Optimal PID-Control on First Order Plus Time Delay Systems - - PowerPoint PPT Presentation

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Optimal PID-Control on First Order Plus Time Delay Systems - - PowerPoint PPT Presentation

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Introduction Optimal Controller Tuning rule Conclusions Optimal PID-Control on First Order Plus Time Delay Systems Verification of the SIMC rules Chriss Grimholt and Sigurd Skogestad Norwegian University of Science and Technology February

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Introduction Optimal Controller Tuning rule Conclusions

Optimal PID-Control on First Order Plus Time Delay Systems

Verification of the SIMC rules Chriss Grimholt and Sigurd Skogestad

Norwegian University of Science and Technology

February 19, 2014

  • C. Grimholt and S. Skogestad (NTNU)

Optimal PID-Control on FOPTD Systems February 19, 2014 1 / 22

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Introduction Optimal Controller Tuning rule Conclusions

INTRODUCTION

The Questions

◮ How much can we gain by using PID instead of PI control? ◮ Is there a simple PID tuning rule that gives close to

  • ptimal performance?
  • C. Grimholt and S. Skogestad (NTNU)

Optimal PID-Control on FOPTD Systems February 19, 2014 2 / 22

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Introduction Optimal Controller Tuning rule Conclusions

THE SYSTEM

We investigating the optimal tuning for the process

G(s) = k (τ1s + 1)e−θs

Considering the cascade form PID controller

K(s) = KC τIs + 1 τIs

  • (τDs + 1)
  • C. Grimholt and S. Skogestad (NTNU)

Optimal PID-Control on FOPTD Systems February 19, 2014 3 / 22

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Introduction Optimal Controller Tuning rule Conclusions

QUANTIFYING THE OPTIMAL CONTROLLER

Trade-off between

◮ Output performance } High controller gain (Tight control) ◮ Robustness

     Low controller gain (Smooth control)

◮ Input usage ◮ Noise sensitivity

We focus on output performance and robustness.

◮ Output Performance: J – weighted average IAE ◮ Robustness: MS – peak sensitivity

  • C. Grimholt and S. Skogestad (NTNU)

Optimal PID-Control on FOPTD Systems February 19, 2014 4 / 22

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Introduction Optimal Controller Tuning rule Conclusions

ROBUSTNESS – MS

S = 1 1 + GK MS = max

ω

|S(jω)| = S∞

10−1 100 101 102 100 101

MS

Frequency, ω Magnitude, |S|

−1.5 −1 −0.5 0.5 −1 −0.5 0.5 1

1/M

S

Re L(jω) Im L(jω)

  • C. Grimholt and S. Skogestad (NTNU)

Optimal PID-Control on FOPTD Systems February 19, 2014 5 / 22

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Introduction Optimal Controller Tuning rule Conclusions

PERFORMANCE – J

We consider unit step disturbances at two different locations:

◮ at plant output ◮ at plant input

di do ysp K(s) G(s) y

−

e u +

+

  • C. Grimholt and S. Skogestad (NTNU)

Optimal PID-Control on FOPTD Systems February 19, 2014 6 / 22

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Introduction Optimal Controller Tuning rule Conclusions

PERFORMANCE – J

IAE = ∞ |e(t)| dt J(K) = 0.5 IAEdo(K) IAE◦

do

+ IAEdi(K) IAE◦

di

  • IAE◦

do PID optimal controller for do (at MS = 1.59)

IAE◦

di PID optimal controller for di (at MS = 1.59)

2 4 6 8 10 −0.5 0.5 1 1.5

IAEdo

do

Time, t

Error, e(t)

2 4 6 8 10 −0.5 0.5 1 1.5

IAEdi

di

Time, t

Error, e(t)

  • C. Grimholt and S. Skogestad (NTNU)

Optimal PID-Control on FOPTD Systems February 19, 2014 7 / 22

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Introduction Optimal Controller Tuning rule Conclusions

PARETO OPTIMAL PLOTS – OPTIMAL J(K) VS. MS

1 1.5 2 2.5 3 1 2 3 4

PO PID

Robustness, MS Performance, J(K)

  • C. Grimholt and S. Skogestad (NTNU)

Optimal PID-Control on FOPTD Systems February 19, 2014 8 / 22

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Introduction Optimal Controller Tuning rule Conclusions

PARETO OPTIMAL PLOTS – OPTIMAL J(K) VS. MS

1 1.5 2 2.5 3 1 2 3 4 Infeasible

PO PID

Robustness, MS Performance, J(K)

  • C. Grimholt and S. Skogestad (NTNU)

Optimal PID-Control on FOPTD Systems February 19, 2014 9 / 22

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Introduction Optimal Controller Tuning rule Conclusions

PARETO OPTIMAL PLOTS – OPTIMAL J(K) VS. MS

1 1.5 2 2.5 3 1 2 3 4 Sub-Optimal

PO PID

Robustness, MS Performance, J(K)

  • C. Grimholt and S. Skogestad (NTNU)

Optimal PID-Control on FOPTD Systems February 19, 2014 10 / 22

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Introduction Optimal Controller Tuning rule Conclusions

PARETO OPTIMAL PLOTS – OPTIMAL J(K) VS. MS

1 1.5 2 2.5 3 1 2 3 4 Uninteresting

PO PID

Robustness, MS Performance, J(K)

  • C. Grimholt and S. Skogestad (NTNU)

Optimal PID-Control on FOPTD Systems February 19, 2014 11 / 22

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Introduction Optimal Controller Tuning rule Conclusions

THE CASES

G1(s) = e−s G2(s) = 1 (s + 1)e−s G3(s) = 1 (8s + 1)e−s G4(s) = 1 s e−s

  • C. Grimholt and S. Skogestad (NTNU)

Optimal PID-Control on FOPTD Systems February 19, 2014 12 / 22

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Introduction Optimal Controller Tuning rule Conclusions

OPTIMAL PI CONTROL

1 1.5 2 2.5 3 1 2 3 4

PO PI

G(s) = e−s Robustness, MS Performance, J(K) 1 1.5 2 2.5 3 1 2 3 4

PO PI

G(s) = 1 (s + 1)e−s Robustness, MS Performance, J(K) 1 1.5 2 2.5 3 1 2 3 4

PO PI

G(s) = 1 (8s + 1)e−s Robustness, MS Performance, J(K) 1 1.5 2 2.5 3 1 2 3 4

PO PI

G(s) = 1 s e−s Robustness, MS Performance, J(K)

  • C. Grimholt and S. Skogestad (NTNU)

Optimal PID-Control on FOPTD Systems February 19, 2014 13 / 22

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Introduction Optimal Controller Tuning rule Conclusions

OPTIMAL PID COMPARED WITH OPTIMAL PI

1 1.5 2 2.5 3 1 2 3 4

PO PI P O P I D

G(s) = e−s

No benefit from using PID

Robustness, MS Performance, J(K) 1 1.5 2 2.5 3 1 2 3 4

PO PI PO PID

G(s) = 1 (s + 1)e−s Robustness, MS Performance, J(K) 1 1.5 2 2.5 3 1 2 3 4

PO PI PO PID

G(s) = 1 (8s + 1)e−s Robustness, MS Performance, J(K) 1 1.5 2 2.5 3 1 2 3 4

PO PI P O P I D

G(s) = 1 s e−s

42% Increase in performance

Robustness, MS Performance, J(K)

  • C. Grimholt and S. Skogestad (NTNU)

Optimal PID-Control on FOPTD Systems February 19, 2014 14 / 22

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Introduction Optimal Controller Tuning rule Conclusions

OPTIMAL PID TUNING

◮ τD is independent of MS and the time constant τ1 for lag

dominated processes (τ1/θ 2.5).

◮ Optimal controller for delay dominated processes

(τ1/θ 2.5) is on the boarder between cascade and ideal controller realization.

5 10 15 20 25 30 0.2 0.4 0.6 0.8 1

MS = 1.20 MS = 1 . 5 MS = 1.59 M

S

= 1 . 7 MS = 2.00

Process time constant τ1/θ Normalized controller gain, KCkθ/τ1 5 10 15 20 25 30 1 2 3

MS = 1.20 M

S

= 1 . 5 MS = 1.59 M

S

= 1 . 7 MS = 2.00 τD ≈ 0.48

τD/θ τI/θ

τD almost independent of MS Two coinciding zeros

Process time constant τ1/θ Integral and derivative time

  • C. Grimholt and S. Skogestad (NTNU)

Optimal PID-Control on FOPTD Systems February 19, 2014 15 / 22

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Introduction Optimal Controller Tuning rule Conclusions

OPTIMAL PID – CASCADE VS. IDEAL

Delay dominated processes τ1/θ < 3:

Performance can be improved by using Ideal controller. KIdeal = K′

C

  • 1 + 1

τ ′

Is + τ ′ Ds

  • 1

1.5 2 2.5 3 1 2 3

Ideal Cascade

G(s) = e−s (s + 1)

Robustness, MS Performance, J(K)

  • C. Grimholt and S. Skogestad (NTNU)

Optimal PID-Control on FOPTD Systems February 19, 2014 16 / 22

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Introduction Optimal Controller Tuning rule Conclusions

THE SIMC PI RULES FOR FOPTD

SIMC: Probably the best simple PID tuning in the world

KC = 1 k τ1 τc + θ (1) τI = min {τ1, 4(τc + θ)} (2) where τc is the tuning constant. τc = θ is recommended for tight control.

However, only PI tuning for FOPTD

  • C. Grimholt and S. Skogestad (NTNU)

Optimal PID-Control on FOPTD Systems February 19, 2014 17 / 22

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Introduction Optimal Controller Tuning rule Conclusions

OPTIMAL PID CONTROL COMPARED WITH SIMC

1 1.5 2 2.5 3 1 2 3 4

PO PI P O P I D SIMC (P)I

G(s) = e−s

τ

c

= . 5 θ τ

c

= 1 θ τ

c

= 1 . 5 θ

Robustness, MS Performance, J(K) 1 1.5 2 2.5 3 1 2 3 4

PO PI PO PID S I M C P I

G(s) = 1 (s + 1)e−s

τ

c

= . 5 θ τ

c

= 1 θ τ

c

= 1 . 5 θ

Robustness, MS Performance, J(K) 1 1.5 2 2.5 3 1 2 3 4

PO PI PO PID SIMC PI

G(s) = 1 (8s + 1)e−s

τ

c

= . 5 θ τ

c

= 1 θ τ

c

= 1 . 5 θ

Robustness, MS Performance, J(K) 1 1.5 2 2.5 3 1 2 3 4

PO PI P O P I D SIMC PI

G(s) = 1 s e−s

τ

c

= . 5 θ τ

c

= 1 θ τ

c

= 1 . 5 θ

Robustness, MS Performance, J(K)

  • C. Grimholt and S. Skogestad (NTNU)

Optimal PID-Control on FOPTD Systems February 19, 2014 18 / 22

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Introduction Optimal Controller Tuning rule Conclusions

PROPOSED: SIMC PID RULES FOR FOPTD

KC = 1 k τ1 τc + θ (3) τI = min {τ1, 4(τc + θ)} (4) τD = θ/3 (5)

  • C. Grimholt and S. Skogestad (NTNU)

Optimal PID-Control on FOPTD Systems February 19, 2014 19 / 22

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Introduction Optimal Controller Tuning rule Conclusions

OPTIMAL PID CONTROL COMPARED WITH SIMC

1 1.5 2 2.5 3 1 2 3 4

PO PI P O P I D SIMC (P)I S I M C P I ( D )

G(s) = e−s

τ

c

= . 5 θ τ

c

= 1 θ τ

c

= 1 . 5 θ

Robustness, MS Performance, J(K) 1 1.5 2 2.5 3 1 2 3 4

PO PI PO PID S I M C P I SIMC PID

G(s) = 1 (s + 1)e−s

τ

c

= . 5 θ τ

c

= 1 θ τ

c

= 1 . 5 θ

Robustness, MS Performance, J(K) 1 1.5 2 2.5 3 1 2 3 4

PO PI PO PID SIMC PI SIMC PID

G(s) = 1 (8s + 1)e−s

τ

c

= . 5 θ τ

c

= 1 θ τ

c

= 1 . 5 θ

Robustness, MS Performance, J(K) 1 1.5 2 2.5 3 1 2 3 4

PO PI P O P I D SIMC PI S I M C P I D

G(s) = 1 s e−s

τ

c

= . 5 θ τ

c

= 1 θ τ

c

= 1 . 5 θ

Robustness, MS Performance, J(K)

  • C. Grimholt and S. Skogestad (NTNU)

Optimal PID-Control on FOPTD Systems February 19, 2014 20 / 22

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Introduction Optimal Controller Tuning rule Conclusions

STEP RESPONSE

10 20 30 40 1 2 3

P O P I D S I M C P I SIMC PID P O P I

do di

G(s) = 1 s e−s Time, t Outputs, y 10 20 30 40 −1.5 −1 −0.5 0.5

PO PID PO PI SIMC PI SIMC PID

Time, t Inputs, u

  • C. Grimholt and S. Skogestad (NTNU)

Optimal PID-Control on FOPTD Systems February 19, 2014 21 / 22

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Introduction Optimal Controller Tuning rule Conclusions

CONCLUSIONS

◮ PID control can substantially increase performance for lag

dominated processes.

◮ No Benefit of using PID control on pure time delay process ◮ SIMC PI with D: τD = θ/3 gives a good PID tuning.

Thank you

  • C. Grimholt and S. Skogestad (NTNU)

Optimal PID-Control on FOPTD Systems February 19, 2014 22 / 22