1 the improved simc method for pi controller tuning chriss grimholt sigurd skogestad ntnu,...
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The improved SIMC method for PI controller tuning
Chriss GrimholtSigurd Skogestad
NTNU, Trondheim, Norway
Reference: C. Grimholt and S. Skogestad, “The improved SIMC method for PI controller tuning”, IFAC-conference PID’12, Brescia, Italy, March 2012
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SIMC PI tuning rule• 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 (Also reprinted in MIC)
(*) “Probably the best simple PID tuning rule in the world”
c ¸ - : Desired closed-loop response time (tuning parameter)•For robustness select: c ¸
Questions:1. How good is really this rule?2. Can it be improved?
Step response
*
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1. How good is really the SIMC rule?
Need to compare with:• Optimal PI-controller
for class of first-order with delay processes
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Optimal controller
• Multiobjective. Tradeoff between– Output performance – Robustness– Input usage– Noise sensitivity
High controller gain (“tight control”)
Low controller gain (“smooth control”)
• Quantification– Output performance:
• Frequency domain: weighted sensitivity ||WpS||
• Time domain: IAE or ISE for setpoint/disturbance
– Robustness: Ms, GM, PM, Delay margin
– Input usage: ||KSGd||, ISE or TV for step response
– Noise sensitivity: ||KS||, etc.
Ms = peak sensitivity
J = avg. IAE for setpoint/disturbance
Our choice:
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IAE = Integrated absolute error = ∫|y-ys|dt, for step change in ys or d
Cost J is independent of:1. process gain k2. disturbance magnitude3. unit for time
Output performance (J)
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Optimal PI-controller: Minimize J for given Ms
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Ms=2
Ms=1.2
Ms=1.59|S|
frequency
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Ms=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
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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
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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
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Optimal performance (J) vs. Ms
Optimal PI-controller
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Input usage (TV) increases with Ms
TVys TVd
Optimal PI-controller
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Setpoint / disturbance tradeoff
Pure time delay process: J=1, No tradeoff (since setpoint and disturbance the same)
Optimal controller: Emphasis on disturbance d
Optimal PI-controller
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Setpoint / disturbance tradeoff
Optimal setpoint: No integral action
Optimal PI-controller
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Comparison with SIMC
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16Comparison 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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2. Can the SIMC-rule be improved?
Yes, possibly for time delay process
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Optimal PI-settings
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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Improved SIMC-rule: Replace 1 by 1+θ/3
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Step response for time delay process
Time delay process: Setpoint and disturbance response same
θ=1
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23Comparison of J vs. Ms for optimal and SIMC for 4 processes
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Conclusion
Questions:1. How good is really the SIMC-rule?
– Answer: Pretty close to optimal, except for time delay process
2. Can it be improved?– Yes, to improve for time delay process: Replace 1 by 1+θ/3 in rule
to get ”Improved-SIMC”
• Not possible to do much better with any other PI-controller!
Reference: C. Grimholt and S. Skogestad, “The improved SIMC method for PI controller tuning”, IFAC-conference PID’12, Brescia, Italy, March 2012
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Model from closed-loop response with P-controller
Kc0=1.5Δys=1
Δyu=0.54
Δyp=0.79
tp=4.4
dyinf = 0.45*(dyp + dyu)Mo =(dyp -dyinf)/dyinfb=dyinf/dys
A = 1.152*Mo^2 - 1.607*Mo + 1.0r = 2*A*abs(b/(1-b))k = (1/Kc0) * abs(b/(1-b))theta = tp*[0.309 + 0.209*exp(-0.61*r)]tau = theta*r
Example: Get k=0.99, theta =1.68, tau=3.03Ref: Shamssuzzoha and Skogestad (JPC, 2010) + modification by C. Grimholt (Project, NTNU, 2010; see also new from PID-book 2011)
Δy∞