Optimality of PID control for process control applications
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Transcript of Optimality of PID control for process control applications
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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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