Model Reduction for Dynamical Systems Lecture 6

Transcription

1 Oo-von-Guericke Universiä Magdeburg Faculy of Mahemaics Summer erm 07 Model Reducion for Dynamical Sysems ecure 6 v eer enner and ihong Feng Max lanck Insiue for Dynamics of Complex echnical Sysems Compuaional Mehods in Sysems and Conrol heory Magdeburg Germany benner@mpi-magdeburg.mpg.de feng@mpi-magdeburg.mpg.de hp://

2 Ouline MOR: alanced runcaion

3 Overlook alanced runcaion: firs balancing hen runcae. Given a I sysem: / x y u x d dx For convenience of discussion we denoe he sysem as a block form: alancing runcae reduced model he unimporan par is runcaed

4 Eigenspaces of and make he wo measures pracically compuable ill now i seems we could do he runcaion by finding subspace spanned by he eigenvecors corresponding o he small eigenvalues of or. However i could happen ha saes which are difficul o reach produce he maximal energy of observaion; saes which produce he smalles energy of observaion are neverheless he easies o reach! For such sysem we do no know which saes o runcae!

5 Example: Consider he following I sysem / x y u x d dx he wo Gramians are: heir eigenvalues and eigenvecors are: λ ξ λ ξ λ ξ λ ξ Eigenspaces of and make he wo measures pracically compuable

6 Eigenspaces of and make he wo measures pracically compuable ξ λ ξ λ ξ he angle beween ξ and is very small. his means if S is he subspace spanned by hen he easily observable saes ξ ξ ξ migh also be in S. ξ αξ α α x α >> - I ells us if we runcae he saes which are difficul o reach he saes locae in S we risk runcaing he saes which are easy o observe produce he maximal energy of observaion because hey migh also be in S.

7 Eigenspaces of and make he wo measures pracically compuable However if and have he same eigenvalues and eigenvecors hen he problems is solved. he saes in he subspace spanned by he eigenvecors of corresponding o he small eigenvalues always in he subspace spanned by he eigenvecors of corresponding o he small eigenvalues because he eigenvalues are he same and eigenvecors are he same herefore he subspaces are he same. Can we achieve his? Yes. e can achieve i by balancing.

8 MOR: alanced runcaion alancing asic idea of balancing ransformaion: x x Use sae space ransformaion o ge anoher realizaion of he same sysem so ha he ransformed Gramians are diagonal marices. Definiion of alancing ransformaion: Finding a nonsingular marix such ha and. Definiion of alanced sysem: he reachable observable and sable I sysem is balanced if is wo Gramians are equal i is principal-axis balanced if. diag σ σ n.

9 MOR: alanced runcaion alancing asic idea of balancing ransformaion: x x Use sae space ransformaion o ge anoher realizaion of he same sysem so ha he ransformed Gramians are equal and are diagonal marices. I.e. How o consruc? Recall ha. Since we have which means. should be he inverse of he marix Y of eigenvecors of.

10 MOR: alanced runcaion Check :? How o make alancing? If UU hen UU U UU U Here we mus have he relaion U. / / If furher U hen U UU How o compue? / Subsiue U he lef hand side / / U I ino we ge / U U / / U UU U U if /. U I. /. ook a he righ hand side we ge / / U U i. e. U U. herefore is he inverse of he marix of eigenvecors of U U. Furunaely he ranspose of U U is a s.p.d. marix. herefore he inverse of he marix of he marix iself.so ha we do no have o compue he inverse. eigenvecors is exacly

11 MOR: alanced runcaion alancing he above analysis clearly shows ha: Exisence of balancing ransformaion: dx / d y x Given a reachable observable and sable I sysem and he corresponding Gramians and a principal axis balancing ransformaion is given as follows: x u K U and UK / / Here UU is he Cholesky facorizaion of. U U K is he eigen-decomposiion of U U. Symmeric posiive semi-definie marix has real non-negive eigenvalues and orhogonal eigenvecors. Here he Eigenvecors in K are aken as orhonormal K

12 MOR: alanced runcaion alancing ha is he corresponding balanced sysem? pply he sae space ansformaion: x x o he original realizaion: dx / d y x x u x x dx / d y x x u

13 alancing : MOR: alanced runcaion alancing Given dx / d x y x u Compue. Compue UU U U K K he eigenvalues are ordered from he larges o he smalles dx / d x u / K U dx / d x u y x / UK y x

14 balanced sysem: dx / d y x x u he uni vecors e are he eigenvecors of : e σ e ssume ha he elemens on he herefore e e easily observable saes. r i i i i i n. diagonal of is already ordered as : σ σ σ. span he subspace conaining he firs r easily conrollable and runcae he difficul - o - observe and difficul - o - conrol saes means : x e x e x e x e x x x 0 0. MOR: alanced runcaion n n I.e. x x 0 0 :. Replace xr x x wih x dx / d x u z : x xr y x r r runcae r in he balanced sysem : z z u d 0 0 u y z 0 n

15 MOR: alanced runcaion runcae z y u u z z d noher realizaion of he same sysem is: is a non-minimal realizaion of a sysem. ˆ z y u z dz he reduced-order model ROM herefore we have he following simple seps for runcaion:

16 alancing: MOR: alanced runcaion runcae dx / d y x x u K U / / UK dx / d y x x u runcae: Small par and Separaed according o he separaion of. dz / d z u yˆ z ROM!

17 MOR: alanced runcaion alancing: dx / d y x x u / K U / UK dx / d y x x u Does i make sense if we do model reducion on he balanced sysem raher han he original sysem? Yes. s a sae ransformaion balancing does no change he ransfer funcion and he HSVs. he balanced sysem is only a differen realizaion of he sysem.

18 Observe: / x y u x d dx / x y u x d dx ˆ / z y u z d dz Y Y Y Yx x x :. produc marix he of eigenvecors he are : in columns he Here.. hen and as as separae If x Y Y x Yx x x x x x Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y MOR: alanced runcaion

19 MOR: alanced runcaion herefore he wo ROMs are he same: dz / d yˆ z z u dx / d Y x y Y x u Conclusion: balanced runcaion is erov-galerkin projecion as below: e x Y x dx / d y x u x erov - Galerkin Y x x dx / d y Y x using Y x u herefore balanced runcaion is equivalen o: finding he invarian subspace rangey of and remaining only he par Y which corresponds o he larges HSVs square roo of he eigenvalues of.

20 MOR: alanced runcaion lgorihm alancing:. Compue. Given dx / d x u y x. Compue UU 3. K U / U U K / UK K 4. alancing and separaing according o he separaion of : runcae: 5. Form he reduced model: dxˆ / d yˆ xˆ xˆ u

21 MOR: alanced runcaion compuaional deails Recall: In M use command: lyap ' lyap '

22 MOR: alanced runcaion Numerical issues he balancing marix is: / K U UU. Compuaion of U may cause numerical insabiliy because U is usually near singular. U is usually near singular because he marix has numerically low-rank i.e. near singular. is near singular because in may cases is eigenvalues decay rapidly o zero some eigenvalues are very close o zero e.g.. and behaves similarly as. λ 0 0 i However in algorihm we need o compue: / / K U UK Can we avoid compuing U?

23 MOR: alanced runcaion p p p p compuaional deails If using Choelsky facorizaion of boh Observe Use SVD insead of eigen- decomposiion U V Comparing wih defined in lgorihm we immediaely ge / U p o avoid compuing he inverse of U V V p U p we have: / V

24 MOR: alanced runcaion Numerical issues he balanced sysem which is balanced by and is: V / / U V V V / / / / / / U U U Furhermore if Define:. V V / V / U V we have: hen V V V U U U

25 MOR: alanced runcaion Numerical issues lgorihm SR mehod Geing he reduced model wihou compuing U :. Do Cholesky facorizaion of he wo Gramians: are lower riangular marices.. Do Singular value decomposiion SVD of marix i.e. here are wo orhonormal marices U V U U I V V I such ha V U V U U. V / 3. e / V V U. 4. e ˆ ˆ ˆ V V. 5. he reduced model is dxˆ / d x ˆ ˆ u ˆ yˆ ˆ xˆ

26 MOR: alanced runcaion Numerical issues lgorihm someimes canno coninue eiher because he Cholesky facorizaion of canno be done. his is because ha in some cases and include oo small eigenvalues like: λ 0 0 which is considered by he algorihm as a singular marix herefore Cholesky facorizaion canno be coninued. aper [enner 05] provides an algorihm compuing he numerically full rank facors n n n n of and which are in he forms ˆ ˆ R R n ˆ << n he full rank facors numerically saisfy:. [enner 05]. enner E.S. uiana-ori Model reducion based on specral projecion mehods. In:. enner V.. Mehrmann D.C. Sorensen eds. "Dimension Reduion of arge-scale Sysems" vol. 45 of ecure Noes in Compuaional Science and Engineering pp Springer-Verlag erlin/heidelberg 005. lgorihm 4 in he paper

27 MOR: alanced runcaion Numerical issues lgorihm 3 Geing he reduced model using full-rank facors [enner 05]:. Compue full-rank facors of he Gramians:. Compue SVD. V V U U V U / V 3. e. / U V 5. he reduced model is ˆ ˆ ˆ ˆ ˆ ˆ / ˆ x y u x d dx 4. e. ˆ ˆ ˆ V V. ˆ ˆ ˆ n n R R n n n n <<

28 MOR: alanced runcaion Error bound heorem [enner 05] If he original I sysem is sable hen he reduced model obained by lgorihm lgorihm lgorihm 3 saisfies: he reduced model is balanced minimal and sable. I s Gramians are equal o he same diagonal marix. he absolue error bound proof in [noulas 05] Chaper 7 holds. H s Hˆ s σ H n k r k