Basics of Systems and Control Theory for pyMOR

transcript

Basics of Systems and ControlTheory for pyMORJens Saak

September 30, 2020

pyMOR Online Course 2020

E x(t) = A x(t) + B u(t)

y(t) = C x(t) + D u(t)

˙x(t) =A

x(t) +B

y(t) = C x(t) + D u(t)

Outline

1 Linear Time-Invariant (LTI) Systems

2 Transfer Function and Realizations

3 System Analysis

4 A Selection of MOR Methods

Restrictions for this lecture

• Only continuous-time systemsDiscrete-time is treated in [1]

• No differential-algebraic systemsFor DAE aspects see [6, 3, 4, 5]

• No non-linearities

• No parameter dependencies

Outline

• Setting for this course

• Examples

3 System Analysis

Linear Time-Invariant (LTI) Systems | Setting for this course

First-order State-space Systems ( : LTIModel)

x(t) = Ax(t) + Bu(t),

y(t) = Cx(t) + Du(t).(Σ)

• x(t) ∈ Rn is called the state,

• u(t) ∈ Rm is called the input,

• y(t) ∈ Rp is called the output

of the LTI system. Correspondingly, we have

A ∈ Rn×n, B ∈ Rn×m, C ∈ Rp×n and D ∈ Rp×m.

We assume t ∈ [0,∞), x(0) = 0.

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x(t) = Ax(t) + Bu(t),

y(t) = Cx(t).(Σ)

A ∈ Rn×n, B ∈ Rn×m, C ∈ Rp×n.

We assume t ∈ [0,∞), x(0) = 0.

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Ex(t) = Ax(t) + Bu(t),

y(t) = Cx(t).(Σ)

E, A ∈ Rn×n, B ∈ Rn×m, C ∈ Rp×n.

We assume t ∈ [0,∞), x(0) = 0 andE invertible.

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Second-order State-space Systems ( : SecondOrderModel)

Mx(t) + Ex(t) + Kx(t) = Bu(t),

y(t) = Cvx(t) + Cpx(t).

• x(t) ∈ Rn is called the position,

• x(t) ∈ Rn is called the velocity,

M, E, K ∈ Rn×n, B ∈ Rn×m, Cv, Cp ∈ Rp×n.

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Linear Time-Invariant (LTI) Systems | Examples

Heat Equation [MORWiki thermal block] I

For t ∈ (0, T ), ξ ∈ Ω and initial values

θ(0, ξ) = 0, for ξ ∈ Ω,

consider

∂tθ(t, ξ) +∇ · (−σ(ξ)∇θ(t, ξ)) = 0,

with boundary conditions

σ(ξ)∇θ(t, ξ) · n(ξ) = u(t) t ∈ (0, T ), ξ ∈ Γin,

σ(ξ)∇θ(t, ξ) · n(ξ) = 0 t ∈ (0, T ), ξ ∈ ΓN ,

θ(t, ξ) = 0 t ∈ (0, T ), ξ ∈ ΓD.

Ω1 Ω2

Ω3Ω4

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Heat Equation [MORWiki thermal block] II

Finite element semi-discretization in space

• pairwise inner products of ansatz functions E

• discretized spatial operator + Dirichlet boundary condition A

• discretized non-zero Neumann boundary condition B

• average temperatures on the inclusions C

• n = 7 488

• m = 1

• p = 4

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An Artificial Fishtail [MORWiki Artificial Fishtail] I

Construction: Fluid Elastomer Actuation:

no pressure

under pressure

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An Artificial Fishtail [MORWiki Artificial Fishtail] II

Variables:

• displacement ~s(t, ~z)

• strain ~ε(~s(t, ~z))

• stress ~σ(~s(t, ~z))

Material parameters:

• density ρ

• Lamé parameters λ, µ

Basic principle:

~ε(~s(t, ~z)) =1

(∇~s(t, ~z) +∇T~s(t, ~z)

)(kinematic equation)

~σ(~s(t, ~z)) = λ tr((~ε(~s(t, ~z))) I + 2µ~ε(~s(t, ~z))) (material equation)

ρ∂2~s(t, ~z)

∂t2= ∇ · ~σ(~s(t, ~z)) + ~f(t, ~z) (equation of motion)

+ initial and boundary conditions

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An Artificial Fishtail [MORWiki Artificial Fishtail] III

FEM semi-discretization:

Mx(t) + Ex(t) + Kx(t) = Bu(t),

y(t) = Cpx(t),

• M,E,K > 0,Cv = 0,

• n = 779 232,m = 1, p = 3.

10−3 10−1 101 10310−12

10−7

10−2

frequency (Hz)

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

3·10−2

time (s)

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Outline

• Laplace Transform

• Transfer Function

• Realizations

• Projection-based MOR

3 System Analysis

Transfer Function and Realizations | Laplace Transform

Definition

Let f : [0,∞)→ Rn be exponentially bounded with bounding exponent α. Then

Lf (s) :=∫ ∞

f(τ)e−sτdτ

for Re(s) > α is called the Laplace transform of f . The process of forming theLaplace transform is called Laplace transformation.

It can be shown that the integral converges uniformly in a domain with Re(s) ≥ β forall β > α.

Allows us to map time signals to frequency signals.

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Definition

Let f : [0,∞)→ Rn be exponentially bounded with bounding exponent α. Then

Lf (s) :=∫ ∞

f(τ)e−sτdτ

for Re(s) > α is called the Laplace transform of f . The process of forming theLaplace transform is called Laplace transformation.

It can be shown that the integral converges uniformly in a domain with Re(s) ≥ β forall β > α.

Allows us to map time signals to frequency signals.

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Theorem

Let f, g, h : [0,∞)→ Rn be given. Then the following two statements hold true:

a) The Laplace transformation is linear, i. e., if f and g are exponentially bounded, thenh := γf + δg is also exponentially bounded and

Lh = γLf+ δLg

holds for all γ, δ ∈ C.b) If f ∈ PC1([0,∞),Rn) and f is exponentially bounded, then f is exponentially

bounded andLf

(s) = sLf(s)− f(0).

• X(s) := Lx(s), U(s) := Lu(s), and Y (s) := Ly(s)• Ax(t) + Bu(t) AX(s) + BU(s)

• y(t) = Cx(t) Y (s) = CX(s)

• sX(s) := Lx(s) since x(0) = 0

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Theorem

Let f, g, h : [0,∞)→ Rn be given. Then the following two statements hold true:

a) The Laplace transformation is linear, i. e., if f and g are exponentially bounded, thenh := γf + δg is also exponentially bounded and

Lh = γLf+ δLg

holds for all γ, δ ∈ C.b) If f ∈ PC1([0,∞),Rn) and f is exponentially bounded, then f is exponentially

bounded andLf

(s) = sLf(s)− f(0).

• X(s) := Lx(s), U(s) := Lu(s), and Y (s) := Ly(s)• Ax(t) + Bu(t) AX(s) + BU(s)

• y(t) = Cx(t) Y (s) = CX(s)

• sX(s) := Lx(s) since x(0) = 0

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Transfer Function and Realizations | Transfer Function

Rational Matrix Function Representation

In summary we have:

• sEX(s) = AX(s) + BU(s)

• Y (s) = CX(s)

Thus the mapping from inputs to outputs in frequency domain can be expressed as

H(s) = C(sE−A)−1B.

Analogously, for second-order systems we get

H(s) = (sCv +Cp)(s2M+ sE+K

)−1B.

H is analytic inC \ Λ(E,A), orC \ Λ(M,E,K), respectively

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In summary we have:

• Y (s) = CX(s)

H(s) = C(sE−A)−1B.

)−1B.

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In summary we have:

• Y (s) = CX(s)

H(s) = C(sE−A)−1B.

)−1B.

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Important Representations of H(s)

(Laurent) series expansion

H(s) =

∞∑

(s− s0)kMk(s0) H(s) =

∞∑

s−kMk(∞)

The matricesMk(s0) are calledmoments ofH. At infinity they are also referred toasMarkov parameters.

Pole Residue Form

Let (λi, wi, vi) be the eigentriplets of the pair (A,E) with no degenerateeigenspaces. Then we have

H(s) =

Ris− λi

whereRi = (Cvi)(wHi B), assumingwH

i vi = 1.

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Transfer Function and Realizations | Realizations

The representation ofH using (E,A,B,C) is not unique.

In fact for any invertible matrixT ∈ Rn×n, we have

H(s) = C(sE−A)−1

= CT−1T(sE−A)−1

T−1TB

= CT−1(sTET−1 −TAT−1

)−1TB

and thus a system given, by (TET−1,TAT−1,TB,CT−1) realizes the exact sameinput/output behavior.

Definition

• All sets of matrices leading to the same functionH are called its realizations.

• The matrixT above is called state-space transformation.

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Important Realizations

• Minimal RealizationsCan we realizeH with less equations?

• Truncated RealizationsCan we introduce a small error to get even less equations?

• Balanced Realizations see here

Can we find state coordinates that allow us to decide what is important?

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McMillan Degree and Minimal Realization

ExampleRealizations can even be of different dimensions. Take for example:

E = I the identity,A =

[−11 0

0 −5

]andC =

Truncating the second state component does not changeH.

Definition

There exists a minimum number of equations necessary to describeH. The statedimension n of this minimal set of equations is calledMcMillan degree of thesystem. A realization ofH with this dimension is calledminimal realization.

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Truncated Realizations via Ritz/Petrov-Galerkin Projection

Ex(t)−Ax(t)−Bu(t) = 0,

y(t)−Cx(t)−Du(t) = 0.

Step I: Use truncated state transformationReplace

x(t) ≈ Vx(t)

withV ∈ Rn×r and x(t) ∈ Rr .

Step II: Mitigate transformation errorSuppress truncation residual through left projection.

• one-sided method: useV again.

• two-sided method: findW ∈ Rn×r .

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EV ˙x(t)−AVx(t)−Bu(t) = eres(t),

y(t)−CVx(t)−Du(t) = eoutput(t).

x(t) ≈ Vx(t)

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VTEV ˙x(t)−VTAVx(t)−VTBu(t) = 0,

x(t) ≈ Vx(t)

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WTEV ˙x(t)−WTAVx(t)−WTBu(t) = 0,

x(t) ≈ Vx(t)

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Transfer Function and Realizations | Projection-based MOR

Reduced order model (ROM) ( : LTIPGReductor)

Define E = WTEV, A = WTAV ∈ Rr×r , B = WTB ∈ Rr×m andC = CV ∈ Rp×r . Then

E ˙x(t) = Ax(t) + Bu(t),

y(t) = Cx(t) + Du(t)(ROM)

approximates the dynamics of the full-order model (Σ) with output error

y(t)− y(t) = eoutput(t).

• We call the corresponding transfer function H.

• Model order reduction (MOR) FindW,V ∈ Rn×r such that eoutput(t) issmall in a suitable sense.

• We will see energy-based and interpolation-based methods today and tomorrow.

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Outline

3 System Analysis

• System Norms and Hardy Spaces

• Frequency-Domain Analysis

System Analysis | System Norms and Hardy Spaces

We haveY (s) = H(s)U(s)

andY (s) = H(s)U(s).

Question

What are suitable norms such that

‖y − y‖ ≤∥∥∥H− H

∥∥∥ ‖u‖?

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The Banach SpaceHp×m∞

Hp×m∞ :=

G : C+ → Cp×m : G is analytic inC+ and sup

s∈C+

‖G(s)‖2 <∞.

Hp×m∞ is a Banach space equipped with theH∞-norm

‖G‖H∞ := supω∈R‖G(iω)‖2 .

Can show: ‖y − y‖L2≤∥∥∥H− H

∥∥∥H∞‖u‖L2

This bound can even be shown to be sharp.

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Hp×m∞ :=

s∈C+

‖G(s)‖2 <∞.

∥∥∥H∞‖u‖L2

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Hp×m∞ :=

s∈C+

‖G(s)‖2 <∞.

∥∥∥H∞‖u‖L2

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The Hilbert SpaceHp×m2

Hp×m2 :=

G : C+ → Cp×m : G is analytic inC+ and

supξ>0

∫ ∞

−∞‖G(ξ + iω)‖2F dω <∞

Hp×m2 is a Hilbert space with the inner product

〈F,G〉H2:=

∫ ∞

−∞tr(F (iω)

HG(iω)

and induced norm

‖G‖H2:= 〈G,G〉1/2H2

∫ ∞

−∞‖G(iω)‖2F dω

Can show: ‖y − y‖L∞ ≤∥∥∥H− H

∥∥∥H2

‖u‖L2.

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Hp×m2 :=

supξ>0

∫ ∞

−∞‖G(ξ + iω)‖2F dω <∞

〈F,G〉H2:=

∫ ∞

−∞tr(F (iω)

HG(iω)

and induced norm

‖G‖H2:= 〈G,G〉1/2H2

∫ ∞

∥∥∥H2

‖u‖L2.

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Hp×m2 :=

supξ>0

∫ ∞

−∞‖G(ξ + iω)‖2F dω <∞

〈F,G〉H2:=

∫ ∞

−∞tr(F (iω)

HG(iω)

and induced norm

‖G‖H2:= 〈G,G〉1/2H2

∫ ∞

∥∥∥H2

‖u‖L2.

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System Gramians andH2-trace-formula

A system (Σ) with Λ(E,A) ⊂ C− is called asymptotically stable. Then, all statetrajectories decay exponentially as t→∞ and

a) the infinite controllability and observability Gramians exist:

∫ ∞

eE−1AtE−1BBTE−TeA

TE−Ttdt

ETQE =

∫ ∞

eATE−TtCTCeE

−1Atdt.

b) P,Q solve the two Lyapunov equations

APET + EPAT = −BBT, ATQE + ETQA = −CTC

c) theH2-norm can be expressed as

‖H‖2H2= tr

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System Analysis | Frequency-Domain Analysis

Bode Plots

The Bode plot forH consists of amagnitude plot and a phase plot.

Bode magnitude plot

• component-wise graph of the function |H(iω)| for frequenciesω ∈ [ωmin, ωmax] ⊂ R.• ω-axis is logarithmic.

• magnitude is given in decibels, i.e., |H(i.)| is plotted as 20 log10(|H(i.)|).

Bode phase plot

• component-wise graph of the function argH(iω) for frequenciesω ∈ [ωmin, ωmax] ⊂ R.• ω-axis is logarithmic.

• phase is given in degrees on a linear scale.

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Bode Plot for the Thermal Block Example

−400

−200

Magnitude

10−3 10−2 10−1 100 101 102 103

−2,000

−1,000

Frequency (Hz)

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(Sigma) Magnitude Plots

Sigma magnitude plot

• 2-norm-wise graph of the functionH(iω) for frequenciesω ∈ [ωmin, ωmax] ⊂ R.• ω-axis is logarithmic.

The name is due to the fact that for a given matrixM the norm ‖M‖2 is given by itslargest singular value.

The real sigma magnitude plot depicts all singular values as functions of ω.

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Sigma Magnitude Plot for the Artificial Fishtail

10−4 10−3 10−2 10−1 100 101 102 103 10410−12

10−9

10−6

10−3

frequency (Hz)

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Outline

3 System Analysis

• Balancing Based MOR

• Moments and Interpolation

A Selection of MOR Methods | Balancing Based MOR

Balanced Truncation aka. Lyapunov Balancing

• The system (Σ), in realization (E = I,A,B,C), is called balanced, if thesolutionsP, Q of the Lyapunov equations

AP + PAT + BBT = 0, ATQ + QA + CTC = 0,

satisfy: P = Q = diag(σ1, . . . , σn) where σ1 ≥ σ2 ≥ · · · ≥ σn > 0.

• σ1, . . . , σn are the Hankel singular values (HSVs) of Σ.• A balanced realization is computed via state space transformation

T : (I,A,B,C) 7→ (I,TAT−1,TB,CT−1)

([A11 A12

A21 A22

],[C1 C2

• Truncation reduced order model: (I, A, B, C) = (I,A11,B1,C1).

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• σ1, . . . , σn are the Hankel singular values (HSVs) of Σ.

• A balanced realization is computed via state space transformation

T : (I,A,B,C) 7→ (I,TAT−1,TB,CT−1)

([A11 A12

A21 A22

],[C1 C2

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T : (I,A,B,C) 7→ (I,TAT−1,TB,CT−1)

([A11 A12

A21 A22

],[C1 C2

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T : (I,A,B,C) 7→ (I,TAT−1,TB,CT−1)

([A11 A12

A21 A22

],[C1 C2

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Implementation: The Square Root Method

The SR Method ( : BTReductor)

1. Compute (Cholesky) factors of the solutions to the Lyapunov equation,

P = STS, Q = RTR.

2. Compute singular value decomposition

SRT = [U1, U2 ]

3. DefineW := RTV1Σ

−1/21 , V := STU1Σ

−1/21 .

4. Then the reduced order model is (WTAV,WTB,CV).

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P = STS, Q = RTR.

SRT = [U1, U2 ]

−1/21 , V := STU1Σ

−1/21 .

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P = STS, Q = RTR.

SRT = [U1, U2 ]

−1/21 , V := STU1Σ

−1/21 .

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Properties

• Lyapunov balancing preserves asymptotic stability.

• We have the a priori error bound:∥∥∥H− H

∥∥∥H∞≤ 2

n∑k=r+1

Variants ( : BRBTReductor, LQGBTReductor)Other versions for special classes of systems or applications exist, such as

• positive-real balancing, (passivity-preserving)

• bounded-real balancing, (contractivity-preserving)

• linear-quadratic Gaussian balancing. (stability preserving)(aims at low-order output feedback controllers)

The given ones all computeP, Q as solutions of algebraic Riccati equations of theform:

0 = APET + EPAT + BBT ± EPCTCPET

0 = ATQE + ETQA+ CTC ± ETQBBTQE.

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A Selection of MOR Methods |Moments and Interpolation

Tools I

Lemma (Neumann series)

LetA ∈ Cn×n with spectral radius ρ(A) < 1 be given. Then I−A is invertible and itholds that

(I−A)−1

∞∑

Will be important to identify the actual shape of Markov parameters and systemmoments.

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Tools II

Definition ((polynomial) Krylov subpace)

Given an invertible matrixA ∈ Rn×n and a vector b ∈ Rn the k-dimensional(polynomial) Krylov subspace is defined as

Kk(A,b) := spanb,Ab,A2b, . . . ,Ak−1b

Definition (rational Krylov subpace)

Given an invertible matrixA ∈ Rn×n a vector b ∈ Rn and a vector of shifts s ∈ Rkthe k-dimensional rational Krylov subspace is defined as

Kk(A,b, s) := span

(s1I−A)−1

b, (s2I−A)−1

b, . . . , (skI−A)−1

Orthonormal bases of these spaces should be computed via the Arnoldi iteration.

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Padé-type approximations

Match the coefficientsMk(s0) orMk(∞) in

H(s) =

∞∑

(s− s0)kMk(s0) H(s) =

∞∑

s−kMk(∞)

Motivation (assume:m = p = 1, s large enough)

H(s) = C(sE−A)−1

B = 1sC

(I− 1

sE−1A

)−1︸︷︷︸=∑∞

sk(E−1A)k

E−1B

∞∑

C(E−1A)k−1

E−1B 1sk.

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Motivation (assume:m = p = 1, s large enough)

H(s) = C(sE−A)−1

B = 1sC

(I− 1

sE−1A

)−1︸︷︷︸=∑∞

sk(E−1A)k

E−1B

∞∑

C(E−1A)k−1

E−1B 1sk.

Therefore, we have

Mk(∞) =

0, if k = 0,

C(E−1A)k−1

E−1B, if k ≥ 1. useV = Kr(E−1A,E−1B)

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Approximation at∞

V = Kr(E−1A,E−1B), W = V orW = Kr(ATE−T,CT)

Approximation at s0 = 0

V = Kr(A−1E,A−1B), W = V orW = Kr(ETA−T,CT)

Approximation at s0 ∈ (0,∞)

V = Kr((s0E−A)−1

E, (s0E−A)−1

B), W = V

orW = Kr(ET(s0E

T −AT)−1,CT)

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Multi-point Moment Matching, Interpolation and IRKA/TSIA

Approximation at s1, . . . , sr

V = Kr(s,E−1A,E−1B), W = V orW = Kr(s,ATE−T,CT).

• W = V as above matches first r moments of (Σ).• W 6= V as above matches first 2r moments of (Σ).• W 6= V as above actually achieves Hermite interpolation ofH, see, e.g., [2].

How do we choose s1, . . . , sr?H2-optimal MOR

Find s = [s1, . . . , sr]T, such that

∥∥∥H− H∥∥∥H2

is minimized.

IRKA iterative improvement of s using Λ(Ej , Aj).( : IRKAReductor)

TSIA run a fixed point iteration on the first order necessary conditions.( : TSIAReductor)

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References

[1] A. C. Antoulas, Approximation of Large-Scale Dynamical Systems, vol. 6 of Adv. Des. Control,SIAM Publications, Philadelphia, PA, 2005,https://doi.org/10.1137/1.9780898718713.

[2] A. C. Antoulas, C. A. Beattie, and S. Gugercin, Interpolatory Methods for Model Reduction,Computational Science & Engineering, Society for Industrial and Applied Mathematics,Philadelphia, PA, 2020, https://doi.org/10.1137/1.9781611976083.

[3] S. Gugercin, T. Stykel, and S. Wyatt, Model reduction of descriptor systems by interpolatoryprojection methods, SIAM J. Sci. Comput., 35 (2013), pp. B1010–B1033,https://doi.org/10.1137/130906635.

[4] V. Mehrmann and T. Stykel, Balanced truncation model reduction for large-scale systems indescriptor form, in Dimension Reduction of Large-Scale Systems, P. Benner, V. Mehrmann, andD. C. Sorensen, eds., vol. 45 of Lect. Notes Comput. Sci. Eng., Springer-Verlag,Berlin/Heidelberg, Germany, 2005, pp. 83–115,https://doi.org/10.1007/3-540-27909-1_3.

[5] T. Stykel, Gramian-based model reduction for descriptor systems, Math. Control Signals Systems,16 (2004), pp. 297–319, https://doi.org/10.1007/s00498-004-0141-4.

[6] M. Voigt, Model reduction, Lecture Notes, Uni Hamburg, 2019,https://www.math.uni-hamburg.de/home/voigt/Modellreduktion_SoSe19/Notes_ModelReduction.pdf.

Questions?

Basics of Systems and Control Theory for pyMOR

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