Model Reduction (Approximation) of Large-Scale Systems - Introduction … - lecture 0… ·...

Lecture outines Motivating examples Norm and linear algebra reminder Introduction Summary

Model Reduction (Approximation) of Large-Scale Systems

Introduction, motivating examples and problem formulationLecture 1

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


Outlines

Lecture outinesPractical aspectsSchedule & outlinesReferences

Motivating examples

Norm and linear algebra reminder

Introduction

Summary



Lecture outinesPractical aspects

Practical aspectsLecture (14 hours) and two Labs (6 hours).The lecture is addressed to people with basic knowledge in dynamical systems and linearalgebra (e.g. Master, Engineer or Ph.D. degree).

Motivation (sketch) & objectiveI Provide an overview of the model reduction problem within the LTI caseI Present theoretical ideas (classical and advanced techniques)I Present practical issuesI Provide practical illustration (by means of 2 Matlab-based Labs)I Introduce recent advances within model approximationI Provide numerical tools

Material provided: slides, references and numerical tools (MORE toolbox).



Lecture outinesSchedule & outlines

Day 1I 14h-17h - Lecture (Introduction)

Introduction, motivating examples, linear algebra and model reduction problemOverview of the approximation methods




Day 2I 9h-12h - Lecture (Realization-based I)

Gramian, SVD and modal techniquesMoment matching and Krylov subspaces techniques

I 14h-17h - Lab 1Application of the SVD techniques, the Arnoldi procedure and Krylov subspace




Day 3I 9h-12h - Lecture (Realization-based II)

Generalized Krylov subspaces, tangential interpolation techniquesH2 model approximation : projection point of viewH2 model approximation : optimization point of view

I 14h-17h - Lecture (Realization free and delay structured reduced order models)The Loewner frameworkApplication to delay and irrational dynamical models

−15

−10

−5

0 0 5 10 15 20 25 30 35 40

0.7

0.75

0.8

0.85

0.9

0.95

1

Im(λ)Real(λ)

H2 n

orm

err

or

100

101

102

103

−100

−90

−80

−70

−60

−50

−40

Mag

nitu

de (

dB)

Bode Diagram

Frequency (rad/sec)




Day 4I 9h-12h - Lab 2

Generalized Krylov subspaces techniques and glimpse of the MORE ToolboxI 14h-16h - Lecture (Further issues, discussion and tools)

Applications on industrial and academic use casesOverview of the lecture, tools and discussions

moremoreΣ

(A,B,C,D)i

Σ

Σ

(A, B, C, D)i


Kr(A,B)

AP + PAT + BBT = 0

WTV



Lecture outinesReferences

Some relevant references (numerical & control communities)This course has been largely inspired from

I Thanos Antoulas, RICE University slidesI David Amsallem & Charbel Farhat, Stanford University slidesI Matthias Kawski, Matlab examplesI Thanos Antoulas, Approximation of Large Scale Dynamical SystemsI Yousef Saad, Iterative methods for sparse linear systems (2nd edition)

PhDs Grimme (1997), Gugercin (2003), Vuillemin (2014)Articles Wilson (1974), Moore (1981), Ruhe (1994), Helmersson (1994), Boley (1994),

Lehoucq (1996), Ebihara (2004), Gallivan (2006), Gugercin (2008), Simoncini(2009), Van Dooren (2010), Poussot-Vassal (2010-2011), Vuillemin (2012-2013)



Lecture outinesReferences1

MOdel REduction toolboxI It’s a MATLAB toolbox (tested on Matlab 2010Rb),I Aim at approximating medium(large)-scale and infinite dimensional LTI modelsI http://w3.onera.fr/more/

1 C. Poussot-Vassal and P. Vuillemin, "Introduction to MORE: a MOdel REduction Toolbox", inProceedings of the IEEE Multi-conference on Systems and Control, Dubrovnik, Croatia, October, 2012, pp.776-781.


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






Outlines

Lecture outines

Motivating examplesProblem statement and scopeBuildingCD playerISS space stationClamped beam systemFluid dynamicsWeather forecasting systemVery large scale integration systemRiver systemBiological systemAerospaceand... connection with control engineer problems


Introduction

Summary



Motivating examplesProblem statement and scope

Digitalization and computer-based modeling and studies are crucial steps for anysystem, concept or physical phenomena understanding.

Problem: numerical dynamical models are too complex and parameter dependent

Finite machine precision, computational burden and memory management:I induces important time consumptionI generate inaccurate results

Actual numerical toolsI limit the use of class and complexity models




Solution: provide robust and efficient numerical tools to simplify dynamical models

The main objectives are to save time and improve quality, by(T) speeding up simulation time and reducing computation burden(Q) enhancing simulation accuracy and in memory managementand extend scope, by(S) tailoring larger and more complex dynamical model class to standard tools


Input-outputfrequency data

Finite orderlarge-scale linear model

Infinite orderlinear model

moremoreΣ

(A,B,C,D)i

Σ

Σ

(A, B, C, D)i


Kr(A,B)

AP + PAT + BBT = 0

WTV A reduced-orderlinear dynamical system


Motivating examplesProblem statement and scope2

How is it possible to approximate a linear dynamical system of large order with alower order one which can be used in place for simulation/control/analysis... ?

10−2

10−1

100

101

102

−20

−10

0

10

20

30

40

50

60

70

80

Mag

nitu

de (

dB)

Bode Diagram

Frequency (rad/sec)

I Widely used in mechanical engineering (e.g. civilian, MEMS, aeronautics...)I Model of 348 statesI

2 F. Leibfritz, "COMPle ib, COnstraint Matrix-optimization Problem LIbrary - a collection of test examplesfor nonlinear semidefinite programs, control system design and related problems", Universitat Trier, Tech. Rep.,2003.



Motivating examplesProblem statement and scope2

How is it possible to approximate a linear dynamical system of large order with alower order one which can be used in place for simulation/control/analysis... ?

10−2

10−1

100

101

102

−20

−10

0

10

20

30

40

50

60

70

80

Mag

nitu

de (

dB)

Bode Diagram

Frequency (rad/s)

Original (n=348)Reduced (r=16)

I Widely used in mechanical engineering (e.g. civilian, MEMS, aeronautics...)I Model of 348 states approximated with a model of 16 states (≈1/20)I Objective: provide methods & tool for engineers & researchers2 F. Leibfritz, "COMPle ib, COnstraint Matrix-optimization Problem LIbrary - a collection of test examples

for nonlinear semidefinite programs, control system design and related problems", Universitat Trier, Tech. Rep.,2003.




% Reduct ion o r d e r o b j e c t i v er = 16 ;% Load COMPleib model and c o n s t r u c t the s t a t e space modelname = ’CBM’ ;[A, B1 ,B, C1 ,C , D11 , D12 , D21 , nx , nw , nu , nz , ny ] = COMPleib (name ) ;s y s = s s (A,B,C , 0 ) ;% Balanced Trunca t i on ( Robust Con t r o l Toolbox , Matlab )sysBT = reduce ( sys , r ) ;% ITIA ( u s i n g the COMPleib name as i npu t )opt . r e s t a r t = 0 ;[ sys IT IA , out ] = moreLTI (name , r , ’ ITIA ’ , opt ) ;

» startExampleLecture1_1



Motivating examplesBuilding

100

101

102

103

−100

−90

−80

−70

−60

−50

−40

Mag

nitu

de (

dB)

Bode Diagram

Frequency (rad/sec)

Obj Earthquake prevention, structure weight reductionI Mass damper like modelsI Los-Angeles Hospital building model: 1 input, 1 output, n = 48



Motivating examplesCD player

−200

−150

−100

−50

0

50

100

150

200From: In(1)

To:

Out

(1)

100

102

104

106

−200

−150

−100

−50

0

50

100

To:

Out

(2)

From: In(2)

100

102

104

106

Bode Diagram

Frequency (rad/sec)

Mag

nitu

de (

dB)

Obj Understand and optimize mechanical systems, control the reading opticsI Electro-mechanical model (DVD, CD-player, HDD,...)I CD model: 2 inputs, 2 outputs, n = 120



Motivating examplesISS space station

−200

−150

−100

−50

0From: In(1)

To:

Out

(1)

−200

−150

−100

−50

0

To:

Out

(2)

10−1

100

101

102

103

−200

−150

−100

−50

0

To:

Out

(3)

From: In(2)

10−1

100

101

102

103

From: In(3)

10−1

100

101

102

103

Bode Diagram

Frequency (rad/sec)

Mag

nitu

de (

dB)

Obj Control a flexible structure of the ISS station modules (due to solar panels...)I Structural model with 60 vibration modesI ISS model: 3 inputs, 3 outputs, n = 270



Motivating examplesClamped beam system

10−2

10−1

100

101

102

−20

−10

0

10

20

30

40

50

60

70

80

Mag

nitu

de (

dB)

Bode Diagram

Frequency (rad/sec)

Obj Control of flexible structuresI Elastic model present in MEMS, industryI Beam model: 1 inputs, 1 outputs, n = 348



Motivating examplesFluid dynamics

100

101

−10

0

10

20

30

40

50

ω [rad/s]

Gai

n [d

B]

Bode diagram

MORE toolbox (20 states, obtained in 1h20)Original model (678735 states)Optimal interpolation points

Obj Control and simulate fluid mechanical systems (video)I Discretisation of Navier Stoke equations at varying Reynolds numbersI LTI model: 1 inputs, 1 outputs, n = 750, 000



Motivating examplesWeather forecasting system

Obj Prediction of natural dynamical events and related security issuesI Model obtained from discretization of PDEsI Model adjustment by simulation and refinement, Riccati equations, optimal

sensor positioning



Motivating examplesVery large scale integration system

Obj Intensive simulations are required to verify that internal electromagnetic fields donot significantly delay or distort circuit signals.

I Discretized Maxwell equations / RLC structureI Interconnected, DAE, passivity



Motivating examplesRiver system

Obj Electricity management, water regulationI Distributed systems, delays, parameter varyingI Saint Venant equations: 2 inputs, 1 output, n =∞



Motivating examplesBiological system

Obj Evaluate and control cell proliferation dynamicsI Markovian processI No input, 1 output, n ≈ 109



Motivating examplesAerospace

Within aircraft manufacturer, models are built using finite elements methods,dynamical properties are adjusted for every mass cases, flight points, and finallyadjustments are achieved with wind tunnel test.

Obj Control, reduce weight, analyze...I Models rigid, flexible, aerodynamical delay dynamicsI Models are scattered and state space is partly/poorly known



Motivating examplesand... connection with control engineer problems

SimulationI Memory management, ODE solversI "Passive" optimization of dynamical systems

Observers / controllersI Observer / Controller design: robust, optimal, predictive, . . . are numerical based

techniques (e.g. involve SDP, LMI, nonlinear optimization, Riccati)I LPV, LFT formulation enhance even more these problemsI H∞ (iterative, Hamiltonian matrix)I H2 (Lyapunov equations) norms computation



Motivating examplesand... connection with control engineer problems3

AnalysisI H∞ computation, . . . no eigenvalues along the imaginary axis of (with D = 0)

H(A,B,C, γ) =[

A γBBT

−γCTC −AT

](1)

I Stability via Lyapunov

and also...I Eigenvalue, Kalman filter designI Data mining (process quality, pertinent data)I LearningI Image compression

3 S. Boyd, V. Balakrishnan and A. Kabamka, "On computing the H∞ norm of a transfer matrix", inProceedings of the American Control Conference, Atlanta, Georgia, June 1988, pp.396-397.



Motivating examplesand... connection with control engineer problems

Original

100 200 300 400 500 600 700

50

100

150

200

250

300

350

400

10−2

10−1

100

10−4

10−3

10−2

10−1

100

k/n

σ k/σm

ax

Truncation ratio k/n: 0.015, with error of 0.049

100 200 300 400 500 600 700

50

100

150

200

250

300

350

400

Truncation ratio k/n: 0.051, with error of 0.013

100 200 300 400 500 600 700

50

100

150

200

250

300

350

400

Trade-off between accuracy and complexity» funImageSVanalysis(’airplane.jpg’,[.05 .1 .2 .5],’gray’)



Outlines

Lecture outines

Motivating examples

Norm and linear algebra reminderH∞-normH2-normH2,Ω-normMatrix factorisation

Introduction

Summary



Norm and linear algebra reminderH∞-norm

Definition: H∞-normThe H∞-norm of a n-th order stable system H(s) is given as,

‖H‖H∞:= sup

ω∈Rσ (H(jω))

:= maxw∈L2

||z||2||w||2

(2)



Norm and linear algebra reminderH∞-norm

Interpretation of the H∞-normI Physically, the H∞-norm of a SISO system describes the maximum value of the

of its frequency response over the entire spectrum.I In other words, it is the largest gain if the system is fed by harmonic input signal.

10−1

100

101

−40

−30

−20

−10

0

10

20

30

Mag

nit

ud

e (d

B)

Bode Diagram

Frequency (rad/s)


H(s) =1

s2 + 0.1s+ 110.0125


Norm and linear algebra reminderH2-norm

Definition: H2-normThe H2-norm of a n-th order strictly proper stable system H(s) is given as,

||H||2H2:= trace

( 12π

∫ ∞−∞

(H(iν)HT (iν)

)dν

):= trace

( 12π

∫ ∞−∞||H(iν)||2F dν

) (3)



Norm and linear algebra reminderH2-norm

Interpretation of the H2-normI Physically, the H2 norm of a SISO system describes the integral of its frequency

response over the entire spectrum.

10−1

100

101

−40

−30

−20

−10

0

10

20

30

Mag

nit

ud

e (d

B)

Bode Diagram

Frequency (rad/s)


H(s) =1

s2 + 0.1s+ 12.2361


Norm and linear algebra reminderH2,Ω-norm

Definition: H2,Ω-norm (e.g. Ω = [−ω ω])The H2,Ω-norm of a n-th order proper stable system H(s) is given as,

||H||2H2,Ω:= trace

( 12π

∫Ω

(H(iν)HT (iν)

)dν

):= trace

( 12π

∫Ω||H(iν)||2F dν

) (4)



Norm and linear algebra reminderH2,Ω-norm

Interpretation of the H2,Ω-normI Physically, the H2,Ω norm of a SISO system describes the integral of its

frequency response over the limited spectrum Ω.

10−1

100

101

−40

−30

−20

−10

0

10

20

30

Mag

nit

ud

e (d

B)

Bode Diagram

Frequency (rad/s)


H(s) =1

s2 + 0.1s+ 12.1914


Norm and linear algebra reminderMatrix factorisation

Frobenius norm, normThe Frobenius norm ||A||F is given as,

||A||F =

√√√√min(m,n)∑i=1

σ2i =√

trace(AAT ) (5)

(quite easy to compute)




SVD factorization, svdDecompose a matrix A such as,

I A = UΣV T ∈ Rn×m

I Σ = diag(σ1, . . . , σn), are the singular values (with σ1 ≥ · · · ≥ σn)I U = (u1, . . . , un), UUT = In are the left singular vectorsI V = (v1, . . . , vm), V V T = Im are the right singular vectors




LU factorization, luDecompose a matrix A into a lower L and upper U matrix.

I A = LU

I Very useful to solve Ax = b, such as,

L(Ux) = b (6)

then, let y = Ux, solveLy = b (7)

y elements are successively founded.I det(A) = det(L) det(U)




Cholesky factorization, cholDecompose a positive definite matrix A such as,

I A = LLT

I L, lowerI Very useful to compute A−1, to solve Ax = b, Lyapunov equations

(AP + PAT +BBT = 0, with P positive definite)




QR factorization, qrDecompose a matrix A such as,

I A = QR

I Q is orthogonal (i.e. QQT = QTQ = I and QT = Q−1) and R is uppertriangular

I Basis of eigenvalue problemsI The first k columns of Q form an orthonormal basis for the span of the first k

columns of A for any 1 ≤ k ≤ n




Eigenvalues factorization, eig or eigs

Given a square matrix A ∈ Rn×n, the left / right eigenvectors and eigenvalues

AR = RΛLA = ΛL

Λ = diag(λ1, . . . , λn)(8)

Present in MANY control engineer problems




... e.g. Sylvester equation solverFind X,

AX +XH +M = 0 (9)

is equivalent to, solve the eigenvalue problem:[A M0 −H

] [V1V2

]=[V1V2

]Z (10)

Indeed, we have: AV1 +MV2 = V1Z

−HV2 = V2Z(11)

thus,AV1 +MV2 = −V1V

−12 HV2

AV1 +MV2 + V1V−12 HV2 = 0

AV1V−12︸︷︷︸X

+ V1V−12︸︷︷︸X

H +M = 0 (12)




Definition: Singular Value DecompositionGiven a matrix An×m, its singular value decomposition is defined as follows:

A = UΣV T where Σ = diag(σ1, . . . , σn) ∈ Rn×m (13)

where σ1(A) ≥ · · · ≥ σn(A) ≥ 0 are the singular values and the columns of theorthogonal matrices U = (u1, . . . , un) and V = (v1, . . . , vm) are the left and rightsingular vector of A respectively.

I SVD provides a measure of the energy (or information) repartition of a matrix.I σ1(A) is the 2-induced norm of A.I The SVD induces the Dyadic decomposition of A:

A = σ1u1vT1 + σ2u2v

T2 + · · ·+ σrurv

Tr (14)

with rank(A) = r.



Outlines

Lecture outines

Motivating examples


IntroductionThe big picture and classification of the linear system problemsCase 1: Data-based model approximationCase 2: The realization-based LTI model approximationCase 3: Realization free model approximationLinear model approximation objective

Summary



IntroductionThe big picture and classification of the linear system problems


DAE/ODE


Data

ReducedDAE/ODE






IntroductionThe big picture and classification of the linear system problems

Dynamical system Hx1(.)x2(.)...

xn(.)

y1(.)y2(.)...

yny (.)

u1(.)u2(.)...

unu (.)

Case 1 Data-driven case: ui(.), and yi(.) are given (marginally treated in this lecture)Case 2 Finite order case: (E,A,B,C,D) is given (first part of the course)Case 3 Infinite order case: H(s) is given (second part of the course)





moremoreΣ

(A,B,C,D)i

Σ

Σ

(A, B, C, D)i


Kr(A,B)

AP + PAT + BBT = 0



IntroductionCase 1: Data-based model approximation

I Given a the ωi, H(ωi) set obtained from experiments or simulationI Find a finite order model that matches the data





moremoreΣ

(A,B,C,D)i

Σ

Σ

(A, B, C, D)i


Kr(A,B)

AP + PAT + BBT = 0


10 20 30 40 50 60 70 80 90 100

−20

−15

−10

−5

0

5

10

15

20

From gust disturbance

Frequency (Hz)

Acc

eler

atio

n (d

B)

WT experimental dataMORE toolbox

(Q) Optimal model found, that interpolates all themeasured points

(S) Tailored to Multi Input Multi Output modelsI Model obtained in few seconds


IntroductionCase 2: The realization-based LTI model approximation

I Given a large-scale linear dynamical realization of a long range aircraftI Find a simpler model that well reproduces the complex one





moremoreΣ

(A,B,C,D)i

Σ

Σ

(A, B, C, D)i


Kr(A,B)

AP + PAT + BBT = 0


10 15 20 25 30 35 40 45 50

10−1

Approximation order, r

Mis

mat

ch e

rror

(ov

er a

lim

ited−

freq

uenc

y ra

nge)

MATLABMORE toolbox

(T) Save engineer time in analysis, controllerdesign, optimization

(Q) Accuracy of 90% with a 97% simpler model(36 instead of 1700 states)

I Outperforms precision of commercial tools


IntroductionCase 3: Realization free model approximation

I Given an infinite dimensional, irrational and delay dependent modelI Find a low order model simpler to simulate and analyse





moremoreΣ

(A,B,C,D)i

Σ

Σ

(A, B, C, D)i


Kr(A,B)

AP + PAT + BBT = 0


10−4

10−2

200700

1400

−70

−65

−60

−55

−50

−45

ω [rad/s]

From qe to z(s,Q)

Q [m3/s]

z [d

B]

10−4

10−2

200700

1400

−80

−70

−60

−50

−40

ω [rad/s]

From qs to z(s,Q)

Q [m3/s]

z [d

B]

10−4

10−2

200700

1400

−70

−65

−60

−55

−50

−45

ω [rad/s]

From qe to z(s,Q)

Q [m3/s]

z [d

B]

10−4

10−2

200700

1400

−80

−70

−60

−50

−40

ω [rad/s]

From qs to z(s,Q)

Q [m3/s]

z [d

B]

(T) Simulation velocity increased with a factor of100, reducing optimization steps

(Q) Accuracy close to 95% with a complexity of 8equations (instead of PDE)

(S) Allows to transform infinite to finite model


IntroductionLinear model approximation objective

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 , (15)



IntroductionLinear model approximation objective


H : C→ Cny×nu , (15)

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

H : C→ Cny×nu , (16)

that well reproduces the input-output behaviour of H.

Dynamical system Hx1(.)x2(.)...

xn(.)

y1(.)y2(.)...

yny (.)

u1(.)u2(.)...

unu (.)

Dynamical system H

x1(.)x2(.)...

xr(.)

y1(.)y2(.)...

yny (.)

u1(.)u2(.)...

unu (.)



IntroductionLinear model approximation objective (with realization)


H : C→ Cny×nu , (17)


H : C→ Cny×nu , (18)

with a given realization e.g.

H :

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

, (19)




IntroductionLinear model approximation objective (with realization)


H : C→ Cny×nu , (17)


H : C→ Cny×nu , (18)

with a given realization e.g.

H :

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

, (19)


"Well reproduce..."?H is a "good" approximation of H if E(t) = y(t)− y(t) is "small"



Outlines

Lecture outines

Motivating examples


Introduction

SummaryThe big pictureApproximation objectives



SummaryThe big picture


DAE/ODE


Data

ReducedDAE/ODE






SummaryApproximation objectives4 5 6 7

Mismatch error

H := argminG ∈ H2

dim(G) = r n

||H −G||H2 (20)

||H||2H2:= trace

( 12π

∫ ∞−∞

(H(iν)HT (iν)

)dν

):= trace

( 12π

∫ ∞−∞||H(iν)||2F dν

) (21)

4 P. Van-Dooren, K. A. Gallivan, and P. A. Absil, "H2-optimal model reduction of MIMO systems",Applied Mathematics Letters, vol. 21(12), December 2008, pp. 53-62.

5 S. Gugercin and A C. Antoulas and C A. Beattie, "H2 Model Reduction for Large Scale LinearDynamical Systems", SIAM Journal on Matrix Analysis and Applications, vol. 30(2), June 2008, pp. 609-638.

6 K. A. Gallivan, A. Vanderope, and P. Van-Dooren, "Model reduction of MIMO systems via tangentialinterpolation", SIAM Journal of Matrix Analysis and Application, vol. 26(2), February 2004, pp. 328-349.

7 C. Poussot-Vassal, "An Iterative SVD-Tangential Interpolation Method for Medium-Scale MIMO SystemsApproximation with Application on Flexible Aircraft", Proceedings of the 50th IEEE CDC - ECC, Orlando, Florida,USA, December, 2011, pp. 7117-7122.



SummaryApproximation objectives8 9 10

Frequency-limited mismatch error

H := argminG ∈ H∞

dim(G) = r n

||H −G||H2,Ω (22)

||H||2H2,Ω:= trace

( 12π

∫Ω

(H(iν)HT (iν)

)dν

):= trace

( 12π

∫Ω||H(iν)||2F dν

) (23)

8 P. Vuillemin, C. Poussot-Vassal and D. Alazard, "H2 optimal and frequency limited approximationmethods for large-scale LTI dynamical systems", in Proceedings of the IFAC Symposium on Systems Structure andControl, Grenoble, France, February, 2013, pp. 719-724.

9 P. Vuillemin, C. Poussot-Vassal and D. Alazard, "A Spectral Expression for the Frequency-LimitedH2-norm", arXiv:1211.1858, 2013.

10 P. Vuillemin, C. Poussot-Vassal and D. Alazard, "Poles Residues Descent Algorithm for OptimalFrequency-Limited H2 Model Approximation", submitted to CDC, Florence, Italy, December, 2013.



SummaryApproximation objectives11

Mismatch objective and eigenvector preservation

H := argminG ∈ H2

dim(G) = r nλk(G) ⊆ λ(H) k = 1, . . . , i1 < r

||H −G||H2 (24)

I More than a H2 (sub-optimal) criteriaI Keep some user defined eigenvalues...

11 C. Poussot-Vassal and P. Vuillemin, "An Iterative Eigenvector Tangential Interpolation Algorithm forLarge-Scale LTI and a Class of LPV Model Approximation", in Proceedings of the European Control Conference,Zurich, Switzerland, July, 2013.



Model Reduction (Approximation) of Large-Scale Systems

Introduction, motivating examples and problem formulationLecture 1

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 Systems - Introduction … - lecture 0… ·...

Documents

Transcript of Model Reduction (Approximation) of Large-Scale Systems - Introduction … - lecture 0… ·...