1. 1 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

2. 2 Outline Motivation Objectives Interpolation constraints Performance limits Comparison with direct control Feedback + Feedforward control

3. 3 General control problem y d

4. 4 “Direct” Control d KG GdGd z u - y = z Unstable (RHP) zeros α i in G limit disturbance rejection: interpolation constraint

5. 5 Problem In many practical problems, –Primary controlled variable z not available Compositions cannot be measured or are available infrequently Need to consider “Indirect control”

6. 6 Indirect Control KGyGy GdGd d z u - G G dy y Primary objective paper: Derive limits on disturbance rejection for indirect control Indirect control: Control y to achieve good performance for z

7. 7 Related work Bounds on various closed loop functions available –S, T – Chen (2000), etc. –KSG d – Kariwala et al. (2005), etc. Special cases of indirect control Secondary objective: Unify treatment of different closed loop functions

8. 8 Main Assumptions (mostly technical) Unstable poles of G and G dy – also appear in G y All signals scalar Unstable poles and zeros are non-repeated G and G dy - no common unstable poles and zeros

9. 9 Nevanlinna-Pick Interpolation Theory Parameterizes all rational functions with Useful for characterizing achievable performance Derivative constraints a Interpolation constraints

10. 10 Indirect control: Interpolation Constraints Need to avoid unstable (RHP) pole-zero cancellations If are unstable zeros of G If are unstable zeros of G dy same as for direct control

11. 11 More new interpolation Constraints If are unstable poles of G y that are shared with G dy If are unstable poles of G y not shared with G and G dy If are unstable poles of G y that are shared with G - stable version (poles mirrored in LHP)

12. 12 Derivative Interpolation Constraint Very conservative: Should be: Special case: Control effort required for stabilization Reason: Derivative is also fixed Bound due to interpolation constraint

13. 13 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:

14. 14 “Perfect” Indirect Control possible when: G and G dy have no unstable zeros –or G d evaluated at these points is zero and G and G dy have no unstable poles –or has transmission zeros at these points and G y has no extra unstable poles

15. 15 Direct Control vs Indirect Control Zeros of G Poles of G + (Possible) derivative constraint Practical consequence: To avoid large T zd, y and z need to be “closely correlated” if the plant is unstable

16. 16 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

17. 17 Simple Example Case DirectIndirect Stable system 0.5 Unstable system 1.515.35 Extra unstable pole of G y -51.95

18. 18 Feedback + Feedforward Control K1K1 GyGy GdGd d z u - G G dy y K2K2 M Disturbance measured (M)

19. 19 Feedback + Feedforward Control Limitation due to –Unstable zeros of G –Extra unstable poles of G y, but no derivative constraint No limitation due to –Unstable zeros of G dy unless M has zeros at same points –Unstable poles of G and G dy + Possible limitation due to uncertainty

20. 20 Simple Example (continued) CaseDirect FB Stable system 0.5 Unstable system1.515.35 Extra unstable pole of G y -51.95 Indirect FB FB+FF 0.5 FB+FF 0.5 -0.68

21. 21 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

22. 22 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