Necessary conditions for consistency of noise-free, closed-loop frequencyresponse data with coprime factor models American Control Conference June 29, 2000 Benoit Boulet, Ph.D., Eng. Industrial Automation Lab McGill Centre for Intelligent Machines Department of Electrical and Computer Engineering McGill University, Montreal 1 Outline 1- Motivation: model validation for robust H / 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 2 1- Motivation: model validation for robust control H / Typical feedback control system Uncertainty Input Dist. Output dist./ Meas. noise Controller reference + - K +

G + Plant Benoit Boulet, June 29, 2000 3 + + Meas. output Motivation: model validation for robust H / control Robust control objective z y P K

w u Design stabilizing LTI K such that CL system is 1 stable H , r 1 robust H stability where is the space of stable Q functions, :sup Q( j ) transfer r and Benoit Boulet, June 29, 2000

( jbound ) ron ( jthe ) uncertainty: isa 4 Motivation: model validation for robust H / control robust control z y P K Condition for robust stability as given by small-gain theorem: w u 1

closed-loop stable H , r iff r F L P, K Benoit Boulet, June 29, 2000 5 1 1 Motivation: model validation for robust H / control robust control z y P K w

Condition for robust stability r F L P, K u Benoit Boulet, June 29, 2000 1 provides the r ( j motivation to) make (the uncertainty) as small as possible through better modeling 6 Motivation: model validation for robust H / control robust control z

y P K Conclusion: w u Robust stability is easier to achieve if the size of the uncertainty is small. Same conclusion for robust performance ( synthesis) Benoit Boulet, June 29, 2000 7 Motivation: model validation for robust H / 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 perturbations e.g., +, ||

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) 1 G p CM p N p J M p R H2323 N p R H2323 Benoit Boulet, June 29, 2000 10 Coprime factor plant models define Factor perturbation Uncertainty set : N

Family of perturbed plants M R H Dr : R H : r 1 1 where r bounds the size of the factor uncertainty: P : G p : Dr Benoit Boulet, June 29, 2000 11 ( j ) r ( j ), Coprime factor plant models block diagram of open-loop perturbed LCF

P(s) rI C M J N 1 where factor uncertainty is normalized r such that, ( j ) 1 ( j ) r ( j ), Benoit Boulet, June 29, 2000 12 Coprime factor plant models Block diagram of closed-loop

perturbed LCF H ( s) rI C M K2 N K1 assumption: K1 , K 2 internally stabilize the plant and provide sufficient damping for FR measurement Benoit Boulet, June 29, 2000 13 3- Consistency of closed-loop

frequency-response data with coprime factor models Model/data consistency problem: Given noise-free, (open-loop,closedN pm loop) frequency-response data i i 1 obtained at frequencies 1 , , N , could the data have been produced by at least one plant model P G p in : Dr ? Benoit Boulet, June 29, 2000 14

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 15 Consistency of closed-loop FR data with coprime factor models closed-loop FR data case consistency equation at frequency i i Hi ri I Ci

M i K 2i N i K1i 0 i FU H i , i i 0 where H i : H ( j i ), Benoit Boulet, June 29, 2000 16

I Consistency of closed-loop FR data with coprime factor models Lemma 1 FU H i , i i 0 rank I F L H i , i 1 i n Lemma 2 (Schmidt-Mirsky Theorem) inf i : rank I F L H i , i 1 i n p F L H i , i 1 Benoit Boulet, June 29, 2000 17

1 Consistency of closed-loop FR data with coprime factor models Lemma 3 (consistency ati ) n( n p ) i , i 1 such that FU H i , i i 0 p F L H i , Benoit Boulet, June 29, 2000 1 i

18 1 1 Consistency of closed-loop FR data with coprime factor models Theorem (consistency with CL FR data) N i i 1 consistent with LCF model only if p F L H i , 1 i 1

1, i 1, , N Proof (using boundary interpolation theorem) ( s) D such that ( j ) r r i i i only if p F L H i , 1 i Benoit Boulet, June 29, 2000 1 i 1, i 1, , N 19

Consistency of closed-loop FR data with coprime factor models This condition is not sufficient. For sufficiency, the perturbation ( s ) Dr would have to be shown to stabilize H ( s) to account for the fact that the closedloop system was stable with the original controller(s) K1 ( s ), K 2 ( s ) We cant just assume this a priori as it would mean that the original controller(s) is already robust! Benoit Boulet, June 29, 2000 20 Example: Daisy LFSS Testbed Example: Daisy LFSS testbed Nominal factorization G CM 1 NJ M , N , J R H2323 , C 2323 Bound on factor uncertainty

0.001s 1.414 2.32 s 1 Ga , one of the plants in family of perturbed plants P was chosen to be the actual plant generating 50 i i1 the 50 closed-loop FR data points r ( s) 23 first-order decentralized SISO lead controllers were used as the original controllerK1 Benoit Boulet, June 29, 2000 21 Example: Daisy LFSS Testbed Example (continued) consistency equation at frequency i i Hi ri I

Ci M i Ji N i K1i 0 i FU H i , i i 0 where H i : H ( j i ), Benoit Boulet, June 29, 2000 22

I Example: Daisy LFSS Testbed Example (continued) Model/data consistency check: p F L H i , i 1 Benoit Boulet, June 29, 2000 1 23 5- Conclusion Necessary condition for consistency of noise-free FR data with uncertain MIMO coprime factor plant model involves the computation of p at theN

measurement frequencies Bound on factor uncertaintyr ( s ) 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 ( s) Dr , proven to exist by the boundary interpolation theorem, also stabilizes the nominal closed-loop system. Benoit Boulet, June 29, 2000 24 Thank you! 25