Model Reduction (Approximation) of Large-Scale Systems. Introduction, motivating examples and problem formulation Lecture 1

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1 Lecture outines Motivating examples Norm and linear algebra reminder Introduction Summary Model Reduction (Approximation) of Large-Scale Systems Introduction, motivating examples and problem formulation Lecture 1 C. Poussot-Vassal, P. Vuillemin & I. Pontes Duff EDSYS, April 4-7th, 2016 (Toulouse, France) Case 1 u(f ) = [u(f1 )... u(fi )] y(f ) = [y(f1 )... y(fi )] Case 2 E x (t) = Ax(t) + Bu(t) y(t) = Cx(t) + Du(t) Case 3 H(s) = e τ s Σ (A, B, C, D)i Data more AP + P AT + BB T = 0 Σ WTV Kr (A, B) model reduction toolbox Σ (A, B, C, D )i C. Poussot-Vassal, P. Vuillemin & I. Pontes Duff [Onera - DCSD] DAE/ODE State x(t) Rn, n large or infinite Reduced DAE/ODE Reduced state x (t) Rr with r n (+) Simulation (+) Analysis (+) Control (+) Optimization Model Reduction (Approximation) of Large-Scale Systems

2 Lecture outines Practical aspects Schedule & outlines References Outlines Motivating examples Norm and linear algebra reminder Introduction Summary

3 Practical aspects Lecture outines Practical aspects Lecture (14 hours) and two Labs (6 hours). The lecture is addressed to people with basic knowledge in dynamical systems and linear algebra (e.g. Master, Engineer or Ph.D. degree). Motivation (sketch) & objective Provide an overview of the model reduction problem within the LTI case Present theoretical ideas (classical and advanced techniques) Present practical issues Provide practical illustration (by means of 2 Matlab-based Labs) Introduce recent advances within model approximation Provide numerical tools Material provided: slides, references and numerical tools (MORE toolbox).

4 Day 1 Lecture outines Schedule & outlines 14h-17h - Lecture (Introduction) Introduction, motivating examples, linear algebra and model reduction problem Overview of the approximation methods

5 Day 2 9h-12h - Lecture (Realization-based I) Gramian, SVD and modal techniques Moment matching and Krylov subspaces techniques Lecture outines Schedule & outlines 14h-17h - Lab 1 Application of the SVD techniques, the Arnoldi procedure and Krylov subspace

6 Bode Diagram Frequency (rad/sec) Day 3 9h-12h - Lecture (Realization-based II) Generalized Krylov subspaces, tangential interpolation techniques H 2 model approximation : projection point of view H 2 model approximation : optimization point of view Lecture outines Schedule & outlines 14h-17h - Lecture (Realization free and delay structured reduced order models) The Loewner framework Application to delay and irrational dynamical models H 2 norm error Magnitude (db) Real(λ) Im(λ)

7 Day 4 Lecture outines Schedule & outlines 9h-12h - Lab 2 Generalized Krylov subspaces techniques and glimpse of the MORE Toolbox 14h-16h - Lecture (Further issues, discussion and tools) Applications on industrial and academic use cases Overview of the lecture, tools and discussions Σ (A, B, C, D)i Σ more AP + P A T + BB T = 0 model reduction toolbox W T V Kr(A, B) ˆΣ (Â, ˆB, Ĉ, ˆD)i

8 Some relevant references (numerical & control communities) This course has been largely inspired from Thanos Antoulas, RICE University slides David Amsallem & Charbel Farhat, Stanford University slides Matthias Kawski, Matlab examples Thanos Antoulas, Approximation of Large Scale Dynamical Systems Yousef Saad, Iterative methods for sparse linear systems (2nd edition) Lecture outines References 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 ( ), Vuillemin ( )

9 Lecture outines Motivating examples Norm and linear algebra reminder Introduction Summary Lecture outines References1 MOdel REduction toolbox I It s a MATLAB toolbox (tested on Matlab 2010Rb), I Aim at approximating medium(large)-scale and infinite dimensional LTI models I Case 1 u(f ) = [u(f1 )... u(fi )] y(f ) = [y(f1 )... y(fi )] Case 2 E x (t) = Ax(t) + Bu(t) y(t) = Cx(t) + Du(t) Case 3 H(s) = e τ s Σ (A, B, C, D)i Data more AP + P AT + BB T = 0 Σ WTV Kr (A, B) model reduction toolbox Σ (A, B, C, D )i DAE/ODE State x(t) Rn, n large or infinite Reduced DAE/ODE Reduced state x (t) Rr with r n (+) Simulation (+) Analysis (+) Control (+) Optimization 1 C. Poussot-Vassal and P. Vuillemin, "Introduction to MORE: a MOdel REduction Toolbox", in Proceedings of the IEEE Multi-conference on Systems and Control, Dubrovnik, Croatia, October, 2012, pp C. Poussot-Vassal, P. Vuillemin & I. Pontes Duff [Onera - DCSD] Model Reduction (Approximation) of Large-Scale Systems

10 Lecture outines Outlines Motivating examples Problem statement and scope Building CD player ISS space station Clamped beam system Fluid dynamics Weather forecasting system Very large scale integration system River system Biological system Aerospace and... connection with control engineer problems Norm and linear algebra reminder Introduction Summary

11 Motivating examples Problem statement and scope Digitalization and computer-based modeling and studies are crucial steps for any system, concept or physical phenomena understanding. Problem: numerical dynamical models are too complex and parameter dependent Finite machine precision, computational burden and memory management: induces important time consumption generate inaccurate results Actual numerical tools limit the use of class and complexity models

12 Motivating examples Problem 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 management and extend scope, by (S) tailoring larger and more complex dynamical model class to standard tools Input-output frequency data Finite order large-scale linear model Σ more AP + P A T + BB T = 0 Σ (A, B, C, D)i model reduction toolbox W T V Kr(A, B) ˆΣ (Â, ˆB, Ĉ, ˆD)i A reduced-order linear dynamical system Infinite order linear model

13 Bode Diagram Frequency (rad/sec) Motivating examples Problem statement and scope 2 How is it possible to approximate a linear dynamical system of large order with a lower order one which can be used in place for simulation/control/analysis...? Magnitude (db) Widely used in mechanical engineering (e.g. civilian, MEMS, aeronautics...) Model of 348 states 2 F. Leibfritz, "COMPl eib, 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.

14 Bode Diagram Frequency (rad/s) Motivating examples Problem statement and scope 2 How is it possible to approximate a linear dynamical system of large order with a lower order one which can be used in place for simulation/control/analysis...? Original (n=348) Reduced (r=16) Magnitude (db) Widely used in mechanical engineering (e.g. civilian, MEMS, aeronautics...) Model of 348 states approximated with a model of 16 states ( 1/20) Objective: provide methods & tool for engineers & researchers 2 F. Leibfritz, "COMPl eib, 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.

15 Motivating examples Problem statement and scope % R e d u c t i o n o r d e r o b j e c t i v e r = 1 6 ; % Load COMPleib model and c o n s t r u c t t h e s t a t e s p a c e model name = 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 T r u n c a t i o n ( Robust C o n t r o l Toolbox, Matlab ) sysbt = r e d u c e ( sys, r ) ; % ITIA ( u s i n g t h e COMPleib name as i n p u t ) opt. r e s t a r t = 0 ; [ s y s I T I A, out ] = morelti ( name, r, ITIA, opt ) ;» startexamplelecture1_1

16 Bode Diagram Frequency (rad/sec) Lecture outines Motivating examples Norm and linear algebra reminder Introduction Summary Motivating examples Building Magnitude (db) Obj Earthquake prevention, structure weight reduction Mass damper like models Los-Angeles Hospital building model: 1 input, 1 output, n = 48

17 From: In(1) Bode Diagram Frequency (rad/sec) From: In(2) Lecture outines Motivating examples Norm and linear algebra reminder Introduction Summary Motivating examples CD player To: Out(2) Magnitude (db) To: Out(1) Obj Understand and optimize mechanical systems, control the reading optics Electro-mechanical model (DVD, CD-player, HDD,...) CD model: 2 inputs, 2 outputs, n = 120

18 Lecture outines Motivating examples Norm and linear algebra reminder Introduction Summary Motivating examples ISS space station Bode Diagram From: In(1) From: In(2) From: In(3) 0 To: Out(1) To: Out(2) Magnitude (db) To: Out(3) Frequency (rad/sec) Obj Control a flexible structure of the ISS station modules (due to solar panels...) I Structural model with 60 vibration modes I 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

19 Bode Diagram Frequency (rad/sec) Lecture outines Motivating examples Norm and linear algebra reminder Introduction Summary Motivating examples Clamped beam system Magnitude (db) Obj Control of flexible structures Elastic model present in MEMS, industry Beam model: 1 inputs, 1 outputs, n = 348

20 Motivating examples Fluid dynamics 50 Bode diagram Gain [db] MORE toolbox (20 states, obtained in 1h20) 10 Original model ( states) Optimal interpolation points ω [rad/s] Obj Control and simulate fluid mechanical systems (video) Discretisation of Navier Stoke equations at varying Reynolds numbers LTI model: 1 inputs, 1 outputs, n = 750, 000

21 Motivating examples Weather forecasting system Obj Prediction of natural dynamical events and related security issues Model obtained from discretization of PDEs Model adjustment by simulation and refinement, Riccati equations, optimal sensor positioning

22 Motivating examples Very large scale integration system Obj Intensive simulations are required to verify that internal electromagnetic fields do not significantly delay or distort circuit signals. Discretized Maxwell equations / RLC structure Interconnected, DAE, passivity

23 Motivating examples River system Obj Electricity management, water regulation Distributed systems, delays, parameter varying Saint Venant equations: 2 inputs, 1 output, n =

24 Motivating examples Biological system Obj Evaluate and control cell proliferation dynamics Markovian process No input, 1 output, n 10 9

25 Motivating examples Within aircraft manufacturer, models are built using finite elements methods, dynamical properties are adjusted for every mass cases, flight points, and finally adjustments are achieved with wind tunnel test. Aerospace Obj Control, reduce weight, analyze... Models rigid, flexible, aerodynamical delay dynamics Models are scattered and state space is partly/poorly known

26 Simulation Memory management, ODE solvers "Passive" optimization of dynamical systems Motivating examples and... connection with control engineer problems Observers / controllers Observer / Controller design: robust, optimal, predictive,... are numerical based techniques (e.g. involve SDP, LMI, nonlinear optimization, Riccati) LPV, LFT formulation enhance even more these problems H (iterative, Hamiltonian matrix) H 2 (Lyapunov equations) norms computation

27 Motivating examples and... connection with control engineer problems 3 Analysis H computation,... no eigenvalues along the imaginary axis of (with D = 0) [ ] A γbb T H(A, B, C, γ) = γc T C A T (1) Stability via Lyapunov and also... Eigenvalue, Kalman filter design Data mining (process quality, pertinent data) Learning Image compression 3 S. Boyd, V. Balakrishnan and A. Kabamka, "On computing the H norm of a transfer matrix", in Proceedings of the American Control Conference, Atlanta, Georgia, June 1988, pp

28 Motivating examples and... connection with control engineer problems Original σ k /σ max k/n Truncation ratio k/n: 0.015, with error of Truncation ratio k/n: 0.051, with error of Trade-off between accuracy and complexity» funimagesvanalysis( airplane.jpg,[ ], gray )

29 Lecture outines Outlines Motivating examples Norm and linear algebra reminder H -norm H 2 -norm H 2,Ω -norm Matrix factorisation Introduction Summary

30 Definition: H -norm The H -norm of a n-th order stable system H(s) is given as, Norm and linear algebra reminder H -norm H H := sup σ (H(jω)) := ω R z 2 (2) max w L 2 w 2

31 Interpretation of the H -norm Norm and linear algebra reminder H -norm Physically, the H -norm of a SISO system describes the maximum value of the of its frequency response over the entire spectrum. In other words, it is the largest gain if the system is fed by harmonic input signal. 30 Bode Diagram Magnitude (db) H(s) = s s Frequency (rad/s)

32 Definition: H 2 -norm Norm and linear algebra reminder H 2-norm The H 2 -norm of a n-th order strictly proper stable system H(s) is given as, ( 1 ( ) H 2 H 2 := trace H(iν)H T (iν) dν) 2π ( ) (3) 1 := trace H(iν) 2 F 2π dν

33 Interpretation of the H 2 -norm Norm and linear algebra reminder H 2-norm Physically, the H 2 norm of a SISO system describes the integral of its frequency response over the entire spectrum. 30 Bode Diagram 20 Magnitude (db) H(s) = s s Frequency (rad/s)

34 Definition: H 2,Ω -norm (e.g. Ω = [ ω ω]) Norm and linear algebra reminder H 2,Ω -norm The H 2,Ω -norm of a n-th order proper stable system H(s) is given as, ( 1 ( ) H 2 H 2,Ω := trace H(iν)H T (iν) dν) 2π ( Ω ) (4) 1 := trace H(iν) 2 F 2π dν Ω

35 Interpretation of the H 2,Ω -norm Norm and linear algebra reminder Physically, the H 2,Ω norm of a SISO system describes the integral of its frequency response over the limited spectrum Ω. H 2,Ω -norm 30 Bode Diagram 20 Magnitude (db) H(s) = s s Frequency (rad/s)

36 Norm and linear algebra reminder Matrix factorisation Frobenius norm, norm The Frobenius norm A F is given as, A F = min(m,n) σi trace(aa 2 = T ) (5) i=1 (quite easy to compute)

37 SVD factorization, svd Decompose a matrix A such as, A = UΣV T R n m Norm and linear algebra reminder Σ = diag(σ 1,..., σ n), are the singular values (with σ 1 σ n) U = (u 1,..., u n), UU T = I n are the left singular vectors V = (v 1,..., v m), V V T = I m are the right singular vectors Matrix factorisation

38 LU factorization, lu Decompose a matrix A into a lower L and upper U matrix. A = LU Very useful to solve Ax = b, such as, Norm and linear algebra reminder Matrix factorisation L(Ux) = b (6) then, let y = Ux, solve Ly = b (7) y elements are successively founded. det(a) = det(l) det(u)

39 Cholesky factorization, chol Decompose a positive definite matrix A such as, A = LL T L, lower Norm and linear algebra reminder Very useful to compute A 1, to solve Ax = b, Lyapunov equations (AP + P A T + BB T = 0, with P positive definite) Matrix factorisation

40 QR factorization, qr Decompose a matrix A such as, A = QR Norm and linear algebra reminder Matrix factorisation Q is orthogonal (i.e. QQ T = Q T Q = I and Q T = Q 1 ) and R is upper triangular Basis of eigenvalue problems 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

41 Norm and linear algebra reminder Matrix factorisation Eigenvalues factorization, eig or eigs Given a square matrix A R n n, the left / right eigenvectors and eigenvalues AR = RΛ LA = ΛL Λ = diag(λ 1,..., λ n) (8) Present in MANY control engineer problems

42 ... e.g. Sylvester equation solver Find X, is equivalent to, solve the eigenvalue problem: [ ] [ ] A M V1 0 H V 2 Norm and linear algebra reminder Matrix factorisation AX + XH + M = 0 (9) [ ] V1 = Z (10) V 2 Indeed, we have: { AV1 + MV 2 = V 1 Z HV 2 = V 2 Z thus, AV 1 + MV 2 = V 1 V 1 2 HV 2 AV 1 + MV 2 + V 1 V 1 2 HV 2 = 0 AV 1 V V 1 V 1 2 H + M }{{}}{{} = 0 X X (11) (12)

43 Definition: Singular Value Decomposition Norm and linear algebra reminder Given a matrix A n m, its singular value decomposition is defined as follows: Matrix factorisation A = UΣV T where Σ = diag(σ 1,..., σ n) R n m (13) where σ 1 (A) σ n(a) 0 are the singular values and the columns of the orthogonal matrices U = (u 1,..., u n) and V = (v 1,..., v m) are the left and right singular vector of A respectively. SVD provides a measure of the energy (or information) repartition of a matrix. σ 1 (A) is the 2-induced norm of A. The SVD induces the Dyadic decomposition of A: A = σ 1 u 1 v1 T + σ 2u 2 v2 T + + σrurvt r (14) with rank(a) = r.

44 Lecture outines Outlines Motivating examples Norm and linear algebra reminder Introduction The big picture and classification of the linear system problems Case 1: Data-based model approximation Case 2: The realization-based LTI model approximation Case 3: Realization free model approximation Linear model approximation objective Summary

45 Lecture outines Motivating examples Norm and linear algebra reminder Introduction Summary Introduction The big picture and classification of the linear system problems Case 1 u(f ) = [u(f1 )... u(fi )] y(f ) = [y(f1 )... y(fi )] Case 2 E x (t) = Ax(t) + Bu(t) y(t) = Cx(t) + Du(t) Case 3 H(s) = e τ s Data DAE/ODE State x(t) Rn, n large or infinite Reduced DAE/ODE Reduced state x (t) Rr with r n (+) Simulation (+) Analysis (+) Control (+) Optimization C. Poussot-Vassal, P. Vuillemin & I. Pontes Duff [Onera - DCSD] Model Reduction (Approximation) of Large-Scale Systems

46 AP + P A T + BB T = 0 W T V Kr(A, B) ˆΣ (Â, ˆB, Ĉ, ˆD)i Introduction The big picture and classification of the linear system problems u 1(.) u 2(.). u nu (.) Dynamical system H x 1(.) x 2(.). x n(.) y 1(.) y 2(.). y ny (.) Case 1 Data-driven case: u i (.), and y i (.) 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) Input-output frequency data Finite order large-scale linear model Σ (A, B, C, D)i Σ more model reduction toolbox A reduced-order linear dynamical system Infinite order linear model

47 AP + P A T + BB T = 0 model reduction toolbox W T V Kr(A, B) ˆΣ (Â, ˆB, Ĉ, ˆD)i Introduction Case 1: Data-based model approximation Input-output frequency data Finite order large-scale linear model Σ (A, B, C, D)i Σ more A reduced-order linear dynamical system Infinite order linear model Given a the {ω i, H(ω i )} set obtained from experiments or simulation Find a finite order model that matches the data From gust disturbance 20 Acceleration (db) (Q) Optimal model found, that interpolates all the measured points (S) Tailored to Multi Input Multi Output models Model obtained in few seconds WT experimental data MORE toolbox Frequency (Hz)

48 AP + P A T + BB T = 0 model reduction toolbox W T V Kr(A, B) ˆΣ (Â, ˆB, Ĉ, ˆD)i Introduction Case 2: The realization-based LTI model approximation Input-output frequency data Finite order large-scale linear model Σ (A, B, C, D)i Σ more A reduced-order linear dynamical system Infinite order linear model Given a large-scale linear dynamical realization of a long range aircraft Find a simpler model that well reproduces the complex one Mismatch error (over a limited frequency range) 10 1 MATLAB MORE toolbox (T) Save engineer time in analysis, controller design, optimization (Q) Accuracy of 90% with a 97% simpler model (36 instead of 1700 states) Outperforms precision of commercial tools Approximation order, r

49 AP + P A T + BB T = 0 model reduction toolbox W T V Kr(A, B) ˆΣ (Â, ˆB, Ĉ, ˆD)i Introduction Case 3: Realization free model approximation Input-output frequency data Finite order large-scale linear model Σ (A, B, C, D)i Σ more A reduced-order linear dynamical system Infinite order linear model Given an infinite dimensional, irrational and delay dependent model Find a low order model simpler to simulate and analyse From q e to z(s,q) From q s to z(s,q) z [db] ω [rad/s] Q [m 3 /s] From q to z(s,q) e 45 z [db] ω [rad/s] Q [m 3 /s] From q to z(s,q) s 40 (T) Simulation velocity increased with a factor of 100, reducing optimization steps (Q) Accuracy close to 95% with a complexity of 8 equations (instead of PDE) (S) Allows to transform infinite to finite model z [db] z [db] Q [m 3 /s] 10 2 ω [rad/s] Q [m 3 /s] 10 2 ω [rad/s] 10 4

50 Introduction Linear model approximation objective Let us consider H, a n u inputs, n y outputs linear dynamical system described by the complex-valued function of order n (n large or ) H : C C ny nu, (15)

51 Introduction Linear model approximation objective Let us consider H, a n u inputs, n y outputs linear dynamical system described by the complex-valued function of order n (n large or ) H : C C ny nu, (15) the model approximation problem consists in finding Ĥ of order r n that well reproduces the input-output behaviour of H. Ĥ : C C ny nu, (16) u1(.) u2(.). unu (.) Dynamical system H x1(.) x2(.). xn(.) y1(.) y2(.). yny (.) u1(.) u2(.). unu (.) Dynamical system Ĥ ˆx1(.) ˆx2(.). ˆxr(.) y1(.) y2(.). yny (.)

52 Introduction Linear model approximation objective (with realization) Let us consider H, a n u inputs, n y outputs linear dynamical system described by the complex-valued function of order n (n large or ) H : C C ny nu, (17) the model approximation problem consists in finding Ĥ of order r n with a given realization e.g. Ĥ : C C ny nu, (18) { Ê ˆx(t) = ˆx(t) + Ĥ : ˆBu(t) ŷ(t) = Ĉ ˆx(t) + ˆDu(t), (19) that well reproduces the input-output behaviour of H.

53 Introduction Linear model approximation objective (with realization) Let us consider H, a n u inputs, n y outputs linear dynamical system described by the complex-valued function of order n (n large or ) H : C C ny nu, (17) the model approximation problem consists in finding Ĥ of order r n with a given realization e.g. Ĥ : C C ny nu, (18) { Ê ˆx(t) = ˆx(t) + Ĥ : ˆBu(t) ŷ(t) = Ĉ ˆx(t) + ˆDu(t), (19) that well reproduces the input-output behaviour of H. "Well reproduce..."? Ĥ is a "good" approximation of H if E(t) = y(t) ŷ(t) is "small"

54 Lecture outines Outlines Motivating examples Norm and linear algebra reminder Introduction Summary The big picture Approximation objectives

55 Lecture outines Motivating examples Norm and linear algebra reminder Introduction Summary Summary The big picture Case 1 u(f ) = [u(f1 )... u(fi )] y(f ) = [y(f1 )... y(fi )] Case 2 E x (t) = Ax(t) + Bu(t) y(t) = Cx(t) + Du(t) Case 3 H(s) = e τ s Data DAE/ODE State x(t) Rn, n large or infinite Reduced DAE/ODE Reduced state x (t) Rr with r n (+) Simulation (+) Analysis (+) Control (+) Optimization C. Poussot-Vassal, P. Vuillemin & I. Pontes Duff [Onera - DCSD] Model Reduction (Approximation) of Large-Scale Systems

56 Mismatch error Summary Approximation objectives Ĥ := argmin G H 2 dim(g) = r n H G H2 (20) ( 1 H 2 H 2 := trace 2π ( 1 := trace 2π ( H(iν)H T (iν) ) dν) ) (21) H(iν) 2 F dν 4 P. Van-Dooren, K. A. Gallivan, and P. A. Absil, "H 2 -optimal model reduction of MIMO systems", Applied Mathematics Letters, vol. 21(12), December 2008, pp S. Gugercin and A C. Antoulas and C A. Beattie, "H 2 Model Reduction for Large Scale Linear Dynamical Systems", SIAM Journal on Matrix Analysis and Applications, vol. 30(2), June 2008, pp K. A. Gallivan, A. Vanderope, and P. Van-Dooren, "Model reduction of MIMO systems via tangential interpolation", SIAM Journal of Matrix Analysis and Application, vol. 26(2), February 2004, pp C. Poussot-Vassal, "An Iterative SVD-Tangential Interpolation Method for Medium-Scale MIMO Systems Approximation with Application on Flexible Aircraft", Proceedings of the 50th IEEE CDC - ECC, Orlando, Florida, USA, December, 2011, pp

57 Frequency-limited mismatch error Summary Approximation objectives Ĥ := argmin G H dim(g) = r n H G H2,Ω (22) ( 1 H 2 H 2,Ω := trace 2π ( 1 := trace 2π Ω Ω ( H(iν)H T (iν) ) dν) H(iν) 2 F dν ) (23) 8 P. Vuillemin, C. Poussot-Vassal and D. Alazard, "H 2 optimal and frequency limited approximation methods for large-scale LTI dynamical systems", in Proceedings of the IFAC Symposium on Systems Structure and Control, Grenoble, France, February, 2013, pp P. Vuillemin, C. Poussot-Vassal and D. Alazard, "A Spectral Expression for the Frequency-Limited H 2 -norm", arxiv: , P. Vuillemin, C. Poussot-Vassal and D. Alazard, "Poles Residues Descent Algorithm for Optimal Frequency-Limited H 2 Model Approximation", submitted to CDC, Florence, Italy, December, 2013.

58 Mismatch objective and eigenvector preservation Summary Approximation objectives 11 Ĥ := argmin G H 2 dim(g) = r n λ k (G) λ(h) k = 1,..., i 1 < r H G H2 (24) More than a H 2 (sub-optimal) criteria Keep some user defined eigenvalues C. Poussot-Vassal and P. Vuillemin, "An Iterative Eigenvector Tangential Interpolation Algorithm for Large-Scale LTI and a Class of LPV Model Approximation", in Proceedings of the European Control Conference, Zurich, Switzerland, July, 2013.

59 Lecture outines Motivating examples Norm and linear algebra reminder Introduction Summary Model Reduction (Approximation) of Large-Scale Systems Introduction, motivating examples and problem formulation Lecture 1 C. Poussot-Vassal, P. Vuillemin & I. Pontes Duff EDSYS, April 4-7th, 2016 (Toulouse, France) Case 1 u(f ) = [u(f1 )... u(fi )] y(f ) = [y(f1 )... y(fi )] Case 2 E x (t) = Ax(t) + Bu(t) y(t) = Cx(t) + Du(t) Case 3 H(s) = e τ s Σ (A, B, C, D)i Data more AP + P AT + BB T = 0 Σ WTV Kr (A, B) model reduction toolbox Σ (A, B, C, D )i C. Poussot-Vassal, P. Vuillemin & I. Pontes Duff [Onera - DCSD] DAE/ODE State x(t) Rn, n large or infinite Reduced DAE/ODE Reduced state x (t) Rr with r n (+) Simulation (+) Analysis (+) Control (+) Optimization Model Reduction (Approximation) of Large-Scale Systems