1
Parasitic ExtractionParasitic ExtractionStep 2: Model Order ReductionStep 2: Model Order Reduction
Luca Daniel
Massachusetts Institute of Technology
Slides available at: http://onigo.mit.edu/~dluca/2009MOMiNE
www.rle.mit.edu/cpg
2
Conventional Design FlowConventional Design Flow
Funct. Spec
Logic Synth.
Gate-level Net.
RTL
Layout
Floorplanning
Place & Route
Front-end
Back-end
Behav. Simul.
Gate-Lev. Sim.
Stat. Wire Model
Parasitic Extrac.
3
Layout parasiticsLayout parasitics
• Wires are not ideal. Wires are not ideal. Parasitics:Parasitics:– ResistanceResistance
– CapacitanceCapacitance
– InductanceInductance
• Why do we care? Why do we care? – Impact on delayImpact on delay
– noisenoise
– energy consumptionenergy consumption
– power distributionpower distribution
Picture from “Digital Integrated Circuits”, Rabaey, Chandrakasan, Nikolic
4
Conventional Design FlowConventional Design Flow
Funct. Spec
Logic Synth.
Gate-level Net.
RTL
Layout
Floorplanning
Place & Route
Front-end
Back-end
Behav. Simul.
Gate-Lev. Sim.
Stat. Wire Model
Parasitic Extrac.
5
)()( tuBtxAdt
dxE
dt
dEH
dt
dHE
dt
dEH
dt
dHE
dt
dEH
dt
dHE
dt
dEH
dt
dHE
Parasitic Extraction: Step 1 Electromagnetic Field Solvers (review from last time)
Step 1. Electromagnetic Field SolversStep 1. Electromagnetic Field Solvers
nanophotonics
carbon nanotubes
RF inductorsinterconnect
6
Overview Step1 : Field SolversOverview Step1 : Field Solvers
• Formulation of Parasitic Extraction Problems:Formulation of Parasitic Extraction Problems:– Capacitance Extraction (electrostatic)Capacitance Extraction (electrostatic)
– Inductance Extraction (Magneto-Quasi-Static MQS)Inductance Extraction (Magneto-Quasi-Static MQS)
– Combined RLC Extraction (EMQS)Combined RLC Extraction (EMQS)
– Electromagnetic Interference Analysis (fullwave)Electromagnetic Interference Analysis (fullwave)
• Electromagnetic solvers: Electromagnetic solvers: – Classification (time vs. frequency, differential vs. integral)Classification (time vs. frequency, differential vs. integral)
– Integral equation solvers in detailsIntegral equation solvers in details• basis functionsbasis functions• equation testing (collocation vs. Galerkin)equation testing (collocation vs. Galerkin)• linear system solution (direct vs. iterative methods)linear system solution (direct vs. iterative methods)• fast matrix-vector products (eg. fastmultipole, pFFT)fast matrix-vector products (eg. fastmultipole, pFFT)• example: FastMaxwell a EMQS/Fullwave solver with pFFTexample: FastMaxwell a EMQS/Fullwave solver with pFFT
– Current Field Solvers Research DirectionsCurrent Field Solvers Research Directions
yesterday
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)()( tuBtxAdt
dxE
dt
dEH
dt
dHE
dt
dEH
dt
dHE
dt
dEH
dt
dHE
dt
dEH
dt
dHE
The two steps Parasitic Extraction process
Step 1. Electromagnetic Field SolversStep 1. Electromagnetic Field Solvers
Step 2. Parameterized Model Order ReductionStep 2. Parameterized Model Order Reduction
)(ˆ)(ˆ),(ˆˆ),(ˆ tuBtxWhA
dt
xdWhE
nanophotonics
carbon nanotubes
RF inductorsW
interconnect
(10(1066 x 10 x 1066)) (10(1066 x 10 x 1066))
),( Wh ),( Wh
• small (10-15 equations)small (10-15 equations)• accuracy of PDE field solvers accuracy of PDE field solvers • geometrically parameterizedgeometrically parameterized• generated automatically!generated automatically!
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ReferencesReferencesfrom Field Solvers to Reduced Order Modelsfrom Field Solvers to Reduced Order Models
[PEEC Ruehli 74] A. E. Ruehli, "Equivalent circuit models for three dimensional multiconductor systems", Trans Microwave Theory and Techniques, 1974, v 22, p 216-221.
[Kamon Trans Packaging 98] M. Kamon, N. Marques, L. M. Silveira, J. K. White, Automatic Generation of Accurate Circuit Models", Trans on Packaging, Aug 1998.
[MIT course 6.336J] http://stellar.mit.edu/S/course/6/fa04/6.336J/
[MIT course 16.920J] http://ocw.mit.edu/OcwWeb/Aeronautics-and-Astronautics/16-920JNumerical-Methods-for-Partial-Differential-EquationsSpring2003/CourseHome/
[Grimme PhD97] Eric Grimme, “Krylov Projection Methods for Model Reduction", PhD Thesis, University of Illinois at Urbana-Champaign", 1997.
[PRIMA TCAD98] A. Odabasioglu, M. Celik, L. Pileggi , "PRIMA: passive reduced-order interconnect macromodeling algorithm", TCAD, v 17, n 8, Aug 1998.
[Phillips96] J. R. Phillips, E. Chiprout, D. D. Ling, "Efficient full-wave electromagnetic analysis via model-order reduction of fast integral transforms", DAC, June 1996.
[Willems72] J. C. Willems, “Dissipative Dynamical Systems”, Arch. Rational Mechanics and Analysis, 1972, v 45, p 321-393.
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ReferencesReferencesParameterized Model Order ReductionParameterized Model Order Reduction
[Pullela97] S. Pullela N. Menezes L.T. Pileggi, "Moment-Sensitivity-Based Wire Sizing for Skew Reduction in On-Chip Clock Nets", Trans on CAD, v 16, n 2, p 210-215, Feb, 1997.
[Weile99] D. S. Weile E. Michielssen Eric Grimme K. Gallivan, "A Method for Generating Rational Interpolant Reduced Order Models of Two-Parameter Linear Systems", Applied Mathematics Letters, v 12, p 93-102, 1999.
[Prud’homme 02] C. Prud'homme, D. Rovas, K. Veroy, Y. Maday, A.T. Patera, G. Turinici, "Reliable real-time solution of parameterized partial differential equations: Reduced-basis output bounds methods", Journal of Fluids Engineering, 2002.
[Liu DAC99] Y Liu, L T. Pileggi, A J. Strojwas", Model Order-Reduction of RCL Interconnect Including Variational Analysis", DAC, p 201-206, 1999.
[Heydari ICCAD01] P. Heydari, M. Pedram, “Model Reduction of Variable-Geometry Interconnects Using Variational Spectrally-Weighted Balanced Truncation", ICCAD, 586—591, 2001.
[Rutenbar DAC02] H Liu, Singhee, A, Rutenbar, R.A., Carley, L.R, “Remembrance of circuits past: macromodeling by data mining in large analog design spaces”, DAC 2002, p 437-42.
[Li DATE05] P Li, F Liu, S Nassif, L Pileggi, Modeling Interconnect Variability using Efficient Parametric Model Order Reduction, DATE 2005.
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ReferencesReferencesParameterized Model Order Reduction (cont.)Parameterized Model Order Reduction (cont.)
[D. TCAD04] L. Daniel, C. S. Ong, S. C. Low, K. H. Lee, J. White, A Multiparameter Moment Matching Model Reduction Approach for Generating Geometrically Parameterized Interconnect Performance Models", IEEE Trans. on Computer-Aided Design of Integrated Circuits and Systems, v 23, n 5, p 678-93, May 2004.
[D. BMAS03] L. Daniel, J. White, "Automatic generation of geometrically parameterized reduced order models for integrated spiral RF-inductors", Proceedings of the 2003 IEEE International Workshop on Behavioral Modeling and Simulation, p 18-23, San Jose, CA, 2003.
[D. APS04] L. Daniel, J. White, “Numerical Techniques for Extracting Geometrically Parameterized Reduced Order Interconnect Models from Full-wave Electromagnetic Analysis”, IEEE AP-S Symposium, June 2004.
[D. DAC02] L. Daniel, J. Phillips, "Model Order Reduction for Strictly Passive and Causal Distributed Systems", IEEE/ACM 39th Design Automation Conference, New Orleans, Jun 2002.
[D. PhD04] Luca Daniel, "Simulation and Modeling Techniques for Signal Integrity and Electromagnetic Interference on High Frequency Electronic Systems," Ph.D. Thesis, University of California at Berkeley, May 2003.
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OverviewOverview
• Problem SetupProblem Setup• Connection between circuits and State Space models Connection between circuits and State Space models • Reduction via eigenmode truncation methodReduction via eigenmode truncation method• Reduction via transfer function fittingReduction via transfer function fitting
– point matchingpoint matching– least squareleast square– quasi-convex optimization methodquasi-convex optimization method
• Reduction via Projection FrameworkReduction via Projection Framework– Truncated Balance RealizationsTruncated Balance Realizations– Krylov Subspace Moment MatchingKrylov Subspace Moment Matching
• need for orthogonalization (Arnoldi process)need for orthogonalization (Arnoldi process)• computational complexitycomputational complexity• passivity preservationpassivity preservation
• Reduction of Non-Linear SystemsReduction of Non-Linear Systems
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State-Space Models State-Space Models
• Linear system of ordinary differential equations Linear system of ordinary differential equations (ABCD form) (ABCD form)
State Input
Output
)()()(
)()(
tDutCxty
tButAxdt
dxE
13
State-Space Model Example:State-Space Model Example:Interconnect Segment Interconnect Segment
• Step 1: Identify internal state variablesStep 1: Identify internal state variables– Example : MNA uses node voltages & inductor current Example : MNA uses node voltages & inductor current
1v 3v
LI
2v
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State-Space Model Example:State-Space Model Example:Interconnect Segment Interconnect Segment
• Step 2: Identify inputs & outputs Step 2: Identify inputs & outputs – Example : For Z-parameter representation, choose port Example : For Z-parameter representation, choose port
currents inputs and port voltage outputs currents inputs and port voltage outputs
in1I
in2I
out2vout
1v
1v 3v
LI
2v
15
State-Space Model Example:State-Space Model Example:Interconnect Segment Interconnect Segment
• Step 3: Write state-space & I/O equations Step 3: Write state-space & I/O equations – Example : KCL + inductor equation Example : KCL + inductor equation
01211 inI
Rvv
dtdv
C
1out1 vv
3out2 vv
012 LI
Rvv 02
3 inL II
dtdv
C
32 vvdt
dIL L
in1I
in2I
out2vout
1v LI
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A linear circuit can be expressed A linear circuit can be expressed as a state space modelas a state space model
• Step 4: Identify state variables & matrices Step 4: Identify state variables & matrices
L
C
C
E0
00
10
00
01
B
LI
v
v
v
x3
2
1
0110
1000
10
0011
11
RR
RR
A
0100
0001C
00
00D
in
in
I
Iu
2
1
2
1
v
vy
inIR
vv
dt
dvC 1
211
LIR
vv
120
inL II
dt
dvC 2
3
32 vvdt
dIL L
)()(
)()(
txCty
tButAxdt
dxE
T
LARGE!LARGE!LARGE!LARGE!
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OverviewOverview
• Problem SetupProblem Setup• Connection between circuits and State Space models Connection between circuits and State Space models • Reduction via eigenmode truncation methodReduction via eigenmode truncation method• Reduction via transfer function fittingReduction via transfer function fitting
– point matchingpoint matching– least squareleast square– quasi-convex optimization methodquasi-convex optimization method
• Reduction via Projection FrameworkReduction via Projection Framework– Truncated Balance RealizationsTruncated Balance Realizations– Krylov Subspace Moment MatchingKrylov Subspace Moment Matching
• need for orthogonalization (Arnoldi process)need for orthogonalization (Arnoldi process)• computational complexitycomputational complexity• passivity preservationpassivity preservation
• Reduction of Non-Linear SystemsReduction of Non-Linear Systems
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Reminder about Eigenanalysis
1Change of variab (le ) ( ) ( ) (s ): Ew t x t w t E x t
1
1 2 1 2
10 0
0 0
0 0
Eigendecomposition: N N
N
E
A E E E E E E
Substituting: ( )
( ) , (0) 0dEw t
AEw t bu t Ewdt
11 1Multiply by : ( )
( )dw t
E AEw t E bu tdt
E
Consider an ODE( )
( ) , (0) 0: dx t
Ax t bu t xdt
E
S S
SS S
S S S1S
1S 2S NS 1S 2S NS
S
19
Reminder about Eigenanalysis Cont.
1
11 1 1
1
0 0
0 0
0 0N N NN
E bw wd
u tdt
w w E b
b
Decoupled Equations
Output Equation
S
S
TT T Ty t c x t c Ew t E c w t
c
S S
20
Reminder about Eigenanalysis Cont.
0
i
tt
i iw t e b u d
Solving Decoupled Equations
1
N
i ii
y t c w t
Output Equation
Assuming Zero Initial Conditions
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Reduced models via mode truncationDynamic Linear Case
1
q
i ii
y t c w t
Output Equation
)(
~
~
~1111
tu
b
b
b
w
w
w
w
w
w
N
q
N
q
N
q
N
q
22
Reduced models via mode TruncationDynamic Linear Case
Why?
• Certain modes are not affected by the input
• Certain modes do not affect the output
• Keep least negative eigenvalues (slowest modes)– Look at response to a constant input
1, , are all smallk Nb b
1, , are all smallk Nc c
0
1
Small if large
i i
tt t
i i i ii
i
w t e b ud b u b ue
23
Reduced models via mode truncationDynamic Linear Case
Single Wire Results
N=100
Exact
q=1
q=3
q=10
Keep qth slowest modes
24
An Aside on Transfer Functions – Laplace Transform
x tBilateral Laplace Transform: stX s e dt
Consider an ODE: ( )
( )dx t
Ax t bu tdt
Key Transform Property: dx t
dtstsX s e dt
Rewrite the ODE in transformed variables
TsX s AX s bU s Y s c X s
1TY s c sI A bU s
H s
Transfer Function
25
An Aside on Transfer Functions – Meaning of H(s)
then ( ) j ty t H j e
For Stable Systems, H(jw) is the frequency response
If ( ) j tu t e
Sinusoid with shifted phase and amplitude
Sinusoid
H j
26
An Aside on Transfer Functions – EigenAnalysis
1TH s c sI A b
Transfer Function
Apply Eigendecomposition
1 1TH s c E sI E b
1
10 0
0 0
10 0
T
N
s
c b
s
1
Ni i
i i
c bH s
s
elimitate eachmode for whichthis term is small
1 SSA S S
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Model Order Reduction via Eigenmode Analysis Model Order Reduction via Eigenmode Analysis Summary Summary
n
i i
ii
s
bcsH
1
~~)(
)(
)()(
1
1
1
i
n
i
i
n
i
s
ssH
Pole-Residue FormPole-Residue FormPole-Zero Form (SISO)Pole-Zero Form (SISO)
• Ideas for reducing order:Ideas for reducing order:– Drop terms with small Drop terms with small
– Drop terms with small residuesDrop terms with small residues
– Drop terms with large negative (“fast” modes)Drop terms with large negative (“fast” modes)
– Remove pole/zero near-cancellations Remove pole/zero near-cancellations
– Cluster poles that are “together”Cluster poles that are “together”
iRe
n
i
tii
iebcth1
~~)(
iibc~~iiibc /~~
28
Eigenmode Analysis Based Reduction SummaryEigenmode Analysis Based Reduction Summary
• Advantages Advantages – Conceptually familiar Conceptually familiar
– Simple physical interpretation : retains dominant Simple physical interpretation : retains dominant system modes/poles system modes/poles
• Drawbacks Drawbacks – Relatively expensive: Relatively expensive: have to find the eigenvalues firsthave to find the eigenvalues first
– Relatively inefficient. For a given model size, many Relatively inefficient. For a given model size, many other approaches can provide better accuracy for the other approaches can provide better accuracy for the same computational costsame computational cost
• e.g. Hankel Model Order Reduction e.g. Hankel Model Order Reduction • e.g. Truncated Balance Realizatione.g. Truncated Balance Realization
O(n3)
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OverviewOverview
• Problem SetupProblem Setup• Connection between circuits and State Space models Connection between circuits and State Space models • Reduction via eigenmode truncation methodReduction via eigenmode truncation method• Reduction via transfer function fittingReduction via transfer function fitting
– point matchingpoint matching– least squareleast square– quasi-convex optimization methodquasi-convex optimization method
• Reduction via Projection FrameworkReduction via Projection Framework– Truncated Balance RealizationsTruncated Balance Realizations– Krylov Subspace Moment MatchingKrylov Subspace Moment Matching
• need for orthogonalization (Arnoldi process)need for orthogonalization (Arnoldi process)• computational complexitycomputational complexity• passivity preservationpassivity preservation
• Reduction of Non-Linear SystemsReduction of Non-Linear Systems
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Dynamical SystemDynamical System
Transfer FunctionTransfer Function
coefficientscoefficients
coefficientscoefficients
Counting Degrees of Freedom...Counting Degrees of Freedom...
q
q
q
q
sasa
sbsbb
ss
ssk
s
r
s
rsH
1
1110
1
11
1
1
1
)()(
)()(
)()()(
2q
q2+2q
)(ˆˆ)(ˆ
)(ˆ)(ˆˆˆ
txcty
tubtxAdt
xd
T
Degrees of Freedom?Degrees of Freedom?
Degrees of Freedom?Degrees of Freedom?
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Reduced Model Transfer FunctionReduced Model Transfer Function
Apply any invertible change of coordinates to the stateApply any invertible change of coordinates to the state
Many Dynamical Systems have the same transfer function!!
Fully Invertible Change of CoordinatesFully Invertible Change of Coordinates
)(ˆˆ)(
)(ˆ)(ˆˆˆ
txcty
tubtxAdt
xd
T
bAsIcsH T ˆˆˆ)(1
)(~)(ˆ txUtx
)(~ˆ)(
)(ˆ)(~ˆ~
txUcty
tubtxUAdt
xdU
T
)(~ˆ)(
)(ˆ)(~ˆ~
111
txUcty
tubUtxUAUdt
xdUU
T
I
)(ˆˆˆ)(~ 1
sHbAsIcsH T
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Original System Transfer Function:Original System Transfer Function:
1
0 1 1
11
NN
NN
b b s b sH s
a s a s
Model Reduction = Find a low order (q << N) Model Reduction = Find a low order (q << N) rational function matchingrational function matching
Model Order Reduction Model Order Reduction via Rational Transfer Function Fittingvia Rational Transfer Function Fitting
rational functionrational function
1
0 1 1
1
ˆ ˆ ˆˆ
ˆ ˆ1
b b s b sH s
a s a s
reduced orderreduced orderrational functionrational function
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H s
Rational Transfer Function Fitting: Rational Transfer Function Fitting: via Point Matchingvia Point Matching
H s
11 0 1 1
ˆ ˆ ˆˆ ˆ1 0q qi q i i i q ia s a s H s b b s b s
For i = 1 to 2q
• cross multiplying generates a linear systemcross multiplying generates a linear system
• Can match 2q pointsCan match 2q points 1
0 1 1
1
ˆ ˆ ˆ
ˆ ˆ1
qi q i
i qi q i
b b s b sH s
a s a s
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• Columns contain progressively higher powers of the test Columns contain progressively higher powers of the test frequencies: problem is numerically ill-conditionedfrequencies: problem is numerically ill-conditioned
• also... missing data can cause severe accuracy problemsalso... missing data can cause severe accuracy problems
2 111 1 1 1 1 1
122 2
2 11 22 2 2 2 2
q
q
qq qq q q q q
H ss H s s H s s a
H ss a
b H ss H s s H s s
Rational Transfer Function Fitting: Rational Transfer Function Fitting: Point Matching matrix can be ill-conditionedPoint Matching matrix can be ill-conditioned
SMA 2005 MIT 35
Hard to Solve Systems
Fitting Example
Polynomial InterpolationTable of Data
t0 f (t0)t1 f (t1)
tN f (tN)
f
tt0 t1 t2 tN
f (t0)
Problem fit data with an Nth order polynomial2
0 1 2( ) NNf t t t t
SMA 2005 MIT 36
Hard to Solve Systems
Example Problem
Matrix Form2
0 0 0 0 0
21 11 1 1
2
interp
1 ( )
( )1
( )1
N
N
NN NN N N
t t t f t
f tt t t
f tt t t
M
SMA 2005 MIT 37
Hard to Solve Systems
Fitting Example
CoefficientValue
Coefficient number
Fitting f(t) = t
38
H s
Rational Transfer Function Fitting via Point MatchingRational Transfer Function Fitting via Point MatchingLeast Square MethodLeast Square Method
H s
• cross multiplying generates a cross multiplying generates a linear TALL SKINNY systemlinear TALL SKINNY system
• solve with Least Square methods, solve with Least Square methods, QR or Gauss-NewtonQR or Gauss-Newton
• get better H(s) accuracy, but get better H(s) accuracy, but produced model can be unstable: produced model can be unstable: needs post-processingneeds post-processing
Use many morepoints than order2q<<m
)(
)(
)(
)( 1
1
1
1
1111
mqqmmm
q
sH
sH
b
a
ssHs
ssHs
m
39
OverviewOverview
• Problem SetupProblem Setup• Connection between circuits and State Space models Connection between circuits and State Space models • Reduction via eigenmode truncation methodReduction via eigenmode truncation method• Reduction via transfer function fittingReduction via transfer function fitting
– point matchingpoint matching– least squareleast square– quasi-convex optimization methodquasi-convex optimization method
• Reduction via Projection FrameworkReduction via Projection Framework– Truncated Balance RealizationsTruncated Balance Realizations– Krylov Subspace Moment MatchingKrylov Subspace Moment Matching
• need for orthogonalization (Arnoldi process)need for orthogonalization (Arnoldi process)• computational complexitycomputational complexity• passivity preservationpassivity preservation
• Reduction of Non-Linear SystemsReduction of Non-Linear Systems
40
Optimization based rational fit Model Order Reduction Setup
p(s),q(s)
( )minimize ( )
( )
p sH s
q s
From field solverOr measurements
Small stable and passivereduced order model
Least square method• Cast as nonlinear least
squares (solved by e.g. Gauss-Newton)
Quasi convex method• Cast as quasi-convex
program (solved by convex optimization algorithm)
• Do not consider stability or passivity while finding poles (need post-processing)
• Explicitly take care of stability and passivity while finding poles
41
Change of variables
• To make our program tractable, we introduce a change offrequency variables (bilinear transform)
Laplace frequency variablez frequency variable
11
zzs
[s] [z]
42
• Desirable MOR setup to solve• Feasible set is not convex if m 3 For example, but
• Problem has not been proved to be NP complete either
Modified optimal H-inf norm MOR setup
331 5q z z 33
2 5q z z
,
( )minimize ( )
( )
deg ,subject to
deg ,
p q
p zH z
q z
q m
p m
31 2( ) ( ) 27
2 2 25
q z q zz z
Stability: q(z) Schur polynomial (roots inside unit circle)
Passivity, and possibly other constraints
43
A standard optimization technique:Relaxation
• Original problem is difficult• Made easier if some constraints are dropped (relaxed)• Solve the relaxed problem • Construct original solution from relaxation
General idea
-c
feasible set…
optimal solution-c
feasible set
optimal relaxed solution
nearest rounding
44 44
Relaxation of the H-inf norm MOR setup [Sou, Megretski, Daniel DAC05 TCAD08]
1
1, ,
( )minimize ( )
( )
deg , deg ,subject to
deg
p q r
r z
q
p zH z
q z z
q m p m
r m
Benefit: Relaxation equivalent to a quasi-convex programDrawback: May obtain suboptimal solutions
Anti-stableterm
Stability: q(z) Schur polynomial (roots inside unit circle)
Passivity, and possibly other constraints
45 45
How bad is the relaxation?
,
1
1,( , , ) arg min
q p r
p zq p r H z
q
r
q zz
z
1m
p zH z m H
q z
Let
such that deg(q) = m, q(z) is Schur polynomial
Then
m+1th Hankel singular value
THEOREM:
47 47
Equivalent quasi-convex setup
This is a quasi-convex program, because
1 02cos( ) 2cos(( 1) ) 0jma e m m a a
defines an intersection of halfspaces
convex set0
1
, ,
( )minimize ( )
( )
deg , deg
deg ,subject to
0, [0, ]
0, [0, ]
j j
jja b c
j
j
b e jc eH e
a e
a m b m
c m
a e
b e
=0
=1
=2
=3
quasi-convex function
convex set
0 0 1 02cos 1 0 1 0mm m a a 1 1 1 02cos 1 0 1 0mm m a a 2 2 1 02cos 1 0 1 0mm m a a 3 3 1 02cos 1 0 1 0mm m a a
Stability:
Passivity:
48 48
Solving the quasi-convex program
, ,
( )minimize ( )
( )
deg , deg
deg ,subject to
0, [0, ]
0, [0, ]
j j
jja b c
j
j
b e jc eH e
a e
a m b m
c m
a e
b e
convex setStability:
Passivity:
quasi-convex set
Standard problem.Use for example by the ellipsoid algorithm
50
Stability?
Solving quasi-convex programs
(a,b,c,) current iteratelocalization set(e.g. ellipsoid)
?b jc
Ha
Passivity?
…
Generate cut
N
N
N
N
Y
Y
Y
Decrease
All Yes
?c
Qb
Termination?
N
target set
localization set
center
cut
min volume covering ellipsoid
new center
new cut
and so on
Updatelocalization set
Objective oracle, stabilityoracle, passivity oracle…
51
Example 1: RF inductor with substrate(from field solver)
0 0.5 1 1.5 2 2.5 3
x 109
-2
-1
0
1
2
3
4
5
6
7
8
frequency (Hz)
qual
ity f
acto
r
training data
test pointsROM
• RF inductor with substrate effect captured by layered Green’s function [Hu Dac 05]• System matrices are frequency dependent• Full model has infinite order• Reduced model has order 6
52
Example 2: RF inductor model (from measurement)
0 1 2 3 4 5 6 7 8 9 10
x 109
0
1000
2000
3000
4000
5000
6000
7000
8000
9000
frequency (Hz)
real
par
t
Fabricated 7 turn spiral inductorBlue: measurementRed: 10th order reduced model (positive real part constraint imposed)
0 0.5 1 1.5 2 2.5 3 3.5
x 109
-5
0
5
10
15
20
25
30
35
40
frequency (Hz)
qual
ity f
acto
r
53
Example 3: Model of graphic card package (from measurement)
• Industry example of a multi-port device (390 frequency samples)• 12th order SISO reduced models are constructed• Bounded realness constraint is imposed• Frequency weight is employed
0 1 2 3 4 5 6
0.4
0.5
0.6
0.7
0.8
0.9
1
mag
nitu
de
frequency (GHz)0 1 2 3 4 5 6
0
0.02
0.04
0.06
0.08
0.1
0.12
mag
nitu
de
frequency (GHz)
S11 S13
Solid: ROMDot: measurement
Solid: ROMDot: measurement
54
Example 4: Large IC power distribution grid(from field solver)
• Power distribution grid (dimension size = 7mm, wire width = 2 µm)• Blue: full model (order 2046)• Red: QCO 40th order reduced model (positive real)
0 10 20 30 40 50 600
500
1000
1500
2000
2500
frequency (GHz)
mag
nitu
de
0 10 20 30 40 50 60-2
-1.5
-1
-0.5
0
0.5
1
1.5
frequency (GHz)
phas
e
2 curves on top of each other
3 curves on top of each other
55
Parameterized stability [Sou TCAD08]
• Checking stability can be done checking positivity of a multivariable trigonometric polynomial• Check positivity by sum of squares argument (semidefinite program)
,min
subject to W ,
0
y W Wy
y a z p
W
*, W W 0a z p y
• Otherwise, claim a(z,p) 0 at some point
• If feasible and y* < 0, then
56
Example: Parameterized RF inductor
0 2 4 6 8 10 12
x 109
-5
0
5
10
15
20
25
30
35
40
frequency (Hz)
qual
ity fa
ctor
Wire width = 16.5 µm,separation = 1,5,18,20µm
----- full model___ QCO PMOR
Worst case errorin amplitude = 4%
57
OverviewOverview
• Problem SetupProblem Setup• Connection between circuits and State Space models Connection between circuits and State Space models • Reduction via eigenmode truncation methodReduction via eigenmode truncation method• Reduction via transfer function fittingReduction via transfer function fitting
– point matchingpoint matching– least squareleast square– quasi-convex optimization methodquasi-convex optimization method
• Reduction via Projection FrameworkReduction via Projection Framework– Truncated Balance RealizationsTruncated Balance Realizations– Krylov Subspace Moment MatchingKrylov Subspace Moment Matching
• need for orthogonalization (Arnoldi process)need for orthogonalization (Arnoldi process)• computational complexitycomputational complexity• passivity preservationpassivity preservation
• Reduction of Non-Linear SystemsReduction of Non-Linear Systems
58
Dynamical SystemDynamical System
Transfer FunctionTransfer Function
coefficientscoefficients
coefficientscoefficients
Remember Remember Counting Degrees of Freedom...Counting Degrees of Freedom...
nn
nn
n
n
n
n
sasa
sbsbb
ss
ssk
s
r
s
rsH
1
1110
1
11
1
1
1
)()(
)()(
)()()(
2n
n2+2n
)()(
)()(
txcty
tubtxAdt
dx
T
Degrees of FreedomDegrees of Freedom
Degrees of FreedomDegrees of Freedom
59
Reduced Model Transfer FunctionReduced Model Transfer Function
Apply any invertible change of coordinates to the stateApply any invertible change of coordinates to the state
Many Dynamical Systems have the same transfer function!!
Remember the idea of theRemember the idea of theFully Invertible Change of Coordinates?Fully Invertible Change of Coordinates?
)()(
)()(
txcty
tubtxAdt
dx
T
bAsIcsH T 1)(
)(~)( txUtx
)(~)(
)()(~~
txUcty
tubtxAUdt
xdU
T
)(~)(
)()(~~
111
txUcty
tubUtxAUUdt
xdUU
T
I
)()(~ 1 sHbAsIcsH T
60
Reminder about Eigenanalysis
1Change of variab (le ) ( ) ( ) (s ): Ew t x t w t E x t
1
1 2 1 2
10 0
0 0
0 0
Eigendecomposition: N N
N
E
A E E E E E E
Substituting: ( )
( ) , (0) 0dEw t
AEw t bu t Ewdt
11 1Multiply by : ( )
( )dw t
E AEw t E bu tdt
E
Consider an ODE( )
( ) , (0) 0: dx t
Ax t bu t xdt
E
S S
SS S
S S S1S
1S 2S NS 1S 2S NS
S
61
Reduced models via mode truncationDynamic Linear Case
)(
~
~
~1111
tu
b
b
b
w
w
w
w
w
w
N
q
N
q
N
q
N
q
62
1
1
1
ˆ
ˆq
q
N
q
x
x
u u
x
x
U
Projection Framework:Projection Framework:Non invertible Change of CoordinatesNon invertible Change of Coordinates
Note: q << NNote: q << N
reduced statereduced state
original stateoriginal state
63
• Original System Original System
Projection FrameworkProjection Framework
• Note: now few variables (q<<N) in the state, but still Note: now few variables (q<<N) in the state, but still thousands of equations (N)thousands of equations (N)
)()(
)()(
txcty
tubtxAdt
dx
T
• SubstituteSubstitute
)(ˆ)(ˆ
)()(ˆ)(ˆ
txUcty
tubtxUAdt
txdU
qT
)(ˆ txUx q
64
)(ˆ)(
)()(ˆ)(ˆ
txUcty
tubVtxUAVdt
txdUV
qT
Tqq
Tqq
Tq
Projection Framework (cont.)Projection Framework (cont.)
• If If VqT and and Uq
T are are
chosen biorthogonal chosen biorthogonal
Tq qV U I
)(ˆ)(
)()(ˆ)(ˆ
txUcty
tubtxUAdt
txdU
qT
Reduction of number of equations: test by multiplying by Vq
T
)(ˆˆ)(
)(ˆ)(ˆˆ)(ˆ
txcty
tubtxAdt
txd
T
65
Projection Framework (graphically)Projection Framework (graphically)
nxnnxn
x TqV bu
nxqnxq
qU xdt
dxA
Eqxqqxq
qxnqxn
TqV
dt
xdˆ
nxqnxq
qU
)(ˆ)(ˆˆˆtubtxA
dt
xd
66
• Use Eigenvectors of the system matrix (modal analysis)Use Eigenvectors of the system matrix (modal analysis)
• Use Frequency Domain DataUse Frequency Domain Data– ComputeCompute
– Use the SVD to pick Use the SVD to pick q < kq < k important vectors important vectors
• Use Time Series DataUse Time Series Data– ComputeCompute
– Use the SVD to pick Use the SVD to pick q < kq < k important vectors important vectors1 2( ), ( ), , ( )kx t x t x t
1 2( ), ( ), , ( )kx s x s x s
Approaches for picking V and UApproaches for picking V and U
II.2.b SVD Singular Value Decomposition other names: POD Proper Orthogonal Decompos.or KLD Karhunen-Lo`eve Decompositionor PCA Principal Component Analysisor Poor Man TBR
Point Matching
67
• Use Eigenvectors of the system matrix (modal analysis)Use Eigenvectors of the system matrix (modal analysis)
• Use Frequency Domain DataUse Frequency Domain Data– ComputeCompute
– Use the SVD to pick Use the SVD to pick q < kq < k important vectors important vectors
• Use Time Series DataUse Time Series Data– ComputeCompute
– Use the SVD to pick Use the SVD to pick q < kq < k important vectors important vectors
Approaches for picking V and UApproaches for picking V and U
• Use Singular Vectors of System Grammians Product Use Singular Vectors of System Grammians Product (Truncated Balance Realizations)(Truncated Balance Realizations)
• Use Krylov Subspace Vectors (Moment Matching)Use Krylov Subspace Vectors (Moment Matching)
1 2( ), ( ), , ( )kx t x t x t
1 2( ), ( ), , ( )kx s x s x s
68
Energy of the output Energy of the output y(t)y(t) starting from state starting from state x x with no input:with no input:
Observability GrammianObservability Grammian
2)(
xty
0
xdtCexCe AtTAt xdtCeCex AtTtAT T
0
0W
0
)()( dttyty T
Observability Grammian:
Note: If Note: If x=xx=xii the i-th eigenvector of the i-th eigenvector of WWo o ::
ioT
ixxWxty
i
2)( io,
Hence: eigenvectors of Hence: eigenvectors of WWoo corresponding to corresponding to smallsmall
eigenvalues do NOT produce much energy at the outputeigenvalues do NOT produce much energy at the output(i.e. they are not very observable):(i.e. they are not very observable): Idea: let’s get rid of them!Idea: let’s get rid of them!
CCAWWA To
T 0Note: it is also the solution of
69
Minimum amount input energy required to drive theMinimum amount input energy required to drive thesystem to a specific state system to a specific state x x ::
Controllability GrammianControllability Grammian
xdteBBex tATAtT T
0
1cWInverse of Controllability Grammian:
Note: If Note: If x=xx=xii the i-th eigenvector of the i-th eigenvector of WWcc::
ic
Tix
xWxtui
12)(min
0
)()(min dttutu T
Hence: eigenvectors of Hence: eigenvectors of WWcc corresponding to corresponding to small small
eigenvalues do require a lot of input energy in ordereigenvalues do require a lot of input energy in orderto be reached (i.e. they are not very controllable): to be reached (i.e. they are not very controllable): Idea: let’s get rid of them!Idea: let’s get rid of them!
TTCC BBAWAW
It is also the solution of
ic,
1
70
Naïve Controllability/Observability MORNaïve Controllability/Observability MOR
• Suppose I could compute a basis for the strongly Suppose I could compute a basis for the strongly observable and/or strongly controllable spaces. observable and/or strongly controllable spaces. Projection-based MOR can give a reduced model that Projection-based MOR can give a reduced model that deletes weakly observable and/or weakly controllable deletes weakly observable and/or weakly controllable modes. modes.
• Problem: Problem: – What if the same mode is strongly controllable, but weakly What if the same mode is strongly controllable, but weakly
observable?observable? – Are the eigenvalues of the respective Gramians even Are the eigenvalues of the respective Gramians even
unique? unique?
71
Changing coordinate systemChanging coordinate system
• Consider an invertible change of coordinates: Consider an invertible change of coordinates: • We know that the input/output relationship will be We know that the input/output relationship will be
unchanged.unchanged.• But what about the the Grammians, and their eigenvalues?But what about the the Grammians, and their eigenvalues?
• Grammians and their eigenvalues change! Hence the Grammians and their eigenvalues change! Hence the relative degrees of observability and controllability are relative degrees of observability and controllability are properties of the coordinate systemproperties of the coordinate system
• A bad choice of coordinates will lead to bad reduced A bad choice of coordinates will lead to bad reduced models if we look at controllability and observability models if we look at controllability and observability separately. separately.
• What coordinate system should we use then?What coordinate system should we use then?
)(~)( txUtx
UWUW oT0
~ Tcc UWUW 1~
72
Fortunately the eigenvalues of the Fortunately the eigenvalues of the product product (Hankel singular (Hankel singular values) do not change when changing coordinates:values) do not change when changing coordinates:
BalancingBalancing
120
SSWWc
)(~)( txUtx
TcUWU 1 UWWU c 0
1 1121 )()( SUSU
Diagonal matrix with eigenvalues of the product
The eigenvectors change
And since And since WWc c and and WWoo are symmetric a are symmetric a
change of coordinate matrix change of coordinate matrix UU can be can be found that diagonalizes both:found that diagonalizes both:
2 In Balanced coordinates the Grammians In Balanced coordinates the Grammians are equal and diagonalare equal and diagonal
But not the eigenvalues
UWU T0
73
Selection of vectors for the columns of Selection of vectors for the columns of the reduced order projection matrix.the reduced order projection matrix.
20
1 UWUUWU TTc
In balanced coordinates it is easy to select the best In balanced coordinates it is easy to select the best vectors for the reduced model: we want the subspace of vectors for the reduced model: we want the subspace of vectors that are at the same time most controllable and vectors that are at the same time most controllable and observable: observable:
In other words the ones corresponding to the largestIn other words the ones corresponding to the largesteigenvalues of the controllability and observability eigenvalues of the controllability and observability Grammians product.Grammians product.
simply pick the eigenvectors simply pick the eigenvectors corresponding to the largest corresponding to the largest entries on the diagonal entries on the diagonal (Hankel singular values).(Hankel singular values).
74
Truncated Balance Realization SummaryTruncated Balance Realization Summary
• The good news:The good news:– we even have we even have bounds for the errorbounds for the error– Can do even a bit better with the optimal Hankel Reduction
• The bad news:The bad news:– it is it is expensiveexpensive::
• need to compute the Gramians (solve Lyapunov equation)need to compute the Gramians (solve Lyapunov equation)• need to compute eigenvalues of the product: need to compute eigenvalues of the product: O(NO(N33))
• The bottom line:The bottom line:– If the size of your system allows you O(NIf the size of your system allows you O(N33) computation, ) computation,
Truncated Balance Realization is a much better choice than the Truncated Balance Realization is a much better choice than the any other reduction methodany other reduction method..
– But if you cannot afford O(NBut if you cannot afford O(N33) computation (e.g. dense matrix ) computation (e.g. dense matrix with N > 5000) then PRIMA or PVL or Quasi-Convex-Optimization with N > 5000) then PRIMA or PVL or Quasi-Convex-Optimization are better choicesare better choices
NNqqq jHjH ,1,1 ...2)()(
CCAWWA To
T 0
75
• Use Eigenvectors of the system matrix (modal analysis)Use Eigenvectors of the system matrix (modal analysis)
• Use Frequency Domain DataUse Frequency Domain Data– ComputeCompute
– Use the SVD to pick Use the SVD to pick q < kq < k important vectors important vectors
• Use Time Series DataUse Time Series Data– ComputeCompute
– Use the SVD to pick Use the SVD to pick q < kq < k important vectors important vectors
Approaches for picking V and UApproaches for picking V and U
• Use Singular Vectors of System Grammians Product Use Singular Vectors of System Grammians Product (Truncated Balance Realizations)(Truncated Balance Realizations)
• Use Krylov Subspace Vectors (Moment Matching)Use Krylov Subspace Vectors (Moment Matching)
1 2( ), ( ), , ( )kx t x t x t
1 2( ), ( ), , ( )kx s x s x s
76
A canonical form for model order reductionA canonical form for model order reduction
Assuming Assuming AA is non-singular is non-singular we can cast the dynamical we can cast the dynamical linear system into a linear system into a canonical form for moment canonical form for moment matching model order matching model order reductionreduction
Note: this step is not Note: this step is not necessary, it just makes the necessary, it just makes the notation simple for notation simple for educational purposeseducational purposes
xcy
ubAxsxT
xcy
ubxsExT
bAb
AE1
1
77
H s
H s
Taylor series expansion:
2 xx b Ab A b
UqUq
The Moment Matching idea [Grimme PhD97]The Moment Matching idea [Grimme PhD97]
2span , , ,x b Eb E b
change basis: use only first q vectors of the Taylor series expansion matching first q derivatives around expansion point
0
k k
k
x s E b u
sEx x bu 1x I sE bu
2b Eb E b
78
Krylov Subspaces - DefinitionKrylov Subspaces - Definition
2 1, , , , kk E b span b Eb E b E b
The order The order kk Krylov subspace generated Krylov subspace generated from matrix E and vector b is defined asfrom matrix E and vector b is defined as
79
Moment matching around non-zero frequenciesMoment matching around non-zero frequencies
In stead of expanding around only s=0 we can expand around another points 1 20 Js s s s s s s
xcy
buAxsxT
xcy
buAxxssT
h
)~(
xcy
ubxxEsT
hh
~
bIsAb
IsAE
hh
hh
1
1
For each expansion point the problem can then be put again in the canonical form
hsss ~
80
IfIf
andand
ThenThen
Projection Framework: Projection Framework: Moment Matching Theorem (E. Grimme 97)Moment Matching Theorem (E. Grimme 97)
hh
J
hkq bEU b
h,)(Range
1
hTh
J
hkq cEV c
h,)(Range
1
Total of 2q moment of the transfer function will match
,...,1
1,...,0for
ˆ
Jh
kkl
s
H
s
H bh
bh
s
l
l
s
l
l
hh
81
Combine point and moment matching: Combine point and moment matching: multipoint moment matchingmultipoint moment matching
H s
H s
• Multipole expansion points give larger bandMultipole expansion points give larger band• Moment (derivates) matching gives more Moment (derivates) matching gives more accurate behavior in between expansion pointsaccurate behavior in between expansion points
82
Moment Matching Moment Matching Parameterized Model Order ReductionParameterized Model Order Reduction
ubxEsWdEWE 1,1,10,2,02
0,0,0 ......
21 2 1 1 2 2 1, , , , , , ,px span b E b E b E b E b E E E E b
ˆq xx U
Uq
It is a p-variables Taylor series expansion
buEsEsEsxm
mpp 2211
Once again change basis:Once again change basis: use first few vectors of the use first few vectors of the
Taylor expansion,Taylor expansion, matching first few matching first few
derivatives with respect to derivatives with respect to each parametereach parameter
complexity O(q N log N)complexity O(q N log N)
83
ubxEsWdEWE ˆˆˆ...ˆ...ˆ1,1,10,2,0
20,0,0
ubxEsWdEWE
1,1,10,2,02
0,0,0 ......
TqU
nxnnxn
qxqqxq
nxqnxq
qxnqxn
Congruence transformations on each of the matrices
kjiE ,,
kjiE ,,ˆ
qU
Moment Matching Moment Matching Parameterized Model Order ReductionParameterized Model Order Reduction
84 84
Example: Parameterized RF inductor[Daniel, White, BMAS03]
Wire width = 5umseparation = 1, 2, 3, 4, 5um
frequency [ x10GHz ]
Worst case errorin amplitude = 4%__ full system (field solver)--- moment matching
Q
85
OverviewOverview
• Problem SetupProblem Setup• Connection between circuits and State Space models Connection between circuits and State Space models • Reduction via eigenmode truncation methodReduction via eigenmode truncation method• Reduction via transfer function fittingReduction via transfer function fitting
– point matchingpoint matching– least squareleast square– quasi-convex optimization methodquasi-convex optimization method
• Reduction via Projection FrameworkReduction via Projection Framework– Truncated Balance RealizationsTruncated Balance Realizations– Krylov Subspace Moment MatchingKrylov Subspace Moment Matching
• need for orthogonalization (Arnoldi process)need for orthogonalization (Arnoldi process)• computational complexitycomputational complexity• passivity preservationpassivity preservation
• Reduction of Non-Linear SystemsReduction of Non-Linear Systems
86
b
Eb
2E b3E b
Vectors will quickly line up with dominant eigenspace!Vectors will quickly line up with dominant eigenspace!
Need for Orthonormalization of UNeed for Orthonormalization of U
Vectors {b,Eb,...,Ek-1b} cannot be computed directly
87
1 1
1
1,
1i i
i
i i
u uu
Normalize new vectorNormalize new vector
1 /u b b
For i = 1 to k
1i iu Eu
Generates new Krylov Generates new Krylov subspace vectorsubspace vector
1 1 1T
i i i j j
ji
u u u u u
Orthogonalize new vectorOrthogonalize new vector
For j = 1 to i
Orthonormalization of U: The Arnoldi AlgorithmOrthonormalization of U: The Arnoldi Algorithm
Normalize first vectorO(n)
O(?????????)O(?????????)
O(kO(k22n)n)
O(n)O(n)
Computational ComplexityComputational Complexity
88
Most of the computation cost is spent in calculating:
Set up and solve a linear system using GCR
If we have a good preconditioners and a fast matrix vector product each new vector is calculated in O(n)
The total complexity for calculating the projection
matrix Uq is O(qn)
Generating vectors for the Krylov subspaceGenerating vectors for the Krylov subspace
ih uIsA1)(
iih uuIsA
1)(
ihi uEu
1
89
What about computing the reduced matrix ?What about computing the reduced matrix ?
iq
i
qi uuuE
,
,1
1 ...
qTq EUUE ˆ
qq uuuuE
...... 11 qq UEU
qTq EUU
E
Orthonormalization of the i-th column of Uq
Orthonormalization of all columns of Uq
So we don’t need to computethe reduced matrix. We have it already:
90
Model Order ReductionComputational Complexity (time and memory)
MOR technique Computational Complexity
TBR (‘81) Hankel (‘84)(have error bounds)
Bottleneck: SVDO(n3)
e.g. 10months, 80GB, for N=100,000
Moment Matching + pFFT matrix-vector product
O(q n log n)e.g. 8hours, 0.3GB for N=100,000 q=10
Moment matching (’97)(no error bounds)
Bottleneck: matrix-vector productO(q n2)
e.g. 7days, 80GB, for N=100,000 q=10
QuasiConvex Optimization + pFFT field solver
O(m n log n)Same as above, but have error bound!
)()(),(),( tuBtxwdAdt
dxwdE
Two Complementary Approaches • Moment Matching Approaches
– Accurate over a narrow band. • Matching function values and
derivatives.
– Cheap: O(qn).– Use it as a FIRST STAGE
REDUCTION
• Truncated Balanced Realization and Hankel Reduction
– Optimal (best accuracy for given size q, and apriori error bound.
– Expensive: O(n3)– USE IT AS A SECOND STAGE
REDUCTION
92
OverviewOverview
• Problem SetupProblem Setup• Connection between circuits and State Space models Connection between circuits and State Space models • Reduction via eigenmode truncation methodReduction via eigenmode truncation method• Reduction via transfer function fittingReduction via transfer function fitting
– point matchingpoint matching– least squareleast square– quasi-convex optimization methodquasi-convex optimization method
• Reduction via Projection FrameworkReduction via Projection Framework– Truncated Balance RealizationsTruncated Balance Realizations– Krylov Subspace Moment MatchingKrylov Subspace Moment Matching
• need for orthogonalization (Arnoldi process)need for orthogonalization (Arnoldi process)• computational complexitycomputational complexity• passivity preservationpassivity preservation
• Reduction of Non-Linear SystemsReduction of Non-Linear Systems
93
Need to preserve passivityNeed to preserve passivityfor models of passive interconnectfor models of passive interconnect
PCB, package, IC PCB, package, IC interconnects interconnects
Analog or digital IP Analog or digital IP blocksblocks
Z(f)Z(f)
Note: passive! Note: passive! Designers will connect models use Designers will connect models use them in ODE time domain simulators.them in ODE time domain simulators.If the models are not passive they If the models are not passive they can generate energy and thecan generate energy and thesimulation may explode!! simulation may explode!!
Picture byPicture byM. ChouM. Chou
QD
C
-
+
+-
Picture byPicture byJ. PhillipsJ. Phillips
94
Interconnecting Passive Systems Interconnecting Passive Systems
QDC
-++-
QDC
-++-
QDC
-++-
QDC
-++-
The interconnection of stable models is not necessarily stable
BUT the interconnection of passive models is a passive model:
95
Sufficient conditions for passivitySufficient conditions for passivity
Sufficient conditions for passivity:
Cxy
BuAxsEx
xAxx
xExx
BC
T
T
T
allfor ,0)3
allfor ,0)2
)1
Note that these are NOT necessary conditions (common misconception)
i.e. E is positive and A is negative semidefinite
Example IC wire discretized into RC line
x
1 1
( )NxN Nx
scalarinp
T
Nxscalarouu tt put
y tdx t
A x t b u t c x tdt
2 1 0 0
1 2
0 0
2 1
0 0 1 1
A
A
1
0
0
b
1
0
0
c
Matrix is negative semidefinite.
97
Congruence Transformation Preserves Definiteness of Congruence Transformation Preserves Definiteness of E and A (hence passivity and stability) [PRIMA97]E and A (hence passivity and stability) [PRIMA97]
If we use Tq
Tq UV
• Then we loose half of the degrees of freedom i.e. we match only q moments instead of 2q• But if the original matrix E is negative semidefinite so is the reduced hence the system is passive and stable
nxnnxn
nxqnxq
qU xs x TqU bu
nxqnxq
qUx xEqxnqxn
TqV
nxnnxn
s Aqxnqxn
TqV
98
Step 2. Model Order Reduction
Fitting Based,System ID
Least SquareCoelho99
PRIMA97(only if A>0)Moment,
MatchingAWE Pillage90,PVL Feldman94
OptimalTBR 81
Hankel 84
Proof Daniel TCAD04
BasicTechnique
Parameterized PassivityPreservation
Guaranteed Passive Quasi-Convex Optimiz. Daniel TCAD08
Daniel ICCAD08any A
best paper
Daniel DAC02 best paper,
Daniel TCAD04
POD, KVL,PCA, SVD
Wilcox Peraire91,PMTBR04 PMTBR04
Weile99,Gunupudi00,
Heydari01
Coelho01if poles given
Daniel ICCAD08any A
best paper
99
dt
dEH
dt
dHE
dt
dEH
dt
dHE
dt
dEH
dt
dHE
dt
dEH
dt
dHE
The two steps modeling process
Step 1. Field SolversStep 1. Field Solvers
Step 2. Model Order ReductionStep 2. Model Order Reduction
)()( tuBtxAdt
dxE
)(ˆ)(ˆˆˆˆ tuBtxAdt
xdE
(10(1066 x 10 x 1066))
nanophotonics
carbon nanotubes
RF inductorswinterconnect
Very difficult to control stability/passivity at this step
Potentiallyunstable and unstructured
Need to enforce stability/passivity at this step
(10(1066 x 10 x 1066))
100
Optimal Stabilizing Projection: Stability Constraints
AVzUzEVU TT E A
Stable Reduced Model
AxxE
V
U
AE,
0U
sUU 0min
Stability Constraints
Field-Solver Model
Any Projection-BasedReduction Technique
Q
f
Reduced ModelsFull model
xxV matches moments
Space of stabilizing
U
Space of U
0U
101
Optimal Stabilizing Projection: Stability Constraints
AVzUzEVU TT E A
Stable Reduced Model
AxxE
V
U
AE,
0U
sUU 0min
Stability Constraints
Field-Solver Model
Any Projection-BasedReduction Technique
Space of stabilizing
U0U
0ˆˆ EVUPUAVAVUPUEV TTTTTT Difficult to solve: Quadratic in U
Stability Constraints0ˆ EVUPUEV TTT
102
Optimal Stabilizing Projection: Stability Constraints
AVzUzEVU TT E A
Stable Reduced Model
AxxE
V
U
AE,
0U
sUU 0min
Stability Constraints
Field-Solver Model
Any Projection-BasedReduction Technique
Space of stabilizing
U0U
Stability Constraints
0UAVAVU TTT 0EVU T
Constraints now linear in U
Force reduced model to be definite
103
109
1010
-50
0
50
100
150
200
250
300
350
Large system should be stable and passive
Extracted with EMQS field solver
0 1 2 3 4 5 6 7
x 10-10
-0.02
0
0.02
0.04
0.06
0.08
0.1
Example: RF Inductor
-2.5 -2 -1.5 -1 -0.5 0 0.5
x 10-9
-1.5
-1
-0.5
0
0.5
1
1.5x 10
-12
-10 -5 0 5
x 10-15
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1x 10
-12
xx
N=961
q=10
Re
Im
Re
Im
)(tyQ
f
Field Solver ModelPRIMA
Proposed Method
1.0 3.0 5.0 time (ns)
910 1010
104
109
1010
101
102
103
Example: Power Grid
N = 1566
q = 10
Large system should be stable and passive
Extracted with EMQS field solver
Re
Imxx
)(ty
f)(Re jH
)(Im jH
Field Solver Model
PRIMA
Proposed Method
time (ns)1 3 5 7
910 1010
105
OverviewOverview
• Problem SetupProblem Setup• Connection between circuits and State Space models Connection between circuits and State Space models • Reduction via eigenmode truncation methodReduction via eigenmode truncation method• Reduction via transfer function fittingReduction via transfer function fitting
– point matchingpoint matching– least squareleast square– quasi-convex optimization methodquasi-convex optimization method
• Reduction via Projection FrameworkReduction via Projection Framework– Truncated Balance RealizationsTruncated Balance Realizations– Krylov Subspace Moment MatchingKrylov Subspace Moment Matching
• need for orthogonalization (Arnoldi process)need for orthogonalization (Arnoldi process)• computational complexitycomputational complexity• passivity preservationpassivity preservation
• Reduction of Non-Linear SystemsReduction of Non-Linear Systems
106
Parameterized Model Order ReductionParameterized Model Order Reductionof Non-Linear Systemsof Non-Linear Systems
Non-Linear analog components
e.g. MEMs, VCO, LNADistributed Amplifiers
)()),(( tuBptxFdt
dx
)(ˆ)),(ˆ(ˆˆtuBptxF
dt
xd
Parameterized NonLinear ReductionParameterized NonLinear Reduction
PDE Field Solvers or Circuit SimulatorsPDE Field Solvers or Circuit Simulators
107
Previous work on Non-ParameterizedPrevious work on Non-ParameterizedMOR for Non-Linear systemsMOR for Non-Linear systems
Representation of Representation of F(x)F(x) using using a polynomiala polynomial (e.g. Taylor’s (e.g. Taylor’s expansions, Volterra Series) [Phillips00]expansions, Volterra Series) [Phillips00]
Representation of Representation of F(x)F(x) with with several polynomialsseveral polynomials (PWP (PWP PieceWise Polynomial) [Dong03]PieceWise Polynomial) [Dong03]
Representation of Representation of F(x)F(x) using using several linearizationsseveral linearizations (Trajectory Piece-Wise Linear TPWL) [Rewienski01](Trajectory Piece-Wise Linear TPWL) [Rewienski01]
...)())(()(
))(()(
000
000
xxxtxxW
xxxJxFxF
...
))(()()(
))(()()(
222222
111111
xxxJxFxw
xxxJxFxwxF
108
x3
x11
x7
x10
x6
x8 x9
x5x4
x0
x1
x2
x1 x20J 3J2J1J
4J 5J 6J7J
8J 9J 10J 11J
#linearization#linearizationss =#samples=#samplesnn
n = 10n = 1044
#samples = 100#samples = 100
Background – TPWL [Rewienski01]Construction of Linearized Systems
109
Background – TPWL [Rewienski01] Background – TPWL [Rewienski01] Construction of Linearized SystemsConstruction of Linearized Systems
x1
x2
t
y(t)
Use training trajectories to pick linearization points
Error
Linearization at current state xi
State SpaceTime Domain Simulation
110
Background – TPWL [Rewienski01] Background – TPWL [Rewienski01] Reduction of the Linearized SystemsReduction of the Linearized Systems
dt
dx tbu
Model from linearization 1
A1A1 K1K1x1w 2w kw
Model from linearization 2
A2A2 K2K2x
Model from linearization k
AkAk KkKkx
dt
xdˆ1A 2A kA
1K 2KkK
tub x x x1w2w kw
=UTU
iA iA
Moment matching projection into a single subspace
111
x0
Background – TPWL [Rewienski01]Background – TPWL [Rewienski01]Possible Behaviors and LimitationsPossible Behaviors and Limitations
x1
x2
x1
x2x3
A
C – poorlyapproximated
B
wellapproximated
112
Non-Linear Parameterized Reduction[Bond, Daniel TCAD07]
x1
x2
time
y(t)
Use training trajectories to pick linearization points
s1
s2
1s
2sTrain again at different points in parameter space
Parameter Space
Repeat at different points in parameter space to populate state space
State Space
113
Non-Linear Parameterized Reduction[Bond, Daniel TCAD07]
1s
2s
4s
5s 3s
Populate relevant regions of state-space with linear models from training at different inputs or different points in parameter-space
Parameter Space
s2
s1
x1
x2
State Space
114
Parameterized Moment Matching Projection into a single subspace
For qth order model with p parameters and k linear models, total number of vectors for U is O(kpm)
Keeping all vectors would results in a huge “reduced” model
Hence we perform an SVD on U and keep only the q most important vectors
SVD
NN
O(kpm) q
U U
115
Model from a Discretized Non-Linear Transmission Line
1exp
Tdd v
VIVi
Threshold voltageOff state current
POSSIBLE PARAMETERS:
Resistance per unit length r
Capacitance per unit length C
116
Model from a Discretized Non-Linear Transmission Line
u(t) = 1 + sin(wt) + sin(5wt)
Our model
117
Micromachined Switch Example
t
pzppzK
t
zhwdypp
z
wv
x
zhwS
x
zwhEI
w
a
1261
2
3
2
2
0
02
20
2
2
04
43
0
Governing Equations
Picture by Michał Rewienski
118
State-Space System
3331
1131
323
2
0
041
43
021
2
0
0
30
31
31
222
31
21
6112
1
3
2
33
3
2
3
xxxxxx
xx
dt
dx
vhx
xwhEI
x
xhwSdypx
hw
x
x
x
dt
dx
x
x
dt
dx
w
a
Beam width
Material Properties
Beam height
Possible Parameters
Micromachined Switch ExampleDiscretization by Finite Difference PDE solver
119
Micromachined Switch Example
Our model
120
SummarySummary
• Problem SetupProblem Setup• Connection between circuits and State Space models Connection between circuits and State Space models • Reduction via eigenmode truncation methodReduction via eigenmode truncation method• Reduction via transfer function fittingReduction via transfer function fitting
– point matchingpoint matching– least squareleast square– quasi-convex optimization methodquasi-convex optimization method
• Reduction via Projection FrameworkReduction via Projection Framework– Truncated Balance RealizationsTruncated Balance Realizations– Krylov Subspace Moment MatchingKrylov Subspace Moment Matching
• need for orthogonalization (Arnoldi process)need for orthogonalization (Arnoldi process)• computational complexitycomputational complexity• passivity preservationpassivity preservation
• Reduction of Non-Linear SystemsReduction of Non-Linear Systems