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Optimality of PID control for process control applications. Sigurd Skogestad Chriss Grimholt NTNU, Trondheim, Norway. AdCONIP, Japan, May 2014. TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: A A A A A A A A A. Trondheim, Norway. Operation hierarchy. - PowerPoint PPT Presentation
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Optimality of PID control for process control applications
Sigurd SkogestadChriss Grimholt
NTNU, Trondheim, Norway
AdCONIP, Japan, May 2014
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Trondheim, Norway
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Operation hierarchy
CV1
MPC
PID
CV2
RTO
u (valves)
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Outline
1.Motivation: Ziegler-Nichols open-loop tuning + IMC
2.SIMC PI(D)-rule
3.Definition of optimality (performance & robustness)
4.Optimal PI control of first-order plus delay process
5.Comparison of SIMC with optimal PI
6.Improved SIMC-PI for time-delay process
7.Non-PID control: Better with IMC / Smith Predictor? (no)
8.Conclusion
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PID controller
•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!
e
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Trans. ASME, 64, 759-768 (Nov. 1942).
Disadvantages Ziegler-Nichols:1.Aggressive settings2.No tuning parameter3.Poor for processes with large time delay (µ)
Comment:Similar to SIMC for integrating process with ¿c=0:Kc = 1/k’ 1/µ¿I = 4 µ
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Disadvantage IMC-PID:1.Many rules2.Poor disturbance response for «slow» processes (with large ¿1/µ)
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Motivation for developing SIMC PID tuning rules
1.The tuning rules should be well motivated, and preferably be model-based and analytically derived.
2.They should be simple and easy to memorize.
3.They should work well on a wide range of processes.
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2. SIMC PI tuning rule1.A
pproximate process as first-order with delay• k = process gain• ¿1 = process time constant• µ = process delay
2.Derive SIMC tuning rule:
Reference: S. Skogestad, “Simple analytic rules for model reduction and PID controller design”, J.Proc.Control, Vol. 13, 291-309, 2003
c ¸ - : Desired closed-loop response time (tuning parameter)
Open-loop step response
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)
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Derivation SIMC tuning rule (setpoints)
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Effect of integral time on closed-loop response
I = 1=30
Setpoint change (ys=1) at t=0 Input disturbance (d=1) at t=20
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SIMC: Integral time correction•S
etpoints: ¿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:
15Typical closed-loop SIMC responses with the choice c=
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SIMC PI tuning rule
c ¸ - : Desired closed-loop response time (tuning parameter)•For robustness select: c ¸
Two questions:• How good is really the SIMC rule?• Can it be improved?
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 rule in the world”
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How good is really the SIMC rule?
Want to compare with:
•Optimal PI-controller
for class of first-order with delay processes
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•Multiobjective. Tradeoff between
– Output performance – Robustness– Input usage– Noise sensitivity
High controller gain (“tight control”)
Low controller gain (“smooth control”)
• 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.
Ms = peak sensitivity
J = avg. IAE for setpoint/disturbance
Our choice:
3. Optimal controller
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IAE = Integrated absolute error = ∫|y-ys|dt, for step change in ys or d
Cost J(c) is independent of:1. process gain (k)2. setpoint (ys or dys) and disturbance (d) magnitude3. unit for time
Output performance (J)
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4. Optimal PI-controller: Minimize J for given Ms
Optimal PI-controller
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, 28-30 March 2012.
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Optimal PI-settings vs. process time constant (1 /θ)
Optimal PI-controller
Ziegler-Nichols
Ziegler-Nichols
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Ms=2
Ms=1.2
Ms=1.59|S|
frequency
Optimal PI-controller
Optimal sensitivity function, S = 1/(gc+1)
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Ms=2
Optimal PI-controller
4 processes, g(s)=k e-θs/(1s+1), Time delay θ=1.Setpoint change at t=0, Input disturbance at t=20,
Optimal closed-loop response
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Ms=1.59
Optimal PI-controller
Setpoint change at t=0, Input disturbance at t=20,g(s)=k e-θs/(1s+1), Time delay θ=1
Optimal closed-loop response
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Ms=1.2
Optimal PI-controller
Setpoint change at t=0, Input disturbance at t=20,g(s)=k e-θs/(1s+1), Time delay θ=1
Optimal closed-loop response
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Infeasible
UninterestingPareto-optimal PI
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Optimal performance (J) vs. Ms
Optimal PI-controller
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5. What about SIMC-PI?
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SIMC: Tuning parameter (¿c) correlates nicely with robustness measures
Ms
GM
PM
¿c=µ ¿c=µ
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What about SIMC-PI performance?
34Comparison of J vs. Ms for optimal and SIMC for 4 processes
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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
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6. Can the SIMC-rule be improved?
Yes, for time delay process
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Optimal PI-settings vs. process time constant (1 /θ)
Optimal PI-controller
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Optimal PI-settings (small 1)
Time-delay processSIMC: I=1=0
0.33
Optimal PI-controller
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Step response for time delay process
θ=1
Optimal PI
NOTE for time delay process: Setpoint response = disturbance responses = input response
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Pure time delay process
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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
42Comparison of J vs. Ms for optimal-PI and SIMC for 4 processes
CONCLUSION PI: SIMC-improved almost «Pareto-optimal»
7. Better with IMC or Smith Predictor?
Surprisingly, the answer is:
NO, worse
Smith Predictor
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
c
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Optimal SP compared with optimal PI
SP = Smith Predictor with PI (K)
¿1=20 since J=1 for SPfor integrating process
¿1=20
¿1=0
¿1=8
¿1=1
Small performance gain with Smith Predictor
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Additional drawbacks with Smith Predictor•N
o integral action for integrating process•S
ensitive to both positive and negative delay error•W
ith tight tuning (Ms approaching 2): Multiple gain and delay margins
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Step response, SP and PI
g(s) = k e¡ s
s+1
µ= 1
y
time time time
Smith Predictor: Sensitive to both positive and negative delay error
SP = Smith Predictor
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Delay margin, SP and PI
SP = Smith Predictor
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8. Conclusion Questions for 1st and 2nd order processes with delay:
1.How good is really PI/PID-control?– Answer: Very good, but it must be tuned properly
2.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
3.Can we do better with Smith Predictor or IMC?– No. Slightly better performance in some cases, but much worse delay margin
4.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”
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11th International IFAC Symposium on Dynamics and Control of Process and
Bioprocess Systems (DYCOPS+CAB). 06-08 June 2016
Location: Trondheim (NTNU)
Organizer: NFA (Norwegian NMO) + NTNU (Sigurd Skogestad, Bjarne Foss, Morten Hovd, Lars Imsland, Heinz Preisig, Magne Hillestad, Nadi Bar),
Norwegian University of Science and Technology (NTNU), Trondheim
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