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Limits of Disturbance Rejection using Indirect Control

Vinay Kariwala* and Sigurd Skogestad

Department of Chemical Engineering

NTNU, Trondheim, Norway

skoge@chemeng.ntnu.no

* From Jan. 2006: Nanyang Technological University (NTU), Singapore

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Outline

• Motivation

• Objectives

• Interpolation constraints

• Performance limits

• Comparison with direct control

• Feedback + Feedforward control

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General control problem

y

d

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“Direct” Controld

K G

Gd

zu- y = z

Unstable (RHP) zeros αi in G limit disturbance rejection:interpolation constraint

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Problem

In many practical problems, – Primary controlled variable z not available

L

V

xD

B

F

zF

D

xB

LC

LR

Compositions cannot be measuredor are available infrequently

Need to consider “Indirect control”

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Indirect Control

K Gy

Gd

d

z

u-

G

Gdy

y

Primary objective paper: Derive limits on disturbance rejection for indirect control

Indirect control: Control y to achieve good performance for z

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Related work

• Bounds on various closed loop functions available– S, T – Chen (2000), etc.

– KSGd – Kariwala et al. (2005), etc.

• Special cases of indirect control

Secondary objective:Unify treatment of different closed loop functions

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Main Assumptions (mostly technical)

• Unstable poles of G and Gdy – also appear in Gy

• All signals scalar

• Unstable poles and zeros are non-repeated

• G and Gdy - no common unstable poles and zeros

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Nevanlinna-Pick Interpolation TheoryParameterizes all rational functions with

Useful for characterizing achievable performance

• Derivative constraints

• a

• Interpolation constraints

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Indirect control: Interpolation ConstraintsNeed to avoid unstable (RHP) pole-zero cancellations

If are unstable zeros of G

If are unstable zeros of Gdy

same as for direct control

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More new interpolation Constraints

If are unstable poles of Gy that are shared with Gdy

If are unstable poles of Gy not shared with G and Gdy

If are unstable poles of Gy that are shared with G

- stable version (poles mirrored in LHP)

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Derivative Interpolation Constraint

Very conservative: Should be:

Special case: Control effort required for stabilization

Reason: Derivative is also fixed

Bound due to interpolation constraint

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Main results: Limit of Performance, indirect control

Derivative constraint neglected, Exact bound in paper

optimal achievable performance

optimal achievable performance

Let v include all unstable poles and zeros:

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“Perfect” Indirect Control possible when:

• G and Gdy have no unstable zeros – or Gd evaluated at these points is zero and

• G and Gdy have no unstable poles

– or has transmission zeros at these points and

• Gy has no extra unstable poles

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Direct Control vs Indirect Control

• Zeros of G

• Poles of G

+ (Possible) derivative constraint

Practical consequence: To avoid large Tzd, y and z need to be “closely correlated” if the plant is unstable

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Indirect control

The required change in u for stabilization may make z sensitive to disturbances

Exception: Tzd(gammak) close to 0 because y and z are “closely correlated”

Example case with no problem : “cascade control”In this case: z = G2 y, so a and y are closely correlated.

Get Gd = G2 Gdy and G = G2 Gy, and we find that the above bound is zero

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Simple Example

Case Direct Indirect

Stable system 0.5 0.5

Unstable system 1.5 15.35

Extra unstable pole of Gy - 51.95

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Feedback + Feedforward Control

K1 Gy

Gd

d

z

u-

G

Gdy

y

K2

M

Disturbance measured (M)

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Feedback + Feedforward Control

• Limitation due to– Unstable zeros of G

– Extra unstable poles of Gy, but no derivative constraint

• No limitation due to– Unstable zeros of Gdy unless M has zeros at same points

– Unstable poles of G and Gdy

• + Possible limitation due to uncertainty

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Simple Example (continued)

Case DirectFB

Stable system 0.5 0.5

Unstable system 1.5 15.35

Extra unstable pole of Gy - 51.95

Indirect

FB FB+FF0.5

FB+FF0.5

0.5 0.5

- 0.68

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Conclusions

• Performance limitations– Interpolation constraint, derivative constraint

– and optimal achievable performance

• Indirect control vs. direct control– No additional fundamental limitation for stable plants

– Unstable plants may impose disturbance sensitivity

• Feedforward controller can overcome limitations– but will add sensitivity to uncertainty

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Limits of Disturbance Rejection using Indirect Control

Vinay Kariwala* and Sigurd Skogestad

Department of Chemical Engineering

NTNU, Trondheim, Norway

skoge@chemeng.ntnu.no

* From Jan. 2006: Nanyang Technological University (NTU), Singapore