Model Reduction (Approximation) of Large-Scale Systems - Introduction … - lecture 0… ·...
Transcript of 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
Lecture outines Motivating examples Norm and linear algebra reminder Introduction Summary
Outlines
Lecture outinesPractical aspectsSchedule & outlinesReferences
Motivating examples
Norm and linear algebra reminder
Introduction
Summary
C. Poussot-Vassal, P. Vuillemin & I. Pontes Duff [Onera - DCSD] Model Reduction (Approximation) of Large-Scale Systems
Lecture outines 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).
C. Poussot-Vassal, P. Vuillemin & I. Pontes Duff [Onera - DCSD] Model Reduction (Approximation) of Large-Scale Systems
Lecture outines Motivating examples Norm and linear algebra reminder Introduction Summary
Lecture outinesSchedule & outlines
Day 1I 14h-17h - Lecture (Introduction)
Introduction, motivating examples, linear algebra and model reduction problemOverview of the approximation methods
C. Poussot-Vassal, P. Vuillemin & I. Pontes Duff [Onera - DCSD] Model Reduction (Approximation) of Large-Scale Systems
Lecture outines Motivating examples Norm and linear algebra reminder Introduction Summary
Lecture outinesSchedule & outlines
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
C. Poussot-Vassal, P. Vuillemin & I. Pontes Duff [Onera - DCSD] Model Reduction (Approximation) of Large-Scale Systems
Lecture outines Motivating examples Norm and linear algebra reminder Introduction Summary
Lecture outinesSchedule & outlines
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
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C. Poussot-Vassal, P. Vuillemin & I. Pontes Duff [Onera - DCSD] Model Reduction (Approximation) of Large-Scale Systems
Lecture outines Motivating examples Norm and linear algebra reminder Introduction Summary
Lecture outinesSchedule & outlines
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
model reduction toolbox
Kr(A,B)
AP + PAT + BBT = 0
WTV
C. Poussot-Vassal, P. Vuillemin & I. Pontes Duff [Onera - DCSD] Model Reduction (Approximation) of Large-Scale Systems
Lecture outines Motivating examples Norm and linear algebra reminder Introduction Summary
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)
C. Poussot-Vassal, P. Vuillemin & I. Pontes Duff [Onera - DCSD] Model Reduction (Approximation) of Large-Scale Systems
Lecture outines Motivating examples Norm and linear algebra reminder Introduction Summary
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.
C. Poussot-Vassal, P. Vuillemin & I. Pontes Duff [Onera - DCSD] Model Reduction (Approximation) of Large-Scale Systems
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
Lecture outines Motivating examples Norm and linear algebra reminder Introduction Summary
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
Norm and linear algebra reminder
Introduction
Summary
C. Poussot-Vassal, P. Vuillemin & I. Pontes Duff [Onera - DCSD] Model Reduction (Approximation) of Large-Scale Systems
Lecture outines Motivating examples Norm and linear algebra reminder 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
C. Poussot-Vassal, P. Vuillemin & I. Pontes Duff [Onera - DCSD] Model Reduction (Approximation) of Large-Scale Systems
Lecture outines Motivating examples Norm and linear algebra reminder Introduction Summary
Motivating examplesProblem statement and scope
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
C. Poussot-Vassal, P. Vuillemin & I. Pontes Duff [Onera - DCSD] Model Reduction (Approximation) of Large-Scale Systems
Input-outputfrequency data
Finite orderlarge-scale linear model
Infinite orderlinear model
moremoreΣ
(A,B,C,D)i
Σ
Σ
(A, B, C, D)i
model reduction toolbox
Kr(A,B)
AP + PAT + BBT = 0
WTV A reduced-orderlinear dynamical system
Lecture outines Motivating examples Norm and linear algebra reminder Introduction Summary
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... ?
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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.
C. Poussot-Vassal, P. Vuillemin & I. Pontes Duff [Onera - DCSD] Model Reduction (Approximation) of Large-Scale Systems
Lecture outines Motivating examples Norm and linear algebra reminder Introduction Summary
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... ?
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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.
C. Poussot-Vassal, P. Vuillemin & I. Pontes Duff [Onera - DCSD] Model Reduction (Approximation) of Large-Scale Systems
Lecture outines Motivating examples Norm and linear algebra reminder Introduction Summary
Motivating examplesProblem statement and scope
% 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
C. Poussot-Vassal, P. Vuillemin & I. Pontes Duff [Onera - DCSD] Model Reduction (Approximation) of Large-Scale Systems
Lecture outines Motivating examples Norm and linear algebra reminder Introduction Summary
Motivating examplesBuilding
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Obj Earthquake prevention, structure weight reductionI Mass damper like modelsI Los-Angeles Hospital building model: 1 input, 1 output, n = 48
C. Poussot-Vassal, P. Vuillemin & I. Pontes Duff [Onera - DCSD] Model Reduction (Approximation) of Large-Scale Systems
Lecture outines Motivating examples Norm and linear algebra reminder Introduction Summary
Motivating examplesCD player
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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
C. Poussot-Vassal, P. Vuillemin & I. Pontes Duff [Onera - DCSD] Model Reduction (Approximation) of Large-Scale Systems
Lecture outines Motivating examples Norm and linear algebra reminder Introduction Summary
Motivating examplesISS space station
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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
C. Poussot-Vassal, P. Vuillemin & I. Pontes Duff [Onera - DCSD] Model Reduction (Approximation) of Large-Scale Systems
Lecture outines Motivating examples Norm and linear algebra reminder Introduction Summary
Motivating examplesClamped beam system
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Obj Control of flexible structuresI Elastic model present in MEMS, industryI Beam model: 1 inputs, 1 outputs, n = 348
C. Poussot-Vassal, P. Vuillemin & I. Pontes Duff [Onera - DCSD] Model Reduction (Approximation) of Large-Scale Systems
Lecture outines Motivating examples Norm and linear algebra reminder Introduction Summary
Motivating examplesFluid dynamics
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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
C. Poussot-Vassal, P. Vuillemin & I. Pontes Duff [Onera - DCSD] Model Reduction (Approximation) of Large-Scale Systems
Lecture outines Motivating examples Norm and linear algebra reminder Introduction Summary
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
C. Poussot-Vassal, P. Vuillemin & I. Pontes Duff [Onera - DCSD] Model Reduction (Approximation) of Large-Scale Systems
Lecture outines Motivating examples Norm and linear algebra reminder Introduction Summary
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
C. Poussot-Vassal, P. Vuillemin & I. Pontes Duff [Onera - DCSD] Model Reduction (Approximation) of Large-Scale Systems
Lecture outines Motivating examples Norm and linear algebra reminder Introduction Summary
Motivating examplesRiver system
Obj Electricity management, water regulationI Distributed systems, delays, parameter varyingI Saint Venant equations: 2 inputs, 1 output, n =∞
C. Poussot-Vassal, P. Vuillemin & I. Pontes Duff [Onera - DCSD] Model Reduction (Approximation) of Large-Scale Systems
Lecture outines Motivating examples Norm and linear algebra reminder Introduction Summary
Motivating examplesBiological system
Obj Evaluate and control cell proliferation dynamicsI Markovian processI No input, 1 output, n ≈ 109
C. Poussot-Vassal, P. Vuillemin & I. Pontes Duff [Onera - DCSD] Model Reduction (Approximation) of Large-Scale Systems
Lecture outines Motivating examples Norm and linear algebra reminder Introduction Summary
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
C. Poussot-Vassal, P. Vuillemin & I. Pontes Duff [Onera - DCSD] Model Reduction (Approximation) of Large-Scale Systems
Lecture outines Motivating examples Norm and linear algebra reminder Introduction Summary
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
C. Poussot-Vassal, P. Vuillemin & I. Pontes Duff [Onera - DCSD] Model Reduction (Approximation) of Large-Scale Systems
Lecture outines Motivating examples Norm and linear algebra reminder Introduction Summary
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.
C. Poussot-Vassal, P. Vuillemin & I. Pontes Duff [Onera - DCSD] Model Reduction (Approximation) of Large-Scale Systems
Lecture outines Motivating examples Norm and linear algebra reminder Introduction Summary
Motivating examplesand... connection with control engineer problems
Original
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Truncation ratio k/n: 0.051, with error of 0.013
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Trade-off between accuracy and complexity» funImageSVanalysis(’airplane.jpg’,[.05 .1 .2 .5],’gray’)
C. Poussot-Vassal, P. Vuillemin & I. Pontes Duff [Onera - DCSD] Model Reduction (Approximation) of Large-Scale Systems
Lecture outines Motivating examples Norm and linear algebra reminder Introduction Summary
Outlines
Lecture outines
Motivating examples
Norm and linear algebra reminderH∞-normH2-normH2,Ω-normMatrix factorisation
Introduction
Summary
C. Poussot-Vassal, P. Vuillemin & I. Pontes Duff [Onera - DCSD] Model Reduction (Approximation) of Large-Scale Systems
Lecture outines Motivating examples Norm and linear algebra reminder 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)
C. Poussot-Vassal, P. Vuillemin & I. Pontes Duff [Onera - DCSD] Model Reduction (Approximation) of Large-Scale Systems
Lecture outines Motivating examples Norm and linear algebra reminder Introduction Summary
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.
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C. Poussot-Vassal, P. Vuillemin & I. Pontes Duff [Onera - DCSD] Model Reduction (Approximation) of Large-Scale Systems
H(s) =1
s2 + 0.1s+ 110.0125
Lecture outines Motivating examples Norm and linear algebra reminder Introduction Summary
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)
C. Poussot-Vassal, P. Vuillemin & I. Pontes Duff [Onera - DCSD] Model Reduction (Approximation) of Large-Scale Systems
Lecture outines Motivating examples Norm and linear algebra reminder Introduction Summary
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.
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C. Poussot-Vassal, P. Vuillemin & I. Pontes Duff [Onera - DCSD] Model Reduction (Approximation) of Large-Scale Systems
H(s) =1
s2 + 0.1s+ 12.2361
Lecture outines Motivating examples Norm and linear algebra reminder Introduction Summary
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)
C. Poussot-Vassal, P. Vuillemin & I. Pontes Duff [Onera - DCSD] Model Reduction (Approximation) of Large-Scale Systems
Lecture outines Motivating examples Norm and linear algebra reminder Introduction Summary
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.
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C. Poussot-Vassal, P. Vuillemin & I. Pontes Duff [Onera - DCSD] Model Reduction (Approximation) of Large-Scale Systems
H(s) =1
s2 + 0.1s+ 12.1914
Lecture outines Motivating examples Norm and linear algebra reminder Introduction Summary
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)
C. Poussot-Vassal, P. Vuillemin & I. Pontes Duff [Onera - DCSD] Model Reduction (Approximation) of Large-Scale Systems
Lecture outines Motivating examples Norm and linear algebra reminder Introduction Summary
Norm and linear algebra reminderMatrix factorisation
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
C. Poussot-Vassal, P. Vuillemin & I. Pontes Duff [Onera - DCSD] Model Reduction (Approximation) of Large-Scale Systems
Lecture outines Motivating examples Norm and linear algebra reminder Introduction Summary
Norm and linear algebra reminderMatrix factorisation
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)
C. Poussot-Vassal, P. Vuillemin & I. Pontes Duff [Onera - DCSD] Model Reduction (Approximation) of Large-Scale Systems
Lecture outines Motivating examples Norm and linear algebra reminder Introduction Summary
Norm and linear algebra reminderMatrix factorisation
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)
C. Poussot-Vassal, P. Vuillemin & I. Pontes Duff [Onera - DCSD] Model Reduction (Approximation) of Large-Scale Systems
Lecture outines Motivating examples Norm and linear algebra reminder Introduction Summary
Norm and linear algebra reminderMatrix factorisation
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
C. Poussot-Vassal, P. Vuillemin & I. Pontes Duff [Onera - DCSD] Model Reduction (Approximation) of Large-Scale Systems
Lecture outines Motivating examples Norm and linear algebra reminder Introduction Summary
Norm and linear algebra reminderMatrix factorisation
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
C. Poussot-Vassal, P. Vuillemin & I. Pontes Duff [Onera - DCSD] Model Reduction (Approximation) of Large-Scale Systems
Lecture outines Motivating examples Norm and linear algebra reminder Introduction Summary
Norm and linear algebra reminderMatrix factorisation
... 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)
C. Poussot-Vassal, P. Vuillemin & I. Pontes Duff [Onera - DCSD] Model Reduction (Approximation) of Large-Scale Systems
Lecture outines Motivating examples Norm and linear algebra reminder Introduction Summary
Norm and linear algebra reminderMatrix factorisation
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.
C. Poussot-Vassal, P. Vuillemin & I. Pontes Duff [Onera - DCSD] Model Reduction (Approximation) of Large-Scale Systems
Lecture outines Motivating examples Norm and linear algebra reminder Introduction Summary
Outlines
Lecture outines
Motivating examples
Norm and linear algebra reminder
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
C. Poussot-Vassal, P. Vuillemin & I. Pontes Duff [Onera - DCSD] Model Reduction (Approximation) of Large-Scale Systems
Lecture outines Motivating examples Norm and linear algebra reminder Introduction Summary
IntroductionThe big picture and classification of the linear system problems
C. Poussot-Vassal, P. Vuillemin & I. Pontes Duff [Onera - DCSD] Model Reduction (Approximation) of Large-Scale Systems
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
Lecture outines Motivating examples Norm and linear algebra reminder Introduction Summary
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)
C. Poussot-Vassal, P. Vuillemin & I. Pontes Duff [Onera - DCSD] Model Reduction (Approximation) of Large-Scale Systems
Input-outputfrequency data
Finite orderlarge-scale linear model
Infinite orderlinear model
moremoreΣ
(A,B,C,D)i
Σ
Σ
(A, B, C, D)i
model reduction toolbox
Kr(A,B)
AP + PAT + BBT = 0
WTV A reduced-orderlinear dynamical system
Lecture outines Motivating examples Norm and linear algebra reminder Introduction Summary
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
C. Poussot-Vassal, P. Vuillemin & I. Pontes Duff [Onera - DCSD] Model Reduction (Approximation) of Large-Scale Systems
Input-outputfrequency data
Finite orderlarge-scale linear model
Infinite orderlinear model
moremoreΣ
(A,B,C,D)i
Σ
Σ
(A, B, C, D)i
model reduction toolbox
Kr(A,B)
AP + PAT + BBT = 0
WTV A reduced-orderlinear dynamical system
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
Lecture outines Motivating examples Norm and linear algebra reminder Introduction Summary
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
C. Poussot-Vassal, P. Vuillemin & I. Pontes Duff [Onera - DCSD] Model Reduction (Approximation) of Large-Scale Systems
Input-outputfrequency data
Finite orderlarge-scale linear model
Infinite orderlinear model
moremoreΣ
(A,B,C,D)i
Σ
Σ
(A, B, C, D)i
model reduction toolbox
Kr(A,B)
AP + PAT + BBT = 0
WTV A reduced-orderlinear dynamical system
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
Lecture outines Motivating examples Norm and linear algebra reminder Introduction Summary
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
C. Poussot-Vassal, P. Vuillemin & I. Pontes Duff [Onera - DCSD] Model Reduction (Approximation) of Large-Scale Systems
Input-outputfrequency data
Finite orderlarge-scale linear model
Infinite orderlinear model
moremoreΣ
(A,B,C,D)i
Σ
Σ
(A, B, C, D)i
model reduction toolbox
Kr(A,B)
AP + PAT + BBT = 0
WTV A reduced-orderlinear dynamical system
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
Lecture outines Motivating examples Norm and linear algebra reminder Introduction Summary
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)
C. Poussot-Vassal, P. Vuillemin & I. Pontes Duff [Onera - DCSD] Model Reduction (Approximation) of Large-Scale Systems
Lecture outines Motivating examples Norm and linear algebra reminder Introduction Summary
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)
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 (.)
C. Poussot-Vassal, P. Vuillemin & I. Pontes Duff [Onera - DCSD] Model Reduction (Approximation) of Large-Scale Systems
Lecture outines Motivating examples Norm and linear algebra reminder Introduction Summary
IntroductionLinear model approximation objective (with realization)
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 , (17)
the model approximation problem consists in finding H of order r n
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)
that well reproduces the input-output behaviour of H.
C. Poussot-Vassal, P. Vuillemin & I. Pontes Duff [Onera - DCSD] Model Reduction (Approximation) of Large-Scale Systems
Lecture outines Motivating examples Norm and linear algebra reminder Introduction Summary
IntroductionLinear model approximation objective (with realization)
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 , (17)
the model approximation problem consists in finding H of order r n
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)
that well reproduces the input-output behaviour of H.
"Well reproduce..."?H is a "good" approximation of H if E(t) = y(t)− y(t) is "small"
C. Poussot-Vassal, P. Vuillemin & I. Pontes Duff [Onera - DCSD] Model Reduction (Approximation) of Large-Scale Systems
Lecture outines Motivating examples Norm and linear algebra reminder Introduction Summary
Outlines
Lecture outines
Motivating examples
Norm and linear algebra reminder
Introduction
SummaryThe big pictureApproximation objectives
C. Poussot-Vassal, P. Vuillemin & I. Pontes Duff [Onera - DCSD] Model Reduction (Approximation) of Large-Scale Systems
Lecture outines Motivating examples Norm and linear algebra reminder Introduction Summary
SummaryThe big picture
C. Poussot-Vassal, P. Vuillemin & I. Pontes Duff [Onera - DCSD] Model Reduction (Approximation) of Large-Scale Systems
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
Lecture outines Motivating examples Norm and linear algebra reminder Introduction Summary
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.
C. Poussot-Vassal, P. Vuillemin & I. Pontes Duff [Onera - DCSD] Model Reduction (Approximation) of Large-Scale Systems
Lecture outines Motivating examples Norm and linear algebra reminder Introduction Summary
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.
C. Poussot-Vassal, P. Vuillemin & I. Pontes Duff [Onera - DCSD] Model Reduction (Approximation) of Large-Scale Systems
Lecture outines Motivating examples Norm and linear algebra reminder Introduction Summary
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.
C. Poussot-Vassal, P. Vuillemin & I. Pontes Duff [Onera - DCSD] Model Reduction (Approximation) of Large-Scale Systems
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