American Control Conference June 29, 2000

1. Necessary conditions for consistency of noise-free, closed-loop frequency-response data with coprime factor models American Control ConferenceJune 29, 2000 Benoit Boulet, Ph.D., Eng.Industrial Automation LabMcGill Centre for Intelligent MachinesDepartment of Electrical and Computer EngineeringMcGill University, Montreal

2. Outline 1- Motivation: model validation for robust control 2- Coprime factor plant models 3- Consistency of closed-loop freq. resp. (FR) data with coprime factor models 4- Example: Daisy LFSS testbed 5- Conclusion Benoit Boulet, June 29, 2000

3.  1- Motivation: model validation for robust control • Typical feedback control system Uncertainty Output dist./ Meas. noise Input Dist. Controller + reference + + Meas. output G K + + - Plant Benoit Boulet, June 29, 2000

4. Motivation: model validation for robust control  P K Robust control objective Design stabilizing LTI K such that CL system is stable “robust stability” where is the space of stable transfer functions, and is a bound on the uncertainty: Benoit Boulet, June 29, 2000

5. Motivation: model validation for robust control robust control  Condition for robust stability as given by small-gain theorem: P K Benoit Boulet, June 29, 2000

6. Motivation: model validation for robust control robust control Condition for robust stability provides the motivation to make (the uncertainty) as small as possible through better modeling  P K Benoit Boulet, June 29, 2000

7. Motivation: model validation for robust control robust control Conclusion: Robust stability is easier to achieve if the size of the uncertainty is small. Same conclusion for robust performance ( -synthesis)  P K Benoit Boulet, June 29, 2000

8. Motivation: model validation for robust control …uncertainty modeling is key to good control • From first principles: Identify nominal values of uncertain gains, time delays, time constants, high freq. dynamics, etc. and bounds on their perturbationse.g., +, ||<b • From experimental I/O data Benoit Boulet, June 29, 2000

9. Coprime factor plant models 2- Coprime factor plant models • Perturbed left-coprime factorization where Benoit Boulet, June 29, 2000

10. Coprime factor plant models Aerospace example: Daisy • Daisy is a large flexible space structure emulator at Univ. of Toronto Institute for Aerospace Studies (46th-order model) Benoit Boulet, June 29, 2000

11. Coprime factor plant models define • Factor perturbation • Uncertainty set • Family of perturbed plants Benoit Boulet, June 29, 2000

12. Coprime factor plant models block diagram of open-loop perturbed LCF Benoit Boulet, June 29, 2000

13. Coprime factor plant models Block diagram of closed-loop perturbed LCF Benoit Boulet, June 29, 2000

14. 3- Consistency of closed-loop frequency-response data with coprime factor models • Model/data consistency problem: Given noise-free, (open-loop,closed-loop) frequency-response data obtained at frequencies , could the data have been produced by at least one plant model in ? Benoit Boulet, June 29, 2000

15. Consistency of closed-loop FR data with coprime factor models open-loop model/data consistency problem solved in: • J. Chen, IEEE T-AC 42(6) June 1997 (general solution for uncertainty in LFT form) • B. Boulet and B.A. Francis, IEEE T-AC 43(12) Dec. 1998 (coprime factor models) • R. Smith and J.C. Doyle, IEEE T-AC 37(7) Jul. 1992 (uncertainty in LFT form, optimization approach) Benoit Boulet, June 29, 2000

16. Consistency of closed-loop FR data with coprime factor models closed-loop FR data case Benoit Boulet, June 29, 2000

17. Consistency of closed-loop FR data with coprime factor models Lemma 1 Lemma 2 (Schmidt-Mirsky Theorem) Benoit Boulet, June 29, 2000

18. Consistency of closed-loop FR data with coprime factor models Lemma 3 (consistency at ) Benoit Boulet, June 29, 2000

19. Consistency of closed-loop FR data with coprime factor models Theorem (consistency with CL FR data) Proof (using boundary interpolation theorem) Benoit Boulet, June 29, 2000

20. Consistency of closed-loop FR data with coprime factor models This condition is not sufficient. For sufficiency, the perturbation would have to be shown to stabilize to account for the fact that the closed-loop system was stable with the original controller(s)We can’t just assume this a priori as it would mean that the original controller(s) is already robust! Benoit Boulet, June 29, 2000

21. Example: Daisy LFSS Testbed Example: Daisy LFSS testbed • Nominal factorization • Bound on factor uncertainty • one of the plants in family of perturbed plants was chosen to be the actual plant generating the 50 closed-loop FR data points • 23 first-order decentralized SISO lead controllers were used as the original controller Benoit Boulet, June 29, 2000

22. Example (continued) Example: Daisy LFSS Testbed Benoit Boulet, June 29, 2000

23. Example: Daisy LFSS Testbed Example (continued) • Model/data consistency check: Benoit Boulet, June 29, 2000

24. 5- Conclusion • Necessary condition for consistency of noise-free FR data with uncertain MIMO coprime factor plant model involves the computation of at the measurement frequencies • Bound on factor uncertainty can be reshaped to account for all FR measurements • Sufficiency of the condition is difficult to obtain as one would have to prove that the factor perturbation , proven to exist by the boundary interpolation theorem, also stabilizes the nominal closed-loop system. Benoit Boulet, June 29, 2000

25. Thank you!