1. PID Tuning and Controllability Sigurd Skogestad NTNU, Trondheim, Norway

2. Tuning of PID controllers 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 One tuning rule! Easily memorized Reference: S. Skogestad, “Simple analytic rules for model reduction and PID controller design”, J.Proc.Control, Vol. 13, 291-309, 2003 (*) “Probably the best simple PID tuning rules in the world”  c ¸ 0: desired closed-loop response time (tuning parameter) For robustness select:  c ¸ 

3. Need a model for tuning Model: Dynamic effect of change in input u (MV) on output y (CV) First-order + delay model for PI-control Second-order model for PID-control

4. Step response experiment Make step change in one u (MV) at a time Record the output (s) y (CV)

5. First-order plus delay process Step response experiment k’=k/  1 STEP IN INPUT u (MV) RESULTING OUTPUT y (CV)  Delay - Time where output does not change  1 : Time constant - Additional time to reach 63% of final change k : steady-state gain =  y(1)/  u k’ : slope after response “takes off” = k/  1

6. Model simplifications For control: Dynamics at “control time scale” (  c ) are important 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 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  ¼ 1 (may be neglected for  c > 5)  1 ¼ 200 (may be neglected for  c < 40) time  c = desired response time

7. “Integrating process” (  c < 5  1 ):  Need only two parameters: k’ and   From step response: Response on stage 70 to step in L 7.5 min 2.62-2.19  Example. Step change in u:  u = 0.1 Initial value for y: y(0) = 2.19 Observed delay:  = 2.5 min At T=10 min: y(T)=2.62 Initial slope: y(t) t [min]

8. First-order with delay process (  c > 5  1 ) Need 3 parameters:  Delay   2 of the following: k, k’,  1 (Note relationship k’ = k/  1) Example: First-order model:  = 0 k= (0.134-0.1)/(-0.5) = -0.068 (steady-state gain)  1 = 16 min (63% of change)  Gives k’ = 0.068/16 = 0.00425  (could alternatively find k’ by observing the initial response)  BOTTOM V: -0.5 xBxB 16 min 63%

9. Step response experiment: How long do we need to wait? FAST TUNING DESIRED (“tight control”,  c =  ):  NORMALLY NO NEED TO RUN THE STEP EXPERIMENT FOR LONGER THAN ABOUT 10 TIMES THE EFFECTIVE DELAY (  )  EXCEPTION: LET IT RUN A LITTLE LONGER IF YOU SEE THAT IT IS ALMOST SETTLING (TO GET  1 RIGHT)  SIMC RULE:   = min (  , 4(  c +  )) with  c =  for tight control SLOW TUNING DESIRED (“smooth control”,  c >  ):  HERE YOU MAY WANT TO WAIT LONGER TO GET  1 RIGHT BECAUSE IT MAY AFFECT THE INTEGRAL TIME  BUT THEN ON THE OTHER HAND, GETTING THE RIGHT INTEGRAL TIME IS NOT ESSENTIAL FOR SLOW TUNING  SO ALSO HERE YOU MAY STOP AT 10 TIMES THE EFFECTIVE DELAY (  )

10. Model reduction of more complicated model Start with complicated stable model on the form Want to get a simplified model on the form Most important parameter is usually the “effective” delay 

13. half rule

14. Approximation of zeros

15. Derivation of SIMC-PID tuning rules PI-controller (based on first-order model) For second-order model add D-action. For our purposes it becomes simplest with the “series” (cascade) PID-form:

16. Basis: Direct synthesis (IMC) Closed-loop response to setpoint change Idea: Specify desired response (y/y s )=T and from this get the controller. Algebra:

17. IMC Tuning = Direct Synthesis

18. Integral time Found: Integral time = dominant time constant (  I =  1 ) Works well for setpoint changes Needs to be modified (reduced) for integrating disturbances Example. “Almost-integrating process” with disturbance at input: G(s) = e -s /(30s+1) Original integral time  I = 30 gives poor disturbance response Try reducing it! gc d y u

19. Integral Time  I =  1 Reduce  I to this value:  I = 4 (  c +  ) = 8 

20. SIMC-PID Tuning Rules One tuning parameter:  c

21. Some special cases One tuning parameter:  c

22. Note: Derivative action is commonly used for temperature control loops. Select  D equal to time constant of temperature sensor

24. Selection of tuning parameter  c Two cases 1. Tight control: Want “fastest possible control” subject to having good robustness 2. Smooth control: Want “slowest possible control” subject to having acceptable disturbance rejection

25. TIGHT CONTROL

26. Example. Integrating process with delay=1. G(s) = e -s /s. Model: k’=1,  =1,  1 =1 SIMC-tunings with  c with =  =1: IMC has  I =1 Ziegler-Nichols is usually a bit aggressive Setpoint change at t=0Input disturbance at t=20 TIGHT CONTROL

27. 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 =  ): K c = 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 =  ): K c = 1 ¢ 1/(2¢ 0.028) = 17.9,  I =min(1,8¢ 0.028) = 0.224,  D =0.22 TIGHT CONTROL

29. Tuning for smooth control Will derive K c,min. From this we can get  c,max using SIMC tuning rule SMOOTH CONTROL Tuning parameter:  c = desired closed-loop response time Selecting  c =  (“tight control”) is reasonable for cases with a relatively large effective delay  Other cases: Select  c >  for  slower control  smoother input usage less disturbing effect on rest of the plant  less sensitivity to measurement noise  better robustness Question: Given that we require some disturbance rejection.  What is the largest possible value for  c ?  Or equivalently: The smallest possible value for K c ?

30. Closed-loop disturbance rejection d0d0 y max -d 0 -y max SMOOTH CONTROL =|u 0 |

31. KcKc u Minimum controller gain for PI-and PID-control: K c ¸ K c,min = |u 0 |/|y max | |u 0 |: Input magnitude required for disturbance rejection |y max |: Allowed output deviation SMOOTH CONTROL

32. Minimum controller gain: Industrial practice: Variables (instrument ranges) often scaled such that Minimum controller gain is then Minimum gain for smooth control ) Common default factory setting K c =1 is reasonable ! SMOOTH CONTROL (span)

33. Exampl e Does not quite reach 1 because d is step disturbance (not not sinusoid) Response to disturbance = 1 at input  c is much larger than  =0.25 SMOOTH CONTROL

34. Application of smooth control Averaging level control V q 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. Let |u 0 | = |  q 0 | – expected flow change [m3/s] (input disturbance) |y max | = |  V max | - largest allowed variation in level [m3] Minimum controller gain for acceptable disturbance rejection: K c ¸ K c,min = |u 0 |/|y max | From the material balance (dV/dt = q – q out ), the model is g(s)=k’/s with k’=1. Select K c =K c,min. SIMC-Integral time for integrating process:  I = 4 / (k’ K c ) = 4 |  V max | / |  q 0 | = 4 ¢ residence time provided tank is nominally half full and  q 0 is equal to the nominal flow. SMOOTH CONTROLLEVEL CONTROL If you insist on integral action then this value avoids cycling

35. Rule: K c ¸ |u 0 |/|y max | =1 (in scaled variables) Exception to rule: Can have K c < 1 if disturbances are handled by the integral action.  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).  Then  I =  1 is small so integral action is “large”  For example, flow control SMOOTH CONTROL K c : Assume variables are scaled with respect to their span

36. Summary: Tuning of easy loops Easy loops: Small effective delay (  ¼ 0), so closed- loop response time  c (>>  ) is selected for “smooth control” Flow control: K c =0.5,  I =  1 = time constant valve (typically, 10s to 30s) Level control: K c =2 (and no integral action) Other easy loops (e.g. pressure control): K c = 2,  I = min(4  c,  1 )  Note: Often want a tight pressure control loop (so may have K c =10 or larger) SMOOTH CONTROL K c : Assume variables are scaled with respect to their span

37. More on level control Level control often causes problems Typical story:  Level loop starts oscillating  Operator detunes by decreasing controller gain  Level loop oscillates even more ...... ??? Explanation: Level is by itself unstable and requires control. LEVEL CONTROL

38. Integrating process: Level control LEVEL CONTROL

39. How avoid oscillating levels? LEVEL CONTROL

40. Case study oscillating level 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 LEVEL CONTROL

42. Conclusion PID tuning

43. Cascade control

44. Tuning of cascade controllers

45. Cascade control serial process d=6 G1G1 u2u2 y1y1 K1K1 ysys G2G2 K2K2 y2y2 y 2s

46. Cascade control serial process d=6 Without cascade With cascade G1G1 u y1y1 K1K1 ysys G2G2 K2K2 y2y2 y 2s

47. Tuning cascade control: serial process Inner fast (secondary) loop:  P or PI-control  Local disturbance rejection  Much smaller effective delay (0.2 s) Outer slower primary loop:  Reduced effective delay (2 s instead of 6 s) Time scale separation  Inner loop can be modelled as gain=1 + 2*effective delay (0.4s) Very effective for control of large-scale systems

48. Controllability (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? CONTROLLABILITY

49. Recall SIMC tuning rules 1. Tight control: Select  c =  corresponding to 2. Smooth control. Select K c ¸ Must require K c,max > K c.min for controllability ) Controllability initial effect of “input” disturbance max. output deviation y reaches k’ ¢ |d 0 |¢ t after time t y reaches y max after t= |y max |/ k’ ¢ |d 0 | CONTROLLABILITY

50. Controllability CONTROLLABILITY

51. Example: Distillation column CONTROLLABILITY

52. Example: Distillation column CONTROLLABILITY

53. Distillation example: Frequency domain analysis CONTROLLABILITY

54. Other factors limiting controllability CONTROLLABILITY Input limitations (saturation or “slow valve”): Can sometimes limit achievable speed of response Unstable process (including levels): Need feedback control for stabilization ) Makes sure inputs do not saturate in stabilizing loops