Introduction State-space formulation Poles-residues formulation OptimalH2 algorithm Summary

Model Reduction (Approximation) of Large-Scale Systems

Optimal H2 approximation: An interpolation point of viewLecture 6

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

April 4th-7th, 2016, École Doctorale Systèmes (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

Introduction State-space formulation Poles-residues formulation OptimalH2 algorithm Summary

IntroductionThe projection-based frameworkObjectives of the lectureGeneral problem formulation

First-order optimality conditions: State-space approachProblem statementGradients of the errorLink with the projection framework

First-order optimality conditions: Poles-residues approachProblem statementGradients of the error

Fixed-point algorithm for optimal H2 reductionIdeaKey elements of the algorithmsIterative Tangential Interpolation Algorithm (ITIA)Example

Summary

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

Introduction State-space formulation Poles-residues formulation OptimalH2 algorithm Summary

Outlines

IntroductionThe projection-based frameworkObjectives of the lectureGeneral problem formulation

First-order optimality conditions: State-space approach

First-order optimality conditions: Poles-residues approach

Fixed-point algorithm for optimal H2 reduction

Summary

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

Introduction State-space formulation Poles-residues formulation OptimalH2 algorithm Summary

IntroductionThe 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

Introduction State-space formulation Poles-residues formulation OptimalH2 algorithm Summary

IntroductionThe 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) (1)

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

E,A B

C

Introduction State-space formulation Poles-residues formulation OptimalH2 algorithm Summary

IntroductionThe 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) (1)

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) (2)

well approximates H.

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

WTEV ,WTAV WTB

CV

ΠV,W =⇒C

E,A B

Introduction State-space formulation Poles-residues formulation OptimalH2 algorithm Summary

IntroductionThe 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) (1)

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) (2)

well approximates H.

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

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

Introduction State-space formulation Poles-residues formulation OptimalH2 algorithm Summary

IntroductionObjectives of the lecture

Previously in Lecture 5:Let us consider a n-th order LTI dynamical model H and σ ∈ C,

I SISO case: one can build V,W ∈ Rn×r such that the r-th order model Hobtained by projection matches some moments of H at multiple points σk,k = 1, . . . , nσ .↪→Rational interpolation at multiple points through generalised Krylov subspaces

I MIMO case: one can build V,W ∈ Rn×r such that the r-th order model Hobtained by projection tangentially interpolates H at multiple points σk in givendirections lk, rk, k = 1, . . . , σ.↪→Tangential interpolation

In this Lecture:I Choice of the interpolation points and tangential directions so that the

reduced-order model is accurate.↪→Optimal H2 model reduction through interpolation

HypothesisI again, one assumes that E = In.

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

Introduction State-space formulation Poles-residues formulation OptimalH2 algorithm Summary

IntroductionObjectives of the lecture

Previously in Lecture 5:Let us consider a n-th order LTI dynamical model H and σ ∈ C,

I SISO case: one can build V,W ∈ Rn×r such that the r-th order model Hobtained by projection matches some moments of H at multiple points σk,k = 1, . . . , nσ .↪→Rational interpolation at multiple points through generalised Krylov subspaces

I MIMO case: one can build V,W ∈ Rn×r such that the r-th order model Hobtained by projection tangentially interpolates H at multiple points σk in givendirections lk, rk, k = 1, . . . , σ.↪→Tangential interpolation

In this Lecture:I Choice of the interpolation points and tangential directions so that the

reduced-order model is accurate.↪→Optimal H2 model reduction through interpolation

HypothesisI again, one assumes that E = In.

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

Introduction State-space formulation Poles-residues formulation OptimalH2 algorithm Summary

IntroductionObjectives of the lecture

Previously in Lecture 5:Let us consider a n-th order LTI dynamical model H and σ ∈ C,

I SISO case: one can build V,W ∈ Rn×r such that the r-th order model Hobtained by projection matches some moments of H at multiple points σk,k = 1, . . . , nσ .↪→Rational interpolation at multiple points through generalised Krylov subspaces

I MIMO case: one can build V,W ∈ Rn×r such that the r-th order model Hobtained by projection tangentially interpolates H at multiple points σk in givendirections lk, rk, k = 1, . . . , σ.↪→Tangential interpolation

In this Lecture:I Choice of the interpolation points and tangential directions so that the

reduced-order model is accurate.↪→Optimal H2 model reduction through interpolation

HypothesisI again, one assumes that E = In.

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

Introduction State-space formulation Poles-residues formulation OptimalH2 algorithm Summary

IntroductionGeneral problem formulation

Reminder: H2-normGiven an asymptotically stable and strictly proper transfer matrix H(s) (i.e. H(s) ∈H2), its H2-norm is defined as

‖H‖H2 :=

√1

2π

∫ ∞−∞

trace (H(jω)H(−jω)T )) dω, (3)

in the frequency-domain.

Some interpretations:I output energy of the system when the input is a white noise,I output energy of the system when the input is an impulse,I integral of the squared gain of the transfer function along the imaginary axis,

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

Introduction State-space formulation Poles-residues formulation OptimalH2 algorithm Summary

IntroductionGeneral problem formulation

Optimal H2 model approximation problemGiven a n-th order MIMO LTI dynamical model H which is assumed to be asymptoticallystable and strictly proper.

The objective is to find a r-th order model H (r � n) s.t.

H = arg minG∈H2

‖H −G‖H2 . (4)

I the problem is non-convex but smooth↪→with the H∞-norm: non-convex andnon-smooth problem + iterative normcomputation

I As ‖H‖H2 = ‖y‖∞‖u‖2

, a time-domaininterpretation of the error is direct

−4

−3.5

−3

−2.5

−2

−1.5

−1

−0.5

0 05

1015

2025

3035

40

70

75

80

85

90

95

100

Im. partReal part

RelativeH

2error(%

)

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

Introduction State-space formulation Poles-residues formulation OptimalH2 algorithm Summary

IntroductionGeneral problem formulation

When facing such an optimisation problem, one wants to characterise the solutions:

I choose a parametrisation for the reduced-order model,I express the first-order (necessary) optimality conditions,I in some cases, the sufficient optimality conditions may be

exploited too.

Computation of the H2-norm:I from the state-space realisation,

‖H‖H2 =√

trace (CPCT ) =√

trace (BTQB), (5)

where P and Q are the controllability and observability gramians.I if H has semi-simple poles only,

‖H‖H2 =

√√√√ n∑i=1

trace (ΦiH(−λi)T ), (6)

where λi ∈ C, Φi ∈ Cny×nu are the poles and associated residues of H(s).

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

Introduction State-space formulation Poles-residues formulation OptimalH2 algorithm Summary

IntroductionGeneral problem formulation

This suggest 2 different parametrisations for H:I the state-space realisation,

H = (A, B, C)I the poles-residues representation.

H(s) =r∑i=1

Φis− λi

=r∑i=1

cibTis− λi

where, if AX = Xdiag(λ1, . . . , λr

),

ci = CXei ∈ Cny , bi =(

eTi X−1B

)T∈ Cnu (7)

↪→assuming H(s) has semi-simple poles only

I they do not ensure stability of H,I both are over-parametrised (the minimal number of parameters is rny + rnu)↪→full state-space realisation: r2 + rnu + rny d.o.f.↪→poles-residues representation: r + rnu + rny d.o.f.

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

Introduction State-space formulation Poles-residues formulation OptimalH2 algorithm Summary

IntroductionGeneral problem formulation

This suggest 2 different parametrisations for H:I the state-space realisation,

H = (A, B, C)I the poles-residues representation.

H(s) =r∑i=1

Φis− λi

=r∑i=1

cibTis− λi

where, if AX = Xdiag(λ1, . . . , λr

),

ci = CXei ∈ Cny , bi =(

eTi X−1B

)T∈ Cnu (7)

↪→assuming H(s) has semi-simple poles only

I they do not ensure stability of H,I both are over-parametrised (the minimal number of parameters is rny + rnu)↪→full state-space realisation: r2 + rnu + rny d.o.f.↪→poles-residues representation: r + rnu + rny d.o.f.

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

Introduction State-space formulation Poles-residues formulation OptimalH2 algorithm Summary

Outlines

Introduction

First-order optimality conditions: State-space approachProblem statementGradients of the errorLink with the projection framework

First-order optimality conditions: Poles-residues approach

Fixed-point algorithm for optimal H2 reduction

Summary

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

Introduction State-space formulation Poles-residues formulation OptimalH2 algorithm Summary

First-order optimality conditions: State-space approachProblem statement

Let us considerH := (A,B,C) and H := (A, B, C), (8)

then the error model E = H− H is given by

E =(Ae BeCe 0

)=

(A 00 A

B

B

C −C 0

). (9)

The H2-norm of the error is

‖E‖2H2

= ‖H − H‖2H2

= trace(CePeC

Te

)= trace

(BTe QeBe

), (10)

where Pe ∈ R(n+r)×(n+r) and Qe ∈ R(n+r)×(n+r) are solutions of

AePe + PeATe +BeBTe = 0QeAe +ATe Qe + CTe Ce = 0 (11)

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

Introduction State-space formulation Poles-residues formulation OptimalH2 algorithm Summary

First-order optimality conditions: State-space approachProblem statement

Let us considerH := (A,B,C) and H := (A, B, C), (8)

then the error model E = H− H is given by

E =(Ae BeCe 0

)=

(A 00 A

B

B

C −C 0

). (9)

The H2-norm of the error is

‖E‖2H2

= ‖H − H‖2H2

= trace(CePeC

Te

)= trace

(BTe QeBe

), (10)

where Pe ∈ R(n+r)×(n+r) and Qe ∈ R(n+r)×(n+r) are solutions of

AePe + PeATe +BeBTe = 0QeAe +ATe Qe + CTe Ce = 0 (11)

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

Introduction State-space formulation Poles-residues formulation OptimalH2 algorithm Summary

First-order optimality conditions: State-space approachProblem statement

Let us considerH := (A,B,C) and H := (A, B, C), (8)

then the error model E = H− H is given by

E =(Ae BeCe 0

)=

(A 00 A

B

B

C −C 0

). (9)

The H2-norm of the error is

‖E‖2H2

= ‖H − H‖2H2

= trace(CePeC

Te

)= trace

(BTe QeBe

), (10)

where Pe ∈ R(n+r)×(n+r) and Qe ∈ R(n+r)×(n+r) are solutions of

AePe + PeATe +BeBTe = 0QeAe +ATe Qe + CTe Ce = 0 (11)

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

Introduction State-space formulation Poles-residues formulation OptimalH2 algorithm Summary

First-order optimality conditions: State-space approachGradients of the error

Definition: Gradient of a real function of a real matrixThe gradient of a real scalar function f(X) of a real matrix variable X ∈ Rm×n is thereal matrix ∇Xf(X) ∈ Rm×n defined by

[∇Xf(X)]i,j =d

dXi,jf(X), i = 1, . . . ,m, j = 1, . . . ,m. (12)

It yields the expansion

f(X + ∆) = f(X) + 〈∇Xf(X),∆〉+O(‖∆‖2

), (13)

where〈M,N〉 = trace

(MTN

). (14)

By noting ‖E‖2H2

= J (A, B, C), the first-order optimality conditions are given by

∇AJ = 0, ∇BJ = 0, ∇CJ = 0. (15)

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

Introduction State-space formulation Poles-residues formulation OptimalH2 algorithm Summary

First-order optimality conditions: State-space approachGradients of the error

Definition: Gradient of a real function of a real matrixThe gradient of a real scalar function f(X) of a real matrix variable X ∈ Rm×n is thereal matrix ∇Xf(X) ∈ Rm×n defined by

[∇Xf(X)]i,j =d

dXi,jf(X), i = 1, . . . ,m, j = 1, . . . ,m. (12)

It yields the expansion

f(X + ∆) = f(X) + 〈∇Xf(X),∆〉+O(‖∆‖2

), (13)

where〈M,N〉 = trace

(MTN

). (14)

By noting ‖E‖2H2

= J (A, B, C), the first-order optimality conditions are given by

∇AJ = 0, ∇BJ = 0, ∇CJ = 0. (15)

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

Introduction State-space formulation Poles-residues formulation OptimalH2 algorithm Summary

First-order optimality conditions: State-space approachGradients of the error

Pe and Qe can be partitioned as

Pe =[

P X

XT P

]and Qe =

[Q Y

Y T Q

]. (16)

Therefore,AePe + PeA

Te +BeB

Te = 0 (17)

can be written as

AP + PAT +BBT = 0 (large-scale Lyap. equation)AP + P AT + BBT = 0 (small-scale Lyap. equation)AX +XAT +BBT = 0 (n× r Sylvester equation)

(18)

andQeAe +ATe Qe + CTe Ce = 0, (19)

asQA+ATQ+ CTC = 0 (large-scale Lyap. equation)QA+ AT Q+ CT C = 0 (small-scale Lyap. equation)ATY + Y A− CT C = 0 (n× r Sylvester equation)

(20)

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

Introduction State-space formulation Poles-residues formulation OptimalH2 algorithm Summary

First-order optimality conditions: State-space approachGradients of the error

Pe and Qe can be partitioned as

Pe =[

P X

XT P

]and Qe =

[Q Y

Y T Q

]. (16)

Therefore,AePe + PeA

Te +BeB

Te = 0 (17)

can be written as

AP + PAT +BBT = 0 (large-scale Lyap. equation)AP + P AT + BBT = 0 (small-scale Lyap. equation)AX +XAT +BBT = 0 (n× r Sylvester equation)

(18)

andQeAe +ATe Qe + CTe Ce = 0, (19)

asQA+ATQ+ CTC = 0 (large-scale Lyap. equation)QA+ AT Q+ CT C = 0 (small-scale Lyap. equation)ATY + Y A− CT C = 0 (n× r Sylvester equation)

(20)

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

Introduction State-space formulation Poles-residues formulation OptimalH2 algorithm Summary

First-order optimality conditions: State-space approachGradients of the error

Pe and Qe can be partitioned as

Pe =[

P X

XT P

]and Qe =

[Q Y

Y T Q

]. (16)

Therefore,AePe + PeA

Te +BeB

Te = 0 (17)

can be written as

AP + PAT +BBT = 0 (large-scale Lyap. equation)AP + P AT + BBT = 0 (small-scale Lyap. equation)AX +XAT +BBT = 0 (n× r Sylvester equation)

(18)

andQeAe +ATe Qe + CTe Ce = 0, (19)

asQA+ATQ+ CTC = 0 (large-scale Lyap. equation)QA+ AT Q+ CT C = 0 (small-scale Lyap. equation)ATY + Y A− CT C = 0 (n× r Sylvester equation)

(20)

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

Introduction State-space formulation Poles-residues formulation OptimalH2 algorithm Summary

First-order optimality conditions: State-space approachGradients of the error

Using the same partitioning of the gramians,

‖E‖2H2

= trace(CePeCTe

)= trace

(CPCT − 2CXCT + CP CT

) (21)

or equivalently with Qe,

‖E‖2H2

= trace(BTe QeBe

)= trace

(BTQB + 2BTY B + BT QB

) (22)

Remark: Note that as P , Q, P and Q are the gramians of H and H respectively,

‖E‖2H2

= ‖H‖2H2

+ ‖H‖2H2− 2trace

(CXCT

)= ‖H‖2

H2+ ‖H‖2

H2+ 2trace

(BTY B

) (23)

⇒ ‖H‖2H2

is constant an can be discarded in the optimisation

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

Introduction State-space formulation Poles-residues formulation OptimalH2 algorithm Summary

First-order optimality conditions: State-space approachGradients of the error

Using the same partitioning of the gramians,

‖E‖2H2

= trace(CePeCTe

)= trace

(CPCT − 2CXCT + CP CT

) (21)

or equivalently with Qe,

‖E‖2H2

= trace(BTe QeBe

)= trace

(BTQB + 2BTY B + BT QB

) (22)

Remark: Note that as P , Q, P and Q are the gramians of H and H respectively,

‖E‖2H2

= ‖H‖2H2

+ ‖H‖2H2− 2trace

(CXCT

)= ‖H‖2

H2+ ‖H‖2

H2+ 2trace

(BTY B

) (23)

⇒ ‖H‖2H2

is constant an can be discarded in the optimisation

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

Introduction State-space formulation Poles-residues formulation OptimalH2 algorithm Summary

First-order optimality conditions: State-space approachGradients of the error

Expression of ∇AJ

Let us considerJ = trace

(BTQB + 2BTY B + BT QB

), (24)

withATY + Y A− CT C = 0, and QA+ AT Q+ CT C = 0 (25)

A perturbation ∆A on A yields

∆J = trace(2BBT∆Y + BBT∆Q

), (26)

where ∆Y and ∆Q are solutions of

AT∆Y + ∆Y A+ Y∆A = 0, (27)

andQ∆A + ∆QA+ AT∆Q + ∆T

AQ = 0. (28)

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

Introduction State-space formulation Poles-residues formulation OptimalH2 algorithm Summary

First-order optimality conditions: State-space approachGradients of the error

Lemma: Sylvester trace

If AM +MB + C = 0 and NA+BN +D = 0, then, trace(CN) = trace(DM)

With this Lemma applied to

AT∆Y + ∆Y A+ Y∆A = 0XTAT + AXT + BBT = 0

(29)

one obtains,trace

(BBT∆Y

)= trace

(Y∆AX

T). (30)

Similarly, with∆QA+ AT∆Q + ∆T

AQ+ Q∆A = 0

AP + P AT + BBT = 0(31)

one obtains,trace

(BBT∆Q

)= trace

((∆TAQ+ Q∆A

)P). (32)

By replacing and reordering in ∆J ,

∆J = 〈2(QP + Y TX),∆A〉 ⇒ ∇AJ = 2(QP + Y TX). (33)

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

Introduction State-space formulation Poles-residues formulation OptimalH2 algorithm Summary

First-order optimality conditions: State-space approachGradients of the error

Lemma: Sylvester trace

If AM +MB + C = 0 and NA+BN +D = 0, then, trace(CN) = trace(DM)

With this Lemma applied to

AT∆Y + ∆Y A+ Y∆A = 0XTAT + AXT + BBT = 0

(29)

one obtains,trace

(BBT∆Y

)= trace

(Y∆AX

T). (30)

Similarly, with∆QA+ AT∆Q + ∆T

AQ+ Q∆A = 0

AP + P AT + BBT = 0(31)

one obtains,trace

(BBT∆Q

)= trace

((∆TAQ+ Q∆A

)P). (32)

By replacing and reordering in ∆J ,

∆J = 〈2(QP + Y TX),∆A〉 ⇒ ∇AJ = 2(QP + Y TX). (33)

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

Introduction State-space formulation Poles-residues formulation OptimalH2 algorithm Summary

First-order optimality conditions: State-space approachGradients of the error

Lemma: Sylvester trace

If AM +MB + C = 0 and NA+BN +D = 0, then, trace(CN) = trace(DM)

With this Lemma applied to

AT∆Y + ∆Y A+ Y∆A = 0XTAT + AXT + BBT = 0

(29)

one obtains,trace

(BBT∆Y

)= trace

(Y∆AX

T). (30)

Similarly, with∆QA+ AT∆Q + ∆T

AQ+ Q∆A = 0

AP + P AT + BBT = 0(31)

one obtains,trace

(BBT∆Q

)= trace

((∆TAQ+ Q∆A

)P). (32)

By replacing and reordering in ∆J ,

∆J = 〈2(QP + Y TX),∆A〉 ⇒ ∇AJ = 2(QP + Y TX). (33)

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

Introduction State-space formulation Poles-residues formulation OptimalH2 algorithm Summary

First-order optimality conditions: State-space approachGradients of the error

Lemma: Sylvester trace

If AM +MB + C = 0 and NA+BN +D = 0, then, trace(CN) = trace(DM)

With this Lemma applied to

AT∆Y + ∆Y A+ Y∆A = 0XTAT + AXT + BBT = 0

(29)

one obtains,trace

(BBT∆Y

)= trace

(Y∆AX

T). (30)

Similarly, with∆QA+ AT∆Q + ∆T

AQ+ Q∆A = 0

AP + P AT + BBT = 0(31)

one obtains,trace

(BBT∆Q

)= trace

((∆TAQ+ Q∆A

)P). (32)

By replacing and reordering in ∆J ,

∆J = 〈2(QP + Y TX),∆A〉 ⇒ ∇AJ = 2(QP + Y TX). (33)

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

Introduction State-space formulation Poles-residues formulation OptimalH2 algorithm Summary

First-order optimality conditions: State-space approachGradients of the error

Expression of ∇BJA perturbation ∆B on B yields

∆J = trace(2BTY∆B + ∆T

BQB + BT Q∆B

)= 〈2

(Y TB + QB

),∆B〉, (34)

hence ∇BJ = 2(Y TB + QB

).

Expression of ∇CJSimilarly,

∇CJ = 2(CP − CX

). (35)

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

Introduction State-space formulation Poles-residues formulation OptimalH2 algorithm Summary

First-order optimality conditions: State-space approachGradients of the error

This theorem is due to Wilson1.Theorem: Gradients of the H2 approximation errorThe gradients ∇AJ , ∇BJ , ∇CJ of J = ‖E‖2

H2are given by

∇AJ = 2(QP + Y TX),∇BJ = 2

(Y TB + QB

),

∇CJ = 2(CP − CX

),

(36)

where P , Q are solutions of the Lyapunov equations,

AP + P AT + BBT = 0,QA+ AT Q+ CT C = 0,

(37)

and Y , X of the Sylvester equations,

ATY + Y A− CT C = 0,AX +XAT +BBT = 0.

(38)

1 D. A. Wilson, "Model reduction for multivariable systems", International Journal of Control, 20(1), 1974,pp. 57-64.

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

Introduction State-space formulation Poles-residues formulation OptimalH2 algorithm Summary

First-order optimality conditions: State-space approachLink with the projection framework2

Theorem: Link with the projection frameworkAt every stationary point of J where P and Q are invertible, we have

A = WTAV, B = WTC, C = CV, (39)

with W , V , WTV = In are defined as,

W =: −Y Q−1 and V := XP−1, (40)

where P , Q are solutions of the Lyapunov equations,

AP + P AT + BBT = 0,QA+ AT Q+ CT C = 0,

(41)

and Y , X of the Sylvester equations,

ATY + Y A− CT C = 0,AX +XAT +BBT = 0.

(42)

2 P. Van-Dooren, K. A. Gallivan, and P. A. Absil, "H2-optimal model reduction of MIMO systems",Applied Mathematics Letters, 2008.

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

Introduction State-space formulation Poles-residues formulation OptimalH2 algorithm Summary

First-order optimality conditions: State-space approachLink with the projection framework

Some remarks:I the first-order optimality conditions can be viewed as coupled equations(

X,Y, P , Q)

= F1(A, B, C

)and

(A, B, C

)= G1

(X,Y, P , Q

), (43)

which admit a fixed point at every stationary point of J .↪→this suggests an iterative algorithm(X,Y, P , Q

)i+1

= F1(A, B, C

)i+1

,(A, B, C

)i+1

= G1(X,Y, P , Q

)i(44)

I the first-order optimality conditions can be interpreted as interpolation conditions↪→easier to see with the poles-residues formulation

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

Introduction State-space formulation Poles-residues formulation OptimalH2 algorithm Summary

Outlines

Introduction

First-order optimality conditions: State-space approach

First-order optimality conditions: Poles-residues approachProblem statementGradients of the error

Fixed-point algorithm for optimal H2 reduction

Summary

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

Introduction State-space formulation Poles-residues formulation OptimalH2 algorithm Summary

First-order optimality conditions: Poles-residues approachProblem statement

Assume that H and H have semi-simple poles,

H(s) =n∑i=1

cibTis− λi

and H(s) =r∑i=1

cibTis− λi

. (45)

Then the transfer function of the error E = H− H is

E(s) =n+r∑i=1

cibTis− λi

, (46)

with {ci = ci, bi = bi, λi = λi i = 1, . . . , nci = −ci, bi = bi, λi = λi i = n+ 1, . . . , n+ r

(47)

The H2-norm of the error is

‖E‖2H2

= J =n+r∑i=1

trace(

cibTi E(−λi)T). (48)

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

Introduction State-space formulation Poles-residues formulation OptimalH2 algorithm Summary

First-order optimality conditions: Poles-residues approachGradients of the error

J is a real function of complex variables and their complex conjugates, therefore itis not analyticI see Cauchy-Riemann conditions,

I with standard complex calculus, one can only differentiate analytic functions.

How can we characterise the stationary points of J ?Wirtinger calculusLet f : Cn → R be a real-valued function of a complex vector z (and implicitly itsconjugate) and g : Cn × Cn → R an analytic function with respect to each of itsargument (by treating the other as a constant) such that

f(z) = g(z, z∗), (49)

then the stationary points of f are characterised by

∇zg = 0 or ∇z∗g = 0. (50)

I various implications for optimisation3,I differentiate J by treating conjugate variables as different variables.

3 L. Sorber, M. Van Barel and L. De Lathauwer, "Unconstrained optimization of real functions in complexvariables", SIAM Journal on Optimization, 2012.

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

Introduction State-space formulation Poles-residues formulation OptimalH2 algorithm Summary

First-order optimality conditions: Poles-residues approachGradients of the error

J is a real function of complex variables and their complex conjugates, therefore itis not analyticI see Cauchy-Riemann conditions,

I with standard complex calculus, one can only differentiate analytic functions.

How can we characterise the stationary points of J ?

Wirtinger calculusLet f : Cn → R be a real-valued function of a complex vector z (and implicitly itsconjugate) and g : Cn × Cn → R an analytic function with respect to each of itsargument (by treating the other as a constant) such that

f(z) = g(z, z∗), (49)

then the stationary points of f are characterised by

∇zg = 0 or ∇z∗g = 0. (50)

I various implications for optimisation3,I differentiate J by treating conjugate variables as different variables.

3 L. Sorber, M. Van Barel and L. De Lathauwer, "Unconstrained optimization of real functions in complexvariables", SIAM Journal on Optimization, 2012.

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

Introduction State-space formulation Poles-residues formulation OptimalH2 algorithm Summary

First-order optimality conditions: Poles-residues approachGradients of the error

J is a real function of complex variables and their complex conjugates, therefore itis not analyticI see Cauchy-Riemann conditions,

I with standard complex calculus, one can only differentiate analytic functions.

How can we characterise the stationary points of J ?Wirtinger calculusLet f : Cn → R be a real-valued function of a complex vector z (and implicitly itsconjugate) and g : Cn × Cn → R an analytic function with respect to each of itsargument (by treating the other as a constant) such that

f(z) = g(z, z∗), (49)

then the stationary points of f are characterised by

∇zg = 0 or ∇z∗g = 0. (50)

I various implications for optimisation3,I differentiate J by treating conjugate variables as different variables.

3 L. Sorber, M. Van Barel and L. De Lathauwer, "Unconstrained optimization of real functions in complexvariables", SIAM Journal on Optimization, 2012.

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

Introduction State-space formulation Poles-residues formulation OptimalH2 algorithm Summary

First-order optimality conditions: Poles-residues approachGradients of the error

The error

J =n+r∑i=1

trace(

cibTi E(−λi)T), (51)

can be expanded asn∑i=1

trace(

cibTi(H(−λi)− H(−λi)

)T)−

r∑i=1

trace(

cibTi(H(−λi)− H(−λi)

)T),

(52)and again as

J =n∑i=1

tr(

cibTi H(−λi)T)

︸ ︷︷ ︸‖H‖2

H2

+r∑i=1

tr(

cibTi H(−λi)T)

︸ ︷︷ ︸‖H‖2

H2

−2r∑i=1

tr(

cibTi H(−λiT )).

(53)The optimality conditions are given by

∇cmJ = 0, ∇bmJ = 0, ∇λm

J = 0 (m = 1, . . . , r) (54)

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

Introduction State-space formulation Poles-residues formulation OptimalH2 algorithm Summary

First-order optimality conditions: Poles-residues approachGradients of the error

The error

J =n+r∑i=1

trace(

cibTi E(−λi)T), (51)

can be expanded asn∑i=1

trace(

cibTi(H(−λi)− H(−λi)

)T)−

r∑i=1

trace(

cibTi(H(−λi)− H(−λi)

)T),

(52)and again as

J =n∑i=1

tr(

cibTi H(−λi)T)

︸ ︷︷ ︸‖H‖2

H2

+r∑i=1

tr(

cibTi H(−λi)T)

︸ ︷︷ ︸‖H‖2

H2

−2r∑i=1

tr(

cibTi H(−λiT )).

(53)The optimality conditions are given by

∇cmJ = 0, ∇bmJ = 0, ∇λm

J = 0 (m = 1, . . . , r) (54)

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

Introduction State-space formulation Poles-residues formulation OptimalH2 algorithm Summary

First-order optimality conditions: Poles-residues approachGradients of the error

The error

J =n+r∑i=1

trace(

cibTi E(−λi)T), (51)

can be expanded asn∑i=1

trace(

cibTi(H(−λi)− H(−λi)

)T)−

r∑i=1

trace(

cibTi(H(−λi)− H(−λi)

)T),

(52)and again as

J =n∑i=1

tr(

cibTi H(−λi)T)

︸ ︷︷ ︸‖H‖2

H2

+r∑i=1

tr(

cibTi H(−λi)T)

︸ ︷︷ ︸‖H‖2

H2

−2r∑i=1

tr(

cibTi H(−λiT )).

(53)The optimality conditions are given by

∇cmJ = 0, ∇bmJ = 0, ∇λm

J = 0 (m = 1, . . . , r) (54)

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

Introduction State-space formulation Poles-residues formulation OptimalH2 algorithm Summary

First-order optimality conditions: Poles-residues approachGradients of the error

The error

J =n+r∑i=1

trace(

cibTi E(−λi)T), (51)

can be expanded asn∑i=1

trace(

cibTi(H(−λi)− H(−λi)

)T)−

r∑i=1

trace(

cibTi(H(−λi)− H(−λi)

)T),

(52)and again as

J =n∑i=1

tr(

cibTi H(−λi)T)

︸ ︷︷ ︸‖H‖2

H2

+r∑i=1

tr(

cibTi H(−λi)T)

︸ ︷︷ ︸‖H‖2

H2

−2r∑i=1

tr(

cibTi H(−λiT )).

(53)The optimality conditions are given by

∇cmJ = 0, ∇bmJ = 0, ∇λm

J = 0 (m = 1, . . . , r) (54)

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

Introduction State-space formulation Poles-residues formulation OptimalH2 algorithm Summary

First-order optimality conditions: Poles-residues approachGradients of the error

Let us expand H(s) and H(s) in J and reorder the terms,

J = ‖H‖2H2

+r∑i=1

cTi

r∑k=1

ckbTk−λi − λk︸ ︷︷ ︸H(−λi)

bi − 2r∑i=1

cTi

n∑k=1

ckbTk−λi − λk︸ ︷︷ ︸H(−λi)

bi. (55)

Equation (55) yields,

∇λmJ = −2cTm

r∑k=1

ckbTk(λm + λk

)2 bm + 2cTm

n∑k=1

ckbTk(λm + λk

)2 bm

= −2cTm(H′(−λm)−H′(−λm)

)bm

∇cmJ = 2r∑k=1

ckbTk−λm − λk

bm − 2H(−λm)bm

= 2(H(−λm)−H(−λm)

)bm

∇bmJ = 2

r∑k=1

bkcTk−λm − λk

cm − 2H(−λm)T cm

= 2(H(−λm)−H(−λm)

)Tcm

(56)

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

Introduction State-space formulation Poles-residues formulation OptimalH2 algorithm Summary

First-order optimality conditions: Poles-residues approachGradients of the error

Let us expand H(s) and H(s) in J and reorder the terms,

J = ‖H‖2H2

+r∑i=1

cTi

r∑k=1

ckbTk−λi − λk︸ ︷︷ ︸H(−λi)

bi − 2r∑i=1

cTi

n∑k=1

ckbTk−λi − λk︸ ︷︷ ︸H(−λi)

bi. (55)

Equation (55) yields,

∇λmJ = −2cTm

r∑k=1

ckbTk(λm + λk

)2 bm + 2cTm

n∑k=1

ckbTk(λm + λk

)2 bm

= −2cTm(H′(−λm)−H′(−λm)

)bm

∇cmJ = 2r∑k=1

ckbTk−λm − λk

bm − 2H(−λm)bm

= 2(H(−λm)−H(−λm)

)bm

∇bmJ = 2

r∑k=1

bkcTk−λm − λk

cm − 2H(−λm)T cm

= 2(H(−λm)−H(−λm)

)Tcm

(56)

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

Introduction State-space formulation Poles-residues formulation OptimalH2 algorithm Summary

First-order optimality conditions: Poles-residues approachGradients of the error

Theorem: First-order optimality conditions for the H2 problemLet H be a r-th order asymptotically stable model with semi-simple poles only. If H issolution of the H2 approximation problem, then

cTi H(−λi) = cTi H(−λi)H(−λi)bi = H(−λi)bi

cTi H′(−λi)bi = cTi H

′(−λi)bi(57)

where λi and {ci, bi} are the poles and associated residues of H(s).

I the reduced-order model is a bi-tangential Hermite interpolant of the large-scalemodel at the opposite of its poles,

I these conditions can be obtained from the state-space formulation (see 4)

4 P. Van-Dooren, K. A. Gallivan, and P. A. Absil, "H2-optimal model reduction of MIMO systems",Applied Mathematics Letters, vol. 21, no. 12, pp. 53-62, December 2008.

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

Introduction State-space formulation Poles-residues formulation OptimalH2 algorithm Summary

First-order optimality conditions: Poles-residues approachGradients of the error

Theorem: First-order optimality conditions for the H2 problemLet H be a r-th order asymptotically stable model with semi-simple poles only. If H issolution of the H2 approximation problem, then

cTi H(−λi) = cTi H(−λi)H(−λi)bi = H(−λi)bi

cTi H′(−λi)bi = cTi H

′(−λi)bi(57)

where λi and {ci, bi} are the poles and associated residues of H(s).

I the reduced-order model is a bi-tangential Hermite interpolant of the large-scalemodel at the opposite of its poles,

I these conditions can be obtained from the state-space formulation (see 4)

4 P. Van-Dooren, K. A. Gallivan, and P. A. Absil, "H2-optimal model reduction of MIMO systems",Applied Mathematics Letters, vol. 21, no. 12, pp. 53-62, December 2008.

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

Introduction State-space formulation Poles-residues formulation OptimalH2 algorithm Summary

First-order optimality conditions: Poles-residues approachGradients of the error

I these conditions are less general than with the state-space approach as H isassumed to have semi-simple poles only,

I once again, the optimality conditions can be viewed as a set of coupled equations,(λi, ci, bi

)= F2

(A, B, C

)and

(A, B, C

)= G2

(λi, ci, bi

). (58)

which admit a fixed point at every stationary point of J .↪→this suggests an iterative procedure(λi, ci, bi

)k+1

= F2(A, B, C

)k+1

,(A, B, C

)k+1

= G2(λi, ci, bi

)k. (59)

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

Introduction State-space formulation Poles-residues formulation OptimalH2 algorithm Summary

Outlines

Introduction

First-order optimality conditions: State-space approach

First-order optimality conditions: Poles-residues approach

Fixed-point algorithm for optimal H2 reductionIdeaKey elements of the algorithmsIterative Tangential Interpolation Algorithm (ITIA)Example

Summary

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

Introduction State-space formulation Poles-residues formulation OptimalH2 algorithm Summary

Fixed-point algorithm for optimal H2 reductionIdea

The first-order optimality conditions can be viewed as coupled equations:I state-space formulation:(

X,Y, P , Q)

= F1(A, B, C

)and

(A, B, C

)= G1

(X,Y, P , Q

), (60)

I poles-residues formulation:(λi, ci, bi

)= F2

(A, B, C

)and

(A, B, C

)= G2

(λi, ci, bi

). (61)

which have a fixed-point at every stationary point of J .This suggest the following iterative procedures

I state-space formulation(X,Y, P , Q

)i+1

= F1(A, B, C

)i+1

,(A, B, C

)i+1

= G1(X,Y, P , Q

)i(62)

I poles-residues formulation(λi, ci, bi

)k+1

= F2(A, B, C

)k+1

,(A, B, C

)k+1

= G2(λi, ci, bi

)k. (63)

↪→Iterative Tangential Interpolation Algorithm (ITIA)

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

Introduction State-space formulation Poles-residues formulation OptimalH2 algorithm Summary

Fixed-point algorithm for optimal H2 reductionIdea

The first-order optimality conditions can be viewed as coupled equations:I state-space formulation:(

X,Y, P , Q)

= F1(A, B, C

)and

(A, B, C

)= G1

(X,Y, P , Q

), (60)

I poles-residues formulation:(λi, ci, bi

)= F2

(A, B, C

)and

(A, B, C

)= G2

(λi, ci, bi

). (61)

which have a fixed-point at every stationary point of J .This suggest the following iterative procedures

I state-space formulation(X,Y, P , Q

)i+1

= F1(A, B, C

)i+1

,(A, B, C

)i+1

= G1(X,Y, P , Q

)i(62)

I poles-residues formulation(λi, ci, bi

)k+1

= F2(A, B, C

)k+1

,(A, B, C

)k+1

= G2(λi, ci, bi

)k. (63)

↪→Iterative Tangential Interpolation Algorithm (ITIA)

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

Introduction State-space formulation Poles-residues formulation OptimalH2 algorithm Summary

Fixed-point algorithm for optimal H2 reductionKey elements of the algorithms

State-space approach:I initially suggested in 5

I the step(X,Y, P , Q

)i+1

= F1(A, B, C

)i+1

is done by solvingLyapunov/Sylvester equations,

I the step(A, B, C

)i+1

= G1(X,Y, P , Q

)iis done by projection.

ITIA:I initially proposed for SISO models as Iterative Rational Krylov Algorithm in 6,I the step

(λi, ci, bi

)k+1

= F2(A, B, C

)k+1

is done by solving a small-scaleeigenvalue problem,↪→assumes that A is diagonalisable

I the step(A, B, C

)k+1

= G2(λi, ci, bi

)kis done by tangential interpolation

through Krylov subspaces (projection).

Solving Lyapunov/Sylvester equations can be slow ⇒ ITIA tends to be faster.

5 P. Van-Dooren, K. A. Gallivan, and P. A. Absil, "H2-optimal model reduction of MIMO systems",Applied Mathematics Letters, 2008.

6 S. Gugercin, A.C. Antoulas and C. Beattie, "A rational Krylov iteration for optimal H2 modelreduction", Proceedings of the International Symposium on Mathematical Theory of Networks and Systems, 2006.

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

Introduction State-space formulation Poles-residues formulation OptimalH2 algorithm Summary

Fixed-point algorithm for optimal H2 reductionKey elements of the algorithms

State-space approach:I initially suggested in 5

I the step(X,Y, P , Q

)i+1

= F1(A, B, C

)i+1

is done by solvingLyapunov/Sylvester equations,

I the step(A, B, C

)i+1

= G1(X,Y, P , Q

)iis done by projection.

ITIA:I initially proposed for SISO models as Iterative Rational Krylov Algorithm in 6,I the step

(λi, ci, bi

)k+1

= F2(A, B, C

)k+1

is done by solving a small-scaleeigenvalue problem,↪→assumes that A is diagonalisable

I the step(A, B, C

)k+1

= G2(λi, ci, bi

)kis done by tangential interpolation

through Krylov subspaces (projection).

Solving Lyapunov/Sylvester equations can be slow ⇒ ITIA tends to be faster.

5 P. Van-Dooren, K. A. Gallivan, and P. A. Absil, "H2-optimal model reduction of MIMO systems",Applied Mathematics Letters, 2008.

6 S. Gugercin, A.C. Antoulas and C. Beattie, "A rational Krylov iteration for optimal H2 modelreduction", Proceedings of the International Symposium on Mathematical Theory of Networks and Systems, 2006.

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

Introduction State-space formulation Poles-residues formulation OptimalH2 algorithm Summary

Fixed-point algorithm for optimal H2 reductionKey elements of the algorithms

State-space approach:I initially suggested in 5

I the step(X,Y, P , Q

)i+1

= F1(A, B, C

)i+1

is done by solvingLyapunov/Sylvester equations,

I the step(A, B, C

)i+1

= G1(X,Y, P , Q

)iis done by projection.

ITIA:I initially proposed for SISO models as Iterative Rational Krylov Algorithm in 6,I the step

(λi, ci, bi

)k+1

= F2(A, B, C

)k+1

is done by solving a small-scaleeigenvalue problem,↪→assumes that A is diagonalisable

I the step(A, B, C

)k+1

= G2(λi, ci, bi

)kis done by tangential interpolation

through Krylov subspaces (projection).

Solving Lyapunov/Sylvester equations can be slow ⇒ ITIA tends to be faster.

5 P. Van-Dooren, K. A. Gallivan, and P. A. Absil, "H2-optimal model reduction of MIMO systems",Applied Mathematics Letters, 2008.

6 S. Gugercin, A.C. Antoulas and C. Beattie, "A rational Krylov iteration for optimal H2 modelreduction", Proceedings of the International Symposium on Mathematical Theory of Networks and Systems, 2006.

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

Introduction State-space formulation Poles-residues formulation OptimalH2 algorithm Summary

Fixed-point algorithm for optimal H2 reductionIterative Tangential Interpolation Algorithm (ITIA)

Require: A ∈ Rn×n, B ∈ Rn×nu , C ∈ Rny×n, {σ(0)k, l(0)k, r(0)k} ∈ C× Cny × Cnu,

k = 1, . . . , r1: Build V,W ∈ Rn×r s.t.

span(V ) = span([(σ(0)1 In −A)−1Br(0)

1 , . . . , (σ(0)kIn −A)−1Br(0)

r ])span(W ) = span([(σ(0)

1 In −A)−TCT l(0)1 , . . . , (σ(0)

kIn −A)−TCT l(0)

r ])2: Ensure bi-orthogonality: W ←W (WTV )−T

3: while |σ(i)k− σ(i−1)

k| > ε do

4: i← i+ 15: Project the model: A = WTAV , B = WTB, C = CV6: Solve the (small) eigenvalue problem: AX = X∆, ∆ = diag(λ1, . . . , λr)7: For k = 1, . . . , r, set σ(i)

k= −λk, r(i)

k=(

eTk X−1B

)T and l(i)k

= CXek8: Build V,W ∈ Rn×r s.t.

span(V ) = span([(σ(i)1 In −A)−1Br(i)

1 , . . . , (σ(0)r In −A)−1Br(0)

r ])span(W ) = span([(σ(i)

1 In −A)−TCT l(i)1 , . . . , (σ(i)

r In −A)−TCT l(i)r ])

9: Ensure bi-orthogonality: W ←W (WTV )−T10: end while11: Apply projectors V and W

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

Introduction State-space formulation Poles-residues formulation OptimalH2 algorithm Summary

Fixed-point algorithm for optimal H2 reductionIterative Tangential Interpolation Algorithm (ITIA)

Some remarks,I convergence has been proven only in some specific cases,↪→in practice, it generally converges

I if the algorithm converges, it is not guaranteed to be a local minimum,↪→in practice, the H2-norm of the error generally decreases during the iteration(so its not a maximum) and the resulting reduced-order model are accurate

To summarize,√

leads to a stationary point (generally min.) of the H2 approximation problem,√

numerically very efficient,↪→has been applied to models with more than 700 000 states

√quite simple to implement,

× no control on the approximation error,× stability not always preserved↪→this can be alleviated in exchange for a loss of performance7

7 S. Gugercin, "Iterative SVD-Krylov based method for model reduction of large-scale dynamical systems",CDC, 2005.

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

Introduction State-space formulation Poles-residues formulation OptimalH2 algorithm Summary

Fixed-point algorithm for optimal H2 reductionIterative Tangential Interpolation Algorithm (ITIA)

Some remarks,I convergence has been proven only in some specific cases,↪→in practice, it generally converges

I if the algorithm converges, it is not guaranteed to be a local minimum,↪→in practice, the H2-norm of the error generally decreases during the iteration(so its not a maximum) and the resulting reduced-order model are accurate

To summarize,√

leads to a stationary point (generally min.) of the H2 approximation problem,√

numerically very efficient,↪→has been applied to models with more than 700 000 states

√quite simple to implement,

× no control on the approximation error,× stability not always preserved↪→this can be alleviated in exchange for a loss of performance7

7 S. Gugercin, "Iterative SVD-Krylov based method for model reduction of large-scale dynamical systems",CDC, 2005.

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

Introduction State-space formulation Poles-residues formulation OptimalH2 algorithm Summary

Fixed-point algorithm for optimal H2 reductionExample

Approximation of the 348-th order model cbm with an order 10:

10−2

10−1

100

101

102

−40

−20

0

20

40

60

80M

agni

tude

(dB

)

Bode Diagram

Frequency (rad/s)

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

Introduction State-space formulation Poles-residues formulation OptimalH2 algorithm Summary

Outlines

Introduction

First-order optimality conditions: State-space approach

First-order optimality conditions: Poles-residues approach

Fixed-point algorithm for optimal H2 reduction

Summary

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

Introduction State-space formulation Poles-residues formulation OptimalH2 algorithm Summary

Summary

In this lecture:

I expression of the first-order optimality conditions for the optimal H2 modelreduction problem↪→state-space and poles-residues

I principle of fixed-point algorithms for optimal H2 reduction↪→Iterative Tangential Interpolation Algorithm

Can we have more control on the H2 error throughout iterations?

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

Introduction State-space formulation Poles-residues formulation OptimalH2 algorithm Summary

Summary

In this lecture:

I expression of the first-order optimality conditions for the optimal H2 modelreduction problem↪→state-space and poles-residues

I principle of fixed-point algorithms for optimal H2 reduction↪→Iterative Tangential Interpolation Algorithm

Can we have more control on the H2 error throughout iterations?

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

Introduction State-space formulation Poles-residues formulation OptimalH2 algorithm Summary

Model Reduction (Approximation) of Large-Scale Systems

Optimal H2 approximation: An interpolation point of viewLecture 6

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

April 4th-7th, 2016, École Doctorale Systèmes (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