Optimal PI-Control & Verification of the SIMC Tuning Rule - - PowerPoint PPT Presentation

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Optimal PI-Control & Verification of the SIMC Tuning Rule - - PowerPoint PPT Presentation

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1 Optimal PI-Control & Verification of the SIMC Tuning Rule Sigurd Skogestad Trondheim, Norway Thanks to Chriss Grimholt IFAC-conference PID12, Brescia, Italy, 29 March 2012 2 Outline 1. Motivation: Ziegler-Nichols open-loop method

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Optimal PI-Control & Verification of the SIMC Tuning Rule

Sigurd Skogestad

Trondheim, Norway

Thanks to Chriss Grimholt IFAC-conference PID’12, Brescia, Italy, 29 March 2012

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Outline

  • 1. Motivation: Ziegler-Nichols open-loop method
  • 2. SIMC PI(D)-rule & derivation
  • 3. Definition of optimality (performance & robustness)
  • 4. Optimal PI control of first-order plus delay processes
  • 5. Comparison of SIMC with optimal PI
  • 6. Improved SIMC-PI for time-delay process
  • 7. Further work and conclusion
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  • Trans. ASME, 64, 759-768 (Nov. 1942).

Disadvantages Ziegler-Nichols: 1.Rather aggressive settings & No tuning parameter 2.Uses only two pieces of information (k’, ) 3.Poor for processes with large time delay (θ)

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Disadvantages IMC-PID: 1.Many rules 2.Poor disturbance response for «slow»/integrating processes (with large τ1/θ)

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Motivation for developing SIMC PID tuning rules (1998)

For teaching & easy practical use, rules should be:

  • Model-based
  • Analytically derived
  • Simple and easy to memorize
  • Work well on a wide range of processes
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  • 2. SIMC PI tuning rule
  • 1. Approximate process as first-order with delay (e.g., use “half rule”)
  • k = process gain
  • τ1 = process time constant
  • θ = 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

IMC ≈ SIMC for small τ1 (τI = τ1) Ziegler-Nichols ≈SIMC for large τ1 if we choose τc= 0 (aggressive!)

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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

  • Setpoints: τI=τ1(“IMC-rule”). Want smaller integral time for disturbance

rejection for “slow” processes (with large τ1), but to avoid “slow oscillations” must require:

  • Derivation:
  • Conclusion SIMC:
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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?
  • 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 PI-rule?

Want to compare with:

  • Optimal PI-controller

for class of first-order with delay processes

Optimal ant SIMC ant

versus

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  • 3. 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, 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:

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Cost J is independent of: 1. process gain (k) 2. setpoint (ys or dys) and disturbance (d) magnitude 3. unit for time

IAE output performance (J)

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  • 4. Optimal PI-controller:

Minimize J for given Ms

Optimal PI-controller

Optimal ant

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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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Optimal IAE-performance (J) vs. Ms

Optimal PI-controller

Optimal ant

1/ = 0 1/ = 8 1/ = 1 1/ = ∞

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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 Ms=1.59

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Setpoint / disturbance tradeoff

Optimal for setpoint: τI=τ1 (except time delay process) Integrating process (τ1=∞): No integral action

Optimal PI-controller

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  • 5. What about SIMC-PI?

SIMC ant

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SIMC: Tuning parameter (τc) correlates nicely with robustness measures

Ms GM PM

τc/θ τc/θ

SIMC a

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What about SIMC-PI performance?

SIMC ant

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27Comparison of J vs. Ms for optimal and SIMC for 4 processes

SIMC ant Optimal ant

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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, possibly 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 process SIMC: I=1=0

0.33

Optimal PI-controller

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

Improved SIMC ant

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Step response for time delay process

Time delay process: Setpoint and disturbance responses same + input response same θ=1

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34Comparison of J vs. Ms for optimal and SIMC-improved

CONCLUSION: SIMC-improved almost «Pareto-optimal»

Optimal ant Improved SIMC ant

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  • 7. Further work
  • More complex controllers than PI:

– Definition of problem becomes more difficult – Not sufficient with only IAE (J) and Ms

  • input usage
  • noise sensitivity
  • robustness
  • Optimal PID

– And comparison with SIMC-PID rule

  • Comparison with truly optimal controller

– Including Smith Predictor controllers

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  • 8. 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”

  • “Probably the best simple PID tuning rule in the

world”

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extra

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38 Model from closed-loop

response with P-controller

Kc0=1.5 Δys=1 Δyu=0.54 Δyp=0.79 tp=4.4

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

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 PID12r paper + new PID-book 2012)

Δy∞