Model order reduction NXP PowerPoint template (Title) via … · 2007-11-22 · NXP PowerPoint...

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Model order reductionvia dominant poles

Joost Rommes [[email protected]]NXP Semiconductors/Corp. I&T/DTF/MathematicsJoint work with Nelson Martins (CEPEL), Gerard Sleijpen (UU)

Symposium on recent advances in MORNovember 23, 2007

Subject/Department, Author, MMMM dd, yyyy

CONFIDENTIAL 3

Introduction

Eigenvalue problems and applications

Dynamical systems and transfer functions

Dominant poles

Dominant Pole Algorithm

Applications

Conclusions

2/38NXP Semiconductors Corp. I&T/DTF, Joost Rommes, November 23, 2007


CONFIDENTIAL 3

Introduction

I Large-scale dynamical systems arise inI electrical circuit simulationI structural engineeringI power system engineeringI . . .

I Transfer function and properties are used forI simulationI behavioral modelingI stability analysisI controller design

I Relatively few transfer function poles of practical importanceI Three key questions:

I Which poles are important (dominant)?I How to compute these poles efficiently?I How to use these poles in model order reduction?



CONFIDENTIAL 3

Motivating example I: Pole-zero analysisFrequency response of regulator IC (1000 unknowns). Which polecauses peak around 6MHz?

1.010.0

100.01.0k

10.0k100.0k

1.0M10.0M

100.0M1.0G

10.0G

(LOG)

-60.0

-50.0

-40.0

-30.0

-20.0

-10.0

0.0

10.0

20.0(LIN)

Oct 17, 200711:35:07

Bode Plot

Analysis: AC

User: nlv18077 Simulation date: 17-10-2007, 10:21:28

File: /home/nlv18077/test/pstar/stability_ne.sdif

F

DB(VN(VREG))

Note dB(x)= 20 ·10 log(x), e.g. -60 dB = 10−3



CONFIDENTIAL 3

Motivating example II: Model order reduction

Transfer function of power system (66 unknowns). How tocompute reduced order model?

0 5 10 15 20 25 30 35−120

−110

−100

−90

−80

−70

−60

−50

−40

−30

Frequency (rad/s)

Gai

n (d

B)

ExactReduced (series) (k=11)Error (orig − reduced)



CONFIDENTIAL 3

The generalized eigenvalue problem

Given A,E ∈ Rn×n, find (λ, x, y) that satisfy

Ax = λEx, x 6= 0

y∗A = λy∗E , y 6= 0

An eigentriplet (λ, x, y) consists of

λ ∈ C eigenvalue

x ∈ Cn right eigenvector

y ∈ Cn left eigenvector

I (A,E ) has n eigenvalues (real / complex conjugated pairs)

I Corresponding eigenspaces need not be n-dimensional

I Bi-orthogonality: λi 6= λj ⇒ y∗j Exi = 0



CONFIDENTIAL 3

Eigenvalue decompositions

Complete eigenvalue decomposition (Λ,X ,Y ):

AX = EXΛ, Y ∗A = ΛY ∗E with Y ∗EX = I ,Y ∗AX = Λ

Λ = diag(λ1, λ2, . . . , λn) ∈ Cn×n

X = [x1, x2, . . . , xn] ∈ Cn×n

Y = [y1, y2, . . . , yn] ∈ Cn×n

In practice only interest in k � n eigentriplets: partial ED

AXk = EXkΛk , Y ∗k A = ΛkY ∗

k E with Y ∗k EXk = I ,Y ∗

k AXk = Λk

Λk = diag(λ1, λ2, . . . , λk) ∈ Ck×k

Xk = [x1, x2, . . . , xk ] ∈ Cn×k

Yk = [y1, y2, . . . , yk ] ∈ Cn×k



CONFIDENTIAL 3

Eigenvalue computations

Methods for complete eigendompositions:

I QR method for AX = XΛ

I QZ method for AX = EXΛ

I Complexity O(n3), practical use up to n ≈ 2000

Methods for partial eigendecompositions:

I Krylov methods (Lanczos, Arnoldi)

I Newton based methods (Jacobi-Davidson [Sleijpen, Van derVorst (1995)])

I No dense matrix computations needed

I Careful selection strategies needed



CONFIDENTIAL 3

First-order dynamical systems

First-order SISO dynamical system{E x(t) = Ax(t) + bu(t)y(t) = c∗x(t) + du(t)

where

u(t), y(t), d ∈ R, input, output, direct i/o

x(t),b, c ∈ Rn, state, input-to-, -to-output

E ∈ Rn×n, system (descriptor) matrix

A ∈ Rn×n, system matrix



CONFIDENTIAL 3

Transfer functionFirst-order SISO dynamical system (d = 0):{

E x(t) = Ax(t) + bu(t)y(t) = c∗x(t)

with transfer function

H(s) = c∗(sE − A)−1b

Poles are λ ∈ C for which

lims→λ

H(s) =∞,

or, equivalently,det(λE − A) = 0,

i.e. the eigenvalues of (A,E )



CONFIDENTIAL 3

Transfer function H(s) = c∗(sE − A)−1b

Can be expressed as

H(s) =n∑

i=1

Ri

s − λi,

where residues Ri are

Ri = (c∗xi )(y∗i b),

and (λi , xi , yi ) are eigentriplets (i = 1, . . . , n)

Axi = λiExi , right eigenpairs

y∗i A = λiy∗i E , left eigenpairs

y∗i Exi = 1, normalization

y∗j Exi = 0 (i 6= j), E -orthogonality



CONFIDENTIAL 3

Dominant poles cause peaks in Bode-plot

H(s) = c∗(sE − A)−1b =n∑

i=1

Ri

s − λiwith Ri = (c∗xi )(y

∗i b)

Bode-plot is graph of (ω, |H(iω)|)I frequency ω ∈ RI magnitude |H(iω)| usually in dB (note dB(x)= 20 ·10 log(x))

Consider pole λ = α + βi with residue R, then

limω→β

H(iω) = limω→β

R

iω − (α + βi)+

n−1∑j=1

Rj

iω − λj

=R

α+ Hn−1(iβ)

Hence pole λ with large∣∣∣ RRe(λ)

∣∣∣ causes peak



CONFIDENTIAL 3

Dominant poles of transfer functions

H(s) =n∑

i=1

Ri

s − λiwith Ri = (c∗xi )(y

∗i b)

I Pole λi dominant if |Ri ||Re(λi )|

large

I Dominant poles cause peaks in Bode-plot (ω, |H(iω)|)I Effective transfer function behavior:

Hk(s) =k∑

i=1

Ri

s − λi,

where k � n and (λi ,Ri ) ordered by decreasing dominance

I Early work modal approximation [Davison, Marschall (1966)]



CONFIDENTIAL 3

Dominant Pole Algorithm (DPA) and extensions

DPA [Martins (1996)] computes dominant poles of

H(s) = c∗(sE − A)−1b

1. Newton scheme

2. Nice convergence behavior [R., Sleijpen (2006)]

3. Subspace acceleration, selection, deflation: SADPA [R.,Martins (2006)]

4. Efficient deflation, extensions, applications [R., Martins,Pellanda (2006)]



CONFIDENTIAL 3

Dominant Pole Algorithm [Martins (1996)]

H(s) = c∗(sE − A)−1b

I Pole λ: lims→λ H(s) =∞, or lims→λ1

H(s) = 0

Apply Newton’s Method to 1/H(s):

sk+1 = sk +1

H(sk)

H2(sk)

H ′(sk)

= sk −c∗(skE − A)−1b

c∗(skE − A)−1E (skE − A)−1b

Note dHds = −c∗(skE − A)−1E (skE − A)−1b



CONFIDENTIAL 3

Dominant Pole Algorithm

1: Initial pole estimate s1, tolerance ε� 12: for k = 1, 2, . . . do3: Solve vk ∈ Cn from (skE − A)vk = b4: Solve wk ∈ Cn from (skE − A)∗wk = c5: Compute the new pole estimate

sk+1 = sk −c∗vk

w∗kEvk

6: The pole λ = sk+1 with x = vk/‖vk‖2 and y = wk/‖wk‖has converged if

‖(sk+1E − A)x‖2 < ε

7: end for



CONFIDENTIAL 3

Twosided Rayleigh quotient iteration

Note that with v ≡ vk and w ≡ wk

sk+1 = sk −c∗(skE − A)−1b

w∗Ev

= skw∗Ev

w∗Ev− c∗(skE − A)−1(skE − A)(skE − A)−1b

w∗Ev

=w∗Av

w∗Ev

Step DPA Twosided RQI

3 solve (skE − A)vk = b solve (skE − A)vk = Evk−1

4 solve (skE − A)∗wk = c solve (skE − A)∗wk = E ∗wk−1

Original work on twosided RQI [Ostrowski (1958), Parlett (1974)]



CONFIDENTIAL 3

Convergence behavior: DPA vs. RQI

Figure: λ = −0.47 + 8.9i : DPA: red + yellow, RQI: red + light blue.



CONFIDENTIAL 3

Convergence behavior: DPA vs. RQI

Typically, with initial pole guess s0,I DPA converges to dominant pole closest to s0

I with ∠(c, x) and ∠(b, y) smallI i.e., large |R| with R = (c∗x)(y∗b)

I Quadratic rate of convergence

I See also [R., Sleijpen (2006)]

while

I RQI converges to pole closest to s0I Originally intended for refinement of eigenpairs

I Cubic rate of convergence

I See also [Ostrowski (1958), Parlett (1974)]



CONFIDENTIAL 3

Extensions of DPA

I DPA is a single pole algorithm

I May have very local behavior

I In practice more than one pole neededI Subspace Accelerated DPA [R., Martins (2006)]

I Subspace AccelerationI Several pole selection strategiesI Deflation techniques



CONFIDENTIAL 3

Subspace acceleration and selectionI Keep approximations vk and wk in search spaces V and W

I Petrov-Galerkin leads to projected eigenproblem

Ax = θE x,

y∗E = θy∗A

where E = W ∗EV ∈ Ck×k and A = W ∗AV ∈ Ck×k

I Gives k approximations (θi , xi = V xi , yi = W yi ) in iter k

I Select approximation with largest residue as next shift:

sk+1 = argmaxi

∣∣∣∣(c∗xi )(y∗i b)

Re(θi )

∣∣∣∣I Similarities with twosided Jacobi-Davidson ([Hochstenbach

(2003), Stathopoulos (2002)])



CONFIDENTIAL 3

Deflation for H(s) = c∗(sE − A)−1b

I Triplet (λ, x, y): Ax = λEx and y∗A = λy∗E

I New search spaces: V ⊥ E ∗y and W ⊥ Ex

I Usual deflation (every iteration):

vk ← (I − xy∗E )vk

wk ← (I − yx∗E ∗)wk

I More efficient: deflate only once

bd ← (I − Exy∗)b ⇒ vk = (skE − A)−1bd ⊥ E ∗y

cd ← (I − E ∗yx∗)c ⇒ wk = (skE − A)−∗cd ⊥ Ex

I Note that y∗bd = c∗dx = 0



CONFIDENTIAL 3

Deflation of dominant poles removes peaks

0 2 4 6 8 10 12 14 16 18 20−90

−80

−70

−60

−50

−40

−30

Frequency (rad/sec)

Gain

(dB

)

Bodeplot

Exact

Modal Equiv.

Figure: Exact transfer function (solid) with removal of dominant poles:−0.467± 8.96i (square), −0.297± 6.96i (asterisk), −0.0649 (diamond),and −0.249± 3.69i (circle).



CONFIDENTIAL 3

Applications of DPA

I Pole-zero analysis in circuit simulationI Applications in model order reduction

I Modal approximationI Dominant poles may lie anywhere in complex planeI Combinations with rational Krylov methods



CONFIDENTIAL 3

Pole-zero analysis

I Large nonlinear Regulator IC (n = 1000)

I Designed to deliver constant output voltage

I Turns unstable for certain loads

I Interested in positive poles and dominant poles

I Linearization around DC solution

Results:

Method Time (s) Poles

QR 450 allDPA 41 994 · 103 ± i5.6 · 106

−8.0 · 106 ± i4 · 106

−337 · 103



CONFIDENTIAL 3

Pole-zero analysisFrequency response of circuit (1000 unknowns). Pole994 · 103 ± i5.6 · 106 causes peak around 6MHz.

1.010.0

100.01.0k

10.0k100.0k

1.0M10.0M

100.0M1.0G

10.0G

(LOG)

-60.0

-50.0

-40.0

-30.0

-20.0

-10.0

0.0

10.0

20.0 (LIN)

Oct 17, 200716:38:18

names: A_* --> 3 stability_ne.qr.sdif (AC)Bode Plot + B_* --> 1 stability_ne_dpa_3.cgap (AC)Bode Plot

Analysis: AC

User: nlv18077 Simulation date: 17-10-2007, 10:21:28

File: /home/nlv18077/test/pstar/stability_ne.qr.sdif

F

- y1-axis -

A_DB(VN(VREG))

B_DB



CONFIDENTIAL 3

Model order reduction

Given large-scale dynamical system{E x(t) = Ax(t) + bu(t)y(t) = c∗x(t) + du(t)

where x(t),b, c ∈ Rn and E ,A ∈ Rn×n, find{Ek xk(t) = Akxk(t) + bku(t)yk(t) = c∗kxk(t) + du(t)

where xk(t),bk , ck ∈ Rk , Ek ,Ak ∈ Rk×k and

I k � n

I approximation error ‖y − yk‖ small



CONFIDENTIAL 3

Model order reduction

Model order reduction via projection:

1. Construct matrices V ,W ∈ Rn×k whose columns form a basisfor the dominant dynamics

2. Project using V and W :

Ek = W ∗EV , Ak = W ∗AV , bk = W ∗b, ck = V ∗c

Various projection based methods:

I Modal truncation: columns V , W are eigenvectors of (A,E )

I Moment matching: columns V , W are bases for Krylov spaces

I Balanced truncation: V , W part of balancing transformation



CONFIDENTIAL 3

Modal approximation and moment matching

0 2 4 6 8 10 12 14 16 18 20−90

−80

−70

−60

−50

−40

−30

−20

Frequency (rad/s)

Gai

n (d

B)

SADPA (k=12)Dual Arnoldi (k=30)Orig (n=66)

Figure: Frequency response of complete system (n = 66), modalapproximation (k = 12), and dual Arnoldi model (k = 30).



CONFIDENTIAL 3

Dominant poles: location in complex plane

−16 −14 −12 −10 −8 −6 −4 −2 0 2−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

real

imag

exact polesSADPA (k=12)Dual Arnoldi (k=30)

region of interest

Figure: Pole spectrum of complete system (n = 66), modalapproximation (k = 12), and dual Arnoldi model (k = 30).



CONFIDENTIAL 3

Dominant poles: location in complex plane (zoom)Dominant poles not necessarily at outside of spectrum

−1 −0.9 −0.8 −0.7 −0.6 −0.5 −0.4 −0.3 −0.2 −0.1 0 0.1

−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

real

imag

exact polesSADPA (k=12)Dual Arnoldi (k=30)

Figure: Pole spectrum (zoom) of complete system (n = 66), modalapproximation (k = 12), and dual Arnoldi model (k = 30).



CONFIDENTIAL 3

Rational Krylov methods [Ruhe (1998)]

General approach:

1. Choose m interpolation points si

2. Construct Vi ,Wi ∈ Cn×ki such that

colspan(Vi ) = Kki ((siE − A)−1E , (siE − A)−1Eb)

colspan(Wi ) = Kki ((siE − A)−∗E ∗, (siE − A)−∗E ∗c)

3. Project with V = [V1, . . . ,Vm] and W = [W1, . . . ,Wm]

Open question:

I How to choose interpolation points si?

I See also PhD thesis Grimme (1997)



CONFIDENTIAL 3

10−1

100

101

−60

−40

−20

0

20

40

60

Frequency (rad/sec)

Gai

n (d

B)

k=40 (RKA)k=10 (QDPA)k=50 (RKA+QDPA10)Exact

Figure: Breathing sphere (n = 17611). Exact transfer function (solid),40th order SOAR RKA model (dot), 10th (dash-dot) order modalequivalent, and 50th order hybrid RKA+QDPA (dash).



CONFIDENTIAL 3

Breathing sphere ([Lampe, Voss (2006)]) (n = 17611)

I Two-sided rational SOAR [Bai and Su (2005)] model (k = 40,shifts 0.1, 0.5, 1, 5) misses peaks

I Small QDPA model (k = 10) matches some peaks, missesglobal response

I 500 s (SOAR, k = 2 · 40) vs. 2800 s (QDPA, 108 iters)

I Hybrid: Y = [YQDPA,YSOAR] and X = [XQDPA,XSOAR]

I Larger SOAR models: no improvement

I More poles with QDPA: expensive

I Use imaginary parts of poles as shifts for SOAR!

I Shifts σ1 = 0.65i , σ2 = 0.78i , σ3 = 0.93i , and σ4 = 0.1

I Two-sided rational SOAR, 10-dimensional bases



CONFIDENTIAL 3

10−1

100

101

−250

−200

−150

−100

−50

0

50

Frequency (rad/sec)

Gai

n (d

B)

k=70 (RKA)ExactRel Error

Figure: Breathing sphere (n = 17611). Exact transfer function (solid),70th order SOAR RKA model (dash) using interpolation points based ondominant poles, and relative error (dash-dot).



CONFIDENTIAL 3

Concluding remarks

I DPA for computation of dominant poles

I Subspace acceleration, selection, and efficient deflation

I Straightforward implementationI Applications:

I Various specialized eigenvalue problemsI Model order reduction:

I Construction of modal approximationsI Interpolation points for rational KrylovI Behavioral modeling

Generalizations:

I Second and higher-order systems

I MIMO systems

I Computation of dominant zeros z : H(z) = 0



CONFIDENTIAL 3

Thank you!

[email protected]

MOOREN I C E!


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