SF & IH Inroducon o Compac Dynamcal Modelng III. Reducng Lnear me Invaran Sysems Luca Danel Massachuses Insue of echnology 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 frequency doman Reducon va ransfer funcon fng o pon machng o leas square 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 I.3-2
d y( Movaons A ( + b u( c ( 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 2. generae a compac dynamcal model 3. re-use he model over and over L9-3 3 Model Order Reducon Hea conducng bar eample lamp power Lamp u Inpu of Ineres Oupu of Ineres end () L9-4
Model Order Reducon Hea conducng bar eample Problem Seup Hea n h(z)u( emperaure Dfferenal Equaon γ specfc hea (, 2 z, κ + 2 z hermal conducvy Δz end h z u scalar npu () Spaal Dscrezaon dˆ κ + + ( ˆ ˆ ˆ ) 2 2 h( z) u( Δz γ + L9-5 Model Order Reducon Hea conducng bar eample Problem Seup Hea n h()u( d Δz A () + bu () y () c () scalar scalar npu oupu end () Spaal Dscrezaon dˆ κ + + ( ˆ ˆ ˆ ) 2 2 h( z) u( Δz γ + L9-6
Model Order Reducon Hea conducng bar eample Problem Seup Hea n h()u( d Δz scalar scalar npu oupu 2 hz ( ) 2 hz ( 2) κ A b γ ( Δz ) 2 γ 2 hz ( ) c end A () + bu () y () c () () L9-7 Model Order Reducon Hea conducng bar eample Problem Seup Orgnal Dynamcal Sysem - Sngle Inpu/Oupu d A () + bu () y () c () scalar scalar npu oupu Reduced Dynamcal Sysem dˆ( Aˆ ˆ( + bˆ u( qq q scalar npu yˆ( cˆ scalar oupu q ˆ( q <<, bu npu/oupu behavor preserved L9-8
Model Order Reducon Oulne Model Order Reducon of Lnear Problems Problem seup Reducon va modal analyss o me doman o frequency doman Reducon va ransfer funcon fng o pon machng o leas square 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- Egendecomposon: Remnder abou Egenvalues d() C onsder an ODE: A( + bu, () A V V2 V V V2 V V le s: V V ( Change of varab () () () dv () Subsun g : AV() + bu(), V( ) d() Mulply by V : V AV () + V bu() () + V bu L9-
L9-2 () V b d u V b b + Decoupled Equaons () () () () y c c V V c c Oupu Equaon Remnder abou Egenvalues L9-3 () [ ] y c c Oupu Equaon ) ( u b b d + Model Order Reducon va Modal Analyss (.e. domnan egenvalues/poles mehod)
L9-4 () [ ] y c c ) ( u b b d + Oupu Equaon Model Order Reducon va Modal Analyss (.e. domnan egenvalues/poles mehod) q q b q q c q L9-5 Ceran modes are no affeced by he npu Ceran modes do no affec he oupu Keep leas negave egenvalues (slowes modes) Look a response o a consan npu,, are all small q b b +,, are all small q c c + () () () Small f large e b u d e b u τ τ Model Order Reducon va Modal Analyss (.e. domnan egenvalues/poles mehod)
Model Order Reducon va Modal Analyss (.e. domnan egenvalues/poles mehod) q q3 q Eac Hea Conducng bar Resuls Keepng q-h slowes modes L9-6 Model Order Reducon Oulne Model Order Reducon of Lnear Problems Problem seup Reducon va modal analyss o me doman o frequency doman Reducon va ransfer funcon fng o pon machng o leas square 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
Modal (egenvalue) analyss n he frequency doman ransfer Funcon ( ) H s c si A b A VV H s c V si V b Apply Egendecomposon s s c b H( s) elmae each mode for whch hs erm s small cb s L9-8 H( s ) Model Order Reducon va Modal Analyss (.e. domnan egenvalues/poles mehod) cb s Π s ζ Π s Pole-Resdue Form Pole-Zero Form h( c b e Impulse Response Ideas for reducng order: Drop erms wh small resdues c b Drop erms wh large negave Re ( fas modes) Remove pole/zero near-cancellaons Cluser poles ha are ogeher L9-9
Model Order Reducon va Modal Analyss (.e. domnan egenvalues/poles mehod) Advanages Concepually famlar Smple physcal nerpreaon : reans domnan sysem modes/poles Drawbacks Relavely epensve : 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 L9-2 Oulne Model Order Reducon Problem seup Reducon va modal analyss o me doman o frequency doman Reducon va ransfer funcon fng o pon machng o leas square Imporance of preservng physcal properes o sably o passvy/dsspavy Projecon Framework o runcaed Balance Realzaons o Krylov Subspace Momen Machng onlnear Sysems L9-2
H( s ) Model Order Reducon va Modal Analyss (.e. domnan egenvalues/poles mehod) cb s Π s ζ Π s Pole-Resdue Form Pole-Zero Form b + bs + + b s + as+ + a s h( c b e Raonal Funcon Form Impulse Response L9-22 Model Order Reducon va ransfer Funcon Fng Orgnal Sysem ransfer Funcon: H ( s) b + bs + + b s + as+ + a s Model Reducon Fnd a low order (q << ) raonal funcon machng ˆ ˆ ˆ q ˆ b + bs+ + bq s reduced order H( s) q raonal funcon + as ˆ + + as ˆ q raonal funcon L9-23
Model Order Reducon va ransfer Funcon Fng Pon machng H( s) Ĥ ( s) Can mach 2q pons H ( s ) cross mulplyng generaes a lnear sysem For o 2q + + + + + + q ( ) ( ˆ ˆ ˆ q as ˆ ˆ as q H s b bs bq s ) ω bˆ + bˆ s + + bˆ s q q q + as ˆ ˆ + + as q L9-24 Model Order Reducon va ransfer Funcon Fng Pon machng mar can become ll-condoned sh s s H s s aˆ H s 2 q q s aˆ 2 2 H s2 s H s s H s s bˆ H s ( 2q ) 2 q 2q 2q 2q 2q 2q q Columns conan progressvely hgher powers of he es frequences: problem s numercally ll-condoned also... mssng daa can cause severe accuracy problems L9-25
Hard o solve problems Polynomal nerpolaon eample able of Daa f ( ) f ( ) f ( ) f f ( ) 2 Problem f daa wh an h order polynomal 2 f () α + α+ α2 + + α L9-26 Hard o solve problems Polynomal nerpolaon eample Mar Form 2 α f ( ) 2 α f ( ) 2 f ( ) α A nerp L9-27
Hard o solve problems Polynomal nerpolaon eample Fng f( Coeffcen Value Coeffcen number L9-28 sh s ) smh ( sm) Model Order Reducon va ransfer Funcon Fng Pon machng usng Leas Square s q m H ( s) H( s) a a 2 b q H( sm) q ( s m Ĥ s Use much lower order han avalable pons 2q<<m ω cross mulplyng generaes a lnear ALL SKIY sysem Leas Square problem: use for nsance QR o solve or Gauss-ewon L9-29
Overvew Problem Seup Connecon beween crcus and Sae Space models Reducon va egenmode runcaon mehod Reducon va ransfer funcon fng pon machng leas square quas-conve opmzaon mehod Reducon va Projecon Framework runcaed Balance Realzaons Krylov Subspace Momen Machng need for orhogonalzaon (Arnold( process) compuaonal compley passvy preservaon Reducon of on-lnear Sysems 3 Opmzaon based raonal f Model Order Reducon Seup ps () mnmze H( s) p(s),q(s) qs () From feld solver Or measuremens Leas square mehod Cas as nonlnear leas squares (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 conve mehod Cas as quas-conve program (solved by conve opmzaon algorhm) Eplcly ake care of sably and passvy whle fndng poles
Relaaon of he H-nf norm MOR seup [Sou, Megresk, Danel DAC5 CAD8] pz r z mnmze H( z) pqr,, qz q subjec o ( z ) deg ( r) < m deg q m, deg p m, An-sable erm Sably: q(z) Schur polynomal (roos nsde un crcle) Passvy, and possbly oher consrans Benef: Relaaon equvalen o a quas-conve program Drawback: May oban subopmal soluons 32 Solvng he quas-conve program jω be + jc e jω mnmze He abc,, jω ae subjec o Sably: Passvy: deg, deg deg ( c) m, jω jω a e b e jω a m b m >, ω [, π] >, ω [, π] quas-conve se conve se Sandard problem. Use for eample by he ellpsod algorhm 35
Eample : RF nducor wh subsrae (from feld solver) RF nducor wh subsrae effec capured by layered Green s funcon [Hu Dac 5] Sysem marces are frequency dependen Full model has nfne order Reduced model has order 6 8 7 6 ranng daa es pons ROM qualy facor 5 4 3 2 - -2.5.5 2 2.5 3 frequency (Hz) 9 Eample 2: RF nducor model (from measuremen Fabrcaed 7 urn spral nducor Blue: measuremen Red: h order reduced model (posve real par consran mposed) 9 4 8 35 real par 7 6 5 4 3 2 qualy facor 3 25 2 5 5 2 3 4 5 6 7 8 9 frequency (Hz) 9-5.5.5 2 2.5 3 3.5 frequency (Hz) 9
Eample 3: Model of graphc card package (from measuremen Indusry eample of a mul-por devce (39 frequency samples) 2 h order SISO reduced models are consruced Bounded realness consran s mposed Frequency wegh s employed.9 S.2. S3 Sold: ROM Do: measuremen magnude.8.7.6.5.4 Sold: ROM Do: measuremen magnude.8.6.4.2 2 3 4 5 6 frequency (GHz) 2 3 4 5 6 frequency (GHz) Eample 4: Large IC power dsrbuon grd (from feld solver) Power dsrbuon grd (dmenson sze 7mm, wre wh 2 µm) Blue: full model (order 246) Red: QCO 4 h order reduced model (posve real) 25.5 magnude 2 5 5 2 curves on op of each oher 2 3 4 5 6 frequency (GHz) phase.5 -.5 - -.5-2 2 3 4 5 6 frequency (GHz) 3 curves on op of each oher