. SIMC-tunings 3. Tight control 4. Smooth control 5. Level control KFUPM-Distillation Control Course 6. Discussion points

Operation: Decision and control layers RTO cs = y 1 s Min J (economics); Operation: Decision and control layers RTO cs = y 1 s Min J (economics); MV=y 1 s CV=y 1; MV=y 2 s MPC y 2 s PID CV=y 2; MV=u u (valves) April 4 -8, 2004 KFUPM-Distillation Control Course 2

Stepwise procedure plantwide control I. TOP-DOWN Step 1. DEGREES OF FREEDOM Step 2. OPERATIONAL Stepwise procedure plantwide control I. TOP-DOWN Step 1. DEGREES OF FREEDOM Step 2. OPERATIONAL OBJECTIVES Step 3. WHAT TO CONTROL? (primary CV’s c=y 1) Step 4. PRODUCTION RATE II. BOTTOM-UP (structure control system): Step 5. REGULATORY CONTROL LAYER (PID) “Stabilization” What more to control? (secondary CV’s y 2) Step 6. SUPERVISORY CONTROL LAYER (MPC) Decentralization Step 7. OPTIMIZATION LAYER (RTO) Can we do without it? April 4 -8, 2004 KFUPM-Distillation Control Course 3

PID controller e n Time domain (“ideal” PID) n Laplace domain (“ideal”/”parallel” form) n PID controller e n Time domain (“ideal” PID) n Laplace domain (“ideal”/”parallel” form) n Usually τD=0. Only two parameters left (Kc and τI)… How difficult can it be? ? ? n q Surprisingly difficult without systematic approach! April 4 -8, 2004 KFUPM-Distillation Control Course 4

Let’s start with the CONCLUSION Tuning of PID controllers n n n SIMC tuning Let’s start with the CONCLUSION Tuning of PID controllers n n n SIMC tuning rules (“Skogestad IMC”)(*) Main message: Can usually do much better by taking a systematic approach Key: Look at initial part of step response Initial slope: k’ = k/ 1 n One tuning rule! Easily memorized • c ¸ - : desired closed-loop response time (tuning parameter) • For robustness select: c ¸ Reference: S. Skogestad, “Simple analytic rules for model reduction and PID controller design”, J. Proc. Control, Vol. 13, 291 -309, 2003 (Also reprinted in MIC) (*) “Probably the best simple PID tuning rules in the world” April 4 -8, 2004 KFUPM-Distillation Control Course 5

MODEL Need a model for tuning n Model: Dynamic effect of change in input MODEL Need a model for tuning n Model: Dynamic effect of change in input u (MV) on output y (CV) First-order + delay model for PI-control n Second-order model for PID-control n April 4 -8, 2004 KFUPM-Distillation Control Course 6

MODEL, Approach 1 A 1. Step response experiment n n Make step change in MODEL, Approach 1 A 1. Step response experiment n n Make step change in one u (MV) at a time Record the output (s) y (CV) April 4 -8, 2004 KFUPM-Distillation Control Course 7

MODEL, Approach 1 A Δy(∞) RESULTING OUTPUT y STEP IN INPUT u Δu April MODEL, Approach 1 A Δy(∞) RESULTING OUTPUT y STEP IN INPUT u Δu April 4 -8, 2004 : Delay - Time where output does not change 1: Time constant - Additional time to reach 63% of final change k = y(∞)/ u : Steady-state gain KFUPM-Distillation Control Course 9

MODEL, Approach 1 A Step response integrating process Δy Δt April 4 -8, 2004 MODEL, Approach 1 A Step response integrating process Δy Δt April 4 -8, 2004 KFUPM-Distillation Control Course 10

MODEL, Approach 1 B Model from closed-loop response with P-controller Kc 0=1. 5 Δys=1 MODEL, Approach 1 B Model from closed-loop response with P-controller Kc 0=1. 5 Δys=1 Δy∞ dyinf = 0. 45*(dyp + dyu) Mo =(dyp -dyinf)/dyinf b=dyinf/dys Δyp=0. 79 Δyu=0. 54 A = 1. 152*Mo^2 - 1. 607*Mo + 1. 0 r = 2*A*abs(b/(1 -b)) k = (1/Kc 0) * abs(b/(1 -b)) theta = tp*[0. 309 + 0. 209*exp(-0. 61*r)] tau = theta*r tp=4. 4 Example 2: Get k=0. 99, theta =1. 68, tau=3. 03 Ref: Shamssuzzoha and Skogestad (JPC, 2010) April 4 -8, 2004 KFUPM-Distillation Control Course + modification by C. Grimholt (Project, NTNU, 2010) 11

MODEL, Approach 2 2. Model reduction of more complicated model n Start with complicated MODEL, Approach 2 2. Model reduction of more complicated model n Start with complicated stable model on the form n Want to get a simplified model on the form n Most important parameter is the “effective” delay April 4 -8, 2004 KFUPM-Distillation Control Course 12

MODEL, Approach 2 April 4 -8, 2004 KFUPM-Distillation Control Course 13 MODEL, Approach 2 April 4 -8, 2004 KFUPM-Distillation Control Course 13

MODEL, Approach 2 Example 1 Half rule April 4 -8, 2004 KFUPM-Distillation Control Course MODEL, Approach 2 Example 1 Half rule April 4 -8, 2004 KFUPM-Distillation Control Course 14

MODEL, Approach 2 original 1 st-order+delay April 4 -8, 2004 KFUPM-Distillation Control Course 15 MODEL, Approach 2 original 1 st-order+delay April 4 -8, 2004 KFUPM-Distillation Control Course 15

MODEL, Approach 2 2 half rule April 4 -8, 2004 KFUPM-Distillation Control Course 16 MODEL, Approach 2 2 half rule April 4 -8, 2004 KFUPM-Distillation Control Course 16

MODEL, Approach 2 original 1 st-order+delay 2 nd-order+delay April 4 -8, 2004 KFUPM-Distillation Control MODEL, Approach 2 original 1 st-order+delay 2 nd-order+delay April 4 -8, 2004 KFUPM-Distillation Control Course 17

MODEL, Approach 2 Approximation of zeros c c c To make these rules more MODEL, Approach 2 Approximation of zeros c c c To make these rules more general (and not only applicable to the choice c= ): Replace (time delay) by c (desired closed-loop response time). (6 places) c Correction: More generally replace θ by τc in the above rules April 4 -8, 2004 KFUPM-Distillation Control Course 20

SIMC-tunings Derivation of SIMC-PID tuning rules n PI-controller (based on first-order model) n For SIMC-tunings Derivation of SIMC-PID tuning rules n PI-controller (based on first-order model) n For second-order model add D-action. For our purposes, simplest with the “series” (cascade) PID-form: April 4 -8, 2004 KFUPM-Distillation Control Course 21

SIMC-tunings Basis: Direct synthesis (IMC) Closed-loop response to setpoint change Idea: Specify desired response: SIMC-tunings Basis: Direct synthesis (IMC) Closed-loop response to setpoint change Idea: Specify desired response: and from this get the controller. ……. Algebra: April 4 -8, 2004 KFUPM-Distillation Control Course 22

SIMC-tunings April 4 -8, 2004 KFUPM-Distillation Control Course 23 SIMC-tunings April 4 -8, 2004 KFUPM-Distillation Control Course 23

SIMC-tunings IMC Tuning = Direct Synthesis Algebra: April 4 -8, 2004 KFUPM-Distillation Control Course SIMC-tunings IMC Tuning = Direct Synthesis Algebra: April 4 -8, 2004 KFUPM-Distillation Control Course 24

SIMC-tunings Integral time n n n Found: Integral time = dominant time constant ( SIMC-tunings Integral time n n n Found: Integral time = dominant time constant ( I = 1) Works well for setpoint changes Needs to be modified (reduced) for integrating disturbances d c u g y Example. “Almost-integrating process” with disturbance at input: G(s) = e-s/(30 s+1) Original integral time I = 30 gives poor disturbance response Try reducing it! April 4 -8, 2004 KFUPM-Distillation Control Course 25

SIMC-tunings Integral Time I = 1 Reduce I to this value: I = 4 SIMC-tunings Integral Time I = 1 Reduce I to this value: I = 4 ( c+ ) = 8 Setpoint change at t=0 April 4 -8, 2004 Input disturbance at t=20 KFUPM-Distillation Control Course 26

SIMC-tunings Integral time n Want to reduce the integral time for “integrating” processes, but SIMC-tunings Integral time n Want to reduce the integral time for “integrating” processes, but to avoid “slow oscillations” we must require: n Derivation: n Setpoint response: Improve (get rid of overshoot) by “prefiltering”, y’s = f(s) ys. Details: See www. ntnu. no/users/skoge/publications/2003/tuning. PID Remark 13 II April 4 -8, 2004 KFUPM-Distillation Control Course 27

SIMC-tunings Conclusion: SIMC-PID Tuning Rules One tuning parameter: c April 4 -8, 2004 KFUPM-Distillation SIMC-tunings Conclusion: SIMC-PID Tuning Rules One tuning parameter: c April 4 -8, 2004 KFUPM-Distillation Control Course 28

SIMC-tunings Some insights from tuning rules 1. 2. 3. 4. The effective delay θ SIMC-tunings Some insights from tuning rules 1. 2. 3. 4. The effective delay θ (which limits the achievable closedloop time constant τc) is independent of the dominant process time constant τ1! n It depends on τ2/2 (PI) or τ3/2 (PID) Use (close to) P-control for integrating process n Beware of large I-action (small τI) for level control Use (close to) I-control for fast process (with small time constant τ1) Parameter variations: For robustness tune at operating point with maximum value of k’ θ = (k/τ1)θ April 4 -8, 2004 KFUPM-Distillation Control Course 29

SIMC-tunings Some special cases One tuning parameter: c April 4 -8, 2004 KFUPM-Distillation Control SIMC-tunings Some special cases One tuning parameter: c April 4 -8, 2004 KFUPM-Distillation Control Course 30

SIMC-tunings Another special case: IPZ process n IPZ-process may represent response from steam flow SIMC-tunings Another special case: IPZ process n IPZ-process may represent response from steam flow to pressure n Rule T 2: SIMC-tunings n These tunings turn out to be almost identical to the tunings given on page 104 -106 in the Ph. D. thesis by O. Slatteke, Lund Univ. , 2006 and K. Forsman, "Reglerteknik for processindustrien", Studentlitteratur, 2005. April 4 -8, 2004 KFUPM-Distillation Control Course 31

SIMC-tunings Note: Derivative action is commonly used for temperature control loops. Select D equal SIMC-tunings Note: Derivative action is commonly used for temperature control loops. Select D equal to 2 = time constant of temperature sensor April 4 -8, 2004 KFUPM-Distillation Control Course 32

SIMC-tunings April 4 -8, 2004 KFUPM-Distillation Control Course 33 SIMC-tunings April 4 -8, 2004 KFUPM-Distillation Control Course 33

SIMC-tunings April 4 -8, 2004 KFUPM-Distillation Control Course 34 SIMC-tunings April 4 -8, 2004 KFUPM-Distillation Control Course 34

SIMC-tunings Quiz: SIMC PI-tunings y y Step response t [s]Time t (a) The Figure SIMC-tunings Quiz: SIMC PI-tunings y y Step response t [s]Time t (a) The Figure shows the response (y) from a test where we made a step change in the input (Δu = 0. 1) at t=0. Suggest PI-tunings for (1) τc=2, . (2) τc=10. (b) Do the same, given that the actual plant is April 4 -8, 2004 KFUPM-Distillation Control Course 35

SIMC-tunings Selection of tuning parameter c Two main cases 1. TIGHT CONTROL: Want “fastest SIMC-tunings Selection of tuning parameter c Two main cases 1. TIGHT CONTROL: Want “fastest possible TIGHT CONTROL: control” subject to having good robustness • 2. SMOOTH CONTROL: Want “slowest possible CONTROL: control” subject to acceptable disturbance rejection • • Want tight control of active constraints (“squeeze and shift”) Want smooth control if fast setpoint tracking is not required, for example, levels and unconstrained (“self-optimizing”) variables THERE ALSO OTHER ISSUES: Input saturation etc. April 4 -8, 2004 KFUPM-Distillation Control Course 36

TIGHT CONTROL April 4 -8, 2004 KFUPM-Distillation Control Course 37 TIGHT CONTROL April 4 -8, 2004 KFUPM-Distillation Control Course 37

TIGHT CONTROL Typical closed-loop SIMC responses with the choice c= April 4 -8, 2004 TIGHT CONTROL Typical closed-loop SIMC responses with the choice c= April 4 -8, 2004 KFUPM-Distillation Control Course 38

TIGHT CONTROL Example. Integrating process with delay=1. G(s) = e-s/s. Model: k’=1, 1=1 SIMC-tunings TIGHT CONTROL Example. Integrating process with delay=1. G(s) = e-s/s. Model: k’=1, 1=1 SIMC-tunings with c with = =1: IMC has I=1 Ziegler-Nichols is usually a bit aggressive Setpoint change at t=0 c April 4 -8, 2004 Input disturbance at t=20 KFUPM-Distillation Control Course 39

TIGHT CONTROL 1. Approximate as first-order model with k=1, 1 = 1+0. 1=1. 1, TIGHT CONTROL 1. Approximate as first-order model with k=1, 1 = 1+0. 1=1. 1, =0. 1+0. 04+0. 008 = 0. 148 Get SIMC PI-tunings ( c= ): Kc = 1 ¢ 1. 1/(2¢ 0. 148) = 3. 71, I=min(1. 1, 8¢ 0. 148) = 1. 1 2. Approximate as second-order model with k=1, 1 = 1, 2=0. 2+0. 02=0. 22, =0. 02+0. 008 = 0. 028 Get SIMC PID-tunings ( c= ): Kc = 1 ¢ 1/(2¢ 0. 028) = 17. 9, I=min(1, 8¢ 0. 028) = 0. 224, D=0. 22 April 4 -8, 2004 KFUPM-Distillation Control Course 40

SMOOTH CONTROL Tuning for smooth control n Tuning parameter: c = desired closed-loop response SMOOTH CONTROL Tuning for smooth control n Tuning parameter: c = desired closed-loop response time n Selecting c= (“tight control”) is reasonable for cases with a relatively large effective delay n Other cases: Select c > for q q slower control smoother input usage n q q n less disturbing effect on rest of the plant less sensitivity to measurement noise better robustness Question: Given that we require some disturbance rejection. q q What is the largest possible value for c ? Or equivalently: The smallest possible value for Kc? Will derive Kc, min. From this we can get c, max using SIMC tuning rule S. Skogestad, ``Tuning for smooth PID control with acceptable disturbance Control Course April 4 -8, 2004 KFUPM-Distillation rejection'', Ind. Eng. Chem. Res, 45 (23), 7817 -7822 (2006). 41

SMOOTH CONTROL Closed-loop disturbance rejection d 0 -d 0 ymax -ymax April 4 -8, SMOOTH CONTROL Closed-loop disturbance rejection d 0 -d 0 ymax -ymax April 4 -8, 2004 KFUPM-Distillation Control Course 42

SMOOTH CONTROL Kc u Minimum controller gain for PI-and PID-control: min |c(j )| = SMOOTH CONTROL Kc u Minimum controller gain for PI-and PID-control: min |c(j )| = Kc April 4 -8, 2004 KFUPM-Distillation Control Course 43

SMOOTH CONTROL Rule: Min. controller gain for acceptable disturbance rejection: Kc ¸ |u 0|/|ymax| SMOOTH CONTROL Rule: Min. controller gain for acceptable disturbance rejection: Kc ¸ |u 0|/|ymax| often ~1 (in span-scaled variables) |ymax| = allowed deviation for output (CV) |u 0| = required change in input (MV) for disturbance rejection (steady state) = observed change (movement) in input from historical data April 4 -8, 2004 KFUPM-Distillation Control Course 44

SMOOTH CONTROL Rule: Kc ¸ |u 0|/|ymax| n Exception to rule: Can have lower SMOOTH CONTROL Rule: Kc ¸ |u 0|/|ymax| n Exception to rule: Can have lower Kc if disturbances are handled by the integral action. q q Disturbances must occur at a frequency lower than 1/ I Applies to: Process with short time constant ( 1 is small) and no delay ( ¼ 0). n q For example, flow control Then I = 1 is small so integral action is “large” April 4 -8, 2004 KFUPM-Distillation Control Course 45

SMOOTH CONTROL Summary: Tuning of easy loops n n n Easy loops: Small effective SMOOTH CONTROL Summary: Tuning of easy loops n n n Easy loops: Small effective delay ( ¼ 0), so closedloop response time c (>> ) is selected for “smooth control” ASSUME VARIABLES HAVE BEEN SCALED WITH RESPECT TO THEIR SPAN SO THAT |u 0/ymax| = 1 (approx. ). Flow control: Kc=0. 2, I = 1 = time constant valve (typically, 2 to 10 s; close to pure integrating!) Level control: Kc=2 (and no integral action) Other easy loops (e. g. pressure): Kc = 2, I = min(4 c, 1) q April 4 -8, 2004 Note: Often want a tight pressure control loop (so may have Kc=10 or larger) KFUPM-Distillation Control Course 46

QUIZ: What are the benefits of adding a flow controller (inner q cascade)? s QUIZ: What are the benefits of adding a flow controller (inner q cascade)? s Extra measurement y 2 = q 1. April 4 -8, 2004 z Counteracts nonlinearity in valve, f(z) • 2. q With fast flow control we can assume q = qs Eliminates effect of disturbances in p 1 and p 2 KFUPM-Distillation Control Course 47

SMOOTH CONTROL LEVEL CONTROL Application of smooth control n Averaging level control q V SMOOTH CONTROL LEVEL CONTROL Application of smooth control n Averaging level control q V If you insist on integral action then this value avoids cycling LC Reason for having tank is to smoothen disturbances in concentration and flow. Tight level control is not desired: gives no “smoothening” of flow disturbances. Proof: 1. Let |u 0| = | q 0| – expected flow change [m 3/s] (input disturbance) |ymax| = | Vmax| - largest allowed variation in level [m 3] Minimum controller gain for acceptable disturbance rejection: Kc ¸ Kc, min = |u 0|/|ymax| = | q 0| / | Vmax| 2. From the material balance (d. V/dt = q – qout), the model is g(s)=k’/s with k’=1. Select Kc=Kc, min. SIMC-Integral time for integrating process: I = 4 / (k’ Kc) = 4 | Vmax| / | q 0| = 4 ¢ residence time provided tank is nominally half full and q 0 is equal to the nominal flow. April 4 -8, 2004 KFUPM-Distillation Control Course 48

LEVEL CONTROL More on level control n n Level control often causes problems Typical LEVEL CONTROL More on level control n n Level control often causes problems Typical story: q q n n Level loop starts oscillating Operator detunes by decreasing controller gain Level loop oscillates even more. . . ? ? ? Explanation: Level is by itself unstable and requires control. April 4 -8, 2004 KFUPM-Distillation Control Course 49

LEVEL CONTROL How avoid oscillating levels? 0. 1 ¼ 1/ 2 April 4 -8, LEVEL CONTROL How avoid oscillating levels? 0. 1 ¼ 1/ 2 April 4 -8, 2004 KFUPM-Distillation Control Course 50

LEVEL CONTROL Case study oscillating level n n n We were called upon to LEVEL CONTROL Case study oscillating level n n n We were called upon to solve a problem with oscillations in a distillation column Closer analysis: Problem was oscillating reboiler level in upstream column Use of Sigurd’s rule solved the problem April 4 -8, 2004 KFUPM-Distillation Control Course 51

LEVEL CONTROL April 4 -8, 2004 KFUPM-Distillation Control Course 52 LEVEL CONTROL April 4 -8, 2004 KFUPM-Distillation Control Course 52

Conclusion PID tuning 3. Derivative time: Only for dominant second-order processes April 4 -8, Conclusion PID tuning 3. Derivative time: Only for dominant second-order processes April 4 -8, 2004 KFUPM-Distillation Control Course 53

Alternative form with detuning factor F April 4 -8, 2004 KFUPM-Distillation Control Course 54 Alternative form with detuning factor F April 4 -8, 2004 KFUPM-Distillation Control Course 54

SIMC-tunings QUIZ Quiz: SIMC PI-tunings y y Step response t [s]Time t (a) The SIMC-tunings QUIZ Quiz: SIMC PI-tunings y y Step response t [s]Time t (a) The Figure shows the response (y) from a test where we made a step change in the input (Δu = 0. 1) at t=0. Suggest PI-tunings for (1) τc=2, . (2) τc=10. (b) Do the same, given that the actual plant is April 4 -8, 2004 KFUPM-Distillation Control Course 55

QUIZ Solution n. Actual plant: April 4 -8, 2004 KFUPM-Distillation Control Course 56 QUIZ Solution n. Actual plant: April 4 -8, 2004 KFUPM-Distillation Control Course 56

QUIZ Approximation of step response Approximation ”bye eye” April 4 -8, 2004 KFUPM-Distillation Control QUIZ Approximation of step response Approximation ”bye eye” April 4 -8, 2004 KFUPM-Distillation Control Course 57

SIMC-tunings Kc=2. 9, tau. I=10 Kc=9. 5, tau. I=10 OUTPUT y INPUT u Tunings SIMC-tunings Kc=2. 9, tau. I=10 Kc=9. 5, tau. I=10 OUTPUT y INPUT u Tunings from Step response “by eye” model n. Setpoint change at t=0, input disturbance = 0. 1 at t=50 Tunings from Half rule (Somewhat better) Kc=2, tau. I=5. 5 Kc=6, tau. I=5. 5 April 4 -8, 2004 KFUPM-Distillation Control Course 58

QUIZ Half-rule approach: Approximation of zeros depends on tauc! April 4 -8, 2004 KFUPM-Distillation QUIZ Half-rule approach: Approximation of zeros depends on tauc! April 4 -8, 2004 KFUPM-Distillation Control Course 59

Some discussion points n n Selection of τc: some other issues Obtaining the model Some discussion points n n Selection of τc: some other issues Obtaining the model from step responses: How long should we run the experiment? Cascade control: Tuning Controllability implications of tuning rules April 4 -8, 2004 KFUPM-Distillation Control Course 60

Selection of c: Other issues n Input saturation. q q Problem. Input may “overshoot” Selection of c: Other issues n Input saturation. q q Problem. Input may “overshoot” if we “speedup” the response too much (here “speedup” = / c). Solution: To avoid input saturation, we must obey max “speedup”: April 4 -8, 2004 KFUPM-Distillation Control Course 61

A little more on obtaining the model from step response experiments n “Factor 5 A little more on obtaining the model from step response experiments n “Factor 5 rule”: Only dynamics within a factor 5 from “control time scale” ( c) are important n 1 ¼ 200 (may be neglected for c < 40) Integrating process ( 1 = 1) Time constant 1 is not important if it is much larger than the desired response time c. More precisely, may use 1 =1 for 1 > 5 c n Delay-free process ( =0) Delay is not important if it is much smaller than the desired response time c. More precisely, may use ¼ 0 for < c/5 April 4 -8, 2004 ¼ 1 (may be neglected for c > 5) time c = desired response time KFUPM-Distillation Control Course 62

Step response experiment: How long do we need to wait? n n RULE: May Step response experiment: How long do we need to wait? n n RULE: May stop at about 10 times effective delay FAST TUNING DESIRED (“tight control”, c = ): q NORMALLY NO NEED TO RUN THE STEP EXPERIMENT FOR LONGER THAN ABOUT 10 TIMES THE EFFECTIVE DELAY ( ) q EXCEPTION: LET IT RUN A LITTLE LONGER IF YOU SEE THAT IT IS ALMOST SETTLING (TO GET 1 RIGHT) q n SIMC RULE: I = min ( 1, 4( c+ )) with c = for tight control SLOW TUNING DESIRED (“smooth control”, c > ): q HERE YOU MAY WANT TO WAIT LONGER TO GET 1 RIGHT BECAUSE IT MAY AFFECT THE INTEGRAL TIME q BUT THEN ON THE OTHER HAND, GETTING THE RIGHT INTEGRAL TIME IS NOT ESSENTIAL FOR SLOW TUNING q SO ALSO HERE YOU MAY STOP AT 10 TIMES THE EFFECTIVE DELAY ( ) April 4 -8, 2004 KFUPM-Distillation Control Course 63

n “Integrating process” ( c < 0. 2 1): q Need only two parameters: n “Integrating process” ( c < 0. 2 1): q Need only two parameters: k’ and q From step response: Response on stage 70 to step in L Example. Step change in u: Initial value for y: Observed delay: At T=10 min: Initial slope: u = 0. 1 y(0) = 2. 19 = 2. 5 min y(T)=2. 62 y(t) 2. 62 -2. 19 7. 5 min =2. 5 April 4 -8, 2004 KFUPM-Distillation Control Course t [min] 64

Example (from quiz) Step response Δu=0. 1 INPUT y OUTPUT y tauc=10 tauc=2 n Example (from quiz) Step response Δu=0. 1 INPUT y OUTPUT y tauc=10 tauc=2 n n n Assume integrating process, theta=1. 5; k’ = 0. 03/(0. 1*11. 5)=0. 026 SIMC-tunings tauc=2: Kc=11, tau. I=14 (OK) SIMC-tunings tauc=10: Kc=3. 3, tau. I = 46 (too long because process is not actually integrating on this time scale!) April 4 -8, 2004 KFUPM-Distillation Control Course 65

Cascade control April 4 -8, 2004 KFUPM-Distillation Control Course 66 Cascade control April 4 -8, 2004 KFUPM-Distillation Control Course 66

Cascade control Tuning of cascade controllers • Want to control y 1 (primary CV), Cascade control Tuning of cascade controllers • Want to control y 1 (primary CV), but have “extra” measurement y 2 • Idea: Secondary variable (y 2) may be tightly controlled and this helps control of y 1. • Implemented using cascade control: Input (MV) of “primary” controller (1) is setpoint (SP) for “secondary” controller (2) • Tuning simple: Start with inner secondary loops (fast) and move upwards • Must usually identify ”new” model ( G 1’ = G 1 G 21 K 2 (I+K 2 G 22)-1 ) experimentally after closing each loop • One exception: Serial process, G 21 = G 22 2 – Inner (secondary-2) loop may be modelled with gain=1 and effective delay=( c+ )2 See next slide April 4 -8, 2004 KFUPM-Distillation Control Course 67

Cascade control Special case: Serial cascade y 2 = T 2 r 2 + Cascade control Special case: Serial cascade y 2 = T 2 r 2 + S 2 d 2, T 2 = G 2 K 2(I+G 2 K 2)-1 n n K 2 is designed based on G 2 (which has effective delay 2) q then y 2 = T 2 r 2 + S 2 d 2 where S 2 ¼ 0 and T 2 ¼ 1 · e-( 2+ c 2)s n T 2: gain = 1 and effective delay = 2+ c 2 q NOTE: If delay is in meas. of y 2 (and not in G 2) then T 2 ¼ 1 ·e- c 2 s n SIMC-rule: c 2 ≥ 2 n Time scale separation: c 2 ≤ c 1/5 (approximately) K 1 is designed based on G 1’ = G 1 T 2 n same as G 1 but with an additional delay 2+ c 2 April 4 -8, 2004 KFUPM-Distillation Control Course 68

Cascade control Example: Cascade control serial process d=6 ys K 1 y 2 s Cascade control Example: Cascade control serial process d=6 ys K 1 y 2 s K 2 u G 2 y 2 G 1 y 1 Use SIMC-rules! Without cascade With cascade April 4 -8, 2004 KFUPM-Distillation Control Course 69

Cascade control Tuning cascade control April 4 -8, 2004 KFUPM-Distillation Control Course 70 Cascade control Tuning cascade control April 4 -8, 2004 KFUPM-Distillation Control Course 70

Cascade control Tuning cascade control : serial process n Inner fast (secondary) loop: q Cascade control Tuning cascade control : serial process n Inner fast (secondary) loop: q q q n Outer slower primary loop: q n Reduced effective delay (2 s instead of 6 s) Time scale separation q n P or PI-control Local disturbance rejection Much smaller effective delay (0. 2 s) Inner loop can be modelled as gain=1 + 2*effective delay (0. 4 s) Very effective for control of large-scale systems April 4 -8, 2004 KFUPM-Distillation Control Course 71

Setpoint overshoot method Alternative closed-loop approach: Setpoint overshoot method Procedure: • Switch to P-only Setpoint overshoot method Alternative closed-loop approach: Setpoint overshoot method Procedure: • Switch to P-only mode and make setpoint change • Adjust controller gain to get overshoot about 0. 30 (30%) Record “key parameters”: 1. Controller gain Kc 0 2. Overshoot = (Δyp-Δy∞)/Δy∞ 3. Time to reach peak (overshoot), tp 4. Steady state change, b = Δy∞/Δys. Estimate of Δy∞ without waiting to settle: Δy∞ = 0. 45(Δyp + Δyu) Advantages compared to Ziegler-Nichols: * Not at limit to instability * Works on a simple second-order process. Closed-loop step setpoint response with P-only control. April 4 -8, 2004 KFUPM-Distillation Control Course M. Shamsuzzoha and S. Skogestad, ``The setpoint overshoot method: A simple and fast method for closed-loop PID tuning'', Journal of Process Control, 20, xxx-xxx (2010) 72

Setpoint overshoot method Summary setpoint overshoot method From P-control setpoint experiment record “key parameters”: Setpoint overshoot method Summary setpoint overshoot method From P-control setpoint experiment record “key parameters”: 1. Controller gain Kc 0 2. Overshoot = (Δyp-Δy∞)/Δy∞ 3. Time to reach peak (overshoot), tp 4. Steady state change, b = Δy∞/Δys Proposed PI settings (including detuning factor F) Choice of detuning factor F: n F=1. Good tradeoff between “fast and robust” (SIMC with τc=θ) n F>1: Smoother control with more robustness n F<1 to speed up the closed-loop response. April 4 -8, 2004 KFUPM-Distillation Control Course 73

Setpoint overshoot method Example: High-order process P-setpoint experiments Closed-loop PI response April 4 -8, Setpoint overshoot method Example: High-order process P-setpoint experiments Closed-loop PI response April 4 -8, 2004 KFUPM-Distillation Control Course 74

Setpoint overshoot method Example: Unstable plant First-order unstable process • No SIMC settings available Setpoint overshoot method Example: Unstable plant First-order unstable process • No SIMC settings available Closed-loop PI response April 4 -8, 2004 KFUPM-Distillation Control Course 75

CONTROLLABILITY A comment on Controllability n n (Input-Output) “Controllability” is the ability to achieve CONTROLLABILITY A comment on Controllability n n (Input-Output) “Controllability” is the ability to achieve acceptable control performance (with any controller) “Controllability” is a property of the process itself Analyze controllability by looking at model G(s) What limits controllability? April 4 -8, 2004 KFUPM-Distillation Control Course 76

CONTROLLABILITY Controllability Recall SIMC tuning rules 1. Tight control: Select c= corresponding to 2. CONTROLLABILITY Controllability Recall SIMC tuning rules 1. Tight control: Select c= corresponding to 2. Smooth control. Select Kc ¸ Must require Kc, max > Kc. min for controllability ) max. output deviation initial effect of “input” disturbance April 4 -8, 2004 KFUPM-Distillation Control Course y reaches k’ ¢ |d 0|¢ t after time t y reaches ymax after t= |ymax|/ k’ ¢ |d 0| 77

CONTROLLABILITY Controllability April 4 -8, 2004 KFUPM-Distillation Control Course 78 CONTROLLABILITY Controllability April 4 -8, 2004 KFUPM-Distillation Control Course 78

CONTROLLABILITY Example: Distillation column April 4 -8, 2004 KFUPM-Distillation Control Course 79 CONTROLLABILITY Example: Distillation column April 4 -8, 2004 KFUPM-Distillation Control Course 79

CONTROLLABILITY Example: Distillation column April 4 -8, 2004 KFUPM-Distillation Control Course 80 CONTROLLABILITY Example: Distillation column April 4 -8, 2004 KFUPM-Distillation Control Course 80

CONTROLLABILITY Conclusion controllability If the plant is not controllable then improved tuning will not CONTROLLABILITY Conclusion controllability If the plant is not controllable then improved tuning will not help Alternatives n n 1. Change the process design to make it more controllable q 2. Better “self-regulation” with respect to disturbances, e. g. insulate your house to make y=Tin less sensitive to d=Tout. Give up some of your performance requirements April 4 -8, 2004 KFUPM-Distillation Control Course 81