1 Limits of Disturbance Rejection using Indirect Control Vinay Kariwala * and Sigurd Skogestad...

transcript

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

Outline

• Motivation

• Objectives

• Interpolation constraints

• Performance limits

• Comparison with direct control

• Feedback + Feedforward control

General control problem

“Direct” Controld

zu- y = z

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

Problem

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

Compositions cannot be measuredor are available infrequently

Need to consider “Indirect control”

Indirect Control

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

Indirect control: Control y to achieve good performance for z

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

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

Nevanlinna-Pick Interpolation TheoryParameterizes all rational functions with

Useful for characterizing achievable performance

• Derivative constraints

• Interpolation constraints

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

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)

Derivative Interpolation Constraint

Very conservative: Should be:

Special case: Control effort required for stabilization

Reason: Derivative is also fixed

Bound due to interpolation constraint

Main results: Limit of Performance, indirect control

Derivative constraint neglected, Exact bound in paper

optimal achievable performance

Let v include all unstable poles and zeros:

“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

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

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

Simple Example

Case Direct Indirect

Stable system 0.5 0.5

Unstable system 1.5 15.35

Extra unstable pole of Gy - 51.95

Feedback + Feedforward Control

Disturbance measured (M)

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

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

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