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The improved SIMC method for PI controller tuning

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25 diapositives · 583 Views · Oct 10, 2014

The improved SIMC method for PI controller tuning. Chriss Grimholt Sigurd 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. SIMC PI tuning rule. *.

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  1. The improved SIMC method for PI controller tuning Chriss GrimholtSigurd SkogestadNTNU, Trondheim, Norway Reference: C. Grimholt and S. Skogestad, “The improved SIMC method for PI controller tuning”, IFAC-conference PID’12, Brescia, Italy, March 2012

  2. SIMC PI tuning rule * • Look at initial part of step response, Initial slope: k’ = k/1 • One tuning rule! Easily memorized • c¸ -: Desired closed-loop response time (tuning parameter) • For robustness select: c¸ Step response • Questions: • How good is really this 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 (Also reprinted in MIC) (*) “Probably the best simple PID tuning rule in the world”

  3. 1. How good is really the SIMC rule? Need to compare with: • Optimal PI-controller for class of first-order with delay processes

  4. Optimal controller • Multiobjective. Tradeoff between • Output performance • Robustness • Input usage • Noise sensitivity High controller gain (“tight control”) Low controller gain (“smooth control”) Our choice: • 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. J = avg. IAE for setpoint/disturbance Ms = peak sensitivity

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

  6. Optimal PI-controller: Minimize J for given Ms

  7. Ms=2 |S| Ms=1.59 Ms=1.2 frequency

  8. Optimal PI-controller Ms=2 Setpoint change at t=0, Input disturbance at t=20, g(s)=k e-θs/(1s+1), Time delay θ=1

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

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

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

  12. Optimal PI-controller Input usage (TV) increases with Ms TVys TVd

  13. Optimal PI-controller Setpoint / disturbance tradeoff Optimal controller: Emphasis on disturbance d Pure time delay process: J=1, No tradeoff (since setpoint and disturbance the same)

  14. Optimal PI-controller Setpoint / disturbance tradeoff Optimal setpoint: No integral action

  15. Comparison with SIMC

  16. Comparison of J vs. Ms for optimal and SIMC for 4 processes

  17. 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

  18. 2. Can the SIMC-rule be improved? Yes, possibly for time delay process

  19. Optimal PI-controller Optimal PI-settings

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

  21. Improved SIMC-rule: Replace 1 by 1+θ/3

  22. Step response for time delay process θ=1 Time delay process: Setpoint and disturbance response same

  23. Comparison of J vs. Ms for optimal and SIMC for 4 processes

  24. Conclusion Questions: • How good is really the SIMC-rule? • Answer: Pretty close to optimal, except for time delay process • 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

  25. Model from closed-loop response with P-controller Kc0=1.5 Δys=1 Δy∞ dyinf = 0.45*(dyp + dyu) Mo =(dyp -dyinf)/dyinf b=dyinf/dys A = 1.152*Mo^2 - 1.607*Mo + 1.0 r = 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 Δyp=0.79 Δyu=0.54 tp=4.4 Example: Get k=0.99, theta =1.68, tau=3.03 Ref: Shamssuzzoha and Skogestad (JPC, 2010) + modification by C. Grimholt (Project, NTNU, 2010; see also new from PID-book 2011)