1 Benoit Boulet, Ph.D., Eng. Industrial Automation Lab McGill Centre for Intelligent Machines Department of Electrical and Computer Engineering McGill University, Montreal Necessary conditions for consistency of noise-free, closed-loop frequency- response data with coprime factor models American Control Conference June 29, 2000

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

Benoit Boulet, June 29, 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: Robust control objective P  K Motivation: model validation for robust control

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

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

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

Benoit Boulet, June 29, …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 perturbations e.g.,  + , |  |<bFrom first principles: Identify nominal values of uncertain gains, time delays, time constants, high freq. dynamics, etc. and bounds on their perturbations e.g.,  + , |  |<b From experimental I/O dataFrom experimental I/O data Motivation: model validation for robust control

Benoit Boulet, June 29, Coprime factor plant models Perturbed left-coprime factorizationPerturbed left-coprime factorization Coprime factor plant modelswhere

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

Benoit Boulet, June 29, define Factor perturbationFactor perturbation Uncertainty setUncertainty set Family of perturbed plantsFamily of perturbed plants Coprime factor plant models

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

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

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

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

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

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

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

Benoit Boulet, June 29, 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! Consistency of closed-loop FR data with coprime factor models

Benoit Boulet, June 29, Example: Daisy LFSS testbed Nominal factorizationNominal factorization Bound on factor uncertaintyBound 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 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 controller23 first-order decentralized SISO lead controllers were used as the original controller Example: Daisy LFSS Testbed

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

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

Benoit Boulet, June 29, Conclusion Necessary condition for consistency of noise-free FR data with uncertain MIMO coprime factor plant model involves the computation of at the measurement frequenciesNecessary 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 measurementsBound 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.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.

25 Thank you!