( ) lamp power. dx dt T. Introduction to Compact Dynamical Modeling. III.1 Reducing Linear Time Invariant Systems

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1 SF & IH Inroducon o Compac Dynamcal Modelng III. Reducng Lnear me Invaran Sysems Luca Danel Massachuses Insue of echnology Movaons dx A x( + b u( y( c x( Suppose: we are jus neresed n ermnal.e. npu/oupu behavor and we need o compue he oupu y( for many many dfferen npu sgnals u( : I mgh be more convenen o:. do some pre-compuaon on he orgnal sysem. generae a compac dynamcal model 3. re-use he model over and over L9-3 3 Course Oulne Quck Sneak Prevew I. Assemblng Models from Physcal Problems II. Smulang Models III. Model Order Reducon for Lnear Sysems Problem seup Reducon va modal analyss o me doman o freuency doman Reducon va ransfer funcon fng o pon machng o leas suare Preservng physcal properes n reduced sysems Projecon Framework o runcaed Balance Realzaons o Krylov Subspace Momen Machng IV. Model Order Reducon for on-lnear Sysems V. Parameerzed Model Order Reducon lamp power Lamp u Inpu of Ineres Oupu of Ineres end I.3- L9-4

2 Problem Seup Hea n h(z)u( emperaure Dfferenal Euaon γ specfc hea ( x, Spaal Dscrezaon dˆ z, κ + z hermal conducvy κ + + h z u scalar npu ( ˆ ˆ ˆ ) h( z) u( γ + end L9-5 Problem Seup Hea n h(x)u( dx x x scalar scalar x npu oupu hz ( ) hz ( ) κ A b γ ( ) γ hz ( ) c end Ax () + bu y () c x () L9-7 Problem Seup Hea n h(x)u( () dx dˆ Ax () + bu y () c x () x x scalar scalar x npu oupu Spaal Dscrezaon κ + + end ( ˆ ˆ ˆ ) h( z) u( γ + L9-6 Problem Seup Orgnal Dynamcal Sysem - Sngle Inpu/Oupu dx Ax () + bu y () c x () x x scalar scalar x npu oupu Reduced Dynamcal Sysem dxˆ( Aˆ xˆ( + bˆ u( x x scalar npu <<, bu npu/oupu behavor preserved yˆ( cˆ xˆ( scalar oupu x L9-8

3 Model Order Reducon Oulne Model Order Reducon of Lnear Problems Problem seup Reducon va modal analyss o me doman o freuency doman Reducon va ransfer funcon fng o pon machng o leas suare Imporance of preservng physcal properes o sably o passvy/dsspavy Projecon Framework o runcaed Balance Realzaons o Krylov Subspace Momen Machng Model Order Reducon of onlnear Problems L9- Decoupled Euaons ( V b) x λ x d u + x λ x ( V b ) Oupu Euaon Remnder abou Egenvalues () () () b () c y c x c Vx V c x () L9- () C onsder an ODE: dx Ax + bu, x() x Egendecomposon: Remnder abou Egenvalues λ A V V V V V V λ V le s: Vx x x V x( Change of varab () () () dvx Subsun g : AVx + bu(), Vx( ) x dx() Mulply by V : V AV x + V bu() λ λ x () + V bu L9- d ~ x ~ x Model Order Reducon va Modal Analyss (.e. domnan egenvalues/poles mehod) λ x y() [ c c ] x ~ ~ x b + u( ~ ~ λ x b Oupu Euaon L9-3

4 d ~ x ~ x Model Order Reducon va Modal Analyss (.e. domnan egenvalues/poles mehod) λ λ x y() [ c c ~ c ] x~ x ~ ~ x b x~ + u( b ~ ~ ~ λ x b Oupu Euaon L9-4 Model Order Reducon va Modal Analyss (.e. domnan egenvalues/poles mehod) 3 Exac Hea Conducng bar Resuls Keepng -h slowes modes L9-6 Model Order Reducon va Modal Analyss (.e. domnan egenvalues/poles mehod) Ceran modes are no affeced by he npu b,, b are all small + Ceran modes do no affec he oupu c +,, c are all small Keep leas negave egenvalues (slowes modes) Look a response o a consan npu λ τ λ () τ ( ) λ x e b u d e b u() Small f λ large L9-5 Model Order Reducon Oulne Model Order Reducon of Lnear Problems Problem seup Reducon va modal analyss o me doman o freuency doman Reducon va ransfer funcon fng o pon machng o leas suare Imporance of preservng physcal properes o sably o passvy/dsspavy Projecon Framework o runcaed Balance Realzaons o Krylov Subspace Momen Machng Model Order Reducon of onlnear Problems L9-7

5 Modal (egenvalue) analyss n he freuency doman ransfer Funcon ( ) H s c si A b Apply Egendecomposon A VλV H s c V si V b λ s λ s λ c b H( s) elmae each mode for whch hs erm s small cb s λ L9-8 Model Order Reducon va Modal Analyss (.e. domnan egenvalues/poles mehod) Advanages Concepually famlar Smple physcal nerpreaon : reans domnan sysem modes/poles Drawbacks Relavely expensve : have o fnd he egenvalues frs Relavely neffcen. For a gven model sze, many oher approaches can provde beer accuracy for he same compuaonal cos O( 3 ) o e.g. Hankel Model Order Reducon o e.g. runcaed Balance Realzaon L9- H( s ) Model Order Reducon va Modal Analyss (.e. domnan egenvalues/poles mehod) cb λ s Π s ζ Π s λ Pole-Resdue Form Pole-Zero Form h( Ideas for reducng order: Drop erms wh small resdues c ~ b ~ Drop erms wh large negave Reλ ( fas modes) Remove pole/zero near-cancellaons Cluser poles ha are ogeher c~ ~ b e λ Impulse Response L9-9 Model Order Reducon Oulne Problem seup Reducon va modal analyss o me doman o freuency doman Reducon va ransfer funcon fng o pon machng o leas suare Imporance of preservng physcal properes o sably o passvy/dsspavy Projecon Framework o runcaed Balance Realzaons o Krylov Subspace Momen Machng onlnear Sysems L9-

6 H( s ) Model Order Reducon va Modal Analyss (.e. domnan egenvalues/poles mehod) cb λ s Π s ζ Π s λ Pole-Resdue Form b + bs + + b s + as+ + a s Pole-Zero Form h( c~ ~ b e Raonal Funcon Form λ Impulse Response Model Order Reducon va ransfer Funcon Fng Pon machng H( s) Ĥ( s) bˆ + bˆ s ˆ + + b s Can mach pons H ( s ) + as ˆ ˆ + + as cross mulplyng generaes a lnear sysem For o ( ) ( ˆ ˆ ˆ as ˆ ˆ as H s b bs b s ) ω L9- L9-4 Orgnal Sysem ransfer Funcon: H Model Order Reducon va ransfer Funcon Fng ( s) b + bs + + b s + as+ + a s Model Reducon Fnd a low order ( << ) raonal funcon machng ˆ ˆ ˆ ˆ b + bs+ + b s reduced order H( s) raonal funcon + as ˆ + + as ˆ raonal funcon Model Order Reducon va ransfer Funcon Fng Pon machng marx can become ll-condoned sh s s H s s aˆ H s s aˆ H s s H s s H s s bˆ H s ( ) Columns conan progressvely hgher powers of he es freuences: problem s numercally ll-condoned also... mssng daa can cause severe accuracy problems L9-3 L9-5

7 Hard o solve problems Polynomal nerpolaon example Hard o solve problems Polynomal nerpolaon example able of Daa f ( ) f ( ) f ( ) f ( ) f Coeffcen Value Fng f( Problem f daa wh an h order polynomal f α + α+ α + + α L9-6 Coeffcen number L9-8 Hard o solve problems Polynomal nerpolaon example Model Order Reducon va ransfer Funcon Fng Pon machng usng Leas Suare Marx Form α f ( ) α f ( ) f ( ) α A nerp L9-7 sh s ) smh ( sm) s s m H( s) H( s a a b H( s ( ) ) m m Ĥ s Use much lower order han avalable pons <<m ω cross mulplyng generaes a lnear ALL SKIY sysem Leas Suare problem: use for nsance QR o solve or Gauss-ewon L9-9

8 Overvew Problem Seup Connecon beween crcus and Sae Space models Reducon va egenmode runcaon mehod Reducon va ransfer funcon fng pon machng leas suare uas-convex opmzaon mehod Reducon va Projecon Framework runcaed Balance Realzaons Krylov Subspace Momen Machng need for orhogonalzaon (Arnold( process) compuaonal complexy passvy preservaon Reducon of on-lnear Sysems Relaxaon of he H-nf norm MOR seup [Sou, Megresk, Danel DAC5 CAD8] ( z ) pz r z mnmze H( z) pr,, z subjec o deg m, deg p m, deg ( r) < m An-sable erm Sably: (z) Schur polynomal (roos nsde un crcle) Passvy, and possbly oher consrans Benef: Relaxaon euvalen o a uas-convex program Drawback: May oban subopmal soluons 3 3 Opmzaon based raonal f Model Order Reducon Seup ps () mnmze H( s) p(s),(s) s () From feld solver Or measuremens Leas suare mehod Cas as nonlnear leas suares (solved by e.g. Gauss-ewon) Do no consder sably or passvy whle fndng poles (need posprocessng) Small sable and passve reduced order model Quas convex mehod Cas as uas-convex program (solved by convex opmzaon algorhm) Explcly ake care of sably and passvy whle fndng poles Solvng he uas-convex program, deg deg ( c) m, be + jc e mnmze He abc,, ae deg a m b m subjec o Sably: Passvy: a e b e >, ω [, π] >, ω [, π] uas-convex se convex se Sandard problem. Use for example by he ellpsod algorhm 35

9 Example : RF nducor wh subsrae (from feld solver) RF nducor wh subsrae effec capured by layered Green s funcon [Hu Dac 5] Sysem marces are freuency dependen Full model has nfne order Reduced model has order 6 ualy facor ranng daa es pons ROM magnude Example 3: Model of graphc card package (from measuremen Indusry example of a mul-por devce (39 freuency samples) h order SISO reduced models are consruced Bounded realness consran s mposed Freuency wegh s employed S Sold: ROM Do: measuremen magnude S3 Sold: ROM Do: measuremen freuency (Hz) x freuency (GHz) freuency (GHz) Example : RF nducor model (from measuremen Fabrcaed 7 urn spral nducor Blue: measuremen Red: h order reduced model (posve real par consran mposed) Example 4: Large IC power dsrbuon grd (from feld solver) Power dsrbuon grd (dmenson sze 7mm, wre wh µm) Blue: full model (order 46) Red: QCO 4 h order reduced model (posve real) 9 4 real par freuency (Hz) x 9 ualy facor freuency (Hz) x 9 magnude curves on op of each oher freuency (GHz) phase freuency (GHz) 3 curves on op of each oher