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Introduction Introduction to the projection-based approaches Projection-based methods Non-projection methods Summary

Model Reduction (Approximation) of Large-Scale Systems

Overview of the model approximation methodsLecture 2

C. Poussot-Vassal, P. Vuillemin & I. Pontes Duff

EDSYS, April 4-7th, 2016 (Toulouse, France)

moremoreΣ

(A,B,C,D)i

Σ

Σ

(A, B, C, D)i

model reduction toolbox

Kr(A,B)

AP + PAT + BBT = 0

WTV

DAE/ODE

State x(t) ∈ Rn, n large orinfinite

Data

ReducedDAE/ODE

Reduced state x(t) ∈ Rrwith r � n(+) Simulation(+) Analysis(+) Control(+) Optimization

Case 1u(f) = [u(f1) . . . u(fi)]y(f) = [y(f1) . . . y(fi)]

Case 2Ex(t) = Ax(t) +Bu(t)y(t) = Cx(t) +Du(t)

Case 3H(s) = e−τs

C. Poussot-Vassal, P. Vuillemin & I. Pontes Duff [Onera - DCSD] Model Reduction (Approximation) of Large-Scale Systems


IntroductionModel approximation problem (realization-based framework)Realization based approximation methods overview

Introduction to the projection-based approachesDynamical systemsPetrov-Galerkin approximationProjectorsApproximation by projectionMismatch error measureThe projection-based framework

Projection-based methodsTruncationKrylov basic iterationProper Orthogonal Decomposition

Non-projection methodsStandard schemeSome LMI methodsSome LMI methodsVector fitting methods

Summary



Outlines

IntroductionModel approximation problem (realization-based framework)Realization based approximation methods overview

Introduction to the projection-based approaches

Projection-based methods

Non-projection methods

Summary



IntroductionThe big picture


DAE/ODE


Data

ReducedDAE/ODE






IntroductionModel approximation problem

Let us consider H, a nu inputs, ny outputs linear dynamical system described by thecomplex-valued function of order n (n large or ∞)

H : C→ Cny×nu , (1)

the model approximation problem consists in finding H of order r � n

H : C→ Cny×nu , (2)

with a given realization e.g.

H :{

E ˙x(t) = Ax(t) + Bu(t)y(t) = Cx(t) + Du(t)

, (3)

that well reproduces the input-output behaviour of H.



IntroductionModel approximation problem (realization-based framework)

Let us consider H, a nu inputs, ny outputs linear dynamical system described by thecomplex-valued function of order n (n large or ∞)

H : C→ Cny×nu , (4)


H :{

Ex(t) = Ax(t) +Bu(t)y(t) = Cx(t) +Du(t) , (5)

the model approximation problem consists in finding H of order r � n

H : C→ Cny×nu , (6)


H :{

E ˙x(t) = Ax(t) + Bu(t)y(t) = Cx(t) + Du(t)

, (7)

that well reproduces the input-output behaviour of H.



IntroductionRealization based approximation methods overview

Realization-based ap-proximation methods


Linear systems• LMIs• Vector fitting• Subspace• Optimization (DARPO)


Linear systems (truncation)• SVD (BT, SPA)• Modal truncation (SDPA)• Hankel approximation (H)

Linear systems (iterative)• Rational interpolation (IRKA)• Tangential interpolation (ITIA)

Linear systems (mixed)• Krylov/SVD (ISRKA, ISTIA)• Krylov/Eigenvectors (IETIA)

Linear / Nonlinear systems• POD• Empirical Gramians



Outlines

Introduction

Introduction to the projection-based approachesDynamical systemsPetrov-Galerkin approximationProjectorsApproximation by projectionMismatch error measureThe projection-based framework



Summary



Introduction to the projection-based approachesDynamical systems

Given H(s), a nu inputs ny outputs, LTI dynamical complex-valued function matrix oforder n:

H(s) = C(sE −A)−1B +D ∈ Cny×nu (8)

with realization H : (E,A,B,C,D)

H :{

Ex(t) = Ax(t) +Bu(t)y(t) = Cx(t) +Du(t) (9)

with x(t) ∈ Rn, n� nu, ny .



Introduction to the projection-based approachesPetrov-Galerkin approximation

Trajectory subspaceState trajectories of high fidelity models are contained in low dimensional subspaces

x(t) = x1(t)v1 + x2(t)v2 + . . . (10)

By denotingI V = [v1, v2, . . . , vr] ∈ Rn×r is a basis for V (span(V ) = V)I x(t) = [x1(t), x2(t), . . . , xr(t)]T ∈ Rr is the reduced state vector.I r the dimension of such subspace V

It comes that x(t) ≈ V x(t)




Trajectory subspaceState trajectories of high fidelity models are contained in low dimensional subspaces

x(t) = x1(t)v1 + x2(t)v2 + . . . (10)

By denotingI V = [v1, v2, . . . , vr] ∈ Rn×r is a basis for V (span(V ) = V)I x(t) = [x1(t), x2(t), . . . , xr(t)]T ∈ Rr is the reduced state vector.I r the dimension of such subspace V

It comes that x(t) ≈ V x(t)




I By setting x(t) ≈ V x(t) and span(V ) = V, the dynamical model becomes,

H :{

EV ˙x(t) = AV x(t) +Bu(t) + r(t)y(t) = CV x(t) +Du(t) (11)

The residual r(t) ∈ Rn accounts for the fact that V x(t) will not be an exactsolution to the dynamical equation.

I The residual r(t) is then constrained to be orthogonal to a subspace W ∈ Rn×r,where span(W ) =W, i.e.:

WT r(t) = 0 (12)

A projection method consists then in seeking for an approximation x(t) of x(t), byimposing the following two conditions:

x(t) ∈ V and(EV ˙x(t)−

(AV x(t) +Bu(t)

))⊥ W (13)





H :{




WT r(t) = 0 (12)



(AV x(t) +Bu(t)

))⊥ W (13)





H :{




WT r(t) = 0 (12)



(AV x(t) +Bu(t)

))⊥ W (13)




By settingx(t) ∈ V and EV ˙x(t)−

(AV x(t) +Bu(t)

)⊥ W (14)

or equivalently

x(t) ∈ V and WT(EV ˙x(t)−

(AV x(t) +Bu(t)

))= 0 (15)

One then obtains,

H :{

WTEV ˙x(t) = WTAV x(t) +WTBu(t) + 0y(t) = CV x(t) +Du(t) (16)

Moreover, the approximated full state vector can be reconstructed if needed as,

x(t) ≈ V x(t) (17)

This is known as the Petrov-Galerkin projection framework



Introduction to the projection-based approachesProjectors

The unifying feature: projectionsLet V,W ∈ Rn×r, such that WTV = Ir, then Π = VWT is a projector.

Definition: ProjectorΠ is said to be a projector if and only if

Π2 = Π (18)

Then, we have Rn = Im(Π)⊕ ker(Π).




Projector propertiesLet us consider a projector Π

I Eigenvalues of Π are only 0 and 1I Let r be the rank of Π, Then there exist a basis X, such that

Π = X

[Ir

0n−r

]X−1 (19)

I Let X = [X1 X2] such that X1 ∈ Rn×r and X2 ∈ Rn×(n−r), then ∀x ∈ Rn,

Πx ∈ span(X1) = Im(Π) = S1x−Πx ∈ span(X2) = ker(Π) = S2

(20)

I Π defines the projection onto S1, parallel to S2,

Rn = Im(Π)⊕ ker(Π) (21)




Orthogonal projectorLet consider a projector Π

I S1 = S⊥2I Let V ∈ Rn×r be an orthogonal matrix, the orthogonal projection of x ∈ Rn

onto S1 is V V T xI The projection matrix is ΠV,V = V V T

I if x ∈ S1, then ΠV,V x = V V T x = x

I if x ⊥ S1, then ΠV,V x = V V T x = 0I ||x−Πx||2 = miny∈Im(Π) ||x− y||2




Oblic projectorLet consider a projector Π Let consider a projector Π

I S1 6= S⊥2I Let V ∈ Rn×r and W ∈ Rn×r be full column rank matrices, spanning S1 andS⊥2 , the projection of x ∈ Rn onto subspace S1 parrallel to S2 isV (WTV )−1WT x

I The projection matrix is ΠV,W = V (WTV )−1WT

I if x ∈ S1, and x = V y, then ΠV,W x = V (WTV )−1WTV y = V y

I if x ⊥ S2, then ΠV,W x = V (WTV )−1WT x = 0



Introduction to the projection-based approachesApproximation by projection

The general caseGiven H, a nu inputs ny outputs, full order nonlinear dynamical system (order n):

H :{

x(t) = f(x(t), u(t))y(t) = g(x(t), u(t)) (22)

by denoting x = WT x, then, the reduced model is

H :{ ˙x(t) = WT f(V x(t), u(t))

y(t) = g(V x(t), u(t)) (23)

with the reduced initial conditions:

x0 = WT x0 ∈ Rr (24)

and in the equivalent full state space initial conditions

V x0 = VWT x0 ∈ Rn (25)




The linear caseGiven H, a nu inputs ny outputs, full order LTI dynamical system (order n):

H :{

Ex(t) = Ax(t) +Bu(t)y(t) = Cx(t) +Du(t) (26)

by denoting x = WT x, then, the reduced model is

H :{

WTEV ˙x(t) = WTAV x(t) +WTBu(t)y(t) = CV x(t) +Du(t) (27)

with the reduced initial conditions:

x0 = WT x0 ∈ Rr (28)

and in the equivalent full state space initial conditions

V x0 = VWT x0 ∈ Rn (29)




Comments about V and W

Let us consider,

H :{


x0 = WT x0 ∈ Rr (31)

LemmaChoosing two different bases V ′ and W ′ that respectively span the same subspaces Vand W result in the same reconstructed solution x(t).

Thus, subspaces are relevant, not basis




Comments about V and W

Let us consider,

H :{


x0 = WT x0 ∈ Rr (31)

LemmaChoosing two different bases V ′ and W ′ that respectively span the same subspaces Vand W result in the same reconstructed solution x(t).

Thus, subspaces are relevant, not basis




ConsequencesI A reduced order model is uniquely defined by its projector ΠV,WI The projector ΠV,W is itself uniquely defined by the two subspaces

span(V ) = Vspan(W ) =W (32)

I V and W belong to the Grassmann manifold G(r, n): known as the set of allsubspaces of dimension r in Rn

Reduced Order Model ↔ (V,W)How to find V and W?




ConsequencesI A reduced order model is uniquely defined by its projector ΠV,WI The projector ΠV,W is itself uniquely defined by the two subspaces

span(V ) = Vspan(W ) =W (32)

I V and W belong to the Grassmann manifold G(r, n): known as the set of allsubspaces of dimension r in Rn

Reduced Order Model ↔ (V,W)How to find V and W?



Introduction to the projection-based approachesMismatch error measure

H is a "good" approximation of H if E(t) = x(t)− V x(t) is "small".

Model error

H :

{Ex(t) = Ax(t) +Bu(t)E ˙x(t) = Ax(t) + Bu(t)e(t) = (Cx(t)− Cx(t)) + (D − D)u(t)

(33)

or, in the frequency domain, represented by its transfer function,

H(s) = H(s)−H(s) =[C −C

] [ sE −A 00 sE − A

]−1 [B

B

]+D−D ∈ Cny×nu

(34)

How to quantify? ... in practice?



Introduction to the projection-based approachesMismatch error measure

H is a "good" approximation of H if E(t) = x(t)− V x(t) is "small".

Model error

H :

{Ex(t) = Ax(t) +Bu(t)E ˙x(t) = Ax(t) + Bu(t)e(t) = (Cx(t)− Cx(t)) + (D − D)u(t)

(33)

or, in the frequency domain, represented by its transfer function,

H(s) = H(s)−H(s) =[C −C

] [ sE −A 00 sE − A

]−1 [B

B

]+D−D ∈ Cny×nu

(34)

How to quantify? ... in practice?



Introduction to the projection-based approachesThe projection-based framework

Let H : C→ Cny×nu be a nu inputs ny outputs, complex-valued function describinga LTI dynamical system as a DAE of order n, with realization H:

H :{

Ex(t) = Ax(t) +Bu(t)y(t) = Cx(t) (35)


E,A B

C




H :{

Ex(t) = Ax(t) +Bu(t)y(t) = Cx(t) (35)

the approximation problem consists in finding V,W ∈ Rn×r (with r � n) spanning Vand W subspaces and forming a projector ΠV,W = VWT , such that

H :{

WTEV ˙x(t) = WTAV x(t) +WTBu(t)y(t) = CV x(t) (36)

well approximates H.


WTEV ,WTAV WTB

CV

ΠV,W =⇒C

E,A B




H :{

Ex(t) = Ax(t) +Bu(t)y(t) = Cx(t) (35)

the approximation problem consists in finding V,W ∈ Rn×r (with r � n) spanning Vand W subspaces and forming a projector ΠV,W = VWT , such that

H :{

WTEV ˙x(t) = WTAV x(t) +WTBu(t)y(t) = CV x(t) (36)

well approximates H.

I Small approximation error and/or global error boundI Stability / passivity preservationI Numerically stable & efficient procedure



Outlines

Introduction


Projection-based methodsTruncationKrylov basic iterationProper Orthogonal Decomposition


Summary



Projection-based methodsApproximation methods: an overview



Linear systems• LMIs• Vector fitting• Complex optimization (DARPO)



Linear systems (iterative)• Rational Interpolation (IRKA)• Tangential (ITIA)


Non linear systems• POD• Empirical Gramians



Projection-based methodsTruncation

The ideaLet divide the realization as follows:

H⊥ =[A⊥ B⊥C⊥ D

]=

[A11 A12 B1A21 A22 B2C1 C2 D

]. (37)

The truncated reduced system is then (r first columns of V,W ),

H =[A11 B1C1 D

]=[WTr AVr WT

r BCVr D

](38)

Performing such a truncation as it is hopeless...One should choose an appropriate basis (subspace indeed)



Projection-based methodsTruncation in the balanced realization

Balanced realizationLet H be a nu inputs ny outputs, nth order model. Its balanced realization H⊥ isobtained as follows,

H⊥ =[A⊥ B⊥C⊥ D

]=[WTAV WTBCV D

](39)

where V,W ∈ Rn×n are defined as:

V = UZΣ−1/2 and W = LY Σ−1/2 (40)

with P = UUT and Q = LLT are the solutions of the two Lyapunov equations andUTL = ZΣY T .

I 2 Lyapunov equations to solve:

P = Q = Σ = diag(σ1, . . . , σn) where σi =√λ(PQ) (41)

I States are sorted by decreasing energy



Projection-based methodsTruncation in the balanced realization

Consequence...

H⊥ =

[A11 A12 B1A21 A22 B2C1 C2 D

](42)

In that case,Σ =

[Σ1 00 Σ2

](43)

The system is said to be "balanced", and the most reachable and most observable statesare kept, only.



Projection-based methodsTruncation in the modal realization

Modal realizationLet H be a nu inputs ny outputs, full order LTI system (order n). Its modal realizationH⊥ is obtained as follows,

H⊥ =[A⊥ B⊥C⊥ D

]=

Λ︷︸︸︷WTAV WTBCV D

(44)

where V,W ∈ Rn×n are defined as:

AV = ΛVWA = ΛW (45)

I 2 eigenvalue problems to solveI States are sorted by user-defined eigensorting/selecting strategyI Eigenvalues are kept



Projection-based methodsTruncation in the modal realization

Consequences...In that case,

H =

[A11 0 B1

0 A22 B2C1 C2 D

](46)

where A11 and A22 corresponds to (block) diagonal matrices with the eigenvalues ofthe original system. The modal approximation consists then in truncating the undesired

modes and then in finding B1 and C1, by a mean of a bi-convex procedure, in order toreduce a given norm.

{B1, C1} = argmin ||H − C1(sIn −A11)−1B1 +D||Hl(47)



Projection-based methodsBrief conclusions

√Stability preserved

√Global error bound (balanced case)

||H− H||H∞ ≤ 2(σk+1 + · · ·+ σn) (48)√

Efficient functions available in MATLAB, SLICOT & LYAPACK√

Possibility to extend to weighted, BR, PR SVD factorizationI Local results may be unsatisfactory× Dense computations, matrix factorizations and inversions may be ill-conditioned× Need whole transformed system in order to truncate× Require a lot of memory and operations O(n3)× Solution of 2 Lyapunov equations (hard for very large-scale systems)× Solution of 2 eigenvalue problems

Well adapted to medium-scale systems but not very large ones(unless approximate Lyapunov or eigenvalue)














Projection-based methodsKrylov basic iteration

The very basic Krylov iterationGiven A ∈ Rn×n and b ∈ Rn, let v1 = b

||b|| , at the kth step of the procedure:

AVk = VkHk + fkeTk (49)

whereI ek ∈ Rn is the canonical unit vectorI Vk = [v1, v2, . . . , vk] ∈ Rk×k, and V Tk Vk = Ik (i.e. is orthogonal)I Hk = V Tk AVk ∈ R

k×k

I vk+1 = fk||fk||

∈ Rn

Note that,I Computational complexity for k steps is O(n2k) and storage is O(nk).I With Arnoldi, Hk is upper Hessenberg (interesting in computational analysis).I With Lanczos, Hk is tridiagonal (interesting in computational analysis).



Projection-based methodsKrylov basic iteration - 3 use of the Krylov iteration

1. Iterative solution of Ax = b: approximate the solution x iteratively.2. Iterative approximation of the eigenvalues of A. In this case b is not fixed a priori.

The eigenvalues of the projected Hr approximate the dominant eigenvalues of A.3. Approximation of linear systems by moment matching.

Aleksey Nikolaevich Krylov (August 15 1863 - October 26, 1945) was aRussian naval engineer, applied mathematician and memoirist.In 1931 he published a paper on what is now called the Krylov subspaceand Krylov subspace methods. The paper deals with eigenvalueproblems, namely, with computation of the characteristic polynomialcoefficients of a given matrix. Krylov was concerned with efficientcomputations and, as a computational scientist, he counts the work as anumber of separate numerical multiplications; something not verytypical for a 1931 mathematical paper. Krylov begins with a carefulcomparison of the existing methods that include the worst-case-scenarioestimate of the computational work in the Jacobi method. Later, hepresents his own method which is superior to the known methods ofthat time and is still widely used. [Source Wikipedia]



Projection-based methodsKrylov basic iteration - Approximation by moment matching

Given H, expand the transfer function around σ

H(s)|σ = η0 + η1(s− σ) + η2(s− σ)2 + . . . (50)

and given H ∈ R(r+nu)×(r+ny), similarly expanded around σ as,

H(s)|σ = η0 + η1(s− σ) + η2(s− σ)2 + . . . (51)

verify,ηi = ηi , ∀i ∈ 1 . . . r at σ (52)

I Problem: Moments are numerically hard to compute (ill conditioned).I Solution: It is possible to achieve moment matching without explicitly computing

them, through the introduction of the Krylov subspaces.



Projection-based methodsKrylov basic iteration - Projectors for Krylov and rational Krylov 1

Given, Π = VWT ,

H =[E,A BC D

]obtain H =

[E, A B

C D

]=[WTEV ,WTAV WTB

CV D

](53)

Krylov (E = In)

span(V ) = [B,AB, . . . , Ak−1B] ∈ Rn×k

span(WT ) = [CT , ATCT , . . . , A(k−1)TCT ]T ∈ Rk×n(54)

then Markovian parameters (impulse response) match:

ηi = H(∞) = CAiB = CAiB = H(∞) = ηi (55)

1 W. E. Arnoldi, "The principle of minimized iterations in the solution of the matrix eigenvalue problem",Quart. Appl. Math., 9, 1951, pp. 17-29.



Projection-based methodsKrylov basic iteration - Projectors for Krylov and rational Krylov

Given, Π = VWT ,

H =[E,A BC D

]obtain H =

[E, A B

C D

]=[WTEV ,WTAV WTB

CV D

](56)

Rational Krylov

span(V ) = [(σ1E −A)−1B, . . . (σkE −A)−1B] ∈ Cn×k

span(WT ) = [(σk+1E −A)−TCT , . . . (σ2kE −A)−TCT ]T ∈ Ck×n(57)

then the moments of H(s) match those of H(s) at σi:

H(σi) = D + C(σiE −A)−1B = D + C(σiE − A)−1B = H(σi) (58)




√Numerically efficient, only matrix-vector multiplications are required (no matrixfactorizations and/or inversion)

√No Lyapunov equation

√Fast computation (no need to compute transformed model and then truncate).O(kn2) or O(k2n) vs. O(n3)

√Applicability to very large-scale systems (e.g. 108 states)

√Possibility to focus on specific poles

√Available numerical tools in MORE

× No global error bound× Stability not always preserved (but mechanisms exist)? How to select the shift points

Well adapted to very large-scale systems














Projection-based methodsProper Orthogonal Decomposition

State snapshotLet,

X = [x(t1), x(t2), . . . , x(tN )] ∈ Rn×N (59)

Then, apply SVD as follows

X = UΣV T ≈ UrΣrV Tk (60)

The, approximate the state as

x(t) = UTr x(t) , x(t) ≈ Urx(t), (61)

where x(t) ∈ Rr. Then project the reduced order system is as:

H :{ ˙x(t) = UTr f(Urx(t), u(t))

y(t) = g(Urx(t), u(t)) (62)

Choice of the snapshots?Singular values non input/output invariant...





√Applicable to nonlinear systems

× Dense computations, matrix factorizations and inversions may be ill-conditioned× Need to simulate the system in order to truncate× Require a lot of memory and operations (at least O(n3))

Well adapted to non-linear systems(but not very large ones, unless with good SVD solver)



Outlines

Introduction



Non-projection methodsStandard schemeSome LMI methodsSome LMI methodsVector fitting methods

Summary














Non-projection methodsStandard scheme

Optimization criteria and parametrizationThe model reduction problem consists in (l defines a norm),

(A, B, C, D) := argmin(A,B,C,D)∈Rx×y

||H −G||Hl(63)

or (N(jω), D(jω)

):= argmin

N(jω),D(jω) polynomial||H(jω)−G(jω)||l (64)

or any other model parametrization



Non-projection methodsStandard scheme

Optimization schemeI Select a criteria J (preferably a system norm)I Define a parametrization of the model (State-space, Pole residue...)I Derive the gradient ∇JI Derive the Hessian ∇2JI Apply a descent algorithm



Non-projection methodsSome LMI methods

Classical ideaGiven the error system,

H :=[A B

C D

]=

[A 0 B

0 A B

C −C D − D

](65)

Find the reduced order model that minimizes theH∞ (orH2 ) norm of the error systemsH. The BRL case: min γ∞, s.t. P > 0 and[

ATP + PA PB CT

BTP −γ2∞I DT

C D −I

]< 0 (66)

Not LMI, hence non solvable is such form ⇒ conservatism is introduced



Non-projection methodsSome LMI methods2

Split the Lyapunov matrix P as follows: ATP11 + P11A (?)T (?)T (?)TATP12 + P12TA ATP22 + P22A (?)T (?)TBTP11 − BTP12T BTP12 − BP22 −γ∞Ip (?)T

C C D − D −γ∞Im

< 0

P =[P11 P12PT12 P22

]> 0

(67)

. . . and iterate1: while ε < γn − γn+1 do2: Fix (A, B)3: min γ∞, subject to LMI with respect to P , C, D4: Fix (P12, P22)5: min γ∞, subject to LMI with respect to P11, A, B, C, D6: γn+1 = γ∞7: end while

2 Helmersson, "Model Reduction using LMIs", in Proceedings of the Conference Decision and Control,Lake Buena Vista, Florida, December 1994, 3217-3222.



Non-projection methodsSome LMI methods

Classical ideaGiven the error system,

H :=[A B

C D

]=

[A 0 B

0 A B

C −C D − D

](68)

Find the reduced order model that minimizes theH∞ (orH2 ) norm of the error systemsH. The BRL case: min γ∞, s.t. P > 0 and[

ATP + PA PB CT

BTP −γ2∞I DT

C D −I

]< 0 (69)

Not LMI, hence non solvable is such form ⇒ conservatism is introduced



Non-projection methodsBrief conclusions


√Global bound

√Easily implementable in MATLAB, YALMIP + SDP solver (e.g. SeDuMi)

× Extended system formulation× Conservatism of the relaxation (or iterative algorithm)× SDP solver limitations (# variable & time!)× Lyapunov function (quadratic increase)

Not adapted to very large-scale systems(can reasonably solve problem of order n=30)



Non-projection methodsVector fitting methods

(An) Algorithm1: Sample original system H frequency response magnitude:

‖H(jω)‖ , {ω0, . . . , ωi} (70)

2: Define H(jω) (of order r) as:

H(s) =brsr + br−1sr−1 + · · ·+ b0

arsr + ar−1sr−1 + · · ·+ a0

or H(s) =brsr + br−1sr−1 + · · ·+ b0∏

i( s2wi

2s+ 2mi

wi+ 1)

∏j

( swj

+ 1)

(71)

3: Nonlinear fitting

min[bi,ai,wi,mi]

N∑i=0

(H(jωi)− H(jωi)

)2(72)



Vector fitting methodsBrief conclusions

√Stability can be preserved

√Handles "bounded" damping and dynamic coefficients

√Quite easily implementable in MATLAB (e.g. lsqcurvefit)

√Very engineering appealing

× Frequency gridding implies complexity (choice of the grid)× Applicability to MIMO× Nonlinear solver limitations (# variable, initial vector & time!)× Problem when state space form are preferred× Usually far to be optimal

Not adapted to very large-scale systems



Outlines

Introduction




Summary



SummaryLecture’s highlights

About this lectureI Norm and classical Linear Algebra reminderI Control engineer point of view: SVD methodsI Numerical engineer point of view: Krylov methodsI Both well adapted to model reductionI Bottlenecks of both methods have to be handled with kind attentionI LMI and vector optimization based approaches not adapted (unless for very low

dimensional models)

Raising questionsI Efficiency, optimalityI Computing (parallel)I Numerical issues



SummaryApproximation methods: an overview



Linear systems• LMIs• Vector fitting• Subspace• Optimization (DARPO)



Linear systems (iterative)• Rational interpolation (IRKA)• Tangential interpolation (ITIA)


Linear / Nonlinear systems• POD• Empirical Gramians



SummaryApproximation methods: an overview and solvers...


Modalmethods

Linear systems• Eigenvalues & Residue• Dominant modes

SVD/Lyapmethods

Linear systems• Balanced truncation• Hankel approximation


Krylovmethods

Linear systems• Arnoldi/Lanczos• Rational Interpolation

Optimizationmethods

Linear systems• DARPO• Vector fitting

Mixedmethods

Linear systems• Krylov/SVD• Krylov/Eigenvectors

• Numerical efficiency• Stability

• Stability• Error bound• n ≈ 103

• Numerical efficiency• n� 103

• Matching points• Optimality

• Numerical efficiency• Identification from data





Modalmethods


SVD/Lyapmethods



Krylovmethods


Optimizationmethods


Mixedmethods







• MATLAB• SLICOT and LyaPack• J. Rommes tools• M.E.S.S (under development)• MORPack (works with ANSYS)• MOREMBS





Modalmethods


SVD/Lyapmethods



Krylovmethods


Optimizationmethods


Mixedmethods







moremoreΣ

(A,B,C,D)i

Σ

Σ

(A, B, C, D)i


Kr(A,B)

AP + PAT + BBT = 0

WTV



Model Reduction (Approximation) of Large-Scale Systems

Overview of the model approximation methodsLecture 2

C. Poussot-Vassal, P. Vuillemin & I. Pontes Duff

EDSYS, April 4-7th, 2016 (Toulouse, France)

moremoreΣ

(A,B,C,D)i

Σ

Σ

(A, B, C, D)i


Kr(A,B)

AP + PAT + BBT = 0

WTV

DAE/ODE


Data

ReducedDAE/ODE






Model Reduction (Approximation) of Large-Scale …w3.onera.fr/more/sites/w3.onera.fr.more/files/2016 - lecture 02... ·...

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