1. Should we forget the Smith Predictor?
Chriss Grimholt
Sigurd Skogestad
Department of Chemical Engineering,
NTNU, Trondheim, Norway
IFAC PID’18
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2. Outline What is a good controller?
Performance, IAE
Robustness, Ms
Delay margin
Smith Predictor (Time delay compensator)
Optimal Smith-Predictor vs. Optimal PI/PI for first-order plus delay plants
Trade-off plots (IAE vs. Ms)
Simulations
Why is Smith Predictor not robust?
Comparison with SIMC PI tuning rules
Conclusion

3. 1. What is a good 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:
Frequency domain: weighted sensitivity ||WpS||
Time domain: 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
IAE=Integrated absolute error

4. Output performance (J): Integrated absolute error (IAE)
IAE = Integrated absolute error = ∫|y-ys|dt, for step change in ys or d

5. Robustness

6. Delay margin Not guaranteed by MST
DM = «How much extra time delay we can add
to the loop before the system goes unstable»
Not guaranteed by MST

7. 2. Smith Predictor (SP) c Our case G0 = k/(τs+1)
K = Kc(1+1/τIs) (PI controller).
Internal model control (IMC): Special case of SP with ¿I=¿

8. 3. Compare SP with PI/PID C(s) SP: G0 = k/(τs+1)
K = Kc(1+1/τIs) (PI controller). Kc and τI:To be optimized
Compare SP with optimally tuned PI and PID
Would expect SP to be better since it uses process information (k, τ, θ)
CPI (s) = Kc(1+1/τIs) Kc and τI:To be optimized
CPID(s) = Kc(1+1/τIs +τDs) Kc, τI and τD :To be optimized

9. Four different first-order plus time delay processes
G(s) = e-s/(τs+1), τ = 0, 1, 8, 20
Each controller: Minimize IAE(J) for given value of MST (and vary MST from about 1.2 to 2.5)
Infeasible
Pareto-optimal
IAE=Integrated absolute error

10. Results for Ms=1.59
Optimal SP
Optimal PI
Optimal PID

11. PERFORMANCE (J-IAE)
Except for pure delay process, PID has best performance!
PI is somewhat worse than SP
IAE=Integrated absolute error

12. Delay margin Not guaranteed by MST
DM = «How much extra time delay we can add
to the loop before the system goes unstable»
Not guaranteed by MST

13. DELAY MARGIN
PI is clearly best. SP has poor delay margin at high performance

14. Hm…..?
Well, the Smith Predictor must at least be better than PI/PID for a pure time delay process with no delay error!

15. Pure time delay process
Ms=1.59
No model error:
G(s) = 1*exp(-1*s) SP
PI/PID
Optimal Tunings for Ms=1.59:
SP: Kc=0.73, taui=0.32
PI/PID: Kc=0.20, taui=0.32
Notes: 1, We show setpoint response (y) for 1-DOF controller, which for all processes is the same as the output disturbance response, but shifted by -1 (on y-axis)
2. Output (y) = Input (u) (delayed) for a pure time delay process
3. Response to input and output disturbance are the same for a pure time delay process.

16. Pure time delay process
Larger gain:
G(s) = 1.5*exp(-1*s) SP SP PI
PI/PID

17. Pure time delay process
Smaller gain:
G(s) = 0.5*exp(-1*s) SP PI SP
PI/PID

18. Hm…..? Well, it’s slightly better but I’m not impressed….
What about time delay error?

19. Pure time delay process
Ms=1.59
No model error:
G(s) = 1*exp(-1*s) SP
PI/PID
Tunings for Ms=1.59:
SP: Kc=0.73, taui=0.32
PI: Kc=0.20, taui=0.32
Note: 1. Output (y) = Input (u) for pure time delay process
2. Setpoint and disturbance response is the same

20. Pure time delay process
Larger time delay:
G(s) = 1*exp(-1.5*s) SP SP PI
PI/PID

21. Pure time delay process
Smaller time delay:
G(s) = 1*exp(-0.5*s) SP SP PI
PI/PID

22. Pure time delay process
Larger gain and larger delay:
G(s) = 1.5*exp(-1.5*s) SP SP PI
PI/PID

23. Pure time delay process
Larger gain and smaller delay:
G(s) = 1.5*exp(-0.5*s) SP SP
PI/PID PI

24. Let’s look at G(s)=e-s/(s+1)
PERFORMANCE (J-IAE)
DELAY MARGIN
When is SP good?
Let’s look at G(s)=e-s/(s+1)

25. Ms=1.59 SETPOINT AND DISTURBANCE RESPONSE PID SP PI SP PID PI
Nominal: G(s)=e-s/(s+1) PI
Larger time delay: G(s)=e-1.5s/(s+1) SP
PID
Smaller delay and smaller time constant:
G(s)=e-0.5s/(0.5*s+1) PI

26. Ms=1.59 INPUT USAGE PID SP PI PID SP PI SP PI PID Nominal:
G(s)=e-s/(s+1) SP PI SP PI
PID
G(s)=e-1.5s/(s+1) SP PI
PID
G(s)=e-0.5s/(0.5*s+1)

27. Summary. G(s)=e-s/(s+1). Ms=1.59
PID best performance, but aggressive input usage
PI slower than SP, but most robust to model errors and less input usage
Overall: PI preferred!

28. PERFORMANCE (J-IAE)
DELAY MARGIN
Ms=1.81. G(s)=e-s/(s+1)
Ms=1.81

29. Ms=1.81: PI is now clearly preferred
SETPOINT and DISTURBANCE RESPONSE
Ms=1.81
Ms=1.81: PI is now clearly preferred

30. 4. What’s the problem with SP
4. What’s the problem with SP? It’s simply a «dangerous» parameterization
Ms=1.81

31. |L|=1
ω=5.
Unstable with phase change of π rad
i.e. delay -0.6 or +0.6

32. Red curve: L with delay reduced from 1 to 0.4

33. 5. So is the Smith Predictor never significantly better than a well-tuned PI/PID?
No, it doesn’t seem so for a first-order plus delay process, especially if we take into account model uncertainty
I was very surprised about how poor SP is
But maybe SP is easier to tune?

34. No, I don’t think so. Try iSIMC-rules
Ingimundarson and Hagglund (2002)
“PID performs best over a large portion of the area but it has been shown that it might be difficult to obtain this optimal performance with manual tuning".
No, I don’t think so. Try iSIMC-rules

35. 1. Improved PID-rule: Add θ/3 to D
Two “Improved SIMC”-rules that give optimal PI/PID for pure time delay process
1. Improved PID-rule: Add θ/3 to D
Tuning parameter: τc
Sigurd Skogestad, ''Simple analytic rules for model reduction and PID controller tuning'' ,
J. Process Control, vol. 13 (2003),

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

37. Comparison of optimal-PID and SIMC
ZN-PID unstable
CONCLUSION PI: SIMC-improved almost «Pareto-optimal»

38. 6. Conclusion
Can we forget the Smith Predictor and similar time-delay compensators?
Yes, it seems so.
A well-tuned PI is better in practice

39. Extra

40. Ms=1.59 Nominal: G(s)=e-s/(s+1) More time delay: G(s)=e-2s/(s+1)
SETPOINT RESPONSE
Ms=1.59
PID SP PI
Nominal: G(s)=e-s/(s+1)
More time delay: G(s)=e-2s/(s+1)
No time delay: G(s)=1/(s+1)

41. Ms=1.59 INPUT USAGE Nominal: G(s)=e-s/(s+1) G(s)=e-2s/(s+1)
PID
Nominal:
G(s)=e-s/(s+1) PI
G(s)=e-2s/(s+1)
PID SP PI
PID
G(s)=1/(s+1) SP PI

42. Ms=1.59 PID SP PI G(s) = exp(-s)/(s+1) G(s) = exp(-0.5*s)/(s+1)

43. Ms=1.59
G(s) = exp(-0.5*s)/(s+1)
PID SP PI
PID
INPUT signal SP PI

44. Ms=1.59
G(s) = exp(-0.5*s)/(0.5*s+1) SP
PID PI

45. G(s) = exp(-0.5*s)/(s+1)
PID SP PI
INPUT