1. PID-tuning using the SIMC rules
Sigurd Skogestad
Department of Chemical Engineering,
NTNU, Trondheim, Norway
Salamanca, Spain, Feb. 2017
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2. Trondheim NORWAY Oslo DENMARK GERMANY UK Arctic circle SWEDEN
North Sea
Trondheim
NORWAY
SWEDEN
Oslo
DENMARK
GERMANY UK

4. Operation hierarchy
RTO
CV1
MPC
CV2
PID
u (valves)

5. Outline First-order plus delay model. PID-control. SIMC PI(D)-rule
Optimal PI controller
Comparison of SIMC with optimal PI
Improved SIMC-PI for time-delay process
Non-PID control of first-order plus delay process: Better with IMC / Smith Predictor / MPC? (no)
Example: Level control
Conclusion

6. 1. Need a model for tuning First-order + delay model for PI-control
Second-order model for PID-control
Recommend: Use second-order model only if ¿2>µ

7. Step response experiment
MODEL, Approach 1
Step response experiment
Make step change in one u (MV) at a time
Record the output (s) y (CV) u y
MV = manipulated variable (input)
CV = controlled variable (output)

8. Identify k, 1 and  from step response
k’=k/1
STEP IN INPUT u (MV)
RESULTING OUTPUT y (CV)
If doesn’t settle: Don’t need k and 1.Stop experiment
at about 8 x  and assume integrating process
: 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

9. Step response integrating processΔy Δt

10. MODEL, Approach 2
Closed-loop setpoint response with P-controller with about 20-40% overshoot
Kc0=1.5
Δys=1
Δy∞
OBTAIN DATA IN RED (first overshoot
and undershoot), and then:
tp=4.4, dyp=0.79; dyu=0.54, Kc0=1.5, dys=1
dyinf = 0.45*(dyp + dyu)
Mo =(dyp -dyinf)/dyinf % Mo=overshoot (about 0.3)
b=dyinf/dys
A = 1.152*Mo^ *Mo + 1.0
r = 2*A*abs(b/(1-b))
%2. OBTAIN FIRST-ORDER MODEL:
k = (1/Kc0) * abs(b/(1-b)) = 0.99
theta = tp*[ *exp(-0.61*r)] = 1.68
tau = theta*r = 3.03
Δyp=0.79
Δyu=0.54
tp=4.4
Similar to Ziegler-Nichols experiment but get more information
Ref: Shamssuzzoha and Skogestad (JPC, 2010)
+ modification by C. Grimholt (PID-book 2012)

11. Model reduction of more complicated model
MODEL, Approach 3
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 the “effective” delay 

12. MODEL, Approach 3. HALF RULE

13. MODEL, Approach 3
Example 1
Half rule

14. MODEL, Approach 3
original
1st-order+delay

15. MODEL, Approach 3 2
half rule

16. MODEL, Approach 3. HALF RULE
original
1st-order+delay
2nd-order+delay

17. 2. PID controller e Only two parameters left (Kc and τI)
Time domain (“ideal” PID)
Laplace domain (“ideal”/”parallel” form)
For our purposes. Simpler with cascade form
Usually τD=0. Then the two forms are identical.
Only two parameters left (Kc and τI)
How difficult can it be to tune???
Surprisingly difficult without systematic approach!

18. Disadvantages Ziegler-Nichols: Aggressive settings No tuning parameter
Trans. ASME, 64, (Nov. 1942).
Comment:
Similar to SIMC for integrating process with ¿c=0:
Kc = 1/k’ 1/µ
¿I = 4 µ
Disadvantages Ziegler-Nichols:
Aggressive settings
No tuning parameter
Poor for processes with large time delay (µ)

19. Disadvantage IMC-PID:
Many rules
Poor disturbance response for «slow» processes (with large ¿1/µ)

20. Motivation for developing SIMC PID tuning rules
The tuning rules should be well motivated, and preferably be model-based and analytically derived.
They should be simple and easy to memorize.
They should work well on a wide range of processes.

21. 3. SIMC PI tuning rule
Approximate process as first-order with delay (e.g., use “half rule”)
k = process gain
¿1 = process time constant
µ = process delay
Derive SIMC tuning rule*:
Open-loop step response
c ¸ - : Desired closed-loop response time (tuning parameter)
Integral time rule combines well-known rules:
IMC (Lamda-tuning): Same as SIMC for small ¿1 (¿I = ¿1)
Ziegler-Nichols: Similar to SIMC for large ¿1 (if we choose ¿c= 0; aggressive!)
Reference: S. Skogestad, “Simple analytic rules for model reduction and PID controller design”, J.Proc.Control, Vol. 13, , 2003
(*) “Probably the best simple PID tuning rules in the world”

22. Derivation SIMC tuning rule.
Basis: Direct synthesis (IMC) for setpoints
Closed-loop response to setpoint change:
Idea: Specify desired response:
and from this get the controller. ……. Algebra:

23. NOTE: Setting T(0)=1 (steady-state gain = 1) gives integral action.

24. Derivation SIMC tuning rule (setpoints)

25. Integral time
Found: Integral time = dominant time constant (I = 1) (IMC-rule)
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! g c d y u

26. Effect of integral time on closed-loop response
IMC-rule:
I = 1=30
Setpoint change (ys=1) at t=0
Input disturbance (d=1) at t=20

27. SIMC: Integral time correction
Setpoints: ¿I=¿1(“IMC-rule”). Want smaller integral time for disturbance rejection for “slow” processes (with large ¿1), but to avoid “slow oscillations” must require:
Derivation:
Conclusion SIMC:

28. Conclusion: SIMC-PID Tuning Rules
One tuning parameter: c
Note:
Recommend derivative action (PID) only for «dominant» second-order proceses» with τ2>θ.
Otherwise, add τ2 to θ and use PI.

29. Selection of tuning parameter c
Tuning parameter: c = desired closed-loop response time
Choice c= (“tight control”) gives good balance between performance (IAE) and robustness
Other cases: Select c >  (“smooth control”) for
slower control
smoother input usage
less disturbing effect on rest of the plant
less sensitivity to measurement noise
better robustness
S. Skogestad, ``Tuning for smooth PID control with acceptable disturbance rejection'', Ind.Eng.Chem.Res, 45 (23), (2006).

30. Typical closed-loop SIMC responses with the choice c=

31. IMC has I=1 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=0c
Input disturbance at t=20

32. Approximate as first-order model:
k=1, 1 = 1+0.1=1.1, = = 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
Approximate as second-order model:
k=1, 1 = 1, 2= =0.22, = = 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

33. SIMC: Tuning parameter (¿c) correlates nicely with robustness measuresMs PM
60o
1.6 1 1
DM=
D / GM 2 3 1 1
c /
c /

34. But: How good is really the SIMC rule?
Want to compare with:
Optimal PI-controller
for class of first-order with delay processes

35. 4. Optimal PID controller
High controller gain (“tight control”)
Low controller gain (“smooth control”)
Multiobjective. Tradeoff between
Output performance
Robustness
Input usage
Noise sensitivity
Our choice:
Quantification
Output performance:
Rise time, overshoot, settling time
IAE or ISE for setpoint/disturbance
Robustness: Ms, Mt, GM, PM, Delay margin, …
Input usage: ||KSGd||, TV(u) for step response
Noise sensitivity: ||KS||, etc.
J = avg. IAE for
setpoint/disturbance
Ms = peak sensitivity

36. Optimal sensitivity function, S = 1/(gc+1)
Optimal PI-controller
Optimal sensitivity function, S = 1/(gc+1)
Ms=2
|S|
Ms=1.59
Ms=1.2
frequency

37. Output performance (J)
IAE = Integrated absolute error = ∫|y-ys|dt, for step change in ys or d
Cost J(c) is independent of:
process gain (k)
setpoint (ys or dys) and disturbance (d) magnitude
unit for time

38. Ms=1.59 Optimal closed-loop response IAE Optimal PI-controller
Setpoint change at t=0, Input disturbance at t=20,
g(s)=k e-θs/(1s+1), Time delay θ=1

39. Ms=1.2 Optimal closed-loop response IAE Optimal PI-controller
Setpoint change at t=0, Input disturbance at t=20,
g(s)=k e-θs/(1s+1), Time delay θ=1

40. Optimal PI-controller: Minimize J=IAE for given Ms
Chriss Grimholt and Sigurd Skogestad. "Optimal PI-Control and Verification of the SIMC Tuning Rule".
Proceedings IFAc conference on Advances in PID control (PID'12), Brescia, Italy, March 2012.

41. Uninteresting
Pareto-optimal PI
Infeasible

42. Optimal performance (J) vs. Ms
Optimal PI-controller
Optimal performance (J) vs. Ms

43. 5. What about SIMC-PI?

44. Comparison of J vs. Ms for optimal and SIMC-PI for 4 processes

45. Optimal PI-settings vs. process time constant (1 /θ)
Optimal PI-controller
Optimal PI-settings vs. process time constant (1 /θ)

46. Conclusion (so far): How good is really the SIMC rule?
Varying C gives (almost) Pareto-optimal tradeoff between performance (J) and robustness (Ms)
C = θ is a good ”default” choice
Not possible to do much better with any other PI-controller!
Exception: Time delay process

47. 6. Can the SIMC-rule be improved?
Yes, for time delay process

48. Optimal PI-settings (small 1)
Optimal PI-controller
Optimal PI-settings (small 1)
0.33
Time-delay process
SIMC: I=1=0

49. Step response for time delay process
Optimal PI
θ=1
NOTE for time delay process: Setpoint response = disturbance responses = input response

50. Two “Improved SIMC”-rules that give optimal for pure time delay process
1. Improved PI-rule: Add θ/3 to 1
1. Improved PID-rule: Add θ/3 to 2

51. Comparison of J vs. Ms for optimal-PI and SIMC for 4 processes
CONCLUSION PI: SIMC-improved almost «Pareto-optimal»

52. 7. Better with IMC, Smith Predictor or MPC?
Surprisingly, the answer is:
NO, often worse

53. Smith Predictor c
K: Typically a PI controller
Internal model control (IMC): Special case with ¿I=¿1
Fundamental problem Smith Predictor: No integral action in c for integrating process

54. Optimal SP compared with optimal PI
¿1=0
¿1=1
¿1=8
¿1=20
¿1=20 since J=1 for SP
for integrating process
Small performance gain with Smith Predictor
SP = Smith Predictor with PI (K)

55. Additional drawbacks with Smith Predictor
No integral action for integrating process
Sensitive to both positive and negative delay error
With tight tuning (Ms approaching 2): Multiple gain and delay margins

56. Step response, SP and PI =1 =0.57 =1.43
(nominal)
=1.43
=0.57 y
time
time
time
Smith Predictor (IMC): Sensitive to both positive and negative delay error
Nominal. 1 =  = 1

57. Delay margin, SP and PI
SP = Smith Predictor

58. 8. Example: Level controlq LC
8. Example: 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.

59. Level control: Can have both fast and slow oscillations
Slow oscillations (Kc too low): Period > 3¿I
Fast oscillations (Kc too high due to delay): Period < 3¿I
Here: Consider the less known slow oscillations

60. How avoid slowly oscillating levels?
Use SIMC with large τc . .
0.1 ¼ = 1/2

61. 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

62. LEVEL CONTROL

63. 8. Conclusion Questions for 1st and 2nd order processes with delay:
How good is really PI/PID-control?
Answer: Very good, but it must be tuned properly
How good is the SIMC PI/PID-rule?
Answer: Pretty close to the optimal PI/PID,
To improve PI for time delay process: Replace 1 by 1+µ/3
Can we do better with Smith Predictor or IMC?
No. Slightly better performance in some cases, but much worse delay margin
Can we do better with other non-PI/PID controllers (MPC)?
Not much (further work needed)
SIMC: “Probably the best simple PID tuning
rule in the world”