model reduction in the loewner framework - some issues€¦ · kit, november 17 - 18 2015 thanos...

Model Reduction in the Loewner FrameworkSome issues

Thanos Antoulas

Rice University, Houston, and Jacobs University, Bremen

e-mail: [email protected]: www.ece.rice.edu/ aca

Workshop: MOR 4 MEMS

KIT, November 17 - 18 2015

Thanos Antoulas (Rice U. & Jacobs U.) The Loewner Framework 1 / 44

Outline

1 The Loewner frameworkIntroduction

Construction of interpolants and generalized inverses

2 Data-driven model reduction: The example of a clamped beam

3 The Loewner framework for bilinear systemsIntroduction

A numerical experiment

4 Challenges and Concluding remarks


Outline








The Loewner matrix

Given:row array (µj , vj), j = 1, · · · , k , and

column array (λi ,wi ), i = 1, · · · , q,

the associated Loewner matrix is:

L =

v1−w1

µ1−λ1· · · v1−wq

µ1−λq

.... . .

...vk−w1

µk−λ1· · · vk−wq

µk−λq

∈ Ck×q

If there is a known underlying function g, then wi = g(λi ), and vj = g(µj).

Main property

Let L be a p × k Loewner matrix. Then p, k ≥ deg g ⇒ rankL = deg g.


Karel Lowner 1893 - 1968

• Born in Bohemia

• Studied in Prague under Georg Pick

• Emigrated to the US in 1939

• adapted his name to Charles Loewner

Seminal paper: Uber monotone Matrixfunctionen, Math. Zeitschrift (1934).


General framework – tangential interpolation

Given: • right data: (λi ; ri ,wi ), i = 1, · · · , k;

• left data: (µj ; `∗j , v∗j ), j = 1, · · · , q.

We assume for simplicity that all points are distinct.

Problem: Find rational p ×m matrices H(s), such that

H(λi )ri = wi `∗j H(µj ) = v∗j

Also define:

Λ =

λ1

. . .

λk

∈ Ck×k ,

R = [r1 r2, · · · rk ] ∈ Cm×k ,

W = [w1 w2 · · · wk ] ∈ Cp×k ,

Left data:

M =

µ1

. . .

µq

∈Cq×q, L =

`∗1...`∗q

∈Cq×p,V =

v∗1...

v∗q

∈ Cq×m


Tangential interpolation: the Loewner pencil

Recall data: H(λi )ri = wi , `∗j H(µj ) = v∗j .

The Loewner matrix L ∈ Cq×k is:

L =

v∗1 r1−`∗1 w1

µ1−λ1· · · v∗1 rk−`∗1 wk

µ1−λk

.... . .

...v∗q r1−`∗q w1

µq−λ1· · ·

v∗q rk−`∗q wk

µq−λk

if we are given H(s) = C(sE−A)−1B:

Ri = (λiE− A)−1Bri ⇒

R: generalized reachability matrix

O∗j = `∗j C(µjE− A)−1 ⇒

O: generalized observability matrix.

L can be factored as

⇒ L = −OER

The Loewner matrix L ∈ Cq×k is:

L =

v∗1 r1−`∗1 w1

µ1−λ1· · · v∗1 rk−`∗1 wk

µ1−λk

.... . .

...v∗q r1−`∗q w1

µq−λ1· · ·

v∗q rk−`∗q wk

µq−λk

L can be factored as

⇒ L = −OER

The shifted Loewner matrix L ∈ Cq×k is:

Lσ =

µ1v∗1 r1−`∗1 w1λ1

µ1−λ1· · · µ1v∗1 rk−`∗1 wkλk

µ1−λk

.... . .

...µqv∗q r1−`∗q w1λ1

µq−λ1· · ·

µqv∗q rk−`∗q wkλk

µq−λk

Lσ can be factored as

⇒ Lσ = −OAR


Structure of the Loewner pencil

Proposition. The controllability and observability matrices satisfy:

AR+ B R = ERΛ, OA + L C = MO E,

and hence:

Lσ = LΛ + VR Lσ = ML + LW

Furthermore L and Lσ satisfy:

ML− LΛ = VR− LW, MLσ − LσΛ = M VR− LWΛ.

Corollary. All systems (C, A, B), with E = I, satisfying given interpolation conditions, can beparametrized as follows:

A = Λ− BR, B =[bj], C =

[ηj], i , j = 1, · · · , k,

where B is free and Λ, R, C are given.

Corollary. Placement of poles and zeros.

N If ri = 0 (wi arbitrary) or `j = 0 (vj arbitrary) ⇒ λi or µj is a pole.

N If wi = 0 (ri arbitrary) or vj = 0 (`j arbitrary) ⇒ λi or µj is a zero.


Outline








Construction of Interpolants (Models)

• If the pencil (Lσ , L) is regular, i.e.

Φ(s) = Lσ − s L, is invertible, then

E = −L, A = −Lσ , B = V, C = W

is a minimal interpolant of the data ⇒ H(s) = W Φ(s)−1 V

• If Φ(s) = Lσ − s L, is singular, let

Φ(s)# be a generalized inverse of Φ(s)

(Drazin or Moore-Penrose).

⇒ H(s) = W Φ(s)# V

• In the latter case, if the numerical rankL = k, compute the rank revealing SVD:

L = YΣX∗ ≈ YkΣkX∗k

Theorem. A realization [C,E,A,B], of an approximate interpolant is given as follows:

E = −Y∗kLXk , A = −Y∗kLσXk , B = Y∗k V, C = WXk .


A simple example Consider the system

x1(t) = x2(t),

x2(t) = −x1(t)− x2(t) + u(t), y(t) = x2(t) ⇒ H(s) =

s

s2 + s + 1.

We now wish to recover state equations equivalent to the ones above from measurements of thetransfer function.

Data: obtained by evaluating the transfer function at λ1 = 12

, λ2 = 1, as well as µ1 = − 12

,µ2 = −1. The corresponding values of H are collected in the matrices

W =(

27

13

), V =

(− 2

3−1

)T.

Furthermore with R = [1 1], L = RT , we construct the Loewner pencil:

L =

[ 2021

23

67

23

], Lσ =

[− 4

210

− 47− 1

3

].

Since the pencil (Lσ , L) is regular, and the rank of both matrices is two:

H(s) = WΦ(s)−1V =s

s2 + s + 1, where Φ(s) = Lσ − s L.


Hence, the measurements above yield a minimal (descriptor) realization of the system in termsof the (state) variables ξ1, ξ2:

2021ξ1(t) + 2

3ξ1(t) = − 4

21ξ1(t) + 2

3u(t),

67ξ1(t) + 2

3ξ2(t) = − 4

7ξ1(t)− 1

3ξ2(t) + u(t),

y(t) = 27ξ1(t) + 1

3ξ2(t).

Question: what happens if we collect more data that necessary:

Λ = diag(

12

1 32

2), M = diag

(− 1

2−1 − 3

2−2

).

In this case, the associated measurements are

W =(

27

13

619

27

), V =

(− 2

3−1 − 6

7− 2

3

)T,

and with R = [1 1 1 1], L = RT , the Loewner pencil is:

L =

2021

23

2857

821

67

23

1019

37

47

1021

52133

1649

821

13

1657

521

, Lσ =

− 4

210 4

572

21

− 47

− 13

− 419

− 17

− 47− 8

21− 36

133− 10

49

− 1021

− 13

− 1457

− 421

.


It turns out that we can choose arbitrary X,Y ∈ R4×2, such that YT X is nonsingular, e.g.

X =

−1 0

0 −10 0−2 1

, YT =

[0 1 0 −11 −1 −1 1

],

so that the projected quantities

W = WX =[− 6

7− 1

21

], L = YTLX =

[− 6

7− 1

7

1849

1147

],

Ls = YTLσX =

[0 1

21

− 4849

− 19147

], V = YT V =

[− 1

3

1121

],

constitute a minimal realization of H(s):

H(s) = W(Ls − sL

)−1V =

s

s2 + s + 1.


There is another way to express the above relationship avoiding arbitrary projectors.

Basic ingredients: the Moore-Penrose generalized inverse and theDrazin generalized inverse.

The Moore-Penrose inverse of the (rectangular) matrix

M ∈ Rq×k , is denoted by MMP ∈ Rk×q , and satisfies:

(a) MMMPM = M, (b) MMPMMMP = MMP ,

(c)[

MMMP]T

= MMMP , (d)[

MMPM]T

= MMPM.

This generalized inverse always exists and is unique.

Given a square matrix M ∈ Rq×q , its index is the least nonneg-ative integer κ such that rank Mκ+1 = rank Mκ.

The Drazin inverse of M is the unique matrix MD satisfying:

(a) Mκ+1MD = Mκ, (b) MDMMD = MD ,

(c) MMD = MDM.

In the sequel we will be concerned with rectangular n ×m polynomial matrices which have anexplicit (rank revealing) factorization as follows:

M = X∆YT ,

where X, ∆, Y have dimension q × n, n × n, n × k, n ≤ q, k, and all have full rank k.

The Moore-Penrose generalized inverse is:

MMP = Y(YT Y)−1∆−1(XT X)−1XT.

If q=k and YTX is invertible, the Drazin generalized inverse is:

MD = X(YT X)−1∆−1(YT X)−1YT.


Example (continued). The quantities needed are the generalized inverses of

Φ(s) = Lσ − s L =

− 20s21− 4

21− 2s

34

57− 28s

572

21− 8s

21

− 6s7− 4

7− 2s

3− 1

3− 10s

19− 4

19− 3s

7− 1

7

− 4s7− 4

7− 10s

21− 8

21− 52s

133− 36

133− 16s

49− 10

49

− 8s21− 10

21− s

3− 1

3− 16s

57− 14

57− 5s

21− 4

21

= X∆(s)YT

.

Let the common range of the columns of L, Lσ be spanned by the columns of X and thecommon range of the rows of the same matrices by the rows of Y; it follows that

X =

1 0

0 1

− 37

87

− 12

1

, Y =

[1 0 − 7

19− 1

2

0 1 2419

97

]⇒ det (YX) 6= 0.

Thus with ∆(s) = Φ(1 : 2, 1 : 2)(s) there holds Φ(s)MP = 180989667

1s2+s+1

·

·

−28 (11610185s + 7274073) 14 (3558666s − 5604037) 6076 (32301s − 391) 14 (15168851s + 1670036)

294 (225182s + 281171) (−147) (192415s − 19668) −2058 (29494s + 15609) −147 (417597s + 261503)

3724 (54617s + 48189) (−1862) (29046s − 17485) −26068 (5715s + 1523) −1862 (83663s + 30704)

98 (2527157s + 2123670) −49 (1250553s − 876439) −98 (1797669s + 409322) −49 (3777710s + 1247231])

,


and Φ(s)D = 14897369

1s2+s+1

·

·

−84 (234677s + 152881) 294 (10652s − 13755) 588 (19079s − 641) 42 (330545s + 29086)

126 (31956s + 42829) −147 (11885s + 4) −882 (4184s + 2255) −63 (67611s + 42841)

684 (19079s + 17063) −798 (4184s − 2171) −4788 (1885s + 441) −342 (31631s + 10550)

42 (330545s + 281368) −147 (22537s − 13751) −294 (31631s + 6124) −21 (533378s + 157609)

.In the rectangular case, where there are two less right measuremnents, i.e we only haveΛ = diag

[12, 1

], while M remains the same, the right values are W = W(:, 1 : 2) and hence

Φ(s) = Ls − s L =

− 20s21− 4

21− 2s

3

− 6s7− 4

7− 2s

3− 1

3

− 4s7− 4

7− 10s

21− 8

21

− 8s21− 10

21− s

3− 1

3

= X ∆(s) YT

,

has dimension 4× 2, where Y = Y(1 : 2, 1 : 2). In this case the Moore-Penrose inverse is

Φ(s)MP =1

737 (s2 + s + 1)

−4767s − 3402 18272

s − 20372

3087s + 294 3297s + 13652

5838s + 5250 −1596s + 903 −4326s − 1218 −4515s − 1722

.

⇒ W Φ(s)MP V = W Φ(s)MP V = W Φ(s)D V = H(s)

Thus, the Loewner framework allows the definition of rectangular and/or singular systems.


Revisit: Construction of Interpolants

• If the pencil (Lσ , L) is regular, i.e.

Φ(s) = Lσ − s L, is invertible, then

E = −L, A = −Lσ , B = V, C = W

is a minimal interpolant of the data ⇒ H(s) = W Φ(s)−1 V

• If Φ(s) = Lσ − s L, is singular, let

Φ(s)# be a generalized inverse of Φ(s)

(Drazin or Moore-Penrose).

⇒ H(s) = W Φ(s)# V

• In the latter case, if the numerical rankL = k, compute the rank revealing SVD:

L = YΣX∗ ≈ YkΣkX∗k

Theorem. A realization [C,E,A,B], of an approximate interpolant is given as follows:

E = −Y∗kLXk , A = −Y∗kLσXk , B = Y∗k V, C = WXk .


Microstrip - data: 1001 S-parameter measurements between 10-18 GHz (CST)

Data frequency response ‖Si,j‖, i, j = 1, 2. Data two singular values.

Singular values of 1001× 1001 Loewner matrix Singular-value fit of model k = 72

S-parameter-error: ∈ [10−6, 10−4] Two singular values of model: ω ∈ [0, 10THz]


Outline








An Euler-Bernoulli beam 11

11/2/2015 https://upload.wikimedia.org/wikipedia/commons/e/e9/Cantilever_Beam.svg

https://upload.wikimedia.org/wikipedia/commons/e/e9/Cantilever_Beam.svg 1/1

Fixed End

Free End

BC

w(0, t) = 0, ∂w∂x

(0, t) = 0, E I ∂2w(L,t)

∂x2 + cd I∂3w(L,t)

∂x2∂t= 0,

−E I ∂3w(L,t)

∂x3 − cd I∂4w(L,t)

∂x3∂t= u(t), y(t) = ∂w(L,t)

∂t,

∂2w(x , t)

∂t2+

∂2

∂x2

[E I

∂2w(x , t)

∂x2+ cd I

∂3w(x , t)

∂x2∂t

]= 0,

where E , I , cd are constants. The transfer function is

G(s) =s N(s)

(E I + s cd I )m3(s)D(s)

where m(s) =

[−s2

E I + cd I s

] 14

,

N(s) = cosh(Lm(s)) sin(Lm(s))− sinh(Lm(s)) cos(Lm(s)) and

D(s) = 1 + cosh(Lm(s)) cos(Lm(s)) .

Parameter values: E = 69, GPa = 6.9 · 1010N/m2 - Young’s modulus elasticity constant, I = (1/12) · 7 · 8.53 · 10−11m4 -

moment of inertia, cd = 5 · 10−4 - damping constant, L = 0.7m, b = 7cm, h = 8.5mm - length, base, height of therectangular cross section.

1R. Curtain, K. Morris, Transfer Functions of Distributed Parameter Systems: A Tutorial, Automatica (2009).


Model Reduction of the beam

Poles: solutions ofs2 + cd I α4

k s + E I α4k = 0,

where αk are the roots in α, of the equation

1 + cosh(Lα) cos(Lα) = 0.

Reduction methods:

Modal truncation.

FEM followed by Loewner.

Loewner based on the transfer function.

Finally, parametric Loewner with damping as a parameter.

Order of reduced models: 32.

Parametric reduced order model 32, 4.


Plots


Poles and Loewner parametric reduction


Outline








Bilinear systems

Bilinear systems with one input and one output are described by:

Σ : Ex(t) = Ax(t) + Nx(t)u(t) + Bu(t), y(t) = Cx(t),

where x ∈ Rn, u, y ∈ R. Σ is equivalent to the infinite set of systems of the form:

Ex1(t) = Ax1(t) + Bu(t),

Ex2(t) = Ax2(t) + Nx1(t)u(t),

Ex3(t) = Ax3(t) + Nx2(t)u(t), · · ·

The solution is given as x(t) =∑∞

i=0 xi (t).Such systems are equivalent in the frequency domainto an inifnite sequence of rational multivariate functions:

H1(s1) = C [s1E− A]−1 B

H2(s1, s2) = C [s1E− A]−1 N [s2E− A]−1 B

H3(s1, s2, s3) = C [s1E− A]−1 N [s2E− A]−1 N [s3E− A]−1 B, · · ·

...Consequence: model reduction can be performed by means of interpolatory methods.


Recall: if the underlying transfer function is

H(s) = C(sE− A)−1B,

let O, R be the generalized controllability and the generalized observability matricesassociated with the data: {(µj , vj ), j = 1, · · · , q}, {(λi ,wi ), i = 1, · · · , k}.

O =

C(µ1E− A)−1

...C(µqE− A)−1

, R =[

(λ1E− A)−1B · · · (λkE− A)−1B]

⇒ L = −OER, Lσ = −OAR, V = CR, W = OB

Property generalized: factorization of L, Lσ, V, W

• Fact: if q = k = N ⇒ 2N moments of H are matched.

• the singular values of the Loewner pencil (Lσ ,L) provide a trade-off between accuracy of fitand complexity of the reduced model.


Nested multi-tuples

The nested right multi-tuples and the nested left multi-tuples

λ ={λ(1),λ(2), · · · ,λ(`)

}, µ =

{µ(1), µ(2), · · · , µ(k)

},

composed of the right ith tuples and the left jth tuples:

λ(i) =

{λ(i)1 },

{λ(i)2 , λ

(i)1 },

...

{λ(i)mi−1, · · · , λ

(i)2 , λ

(i)1 },

{λ(i)mi, λ

(i)mi−1, · · · , λ

(i)2 , λ

(i)1 },

, µ(j) =

{µ(j)1 },

{µ(j)2 , µ

(j)1 },

...

{µ(j)pj−1, · · · , µ

(j)2 , µ

(j)1 },

{µ(j)pj , µ

(j)pj−1, · · · , µ

(j)2 , µ

(j)1 },

where λ(i)j , µ

(j)j ∈ C and m1 + · · · + mk = k, p1 + · · · + pk = k.

Notice: the nestedness property of these tuples is reflected in the fact that each row in λ(i)

(µ(j)) is contained in the subsequent ones.


To λ(i) we associate:

R(i) = Φ(λ(i)mi

) N · · · N Φ(λ(i)2 ) N Φ(λ

(i)1 ) B, i = 1, · · · , k.

The matrixR =

[R(1), R(2), · · · , R(k)

]∈ Cn×k ,

is the generalized controllability matrix of the bilinear system Σ, associated with the rightmutli-tuple λ.Similarly, to the left tuples we associate the matrices

O(j) = C Φ(µ(j)1 ) N Φ(µ

(j)2 ) N · · · N Φ(µ

(j)pj ), i = 1, · · · , k,

and the generalized observability matrix:

O =

O(1)

...

O(k)

∈ Ck×n.


Interpolation of bilinear systems

Lemma. Let Σ = (C,E,A,N,B) be a scalar bilinear system of order n. Assume that it isprojected to a kth order systems by means of

V = R and WT = O:

E = WT EV, A = WT AV, N = WT NV, B = WT B, C = CV,

The system Σ = (C, E, A, N, B), of order k, satisfies the 2k + k2 interpolation conditions:

Hj (µ1, · · · , µj ) = Hj (µ1, · · · , µj ), Hi (λi , · · · , λ1) = Hi (λj , · · · , λ1),

Hj+i (µ1, · · · , µj , λi , · · · , λ1) = Hj+i (µ1, · · · , µj , λi , · · · , λ1), i , j = 1, · · · , k.


The generalized Loewner pencil

We define:L = O ER, Lσ = OAR

In addition we define the quantities

Ψ = ONR, V = OB, W = CR

Proposition. The following relationships hold:

L(k, `) =

Hk+`−1(µ1, · · · , µk , λ`−1, · · · , λ1)−Hk+`−1(µ1, · · · , µk−1, λ`, · · · , λ1)µk − λ`

Lσ(k, `) =

µkHk+`−1(µ1, · · · , µk , λ`−1, · · · , λ1)− λiHk+`−1(µ1, · · · , µk−1, λ`, · · · , λ1)µk − λ`

while V(k, 1) = Hk (µ1, · · · , µk−1, µk ),

W(1, `) = H`(λ`, λ`−1, · · · , λ1),

Ψ(k, `) = Hk+`(µ1, · · · , µk−1, µk , λ`, λ`−1, · · · , λ1).


Example. Given the SISO bilinear system (C,E,A,N,B), where A is n × n, consider the

ordered tuples of left and right interpolation points:

[{µ1}{µ1, µ2}

], [{λ1}, {λ2, λ1}].

The associated generalized observability and controllability matrices are

O =

[C(µ1E− A)−1

C(µ1E− A)−1N(µ2E− A)−1

], R =

[(λ1E− A)−1B, (λ2E− A)−1N(λ1E− A)−1B

].

Then:

L =

[H1(µ1)−H1(λ1)

µ1−λ1

H2(µ1,λ1)−H2(λ2,λ1)µ1−λ2

H2(µ1,µ2)−H2(µ1,λ1)µ2−λ1

H3(µ1,µ2,λ1)−H3(µ1,λ2,λ1)µ2−λ2

]= −OER,

Lσ =

[µ1H1(µ1)−λ1H1(λ1)

µ1−λ1

µ1H2(µ1,λ1)−λ2H2(λ2,λ1)µ1−λ2

µ2H2(µ1,µ2)−λ1H2(µ1,λ1)µ2−λ1

µ2H3(µ1,µ2,λ1)−λ2H3(µ1,λ2,λ1)µ2−λ2

]= −OAR,

Ψ =

[H2(µ1, µ2) H3(µ1, λ2, λ1)

H3(µ1, µ2, λ1) H4(µ1, µ2, λ2, λ1)

]= ONR,

V =

[H1(µ1)

H2(µ1, µ2)

]= OB,

W =[

H1(λ1) H2(λ2, λ1)]

= CR.


It readily follows that given the bilinear system (C, E, A, N, B) a reduced bilinear system oforder two, can be obtained without computation (matrix factorizations or solves) as:

E = OER, A = OAR, N = ONR, B = OB, C = CR.

• The following measurements are needed to construct the reduced second order system:

Linear Bilinear

H(µ1) H1(µ1)

H(λ1) H1(λ1)

H(µ2) H2(µ1, µ2)

H(λ2) H2(λ2, λ1)

H2(µ1, λ1)

H3(µ1, µ2, λ1)

H3(µ1, λ2, λ1)

H4(µ1, µ2, λ2, λ1)

2n moments n2 + 2n moments

matched matched


Construction of interpolants

Lemma. If k = `, and (Lσ , L), is a regular pencil, then

E = −L, A = −Lσ , N = Ψ, B = V, C = W,

is a minimal realization of an interpolant of the data, i.e., the rational functions:

Hk (s1, · · · ., sk ) = W(Lσ − s1L)−1Ψ · · ·Ψ(Lσ − skL)−1V, k ≥ 1,

interpolate the data.

Theorem. In the case of redundant data, the pencil (Lσ , L) is singular, and we constructX,Y ∈ Rρ×k as before. The quintuple (C,E,A,N,B) given by:

E = −Y∗LX, A = −Y∗LσX, N = Y∗ΨX, B = Y∗V, C = WX,

is the realization of an (approximate) interpolant of the data.

Remark. As in the linear case, if we have more data than necessary, we can either consider(W,−L,−Lσ , Ψ,V) as an exact but singular model of the data or

(WX,−Y∗LX,−Y∗LσX, Y∗ΨX, Y∗V),

as an approximate (nonsingular) model of the data.


Outline








Numerical experiment: Bilinear controlled heat transfer system

Consider the heat equation∂x

∂t= ∆x on [0, 1]× [0, 1]

with the mixed Dirichlet and Robin boundary conditions:

n · ∇x = u1(x − 1), on Γ1 = {0}×c0, 1b,

n · ∇x = u2(x − 1), on Γ2 =c0, 1b×{0},

x = 0, on Γ3 = {1} × [0, 1] and Γ4 = [0, 1]× {1}.

The heat transfer coefficients u1 and u2 and the lower boundaries Γ1 and Γ2 are the inputvariables (spraying-intensities of a cooling- fluid acting on these boundaries). By a finitedifference discretization of the Poisson equation on an equidistant k × k mesh (with meshsizeh = 1

k+1) with nodes xij , we obtain the Poisson matrix:

P = I⊗ Tk + Tk ⊗ I, Tk = tridiag [1, −2, 1].

The dynamics of the heat flow yield a bilinear system described by

x = Ax + u1N1x + u2N2x + Bu,


where the system matrices are:

A =1

h2(I⊗ Tk + Tk ⊗ I + E1 ⊗ I + I⊗ Ek ), Ej = eje

Tj ,

N1 =1

hE1 ⊗ I, N2 =

1

hI⊗ Ek , b1 =

1

hE1 ⊗ e, b2 =

1

he⊗ Ek , e = [1, 1, · · · , 1]T ,

C =1

k2(e⊗ e)T .

ΣB is the 2500th order system obtained by discretiz-ing the heat equation.

1 ΣB is reduced using Loewner to obtain Σ1 oforder 28.

2 ΣB is reduced using BIRKA to obtain Σ2 oforder 28.

Singular values ofthe Loewner pencil

The first step is to collect samples from generalized bilinear transfer functions up to order 2, andplot the singular values of the ensuing Loewner pencil. We notice that σ1 = 1, σ28 ≈ 10−15

⇒ k = 28. Next, we compare the time-domain output of Σ0, and the outputs the reducedsystems when the input signals are:

u1(t) = sin(4t)exp(−t/2) + (0.5)exp(−t/2), u2(t) =1

2cos(πt + 1).

The tangential directions are chosen randomly.


Time domain simulation - Outputs and Approximation errors


Outline








Challenge: Earthquake prevention for high-rise buildings

Taipei 101 damper

Taipei 101: 508m Damper between 87-91 floors 730 ton damper


Challenge: Damping of lightweight bridges

Lamot footbridge damper optimization

Lamot bridge finite elementmodel (n = 25,962)The goal is to determine theoptimal stiffness and dampingcoefficient of four bridgedampers (=8 parameters).

K. Meerbergen (KU Leuven) MODRED 2013 December 11–13, 2013 6 / 53

The Lamot footbridge Parametrized freq. response

• Lamot lightweight bridge in Mechelen: finite element model n = 25, 962.

• The goal: determine optimal stiffness and damping coefficients of the

four bridge dampers (8 parameters).

• Example due to Karl Meerbergen (KU Leuven).


Conclusions

Given data (`i , vi ), (λj ,wj ), construct the Loewner pencil (Lσ ,L).

The quadruple (W,L,Lσ ,V), where the pencil (Lσ ,L) may be singular,

is a natural model of the data. The construction involves no computation.

(Lσ ,L) and the underlying (A,E) have the same non-trivial eigenvalues.

The projection to a minimal realization can be chosen arbitrarily.

New: The Moore-Penrose or Drazin inverses of Φ(s) = Lσ − s L, satisfy:

H(s) = W Φ(s)# V


References

Overview article:

A.C. Antoulas, S. Lefteriu and A.C. Ionita,

A tutorial introduction to the Loewner framework for model reduction,

in Model Reduction and Approximation for Complex Systems, edited by P. Benner,

A. Cohen, M. Ohlberger, and K. Willcox, Birkhauser Verlag, ISNM Series (2015).

Just accepted:

A.C. Antoulas,

The Loewner Framework and Transfer Functions of Singular and Rectangular Systems,

Applied Mathematics Letters (2016).


model reduction in the loewner framework - some issues€¦ · kit, november 17 - 18 2015 thanos...

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