SiMpLIfy: A Toolbox for Structured Model Reduction

Transcription

1 SiMpLIfy: A Toolbox for Structured Model Reduction Martin Biel, Farhad Farokhi, and Henrik Sandberg ACCESS Linnaeus Center, KTH Royal Institute of Technology Department of Electrical and Electronic Engineering, University of Melbourne Wednesday July 15, 2015 Session WeC11: Model Reduction II European Control Conference (ECC 2015) M. Biel, et al (KTH/UniMelb) SiMpLIfy Wednesday July 15, / 13

2 summer projects In summers, ACCESS hires bachelor students to work closely with academics for two months Provides an exciting and valuable research experience to the students Provides a nice opportunity to the academics to recruit the next generation of students and engineers M. Biel, et al (KTH/UniMelb) SiMpLIfy Wednesday July 15, / 13

3 Large-scale Control Applications Motivation The systems are typically highly structured Traditional model reduction methods do not respect the structure Structured model reduction addresses this issue To the best of our knowledge, there is no MATLAB toolbox for structured model reduction 1 of 18 M. Biel, et al (KTH/UniMelb) SiMpLIfy Wednesday July 15, / 13

4 Interconnected systems z(t) Network [ ] DE D N = F D H D K w(t) u(t) G 1 (s) G 2 (s) 0 G(s) = G q (s) Subsystems y(t) Interconnected system is given by the lower linear fractional transformation F(N, G(s)) G i (s), 1 i q, is a subsystem of order n i F(N, G(s)) is stable M. Biel, et al (KTH/UniMelb) SiMpLIfy Wednesday July 15, / 13

5 Structured model reduction z(t) F(N, G(s)) Network [ ] DE D N = F D H D K w(t) u(t) G 1 (s) G 2 (s) 0 G(s) = G q (s) Subsystems y(t) M. Biel, et al (KTH/UniMelb) SiMpLIfy Wednesday July 15, / 13

6 Structured model reduction z(t) F(N, G(s)) Network [ ] DE D N = F D H D K w(t) u(t) Ĝ 1 (s) Ĝ 2 (s) 0 Ĝ(s) = Ĝ q (s) Subsystems y(t) Ĝ i (s), 1 i q, is a subsystem of order r i (r i n i ) minĝ(s) F(N, G(s)) F(N, Ĝ(s)) M. Biel, et al (KTH/UniMelb) SiMpLIfy Wednesday July 15, / 13

7 Unstructured model reduction z(t) F(N, G(s)) Network [ ] DE D N = F D H D K w(t) u(t) H(s) G 1 (s) G 2 (s) 0 G(s) = G q (s) Subsystems y(t) H(s), 1 i q, is a subsystem of order i r i min H(s) F(N, G(s)) H(s) M. Biel, et al (KTH/UniMelb) SiMpLIfy Wednesday July 15, / 13

8 Structured Gramians (A, B, C, D) is a minimal state-space representation of G(s) Let P 11 P 1q P =..... P q1 P qq, Q = Q 11 Q 1q..... Q q1 Q qq be the unique positive definite solutions of the Lyapunov equations Define the structured Gramians (Vandendorpe and Van Dooren, 2004) as P 11 0 Q 11 0 P =....., Q = P qq 0 Q qq M. Biel, et al (KTH/UniMelb) SiMpLIfy Wednesday July 15, / 13

9 Generalized structured Gramians Define the generalized structured Gramians P arg min trace(p), P=diag((P ii ) q i=1 ) s. t. AP + PA + BB 0, P ii S n i +, i {1,..., q}, Q arg min trace(q), Q=diag((Q ii ) q i=1 ) s. t. A Q + QA + C C 0, Q ii S n i +, i {1,..., q}, The generalized structured Gramians may not exist in general unless focusing on specific systems, e.g., the subsystems are strictly positive real M. Biel, et al (KTH/UniMelb) SiMpLIfy Wednesday July 15, / 13

10 Heuristics for structured model reduction Use the structured Gramians or the generalized structured to find the balanced realization of G(s) Balanced truncation Theorem (Green & Limebeer, 95) F(N, Ĝ( )) = F(N, G( )) Theorem (Sandberg & Murray, 09) For balanced truncation with the generalized structured Gramians, F(N, Ĝ(s)) F(N, G(s)) 2 q ni i=1 k=r i +1 σ i,k Singular perturbation Theorem (Green & Limebeer, 95) F(N, Ĝ(0)) = F(N, G(0)) Both methods are heuristics and there give no optimality guarantees M. Biel, et al (KTH/UniMelb) SiMpLIfy Wednesday July 15, / 13

11 Subgradient optimization P(s) z(t) w(t) N u(t) y(t) G(s) + z (t) w(t) N The realization of the transfer function F(P(s), Φ) = F(N, G(s)) F(N, Ĝ(s)) Use the results of (Apkarian & Noll, 2006) to evaluate Φ F(P(s), Φ) Implement projected subgradient algorithm to improve the quality of the reduced-order model 1 s I r [ Â ˆB Φ = Ĉ ˆD ] M. Biel, et al (KTH/UniMelb) SiMpLIfy Wednesday July 15, / 13

12 Structured ModeL reduction (SiMpLIfy) SiMpLIfy is a MATLAB toolbox for structured model reduction A complete step-by-step manual with many examples can be found in M. Biel, et al (KTH/UniMelb) SiMpLIfy Wednesday July 15, / 13

13 Structured ModeL reduction (SiMpLIfy) w = u 1,1 u 1,2 u 2 G 1 (s) k G 2 (s) z 1 = y 1 z 2 = y 2 Flexible masses modeled by linear systems of orders 8 and 10 Bode Diagram From: In(1) From: In(2) Magnitude (db) To: Out(1) Magnitude (db) Frequency (rad/s) Bode Diagram Frequency (rad/s) M. Biel, et al (KTH/UniMelb) SiMpLIfy Wednesday July 15, / 13

14 Structured ModeL reduction (SiMpLIfy) u 1,1 1 1 y 1 y u 2 u 1,2 2 G1 G2 k +k k +k M. Biel, et al (KTH/UniMelb) SiMpLIfy Wednesday July 15, / 13

15 Structured ModeL reduction (SiMpLIfy) u 1,1 1 1 y 1 y u 2 u 1,2 2 G1 G2 k +k k +k Define the structured system as >> iedges=[1 2 -k; 2 2 k; 1 3 k; 2 3 -k]; >> einedges=[1 1]; >> eoutedges=[1 1; 2 2]; >> eedges=[ ]; >> systemnetwork=systemnetwork(iedges,einedges,eoutedges,eedges,g1,g2) M. Biel, et al (KTH/UniMelb) SiMpLIfy Wednesday July 15, / 13

16 Structured ModeL reduction (SiMpLIfy) >> comparehankels(systemnetwork) 0.25 Structured Hankel Singular Values The singular values corresponding to G 1 (s) and G 2 (s) are, respectively, marked with and. M. Biel, et al (KTH/UniMelb) SiMpLIfy Wednesday July 15, / 13

17 Structured ModeL reduction (SiMpLIfy) >> red = balancednetworkreduction(systemnetwork,[6 3],... ReductionMethod, perturbation ) >> red.extractsubsystem(1) >> red.extractsubsystem(2) Balanced Truncation Singular Perturbation r 1 = 8 r 1 = 6 r 1 = 4 r 1 = 2 r 2 = r 2 = r 2 = r 2 = r 2 = r 1 = 8 r 1 = 6 r 1 = 4 r 1 = 2 r 2 = r 2 = r 2 = r 2 = r 2 = M. Biel, et al (KTH/UniMelb) SiMpLIfy Wednesday July 15, / 13

18 Structured ModeL reduction (SiMpLIfy) >> optred=improvenetworkreduction(systemnetwork,red) 0.35 F(N,G(s)) F (N,Ĝ(s)) r r Subgradient optimization method initialized with the balanced truncation result Although the error is mostly decreasing with increasing the orders, this not true for all cases, which is because the proposed algorithm at best recovers a local optimum M. Biel, et al (KTH/UniMelb) SiMpLIfy Wednesday July 15, / 13

19 Conclusions and future work Conclusions SiMpLIfy: A toolbox for structured model reduction developed for MATLAB Several methods are implemented and compared with each other A complete step-by-step manual with many examples can be found in Future work Improve the toolbox based on your feedback Extend the framework to dynamic networks M. Biel, et al (KTH/UniMelb) SiMpLIfy Wednesday July 15, / 13

20 Please report bugs, complaints, suggestions, and encouragements (if any) to M. Biel, et al (KTH/UniMelb) SiMpLIfy Wednesday July 15, / 13