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Automatic Generation of Minimal and Reduced Models for Structured - - PowerPoint PPT Presentation

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Automatic Generation of Minimal and Reduced Models for Structured Parametric Dynamical Systems Igor Pontes Duff Joint work with Peter Benner (MPI, Magdeburg) Pawan Goyal (MPI, Magdeburg) ICERM Workshop - Mathematics of Reduced Order Models

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Automatic Generation of Minimal and Reduced Models for Structured Parametric Dynamical Systems

Igor Pontes Duff

Joint work with Peter Benner (MPI, Magdeburg) Pawan Goyal (MPI, Magdeburg)

ICERM Workshop - Mathematics of Reduced Order Models Providence, RI, USA February 20, 2020

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Table of Contents

  • 1. Introduction
  • 2. First-order systems
  • 3. Structured Transfer Function
  • 4. Parametric extension
  • 5. Numerical Examples
  • 6. Outlook

Igor Pontes Duff, pontes@mpi-magdeburg.mpg.de Automatic Generation of Minimal and Reduced Models for Structured and Nonlinear Parametric Dynamical Systems 2/34

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Introduction

–Large-Scale Dynamical Systems–

− → ∇ · σ = ρ ∂2s ∂t2 ε = 1 2

  • ∇s + ∇T s
  • ,

σ = λ tr (ε) I + 2µε Continuous mechanics

Igor Pontes Duff, pontes@mpi-magdeburg.mpg.de Automatic Generation of Minimal and Reduced Models for Structured and Nonlinear Parametric Dynamical Systems 3/34

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Introduction

–Large-Scale Dynamical Systems–

− → ∇ · σ = ρ ∂2s ∂t2 ε = 1 2

  • ∇s + ∇T s
  • ,

σ = λ tr (ε) I + 2µε Continuous mechanics M¨ x(t) + D ˙ x(t) + Kx(t) = Bu(t) y(t) = Cx(t) High order ODE

semi-discretization in space Igor Pontes Duff, pontes@mpi-magdeburg.mpg.de Automatic Generation of Minimal and Reduced Models for Structured and Nonlinear Parametric Dynamical Systems 3/34

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Introduction

–Large-Scale Dynamical Systems–

− → ∇ · σ = ρ ∂2s ∂t2 ε = 1 2

  • ∇s + ∇T s
  • ,

σ = λ tr (ε) I + 2µε Continuous mechanics M¨ x(t) + D ˙ x(t) + Kx(t) = Bu(t) y(t) = Cx(t) High order ODE

semi-discretization in space

∂v ∂t = ∂2v ∂x2 (x, t) + a0(x)v(x, t) + a1(x)v(x, t − 1) v(0, t) = u(t) t ≥ 0 Heated rod with delay

Igor Pontes Duff, pontes@mpi-magdeburg.mpg.de Automatic Generation of Minimal and Reduced Models for Structured and Nonlinear Parametric Dynamical Systems 3/34

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Introduction

–Large-Scale Dynamical Systems–

− → ∇ · σ = ρ ∂2s ∂t2 ε = 1 2

  • ∇s + ∇T s
  • ,

σ = λ tr (ε) I + 2µε Continuous mechanics M¨ x(t) + D ˙ x(t) + Kx(t) = Bu(t) y(t) = Cx(t) High order ODE

semi-discretization in space

∂v ∂t = ∂2v ∂x2 (x, t) + a0(x)v(x, t) + a1(x)v(x, t − 1) v(0, t) = u(t) t ≥ 0 Heated rod with delay ˙ x(t) = A0x(t) + Aτx(t − τ) + Bu(t) y(t) = Cx(t) Functional differencial equation

semi-discretization in space Igor Pontes Duff, pontes@mpi-magdeburg.mpg.de Automatic Generation of Minimal and Reduced Models for Structured and Nonlinear Parametric Dynamical Systems 3/34

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Introduction

–Large-Scale Dynamical Systems–

− → ∇ · σ = ρ ∂2s ∂t2 ε = 1 2

  • ∇s + ∇T s
  • ,

σ = λ tr (ε) I + 2µε Continuous mechanics M¨ x(t) + D ˙ x(t) + Kx(t) = Bu(t) y(t) = Cx(t) High order ODE

semi-discretization in space

∂v ∂t = ∂2v ∂x2 (x, t) + a0(x)v(x, t) + a1(x)v(x, t − 1) v(0, t) = u(t) t ≥ 0 Heated rod with delay ˙ x(t) = A0x(t) + Aτx(t − τ) + Bu(t) y(t) = Cx(t) Functional differencial equation

semi-discretization in space

Semi-discretization in space leads to large-scale dynamical systems. Dynamical Structures: higher order ODE and functional differential equations. Other examples: Fading memories, fractional order systems, . . .

Igor Pontes Duff, pontes@mpi-magdeburg.mpg.de Automatic Generation of Minimal and Reduced Models for Structured and Nonlinear Parametric Dynamical Systems 3/34

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Introduction

–Structured Linear Dynamical Systems–

Linear Structured Dynamical systems

E ˙ x(t) = f(x(t), u(t))), x(0) = 0, y(t) = Cx(t), where (generalized) states x(t) ∈ Rn (invertible E ∈ Rn×n), f is linear. inputs (controls) u(t) ∈ Rm,

  • utputs (measurements, quantity of interest) y(t) ∈ Rq. In this talk, m = q = 1 (SISO).

Igor Pontes Duff, pontes@mpi-magdeburg.mpg.de Automatic Generation of Minimal and Reduced Models for Structured and Nonlinear Parametric Dynamical Systems 4/34

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Introduction

–Structured Linear Dynamical Systems–

Linear Structured Dynamical systems

E ˙ x(t) = f(x(t), u(t))), x(0) = 0, y(t) = Cx(t), where (generalized) states x(t) ∈ Rn (invertible E ∈ Rn×n), f is linear. inputs (controls) u(t) ∈ Rm,

  • utputs (measurements, quantity of interest) y(t) ∈ Rq. In this talk, m = q = 1 (SISO).

System Classes

Classical linear systems: f(x) := Ax(t) + Bu(t), Delay systems: f(x) := Ax(t) + Aτx(t − τ) + Bu(t), Second-order system f(x) := Ax(t) + A1 t x(τ)dτ + t Bu(τ)τ, . . . , Integro-differential systems: f(x) := t µ(ds)x(t − s) + Bu(t),

Igor Pontes Duff, pontes@mpi-magdeburg.mpg.de Automatic Generation of Minimal and Reduced Models for Structured and Nonlinear Parametric Dynamical Systems 4/34

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Introduction

–Structured Linear Dynamical Systems– A linear system Input Output

Igor Pontes Duff, pontes@mpi-magdeburg.mpg.de Automatic Generation of Minimal and Reduced Models for Structured and Nonlinear Parametric Dynamical Systems 5/34

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Introduction

–Structured Linear Dynamical Systems– A linear system Input Output I n p u t

  • u

t p u t m a p p i n g

Input-output representation or transfer function

Igor Pontes Duff, pontes@mpi-magdeburg.mpg.de Automatic Generation of Minimal and Reduced Models for Structured and Nonlinear Parametric Dynamical Systems 5/34

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Introduction

–Structured Linear Dynamical Systems– A linear system Input Output I n p u t

  • u

t p u t m a p p i n g

Input-output representation or transfer function

Frequency domain representation of the system (Laplace transform) x(t) → X(s), ˙ x(t) → sX(s).

E ˙ x(t) = Ax(t) + Bu(t), y(t) = Cx(t), x(0) = 0.

Linear System

Igor Pontes Duff, pontes@mpi-magdeburg.mpg.de Automatic Generation of Minimal and Reduced Models for Structured and Nonlinear Parametric Dynamical Systems 5/34

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Introduction

–Structured Linear Dynamical Systems– A linear system Input Output I n p u t

  • u

t p u t m a p p i n g

Input-output representation or transfer function

Frequency domain representation of the system (Laplace transform) x(t) → X(s), ˙ x(t) → sX(s). As a result, we have sEX(s) = AX(s) + BU(s), Y(s) = CX(s), yielding I/O mapping as Y(s) =

  • C(sE − A)−1B
  • =:G(s)
  • U(s).

E ˙ x(t) = Ax(t) + Bu(t), y(t) = Cx(t), x(0) = 0.

Linear System

Igor Pontes Duff, pontes@mpi-magdeburg.mpg.de Automatic Generation of Minimal and Reduced Models for Structured and Nonlinear Parametric Dynamical Systems 5/34

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Introduction

–Structured Linear Dynamical Systems– A linear system Input Output I n p u t

  • u

t p u t m a p p i n g

Input-output representation or transfer function

Frequency domain representation of the system (Laplace transform) x(t) → X(s), ˙ x(t) → sX(s). As a result, we have sEX(s) = AX(s) + BU(s), Y(s) = CX(s), yielding I/O mapping as Y(s) =

  • C(sE − A)−1B
  • =:G(s)
  • U(s).

E ˙ x(t) = Ax(t) + Bu(t), y(t) = Cx(t), x(0) = 0.

Linear System

G(s) := C(sE − A)−1B. also known as transfer function

Input-Output Mapping

Igor Pontes Duff, pontes@mpi-magdeburg.mpg.de Automatic Generation of Minimal and Reduced Models for Structured and Nonlinear Parametric Dynamical Systems 5/34

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SLIDE 15

Introduction

–Structured Linear Dynamical Systems–

E ˙ x(t) = Ax(t) + Bu(t), y(t) = Cx(t). Linear system (standard) G(s) = C(sE − A)−1B. Linear system (standard)

Time → Frequency Igor Pontes Duff, pontes@mpi-magdeburg.mpg.de Automatic Generation of Minimal and Reduced Models for Structured and Nonlinear Parametric Dynamical Systems 6/34

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Introduction

–Structured Linear Dynamical Systems–

E ˙ x(t) = Ax(t) + Bu(t), y(t) = Cx(t). Linear system (standard) M¨ x(t) + D ˙ x(t) + Kx(t) = Bu(t), y(t) = Cx(t). Second-order system G(s) = C(sE − A)−1B. Linear system (standard)

Time → Frequency

G(s) = C(s2M + sD + K)−1B. Second-order system

Time → Frequency Igor Pontes Duff, pontes@mpi-magdeburg.mpg.de Automatic Generation of Minimal and Reduced Models for Structured and Nonlinear Parametric Dynamical Systems 6/34

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Introduction

–Structured Linear Dynamical Systems–

E ˙ x(t) = Ax(t) + Bu(t), y(t) = Cx(t). Linear system (standard) M¨ x(t) + D ˙ x(t) + Kx(t) = Bu(t), y(t) = Cx(t). Second-order system E ˙ x(t) = Ax(t) + Aτx(t−τ) + Bu(t), y(t) = Cx(t). Delay system G(s) = C(sE − A)−1B. Linear system (standard)

Time → Frequency

G(s) = C(s2M + sD + K)−1B. Second-order system

Time → Frequency

G(s) = C

  • sE − A − Aτe−sτ−1 B.

Delay system

Time → Frequency Igor Pontes Duff, pontes@mpi-magdeburg.mpg.de Automatic Generation of Minimal and Reduced Models for Structured and Nonlinear Parametric Dynamical Systems 6/34

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SLIDE 18

Introduction

–Structured Linear Dynamical Systems–

E ˙ x(t) = Ax(t) + Bu(t), y(t) = Cx(t). Linear system (standard) M¨ x(t) + D ˙ x(t) + Kx(t) = Bu(t), y(t) = Cx(t). Second-order system E ˙ x(t) = Ax(t) + Aτx(t−τ) + Bu(t), y(t) = Cx(t). Delay system E ˙ x(t) = Ax(t) + Aτ t x(τ)dτ + Bu(t), y(t) = Cx(t). Integro system G(s) = C(sE − A)−1B. Linear system (standard)

Time → Frequency

G(s) = C(s2M + sD + K)−1B. Second-order system

Time → Frequency

G(s) = C

  • sE − A − Aτe−sτ−1 B.

Delay system

Time → Frequency

G(s) = C

  • sE − A − 1

s Aτ −1 B. Integro system

Time → Frequency Igor Pontes Duff, pontes@mpi-magdeburg.mpg.de Automatic Generation of Minimal and Reduced Models for Structured and Nonlinear Parametric Dynamical Systems 6/34

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Problem Formulation

Problem Formulation

Approximate the transfer function of an n-dimensional system, H(s) = C(s)K(s)−1B(s),

Igor Pontes Duff, pontes@mpi-magdeburg.mpg.de Automatic Generation of Minimal and Reduced Models for Structured and Nonlinear Parametric Dynamical Systems 7/34

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SLIDE 20

Problem Formulation

Problem Formulation

Approximate the transfer function of an n-dimensional system, H(s) = C(s)K(s)−1B(s), by the transfer function of a system ˆ H(s) = ˆ C(s) ˆ K(s)−1 ˆ B(s),

Igor Pontes Duff, pontes@mpi-magdeburg.mpg.de Automatic Generation of Minimal and Reduced Models for Structured and Nonlinear Parametric Dynamical Systems 7/34

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Problem Formulation

Problem Formulation

Approximate the transfer function of an n-dimensional system, H(s) = C(s)K(s)−1B(s), by the transfer function of a system ˆ H(s) = ˆ C(s) ˆ K(s)−1 ˆ B(s),

  • f order r ≪ n, such that

H(s) − ˆ H(s) < tolerance ∀s. = ⇒ Optimization problem: min

  • rder( ˆ

H)≤r

H − ˆ H. K(s) B(s) C(s)

⇓

ˆ K(s) ˆ B(s) ˆ C(s)

Igor Pontes Duff, pontes@mpi-magdeburg.mpg.de Automatic Generation of Minimal and Reduced Models for Structured and Nonlinear Parametric Dynamical Systems 7/34

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Problem Formulation

–Structure Preservation–

Structured system H(s) = C(s)K(s)−1B(s), C(s) = k

i=1 αi(s)Ci ∈ Rq×n,

K(s) = l

i=1 βi(s)Ai ∈ Rn×n,

B(s) = m

i=1 γi(s)Bi ∈ Rn×m,

αi(s), βi(s) and γi(s) are meromorphic functions (dynamical structures).

Igor Pontes Duff, pontes@mpi-magdeburg.mpg.de Automatic Generation of Minimal and Reduced Models for Structured and Nonlinear Parametric Dynamical Systems 8/34

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Problem Formulation

–Structure Preservation–

Structured system H(s) = C(s)K(s)−1B(s), C(s) = k

i=1 αi(s)Ci ∈ Rq×n,

K(s) = l

i=1 βi(s)Ai ∈ Rn×n,

B(s) = m

i=1 γi(s)Bi ∈ Rn×m,

αi(s), βi(s) and γi(s) are meromorphic functions (dynamical structures). Structured reduced system ˆ H(s) = ˆ C(s) ˆ K(s)−1 ˆ B(s), ˆ C(s) = k

i=1 αi(s) ˆ

Ci ∈ Rq×r, ˆ K(s) = g

i=1 βi(s) ˆ

Ai ∈ Rr×r, ˆ B(s) = m

i=1 γi(s) ˆ

Bi ∈ Rr×m. Hence, preserve meromorphic functions, and order r ≪ n. – How to construct reduced systems, satisfying the desired goals, i.e., H(s) − ˆ H(s) ≤ tol.

Igor Pontes Duff, pontes@mpi-magdeburg.mpg.de Automatic Generation of Minimal and Reduced Models for Structured and Nonlinear Parametric Dynamical Systems 8/34

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Construction of reduced-order systems

Petrov-Galerkin-type projection

For given projection matrices V, W ∈ Rn×r, leading to

Igor Pontes Duff, pontes@mpi-magdeburg.mpg.de Automatic Generation of Minimal and Reduced Models for Structured and Nonlinear Parametric Dynamical Systems 9/34

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Construction of reduced-order systems

Petrov-Galerkin-type projection

For given projection matrices V, W ∈ Rn×r, leading to ˆ C(s) = α1(s)C1V + α2(s)C2V + · · · + αk(s)CkV, = α1(s) ˆ C1 + α2(s) ˆ C2 + · · · + α(s) ˆ Ck, ˆ K(s) = β1(s)WT A1V + · · · + βg(s)WT AgV, = +β1(s) ˆ A1 + · · · + βg(s) ˆ Ag, ˆ B(s) = γ1(s)WT B1 + γ2(s)WT B2 + · · · + γm(s)WT Bm, = γ1(s) ˆ B1 + γ2(s) ˆ B2 + · · · + γm(s) ˆ Bm,

Igor Pontes Duff, pontes@mpi-magdeburg.mpg.de Automatic Generation of Minimal and Reduced Models for Structured and Nonlinear Parametric Dynamical Systems 9/34

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Construction of reduced-order systems

Petrov-Galerkin-type projection

For given projection matrices V, W ∈ Rn×r, leading to ˆ C(s) = α1(s)C1V + α2(s)C2V + · · · + αk(s)CkV, = α1(s) ˆ C1 + α2(s) ˆ C2 + · · · + α(s) ˆ Ck, ˆ K(s) = β1(s)WT A1V + · · · + βg(s)WT AgV, = +β1(s) ˆ A1 + · · · + βg(s) ˆ Ag, ˆ B(s) = γ1(s)WT B1 + γ2(s)WT B2 + · · · + γm(s)WT Bm, = γ1(s) ˆ B1 + γ2(s) ˆ B2 + · · · + γm(s) ˆ Bm, Choice of the projection matrices?

Igor Pontes Duff, pontes@mpi-magdeburg.mpg.de Automatic Generation of Minimal and Reduced Models for Structured and Nonlinear Parametric Dynamical Systems 9/34

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Some Literature

Common existing approaches

  • 1. Interpolating reduced-order systems

[Beattie/Gugercin ’09]

H(σi) = ˆ H(σi), for i = 1, . . . , 2r

– How to choose σi??

Igor Pontes Duff, pontes@mpi-magdeburg.mpg.de Automatic Generation of Minimal and Reduced Models for Structured and Nonlinear Parametric Dynamical Systems 10/34

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Some Literature

Common existing approaches

  • 1. Interpolating reduced-order systems

[Beattie/Gugercin ’09]

H(σi) = ˆ H(σi), for i = 1, . . . , 2r

– How to choose σi??

  • 2. Reduced-order modeling via balancing truncation

[Breiten ’16]

– aims at removing the subspaces those are less important for the dynamics – Expensive to solve Lyapunov equation

Igor Pontes Duff, pontes@mpi-magdeburg.mpg.de Automatic Generation of Minimal and Reduced Models for Structured and Nonlinear Parametric Dynamical Systems 10/34

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SLIDE 29

Some Literature

Common existing approaches

  • 1. Interpolating reduced-order systems

[Beattie/Gugercin ’09]

H(σi) = ˆ H(σi), for i = 1, . . . , 2r

– How to choose σi??

  • 2. Reduced-order modeling via balancing truncation

[Breiten ’16]

– aims at removing the subspaces those are less important for the dynamics – Expensive to solve Lyapunov equation

  • 3. Data-driven structured realization (non-intrusive way)

[Schulze et. al ’18]

– Required expert knowledge and not straightforward to implement.

Igor Pontes Duff, pontes@mpi-magdeburg.mpg.de Automatic Generation of Minimal and Reduced Models for Structured and Nonlinear Parametric Dynamical Systems 10/34

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SLIDE 30

Some Literature

Common existing approaches

  • 1. Interpolating reduced-order systems

[Beattie/Gugercin ’09]

H(σi) = ˆ H(σi), for i = 1, . . . , 2r

– How to choose σi??

  • 2. Reduced-order modeling via balancing truncation

[Breiten ’16]

– aims at removing the subspaces those are less important for the dynamics – Expensive to solve Lyapunov equation

  • 3. Data-driven structured realization (non-intrusive way)

[Schulze et. al ’18]

– Required expert knowledge and not straightforward to implement.

Igor Pontes Duff, pontes@mpi-magdeburg.mpg.de Automatic Generation of Minimal and Reduced Models for Structured and Nonlinear Parametric Dynamical Systems 10/34

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SLIDE 31

First-order systems

–Interpolation-based MOR–

An n-dimensional linear system

Σ :=

  • ˙

x(t) = Ax(t) + Bu(t), y(t) = Cx(t).

⇒

Transfer function H(s) := C(sI − A)−1B.

Igor Pontes Duff, pontes@mpi-magdeburg.mpg.de Automatic Generation of Minimal and Reduced Models for Structured and Nonlinear Parametric Dynamical Systems 11/34

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SLIDE 32

First-order systems

–Interpolation-based MOR–

An n-dimensional linear system

Σ :=

  • ˙

x(t) = Ax(t) + Bu(t), y(t) = Cx(t).

⇒

Transfer function H(s) := C(sI − A)−1B.

Theorem (simplified)

[Villemagne/Skelton 1987,Grimme 1997]

If range (V) ⊇ span

  • (σ1I − A)−1B, . . . , (σrI − A)−1B
  • ,

range (W) ⊇ span

  • (µ1I − A)−1C, . . . , (µrI − A)−T CT

,

Igor Pontes Duff, pontes@mpi-magdeburg.mpg.de Automatic Generation of Minimal and Reduced Models for Structured and Nonlinear Parametric Dynamical Systems 11/34

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SLIDE 33

First-order systems

–Interpolation-based MOR–

An n-dimensional linear system

Σ :=

  • ˙

x(t) = Ax(t) + Bu(t), y(t) = Cx(t).

⇒

Transfer function H(s) := C(sI − A)−1B.

Theorem (simplified)

[Villemagne/Skelton 1987,Grimme 1997]

If range (V) ⊇ span

  • (σ1I − A)−1B, . . . , (σrI − A)−1B
  • ,

range (W) ⊇ span

  • (µ1I − A)−1C, . . . , (µrI − A)−T CT

, then H(s) = ˆ H(s), s ∈ {σ1, σr, µ1 . . . , µr}.

Igor Pontes Duff, pontes@mpi-magdeburg.mpg.de Automatic Generation of Minimal and Reduced Models for Structured and Nonlinear Parametric Dynamical Systems 11/34

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SLIDE 34

First-order systems

–Reachable and observable subspaces–

An n-dimensional linear system

Σ :=

  • ˙

x(t) = Ax(t) + Bu(t), y(t) = Cx(t).

⇒

Input to state map: x(t) = t eAσBu(t − σ)dσ State to output map: y(t) = CeAtx0.

Igor Pontes Duff, pontes@mpi-magdeburg.mpg.de Automatic Generation of Minimal and Reduced Models for Structured and Nonlinear Parametric Dynamical Systems 12/34

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SLIDE 35

First-order systems

–Reachable and observable subspaces–

An n-dimensional linear system

Σ :=

  • ˙

x(t) = Ax(t) + Bu(t), y(t) = Cx(t).

⇒

Input to state map: x(t) = t eAσBu(t − σ)dσ State to output map: y(t) = CeAtx0.

Reachable and observable subspaces

The reachable subspace R and the observable subspace O are the smallest subspaces of Cn such that eAtB ∈ R and eAT tCT ∈ O for every t ≥ 0.

Igor Pontes Duff, pontes@mpi-magdeburg.mpg.de Automatic Generation of Minimal and Reduced Models for Structured and Nonlinear Parametric Dynamical Systems 12/34

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SLIDE 36

First-order systems

–Reachable and observable subspaces–

An n-dimensional linear system

Σ :=

  • ˙

x(t) = Ax(t) + Bu(t), y(t) = Cx(t).

⇒

Input to state map: x(t) = t eAσBu(t − σ)dσ State to output map: y(t) = CeAtx0.

Reachable and observable subspaces

The reachable subspace R and the observable subspace O are the smallest subspaces of Cn such that eAtB ∈ R and eAT tCT ∈ O for every t ≥ 0.

  • r, in the frequency domain,

(sI − A)−1B ∈ R and (sI − A)−T CT ∈ O for every s ∈ iR.

Igor Pontes Duff, pontes@mpi-magdeburg.mpg.de Automatic Generation of Minimal and Reduced Models for Structured and Nonlinear Parametric Dynamical Systems 12/34

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SLIDE 37

First-order systems

–Reachable and observable subspaces–

An n-dimensional linear system

Σ :=

  • ˙

x(t) = Ax(t) + Bu(t), y(t) = Cx(t).

⇒

Input to state map: x(t) = t eAσBu(t − σ)dσ State to output map: y(t) = CeAtx0.

Reachable and observable subspaces

The reachable subspace R and the observable subspace O are the smallest subspaces of Cn such that eAtB ∈ R and eAT tCT ∈ O for every t ≥ 0.

  • r, in the frequency domain,

(sI − A)−1B ∈ R and (sI − A)−T CT ∈ O for every s ∈ iR. Moreover, R⊥ and O⊥ are respectively, the unreachable and unobservable subspaces.

Igor Pontes Duff, pontes@mpi-magdeburg.mpg.de Automatic Generation of Minimal and Reduced Models for Structured and Nonlinear Parametric Dynamical Systems 12/34

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SLIDE 38

First-order systems

–Reachable and observable subspaces– A classical result in system theory: the unreachable (R⊥) or unobservable states (O⊥) can be removed from the dynamics, without changing the transfer function.

Igor Pontes Duff, pontes@mpi-magdeburg.mpg.de Automatic Generation of Minimal and Reduced Models for Structured and Nonlinear Parametric Dynamical Systems 13/34

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SLIDE 39

First-order systems

–Reachable and observable subspaces– A classical result in system theory: the unreachable (R⊥) or unobservable states (O⊥) can be removed from the dynamics, without changing the transfer function.

Characterization subspaces

Krylov subspaces (R. Kalman): R = range B AB A2B . . . An−1B , and O = range

  • CT

AT CT (A2)T CT . . . (An−1)T CT ,

Igor Pontes Duff, pontes@mpi-magdeburg.mpg.de Automatic Generation of Minimal and Reduced Models for Structured and Nonlinear Parametric Dynamical Systems 13/34

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SLIDE 40

First-order systems

–Reachable and observable subspaces– A classical result in system theory: the unreachable (R⊥) or unobservable states (O⊥) can be removed from the dynamics, without changing the transfer function.

Characterization subspaces

Krylov subspaces (R. Kalman): R = range B AB A2B . . . An−1B , and O = range

  • CT

AT CT (A2)T CT . . . (An−1)T CT , Rational Krylov subspaces

[Anderson/Antoulas 90’]

R = range (R) = range (σ1I − A)−1B (σ2I − A)−1B . . . (σnI − A)−1B , and O = range (O) = range

  • (σ1I − A)−T CT

(σ2I − A)−T CT . . . (σnI − A)−T CT

Igor Pontes Duff, pontes@mpi-magdeburg.mpg.de Automatic Generation of Minimal and Reduced Models for Structured and Nonlinear Parametric Dynamical Systems 13/34

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SLIDE 41

First-order systems

–Reachable and observable subspaces– A classical result in system theory: the unreachable (R⊥) or unobservable states (O⊥) can be removed from the dynamics, without changing the transfer function.

Characterization subspaces

Krylov subspaces (R. Kalman): R = range B AB A2B . . . An−1B , and O = range

  • CT

AT CT (A2)T CT . . . (An−1)T CT , Rational Krylov subspaces

[Anderson/Antoulas 90’]

R = range (R) = range (σ1I − A)−1B (σ2I − A)−1B . . . (σnI − A)−1B , and O = range (O) = range

  • (σ1I − A)−T CT

(σ2I − A)−T CT . . . (σnI − A)−T CT Notice: R = V and O = W are the same matrices for interpolation based MOR.

Igor Pontes Duff, pontes@mpi-magdeburg.mpg.de Automatic Generation of Minimal and Reduced Models for Structured and Nonlinear Parametric Dynamical Systems 13/34

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SLIDE 42

First-order systems

–Minimal realization– R = (σ1I − A)−1B (σ2I − A)−1B . . . (σnI − A)−1B , and O =

  • (σ1I − A)−T CT

(σ2I − A)−T CT . . . (σnI − A)−T CT

Minimal order

[Anderson/Antoulas 90’]

rank

  • OT R

OT AR

  • =
  • rder of the minimal realization obtained by

removing unreachable and unobservable states

Igor Pontes Duff, pontes@mpi-magdeburg.mpg.de Automatic Generation of Minimal and Reduced Models for Structured and Nonlinear Parametric Dynamical Systems 14/34

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SLIDE 43

First-order systems

–Minimal realization– R = (σ1I − A)−1B (σ2I − A)−1B . . . (σnI − A)−1B , and O =

  • (σ1I − A)−T CT

(σ2I − A)−T CT . . . (σnI − A)−T CT

Minimal order

[Anderson/Antoulas 90’]

rank

  • OT R

OT AR

  • =
  • rder of the minimal realization obtained by

removing unreachable and unobservable states

Construction of minimal or reduced-order approximation e.g.,[Mayo/Antoulas ’07]

Matrices OT R and OT AR allow us to find appropriate projection subspaces V and W Construction of a minimal system or reduced-order system.

Igor Pontes Duff, pontes@mpi-magdeburg.mpg.de Automatic Generation of Minimal and Reduced Models for Structured and Nonlinear Parametric Dynamical Systems 14/34

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SLIDE 44

An Illustrative Example

A demo example

H(s) = C(sE − A)−1B, E =   1 1 1   , A =   −1 −1 1 −2 −1 −3   , B =   1 2 1   , CT =   −1   ,

Igor Pontes Duff, pontes@mpi-magdeburg.mpg.de Automatic Generation of Minimal and Reduced Models for Structured and Nonlinear Parametric Dynamical Systems 15/34

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SLIDE 45

An Illustrative Example

A demo example

H(s) = C(sE − A)−1B, E =   1 1 1   , A =   −1 −1 1 −2 −1 −3   , B =   1 2 1   , CT =   −1   ,

Decay of singular values

2 4 6 10−25 10−10 105 k Singular values

Igor Pontes Duff, pontes@mpi-magdeburg.mpg.de Automatic Generation of Minimal and Reduced Models for Structured and Nonlinear Parametric Dynamical Systems 15/34

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SLIDE 46

An Illustrative Example

A demo example

H(s) = C(sE − A)−1B, E =   1 1 1   , A =   −1 −1 1 −2 −1 −3   , B =   1 2 1   , CT =   −1   ,

Construction of a minimal system

H(s), n = 3 ˆ H(s), r = 2 H(s) − ˆ H(s) 10−2 10−1 100 101 102 10−2 10−1 100 10−2 10−1 100 101 102 10−18 10−16

Igor Pontes Duff, pontes@mpi-magdeburg.mpg.de Automatic Generation of Minimal and Reduced Models for Structured and Nonlinear Parametric Dynamical Systems 15/34

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SLIDE 47

Structured Transfer Function

–Interpolation– Can we extend these ideas to structure linear systems?

An n-dimensional structured linear system

Transfer function H(s) = C(s)K(s)−1B(s), .

Igor Pontes Duff, pontes@mpi-magdeburg.mpg.de Automatic Generation of Minimal and Reduced Models for Structured and Nonlinear Parametric Dynamical Systems 16/34

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SLIDE 48

Structured Transfer Function

–Interpolation– Can we extend these ideas to structure linear systems?

An n-dimensional structured linear system

Transfer function H(s) = C(s)K(s)−1B(s), .

Theorem (simplified)

[Beattie/Gugercin ’09]

If range (V) ⊇ span

  • K(σ1)−1B(σ1), . . . , K(σr)−1B(σr)
  • ,

range (W) ⊇ span

  • K(µ1)−T C(µ1)T , . . . , K(µr)−T C(µr)T

,

Igor Pontes Duff, pontes@mpi-magdeburg.mpg.de Automatic Generation of Minimal and Reduced Models for Structured and Nonlinear Parametric Dynamical Systems 16/34

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SLIDE 49

Structured Transfer Function

–Interpolation– Can we extend these ideas to structure linear systems?

An n-dimensional structured linear system

Transfer function H(s) = C(s)K(s)−1B(s), .

Theorem (simplified)

[Beattie/Gugercin ’09]

If range (V) ⊇ span

  • K(σ1)−1B(σ1), . . . , K(σr)−1B(σr)
  • ,

range (W) ⊇ span

  • K(µ1)−T C(µ1)T , . . . , K(µr)−T C(µr)T

, then H(s) = ˆ H(s), s ∈ {σ1, σr, µ1 . . . , µr}.

Igor Pontes Duff, pontes@mpi-magdeburg.mpg.de Automatic Generation of Minimal and Reduced Models for Structured and Nonlinear Parametric Dynamical Systems 16/34

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SLIDE 50

Structured Transfer Function

–Reachability and observability–

An n-dimensional linear system

Transfer function H(s) = C(s)K(s)−1B(s), .

⇒

Input to state map: X(s) = K(s)−1B(s)U(s) State to output map: Y(s) = C(s)K(s)−1X(s).

Igor Pontes Duff, pontes@mpi-magdeburg.mpg.de Automatic Generation of Minimal and Reduced Models for Structured and Nonlinear Parametric Dynamical Systems 17/34

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SLIDE 51

Structured Transfer Function

–Reachability and observability–

An n-dimensional linear system

Transfer function H(s) = C(s)K(s)−1B(s), .

⇒

Input to state map: X(s) = K(s)−1B(s)U(s) State to output map: Y(s) = C(s)K(s)−1X(s).

Reachable and observable subspaces for structured systems

The reachable subspace R and the observable subspace O are the smallest subspaces of Cn such that K(s)−1B(s) ∈ R and K(s)−T C(s)T ∈ O for every s ∈ iR.

Igor Pontes Duff, pontes@mpi-magdeburg.mpg.de Automatic Generation of Minimal and Reduced Models for Structured and Nonlinear Parametric Dynamical Systems 17/34

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SLIDE 52

Structured Transfer Function

–Reachability and observability–

An n-dimensional linear system

Transfer function H(s) = C(s)K(s)−1B(s), .

⇒

Input to state map: X(s) = K(s)−1B(s)U(s) State to output map: Y(s) = C(s)K(s)−1X(s).

Reachable and observable subspaces for structured systems

The reachable subspace R and the observable subspace O are the smallest subspaces of Cn such that K(s)−1B(s) ∈ R and K(s)−T C(s)T ∈ O for every s ∈ iR. Moreover, R⊥ and O⊥ are respectively, the unreachable and unobservable subspaces.

Igor Pontes Duff, pontes@mpi-magdeburg.mpg.de Automatic Generation of Minimal and Reduced Models for Structured and Nonlinear Parametric Dynamical Systems 17/34

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SLIDE 53

Structured Transfer Function

–Reachability and observability–

An n-dimensional linear system

Transfer function H(s) = C(s)K(s)−1B(s), .

⇒

Input to state map: X(s) = K(s)−1B(s)U(s) State to output map: Y(s) = C(s)K(s)−1X(s).

Reachable and observable subspaces for structured systems

The reachable subspace R and the observable subspace O are the smallest subspaces of Cn such that K(s)−1B(s) ∈ R and K(s)−T C(s)T ∈ O for every s ∈ iR. Moreover, R⊥ and O⊥ are respectively, the unreachable and unobservable subspaces. A result in for structured linear systems: the unreachable (R⊥) or unobservable states (O⊥) can be removed from the dynamics, without changing the transfer function.

Igor Pontes Duff, pontes@mpi-magdeburg.mpg.de Automatic Generation of Minimal and Reduced Models for Structured and Nonlinear Parametric Dynamical Systems 17/34

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SLIDE 54

Structured Transfer Function

–Reachability and observability–

Structured transfer function

Consider an n-dimensional linear system, whose structure transfer function is given by H(s) := C(s)K(s)−1B(s).

Characterization Controllable and Observable Subspaces (simplified)

[Benner/Goyal/P. ’19]

Let R = K(σ1)−1B(σ1) K(σ2)−1B(σ2) . . . K(σg)−1B(σg) , O =

  • K(σ1)−T C(σ1)T

K(σ2)−T C(σ2)T . . . K(σg)−T C(σg)T .

Igor Pontes Duff, pontes@mpi-magdeburg.mpg.de Automatic Generation of Minimal and Reduced Models for Structured and Nonlinear Parametric Dynamical Systems 18/34

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SLIDE 55

Structured Transfer Function

–Reachability and observability–

Structured transfer function

Consider an n-dimensional linear system, whose structure transfer function is given by H(s) := C(s)K(s)−1B(s).

Characterization Controllable and Observable Subspaces (simplified)

[Benner/Goyal/P. ’19]

Let R = K(σ1)−1B(σ1) K(σ2)−1B(σ2) . . . K(σg)−1B(σg) , O =

  • K(σ1)−T C(σ1)T

K(σ2)−T C(σ2)T . . . K(σg)−T C(σg)T . Then Reachable subspace: R = range (R) . Observable subspace: O = range (O) .

Igor Pontes Duff, pontes@mpi-magdeburg.mpg.de Automatic Generation of Minimal and Reduced Models for Structured and Nonlinear Parametric Dynamical Systems 18/34

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SLIDE 56

Structured Transfer Function

–Reachability and observability–

Structured transfer function

Consider an n-dimensional linear system, whose structure transfer function is given by H(s) := C(s)K(s)−1B(s).

Characterization Controllable and Observable Subspaces (simplified)

[Benner/Goyal/P. ’19]

Let R = K(σ1)−1B(σ1) K(σ2)−1B(σ2) . . . K(σg)−1B(σg) , O =

  • K(σ1)−T C(σ1)T

K(σ2)−T C(σ2)T . . . K(σg)−T C(σg)T . Then Reachable subspace: R = range (R) . Observable subspace: O = range (O) . Notice: R = V and O = W are the same matrices for interpolation based MOR.

Igor Pontes Duff, pontes@mpi-magdeburg.mpg.de Automatic Generation of Minimal and Reduced Models for Structured and Nonlinear Parametric Dynamical Systems 18/34

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SLIDE 57

Structured Transfer Function

–Minimal realization–

Structured transfer function

Consider an n-dimensional linear system, whose structure transfer function is given by H(s) := C(s)K(s)−1B(s), with K(s) =

l

  • i=1

βi(s)Ai, and let R =

  • K(σ1)−1B(σ1)

K(σ2)−1B(σ2) . . . K(σN)−1B(σN )

  • ,

O =

  • K(σ1)−T C(σ1)T

K(σ2)−T C(σ2)T . . . K(σN)−T C(σN )T . such that R = range (R) and O = range (O).

Igor Pontes Duff, pontes@mpi-magdeburg.mpg.de Automatic Generation of Minimal and Reduced Models for Structured and Nonlinear Parametric Dynamical Systems 19/34

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SLIDE 58

Structured Transfer Function

–Minimal realization–

Structured transfer function

Consider an n-dimensional linear system, whose structure transfer function is given by H(s) := C(s)K(s)−1B(s), with K(s) =

l

  • i=1

βi(s)Ai, and let R =

  • K(σ1)−1B(σ1)

K(σ2)−1B(σ2) . . . K(σN)−1B(σN )

  • ,

O =

  • K(σ1)−T C(σ1)T

K(σ2)−T C(σ2)T . . . K(σN)−T C(σN )T . such that R = range (R) and O = range (O).

Minimal order (simplified)

[Benner/Goyal/P. ’19]

rank

  • OT A1R

. . . OT AlR

  • =
  • rder of the minimal realization obtained by

removing unreachable and unobservable states

Igor Pontes Duff, pontes@mpi-magdeburg.mpg.de Automatic Generation of Minimal and Reduced Models for Structured and Nonlinear Parametric Dynamical Systems 19/34

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SLIDE 59

Structured Transfer Function

–Towards model reduction– We propose a method enabling to identify simultaneously the states that are unreachable and

  • unobservable. Assume

rank

  • OT A1R

. . . OT AlR

  • = rank

      OT A1R . . . OT AlR       = r, with range (R) = R and range (O) = O.

Igor Pontes Duff, pontes@mpi-magdeburg.mpg.de Automatic Generation of Minimal and Reduced Models for Structured and Nonlinear Parametric Dynamical Systems 20/34

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SLIDE 60

Structured Transfer Function

–Towards model reduction– We propose a method enabling to identify simultaneously the states that are unreachable and

  • unobservable. Assume

rank

  • OT A1R

. . . OT AlR

  • = rank

      OT A1R . . . OT AlR       = r, with range (R) = R and range (O) = O. Then, we consider the compact SVDs

  • OT A1R

. . . OT AlR

  • = W1Σl ˜

VT and    OT A1R . . . OT AlR    = ˜ WΣrVT

1 .

Let W := OW1 and V := RV1 be two projection matrices and let us consider the lower-order realization ˆ C(s) ˆ K(s)−1 ˆ B(s) constructed by Petrov-Galerkin projection.

Igor Pontes Duff, pontes@mpi-magdeburg.mpg.de Automatic Generation of Minimal and Reduced Models for Structured and Nonlinear Parametric Dynamical Systems 20/34

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SLIDE 61

Structured Transfer Function

–Towards model reduction– Petrov-Galerkin projections: W := OW1 and V := RV1

Theorem

[Benner/Goyal/P. ’19]

The lower-order system ˆ C(s) ˆ K(s)−1 ˆ B(s) of order r, obtained by Petrov-Galerkin projection with V and W, realizes the original transfer function, i.e., ˆ C(s) ˆ K(s)−1 ˆ B(s) = C(s)K(s)−1B(s) for every s ∈ iR.

Igor Pontes Duff, pontes@mpi-magdeburg.mpg.de Automatic Generation of Minimal and Reduced Models for Structured and Nonlinear Parametric Dynamical Systems 21/34

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SLIDE 62

Structured Transfer Function

–Towards model reduction– Petrov-Galerkin projections: W := OW1 and V := RV1

Theorem

[Benner/Goyal/P. ’19]

The lower-order system ˆ C(s) ˆ K(s)−1 ˆ B(s) of order r, obtained by Petrov-Galerkin projection with V and W, realizes the original transfer function, i.e., ˆ C(s) ˆ K(s)−1 ˆ B(s) = C(s)K(s)−1B(s) for every s ∈ iR.

Determine dominate reachable and observable subspaces

The proposed procedure remove uncontrollable and unobservable subspaces simultaneously.

Igor Pontes Duff, pontes@mpi-magdeburg.mpg.de Automatic Generation of Minimal and Reduced Models for Structured and Nonlinear Parametric Dynamical Systems 21/34

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SLIDE 63

Structured Transfer Function

–Towards model reduction– Petrov-Galerkin projections: W := OW1 and V := RV1

Theorem

[Benner/Goyal/P. ’19]

The lower-order system ˆ C(s) ˆ K(s)−1 ˆ B(s) of order r, obtained by Petrov-Galerkin projection with V and W, realizes the original transfer function, i.e., ˆ C(s) ˆ K(s)−1 ˆ B(s) = C(s)K(s)−1B(s) for every s ∈ iR.

Determine dominate reachable and observable subspaces

The proposed procedure remove uncontrollable and unobservable subspaces simultaneously. Neglecting small singular values leads to reduced-order models.

Igor Pontes Duff, pontes@mpi-magdeburg.mpg.de Automatic Generation of Minimal and Reduced Models for Structured and Nonlinear Parametric Dynamical Systems 21/34

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SLIDE 64

Structured Transfer Function

–Towards model reduction– Petrov-Galerkin projections: W := OW1 and V := RV1

Theorem

[Benner/Goyal/P. ’19]

The lower-order system ˆ C(s) ˆ K(s)−1 ˆ B(s) of order r, obtained by Petrov-Galerkin projection with V and W, realizes the original transfer function, i.e., ˆ C(s) ˆ K(s)−1 ˆ B(s) = C(s)K(s)−1B(s) for every s ∈ iR.

Determine dominate reachable and observable subspaces

The proposed procedure remove uncontrollable and unobservable subspaces simultaneously. Neglecting small singular values leads to reduced-order models. Like balanced truncation, order the vectors in order of their importance.

Igor Pontes Duff, pontes@mpi-magdeburg.mpg.de Automatic Generation of Minimal and Reduced Models for Structured and Nonlinear Parametric Dynamical Systems 21/34

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SLIDE 65

Algorithm to Construct Structured ROMs

Algorithm: Dominant Reachable and Observable Projection (DROP)

  • 1. Take σi, µi, i = 1, . . . , N.

Igor Pontes Duff, pontes@mpi-magdeburg.mpg.de Automatic Generation of Minimal and Reduced Models for Structured and Nonlinear Parametric Dynamical Systems 22/34

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SLIDE 66

Algorithm to Construct Structured ROMs

Algorithm: Dominant Reachable and Observable Projection (DROP)

  • 1. Take σi, µi, i = 1, . . . , N.
  • 2. Compute R =
  • K(σ1)−1B(σ1), . . . , K(σN )−1B(σN )
  • .
  • 3. Compute O =
  • K(µ1)−T C(µ1)T , . . . , K(µN )−T C(µN )T

.

Igor Pontes Duff, pontes@mpi-magdeburg.mpg.de Automatic Generation of Minimal and Reduced Models for Structured and Nonlinear Parametric Dynamical Systems 22/34

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SLIDE 67

Algorithm to Construct Structured ROMs

Algorithm: Dominant Reachable and Observable Projection (DROP)

  • 1. Take σi, µi, i = 1, . . . , N.
  • 2. Compute R =
  • K(σ1)−1B(σ1), . . . , K(σN )−1B(σN )
  • .
  • 3. Compute O =
  • K(µ1)−T C(µ1)T , . . . , K(µN )−T C(µN )T

.

  • 4. Determine L(i) = OT AiR.

Igor Pontes Duff, pontes@mpi-magdeburg.mpg.de Automatic Generation of Minimal and Reduced Models for Structured and Nonlinear Parametric Dynamical Systems 22/34

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SLIDE 68

Algorithm to Construct Structured ROMs

Algorithm: Dominant Reachable and Observable Projection (DROP)

  • 1. Take σi, µi, i = 1, . . . , N.
  • 2. Compute R =
  • K(σ1)−1B(σ1), . . . , K(σN )−1B(σN )
  • .
  • 3. Compute O =
  • K(µ1)−T C(µ1)T , . . . , K(µN )−T C(µN )T

.

  • 4. Determine L(i) = OT AiR.
  • 5. Compute singular value decomposition:

Y1, Σ1, X1

  • = svd
  • L(1), . . . , L(l)

, Y2, Σ2, X2

  • = svd

        L(1) . . . L(l)         .

Igor Pontes Duff, pontes@mpi-magdeburg.mpg.de Automatic Generation of Minimal and Reduced Models for Structured and Nonlinear Parametric Dynamical Systems 22/34

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SLIDE 69

Algorithm to Construct Structured ROMs

Algorithm: Dominant Reachable and Observable Projection (DROP)

  • 1. Take σi, µi, i = 1, . . . , N.
  • 2. Compute R =
  • K(σ1)−1B(σ1), . . . , K(σN )−1B(σN )
  • .
  • 3. Compute O =
  • K(µ1)−T C(µ1)T , . . . , K(µN )−T C(µN )T

.

  • 4. Determine L(i) = OT AiR.
  • 5. Compute singular value decomposition:

Y1, Σ1, X1

  • = svd
  • L(1), . . . , L(l)

, Y2, Σ2, X2

  • = svd

        L(1) . . . L(l)         .

  • 6. Determine projection matrices: V := RX2(:, 1 : r),

W := OY1(:, 1 : r).

Igor Pontes Duff, pontes@mpi-magdeburg.mpg.de Automatic Generation of Minimal and Reduced Models for Structured and Nonlinear Parametric Dynamical Systems 22/34

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SLIDE 70

Algorithm to Construct Structured ROMs

Algorithm: Dominant Reachable and Observable Projection (DROP)

  • 1. Take σi, µi, i = 1, . . . , N.
  • 2. Compute R =
  • K(σ1)−1B(σ1), . . . , K(σN )−1B(σN )
  • .
  • 3. Compute O =
  • K(µ1)−T C(µ1)T , . . . , K(µN )−T C(µN )T

.

  • 4. Determine L(i) = OT AiR.
  • 5. Compute singular value decomposition:

Y1, Σ1, X1

  • = svd
  • L(1), . . . , L(l)

, Y2, Σ2, X2

  • = svd

        L(1) . . . L(l)         .

  • 6. Determine projection matrices: V := RX2(:, 1 : r),

W := OY1(:, 1 : r).

  • 7. Determine reduced-order system

ˆ K(s) = WT K(s)V, ˆ B(s) = WT B(s), ˆ C(s) = C(s)V.

Igor Pontes Duff, pontes@mpi-magdeburg.mpg.de Automatic Generation of Minimal and Reduced Models for Structured and Nonlinear Parametric Dynamical Systems 22/34

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SLIDE 71

Algorithm to Construct Structured ROMs

Algorithm: Dominant Reachable and Observable Projection (DROP)

  • 1. Take σi, µi, i = 1, . . . , N.
  • 2. Compute R =
  • K(σ1)−1B(σ1), . . . , K(σN )−1B(σN )
  • .
  • 3. Compute O =
  • K(µ1)−T C(µ1)T , . . . , K(µN )−T C(µN )T

.

  • 4. Determine L(i) = OT AiR.
  • 5. Compute singular value decomposition:

Y1, Σ1, X1

  • = svd
  • L(1), . . . , L(l)

, Y2, Σ2, X2

  • = svd

        L(1) . . . L(l)         .

  • 6. Determine projection matrices: V := RX2(:, 1 : r),

W := OY1(:, 1 : r).

  • 7. Determine reduced-order system

ˆ K(s) = WT K(s)V, ˆ B(s) = WT B(s), ˆ C(s) = C(s)V.

Can be easily parallarized not need to solve all shifted systems Make use of Low-rank solvers.

Igor Pontes Duff, pontes@mpi-magdeburg.mpg.de Automatic Generation of Minimal and Reduced Models for Structured and Nonlinear Parametric Dynamical Systems 22/34

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SLIDE 72

Algorithm to Construct Structured ROMs

Algorithm: Dominant Reachable and Observable Projection (DROP)

  • 1. Take σi, µi, i = 1, . . . , N.
  • 2. Compute R =
  • K(σ1)−1B(σ1), . . . , K(σN )−1B(σN )
  • .
  • 3. Compute O =
  • K(µ1)−T C(µ1)T , . . . , K(µN )−T C(µN )T

.

  • 4. Determine L(i) = OT AiR.
  • 5. Compute singular value decomposition:

Y1, Σ1, X1

  • = svd
  • L(1), . . . , L(l)

, Y2, Σ2, X2

  • = svd

        L(1) . . . L(l)         .

  • 6. Determine projection matrices: V := RX2(:, 1 : r),

W := OY1(:, 1 : r).

  • 7. Determine reduced-order system

ˆ K(s) = WT K(s)V, ˆ B(s) = WT B(s), ˆ C(s) = C(s)V.

Can be easily parallarized not need to solve all shifted systems Make use of Low-rank solvers. Efficient variant of SVDs can be applied including randomized SVD.

Igor Pontes Duff, pontes@mpi-magdeburg.mpg.de Automatic Generation of Minimal and Reduced Models for Structured and Nonlinear Parametric Dynamical Systems 22/34

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SLIDE 73

Low Rank Solvers for Sylvester Equations

If we take enough points (σi), the matrix R =

  • K(σ1)−1B(σ1)

. . . K(σN)−1B(σN )

  • ,

encodes the Cn reachable subspace.

Igor Pontes Duff, pontes@mpi-magdeburg.mpg.de Automatic Generation of Minimal and Reduced Models for Structured and Nonlinear Parametric Dynamical Systems 23/34

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SLIDE 74

Low Rank Solvers for Sylvester Equations

If we take enough points (σi), the matrix R =

  • K(σ1)−1B(σ1)

. . . K(σN)−1B(σN )

  • ,

encodes the Cn reachable subspace. Notice that R solves

l

  • i=1

AiRMi =

m

  • i=1

Bibi, where Mi = diag (βi(σ1), . . . , βi(σN )) and bi = [γi(σ1), . . . , γi(σN )] .

Igor Pontes Duff, pontes@mpi-magdeburg.mpg.de Automatic Generation of Minimal and Reduced Models for Structured and Nonlinear Parametric Dynamical Systems 23/34

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SLIDE 75

Low Rank Solvers for Sylvester Equations

If we take enough points (σi), the matrix R =

  • K(σ1)−1B(σ1)

. . . K(σN)−1B(σN )

  • ,

encodes the Cn reachable subspace. Notice that R solves

l

  • i=1

AiRMi =

m

  • i=1

Bibi, where Mi = diag (βi(σ1), . . . , βi(σN )) and bi = [γi(σ1), . . . , γi(σN )] . It is a generalized Sylvester equation. Low-rank solution is suitable. Truncated low-rank methods for generalized Sylveter equation.

[Kressner, Sirkovic 15’]

Igor Pontes Duff, pontes@mpi-magdeburg.mpg.de Automatic Generation of Minimal and Reduced Models for Structured and Nonlinear Parametric Dynamical Systems 23/34

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SLIDE 76

Parametric extension

–Parametric Structured Linear Systems– We also consider dynamical systems that are linear in state and parameterized.

Igor Pontes Duff, pontes@mpi-magdeburg.mpg.de Automatic Generation of Minimal and Reduced Models for Structured and Nonlinear Parametric Dynamical Systems 24/34

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SLIDE 77

Parametric extension

–Parametric Structured Linear Systems– We also consider dynamical systems that are linear in state and parameterized.

Parametric Butterfly Gyroscope

[morwiki, Modified Gyroscope]

M(d)¨ x(t) + D(d, θ) ˙ x(t) + K(θ) = Bu(t), y(t) = Cx(t), where M(d) = M1 + dM2 ∈ Rn, D(d, θ) = θ (D1 + dD2) , K(d) = T1 + 1 dT2 + dT3. Parameters and frequency range: θ ∈

  • 10−5, 10−7

, d ∈

  • 1, 2
  • f ∈
  • 0.025, 40
  • .

Order of the system: 17, 913. Figure: Semantic gyroscope diagram.

Igor Pontes Duff, pontes@mpi-magdeburg.mpg.de Automatic Generation of Minimal and Reduced Models for Structured and Nonlinear Parametric Dynamical Systems 24/34

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SLIDE 78

Parametric extension

–Problem formulation–

Parametric Problem Formulation

Approximate the transfer function of an n-dimensional system, H(s, p) = C(s, p)K(s, p)−1B(s, p), by the transfer function of a system ˆ H(s, p) = ˆ C(s, p) ˆ K(s, p)−1 ˆ B(s, p),

  • f order r ≪ n, such that

H(s, p) − ˆ H(s, p) < tolerance ∀s and ∀p. K(s, p) B(s, p) C(s, p)

⇓

ˆ K(s, p) ˆ B(s, p) ˆ C(s, p)

Igor Pontes Duff, pontes@mpi-magdeburg.mpg.de Automatic Generation of Minimal and Reduced Models for Structured and Nonlinear Parametric Dynamical Systems 25/34

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SLIDE 79

Parametric extension

–Parametric transfer function–

Parametric structured linear system

H(s) = C(s, p)K(s, p)−1B(s, p), C(s, p) = k

i=1 αi(s, p)Ci ∈ Rq×n,

K(s, p) = l

i=1 βi(s, p)Ai ∈ Rn×n,

B(s, p) = m

i=1 γi(s, p)Bi ∈ Rn×m,

Igor Pontes Duff, pontes@mpi-magdeburg.mpg.de Automatic Generation of Minimal and Reduced Models for Structured and Nonlinear Parametric Dynamical Systems 26/34

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SLIDE 80

Parametric extension

–Parametric transfer function–

Parametric structured linear system

H(s) = C(s, p)K(s, p)−1B(s, p), C(s, p) = k

i=1 αi(s, p)Ci ∈ Rq×n,

K(s, p) = l

i=1 βi(s, p)Ai ∈ Rn×n,

B(s, p) = m

i=1 γi(s, p)Bi ∈ Rn×m,

Reachable and observable subspaces for parametric structured systems

The reachable subspace R and the observable subspace O are the smallest subspaces of Cn such that K(s, p)−1B(s, p) ∈ R and K(s, p)−T C(s, p)T ∈ O for every s ∈ iR and p ∈ Ω .

Igor Pontes Duff, pontes@mpi-magdeburg.mpg.de Automatic Generation of Minimal and Reduced Models for Structured and Nonlinear Parametric Dynamical Systems 26/34

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SLIDE 81

Parametric extension

–Parametric transfer function–

Parametric structured linear system

H(s) = C(s, p)K(s, p)−1B(s, p), C(s, p) = k

i=1 αi(s, p)Ci ∈ Rq×n,

K(s, p) = l

i=1 βi(s, p)Ai ∈ Rn×n,

B(s, p) = m

i=1 γi(s, p)Bi ∈ Rn×m,

Reachable and observable subspaces for parametric structured systems

The reachable subspace R and the observable subspace O are the smallest subspaces of Cn such that K(s, p)−1B(s, p) ∈ R and K(s, p)−T C(s, p)T ∈ O for every s ∈ iR and p ∈ Ω . R =

  • K(σ1, p1)−1B(σ1, p1)

K(σ2.p2)−1B(σ2, p2) . . . K(σg, pg)−1B(σg, pg)

  • ,

O =

  • K(σ1, p1)−T C(σ1, p1)T

K(σ2, p2)−T C(σ2, p2)T . . . K(σg, pg)−T C(σg, pg)T . Then, if we have enough interpolation points, R = range (R) and O = range (O).

Igor Pontes Duff, pontes@mpi-magdeburg.mpg.de Automatic Generation of Minimal and Reduced Models for Structured and Nonlinear Parametric Dynamical Systems 26/34

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SLIDE 82

Parametric extension

–Minimal and reduced order model–

Structured transfer function

Consider an n-dimensional linear system, whose structure transfer function is given by H(s, p) := C(s, p)K(s, p)−1B(s, p), with K(s, p) =

l

  • i=1

βi(s, p)Ai,

Igor Pontes Duff, pontes@mpi-magdeburg.mpg.de Automatic Generation of Minimal and Reduced Models for Structured and Nonlinear Parametric Dynamical Systems 27/34

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SLIDE 83

Parametric extension

–Minimal and reduced order model–

Structured transfer function

Consider an n-dimensional linear system, whose structure transfer function is given by H(s, p) := C(s, p)K(s, p)−1B(s, p), with K(s, p) =

l

  • i=1

βi(s, p)Ai,

Minimal order (simplified)

[Benner/Goyal/P. ’19]

rank

  • OT A1R

. . . OT AlR

  • =
  • rder of the minimal realization obtained by

removing unreachable and unobservable states

Igor Pontes Duff, pontes@mpi-magdeburg.mpg.de Automatic Generation of Minimal and Reduced Models for Structured and Nonlinear Parametric Dynamical Systems 27/34

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SLIDE 84

Parametric extension

–Minimal and reduced order model–

Structured transfer function

Consider an n-dimensional linear system, whose structure transfer function is given by H(s, p) := C(s, p)K(s, p)−1B(s, p), with K(s, p) =

l

  • i=1

βi(s, p)Ai,

Minimal order (simplified)

[Benner/Goyal/P. ’19]

rank

  • OT A1R

. . . OT AlR

  • =
  • rder of the minimal realization obtained by

removing unreachable and unobservable states

Dominant Reachable and Observable Projection (DROP)

The proposed procedure remove uncontrollable and unobservable subspaces simultaneously. Neglecting small singular values leads to reduced-order models.

Igor Pontes Duff, pontes@mpi-magdeburg.mpg.de Automatic Generation of Minimal and Reduced Models for Structured and Nonlinear Parametric Dynamical Systems 27/34

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SLIDE 85

Numerical Examples

–Delay demo example–

A time-delay demo system

H(s) = C

  • sI − A1 − A2e−s−1 B

A1 =   −1 −1 −1   , A2 =   1 1 1   , B =   1   , C =   1 1  

T

,

Igor Pontes Duff, pontes@mpi-magdeburg.mpg.de Automatic Generation of Minimal and Reduced Models for Structured and Nonlinear Parametric Dynamical Systems 28/34

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SLIDE 86

Numerical Examples

–Delay demo example–

A time-delay demo system

H(s) = C

  • sI − A1 − A2e−s−1 B

A1 =   −1 −1 −1   , A2 =   1 1 1   , B =   1   , C =   1 1  

T

,

Let us construct, for σi = [1, 2, 3, 4, 5, 6], R =

  • K(σ1)−1B

. . . K(σ6)−1B

  • ,

O =

  • K(σ1)−T CT

. . . K(σ6)−T CT .

Igor Pontes Duff, pontes@mpi-magdeburg.mpg.de Automatic Generation of Minimal and Reduced Models for Structured and Nonlinear Parametric Dynamical Systems 28/34

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SLIDE 87

Numerical Examples

–Delay demo example–

A time-delay demo system

H(s) = C

  • sI − A1 − A2e−s−1 B

A1 =   −1 −1 −1   , A2 =   1 1 1   , B =   1   , C =   1 1  

T

,

Let us construct, for σi = [1, 2, 3, 4, 5, 6], R =

  • K(σ1)−1B

. . . K(σ6)−1B

  • ,

O =

  • K(σ1)−T CT

. . . K(σ6)−T CT . rank (R) = 2, rank (O) = 1.

nonreachable nonobservable

  • Igor Pontes Duff, pontes@mpi-magdeburg.mpg.de

Automatic Generation of Minimal and Reduced Models for Structured and Nonlinear Parametric Dynamical Systems 28/34

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SLIDE 88

Numerical Examples

–Delay demo example–

A time-delay demo system

H(s) = C

  • sI − A1 − A2e−s−1 B

A1 =   −1 −1 −1   , A2 =   1 1 1   , B =   1   , C =   1 1  

T

,

Let us construct, for σi = [1, 2, 3, 4, 5, 6], R =

  • K(σ1)−1B

. . . K(σ6)−1B

  • ,

O =

  • K(σ1)−T CT

. . . K(σ6)−T CT . rank (R) = 2, rank (O) = 1.

nonreachable nonobservable

  • rank
  • OT R

OT A1R OT A2R

  • = 1. (minimal

realization order)

Then, using DROP, we get the projection matrices V = RX(:, 1) and W = OY(:, 1).

Igor Pontes Duff, pontes@mpi-magdeburg.mpg.de Automatic Generation of Minimal and Reduced Models for Structured and Nonlinear Parametric Dynamical Systems 28/34

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SLIDE 89

Numerical Examples

–Delay demo example–

A time-delay demo system

H(s) = C

  • sI − A1 − A2e−s−1 B

A1 =   −1 −1 −1   , A2 =   1 1 1   , B =   1   , C =   1 1  

T

,

Let us construct, for σi = [1, 2, 3, 4, 5, 6], R =

  • K(σ1)−1B

. . . K(σ6)−1B

  • ,

O =

  • K(σ1)−T CT

. . . K(σ6)−T CT . rank (R) = 2, rank (O) = 1.

nonreachable nonobservable

  • rank
  • OT R

OT A1R OT A2R

  • = 1. (minimal

realization order)

Then, using DROP, we get the projection matrices V = RX(:, 1) and W = OY(:, 1). The ˆ H obtained using V and W satisfies H(s) = ˆ H(s), ∀s.

Igor Pontes Duff, pontes@mpi-magdeburg.mpg.de Automatic Generation of Minimal and Reduced Models for Structured and Nonlinear Parametric Dynamical Systems 28/34

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SLIDE 90

Numerical Examples

–Delay demo example–

A time-delay demo system

H(s) = C

  • sI − A1 − A2e−s−1 B

A1 =   −1 −1 −1   , A2 =   1 1 1   , B =   1   , C =   1 1  

T

,

Let us construct, for σi = [1, 2, 3, 4, 5, 6], R =

  • K(σ1)−1B

. . . K(σ6)−1B

  • ,

O =

  • K(σ1)−T CT

. . . K(σ6)−T CT . rank (R) = 2, rank (O) = 1.

nonreachable nonobservable

  • rank
  • OT R

OT A1R OT A2R

  • = 1. (minimal

realization order)

Then, using DROP, we get the projection matrices V = RX(:, 1) and W = OY(:, 1). The ˆ H obtained using V and W satisfies H(s) = ˆ H(s), ∀s. Decay of singular values

2 4 6 10−25 10−10 105 k Singular values

Igor Pontes Duff, pontes@mpi-magdeburg.mpg.de Automatic Generation of Minimal and Reduced Models for Structured and Nonlinear Parametric Dynamical Systems 28/34

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SLIDE 91

Numerical Examples

–Delay demo example–

A time-delay demo system

H(s) = C

  • sI − A1 − A2e−s−1 B

A1 =   −1 −1 −1   , A2 =   1 1 1   , B =   1   , C =   1 1  

T

,

Construction of a minimal system

H(s)(n = 3) ˆ H(s)(r = 1) H(s) − ˆ H(s) 10−1 100 101 10−2 10−1 100 101 10−2 10−1 100 101 102 10−17 10−15

Igor Pontes Duff, pontes@mpi-magdeburg.mpg.de Automatic Generation of Minimal and Reduced Models for Structured and Nonlinear Parametric Dynamical Systems 28/34

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SLIDE 92

Numerical Examples

–Parametric demo example–

A parametric demo dynamical system

H(s, p) = C (sI − A1 − pA2)−1 B,

A1 =   −2 −1 −2   , A2 =   1 −1 1   , B =   1 1   , CT =   1 1   ,

Igor Pontes Duff, pontes@mpi-magdeburg.mpg.de Automatic Generation of Minimal and Reduced Models for Structured and Nonlinear Parametric Dynamical Systems 29/34

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SLIDE 93

Numerical Examples

–Parametric demo example–

A parametric demo dynamical system

H(s, p) = C (sI − A1 − pA2)−1 B,

A1 =   −2 −1 −2   , A2 =   1 −1 1   , B =   1 1   , CT =   1 1   ,

For l = 20 points (σi, pi), let R = K(σ1, p1)−1B . . . K(σl, pl)−1B , O =

  • K(σ1, p1)−T CT

. . . K(σl, pl)−T CT .

Igor Pontes Duff, pontes@mpi-magdeburg.mpg.de Automatic Generation of Minimal and Reduced Models for Structured and Nonlinear Parametric Dynamical Systems 29/34

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SLIDE 94

Numerical Examples

–Parametric demo example–

A parametric demo dynamical system

H(s, p) = C (sI − A1 − pA2)−1 B,

A1 =   −2 −1 −2   , A2 =   1 −1 1   , B =   1 1   , CT =   1 1   ,

For l = 20 points (σi, pi), let R = K(σ1, p1)−1B . . . K(σl, pl)−1B , O =

  • K(σ1, p1)−T CT

. . . K(σl, pl)−T CT . rank

  • OT R

OT A1R OT A2R

  • = 2.

Igor Pontes Duff, pontes@mpi-magdeburg.mpg.de Automatic Generation of Minimal and Reduced Models for Structured and Nonlinear Parametric Dynamical Systems 29/34

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SLIDE 95

Numerical Examples

–Parametric demo example–

A parametric demo dynamical system

H(s, p) = C (sI − A1 − pA2)−1 B,

A1 =   −2 −1 −2   , A2 =   1 −1 1   , B =   1 1   , CT =   1 1   ,

For l = 20 points (σi, pi), let R = K(σ1, p1)−1B . . . K(σl, pl)−1B , O =

  • K(σ1, p1)−T CT

. . . K(σl, pl)−T CT . rank

  • OT R

OT A1R OT A2R

  • = 2.

Compute projectors V and W and ˆ H(s, p). Then, H(s, p) = ˆ H(s, p). 5 10 15 20 10−25 10−10 105 Decay of Singular values

Igor Pontes Duff, pontes@mpi-magdeburg.mpg.de Automatic Generation of Minimal and Reduced Models for Structured and Nonlinear Parametric Dynamical Systems 29/34

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SLIDE 96

Numerical Examples

–Parametric demo example–

A parametric demo dynamical system

H(s, p) = C (sI − A1 − pA2)−1 B,

A1 =   −2 −1 −2   , A2 =   1 −1 1   , B =   1 1   , CT =   1 1   ,

For l = 20 points (σi, pi), let R = K(σ1, p1)−1B . . . K(σl, pl)−1B , O =

  • K(σ1, p1)−T CT

. . . K(σl, pl)−T CT . rank

  • OT R

OT A1R OT A2R

  • = 2.

Compute projectors V and W and ˆ H(s, p). Then, H(s, p) = ˆ H(s, p). 10−3 10−2 10−1 100 101 102 10−22 10−18 10−14 Absolute error

Igor Pontes Duff, pontes@mpi-magdeburg.mpg.de Automatic Generation of Minimal and Reduced Models for Structured and Nonlinear Parametric Dynamical Systems 29/34

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SLIDE 97

Numerical Examples

–Time-delay example–

Delay example

[Beattie/Gugercin ’09]

E ˙ x(t) = Ax(t) + Aτx(t − τ) + Bu(t), y(t) = Cx(t). H(s) = C(sE − A1 − Aτe−sτ)B Full order model n = 500 and τ = 1. To employ the proposed methods, we consider 100 points on the imaginary axis.

Igor Pontes Duff, pontes@mpi-magdeburg.mpg.de Automatic Generation of Minimal and Reduced Models for Structured and Nonlinear Parametric Dynamical Systems 30/34

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SLIDE 98

Numerical Examples

–Time-delay example–

Delay example

[Beattie/Gugercin ’09]

E ˙ x(t) = Ax(t) + Aτx(t − τ) + Bu(t), y(t) = Cx(t). H(s) = C(sE − A1 − Aτe−sτ)B Full order model n = 500 and τ = 1. To employ the proposed methods, we consider 100 points on the imaginary axis. Decay of singular values DROP Balanced truncation [Breiten ’16] 12 25 50 75 100 100 10−7 10−14 10−20 Figure: Delay example: relative decay of the singular values using the proposed method and structured balanced truncation.

Igor Pontes Duff, pontes@mpi-magdeburg.mpg.de Automatic Generation of Minimal and Reduced Models for Structured and Nonlinear Parametric Dynamical Systems 30/34

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SLIDE 99

Numerical Examples

–Time-delay example–

Delay example

[Beattie/Gugercin ’09]

E ˙ x(t) = Ax(t) + Aτx(t − τ) + Bu(t), y(t) = Cx(t). H(s) = C(sE − A1 − Aτe−sτ)B Full order model n = 500 and τ = 1. To employ the proposed methods, we consider 100 points on the imaginary axis. Reduced system of order r = 12

  • Ori. sys.

DROP BT [Breiten ’16]

10−2 10−1 100 101 102 103 104 10−4 10−2 100 freq (s) 10−2 10−1 100 101 102 103 104 10−12 10−7 10−2 freq (s) Igor Pontes Duff, pontes@mpi-magdeburg.mpg.de Automatic Generation of Minimal and Reduced Models for Structured and Nonlinear Parametric Dynamical Systems 30/34

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SLIDE 100

Numerical Examples

–Fractional derivative example–

Fractional Maxwell equations.

[Feng/Benner ’08]

H(s) = sBT

  • s2I − 1

√sD + A −1 B, Full order model n = 29, 295. Frequency range is F :=

  • 4e9, 8e9
  • Hz.

To employ the proposed methods, we consider 50 points on the imaginary axis.

Igor Pontes Duff, pontes@mpi-magdeburg.mpg.de Automatic Generation of Minimal and Reduced Models for Structured and Nonlinear Parametric Dynamical Systems 31/34

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SLIDE 101

Numerical Examples

–Fractional derivative example–

Fractional Maxwell equations.

[Feng/Benner ’08]

H(s) = sBT

  • s2I − 1

√sD + A −1 B, Full order model n = 29, 295. Frequency range is F :=

  • 4e9, 8e9
  • Hz.

To employ the proposed methods, we consider 50 points on the imaginary axis. Decay of singular values DROP 8 25 50 75 100 100 10−7 10−14 10−20

Igor Pontes Duff, pontes@mpi-magdeburg.mpg.de Automatic Generation of Minimal and Reduced Models for Structured and Nonlinear Parametric Dynamical Systems 31/34

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SLIDE 102

Numerical Examples

–Fractional derivative example–

Fractional Maxwell equations.

[Feng/Benner ’08]

H(s) = sBT

  • s2I − 1

√sD + A −1 B, Full order model n = 29, 295. Frequency range is F :=

  • 4e9, 8e9
  • Hz.

To employ the proposed methods, we consider 50 points on the imaginary axis. Reduced system of order r = 12

  • Ori. sys.

DROP (r = 8) Method in [Feng/Benner ’08] (r = 38)

4 5 6 7 8 ·109 1.5 2 2.5 3 Freq (s) H(s) 4 5 6 7 8 ·109 10−13 10−10 10−7 Freq (s) H − Hr Igor Pontes Duff, pontes@mpi-magdeburg.mpg.de Automatic Generation of Minimal and Reduced Models for Structured and Nonlinear Parametric Dynamical Systems 31/34

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SLIDE 103

Numerical Examples

–Parametric Butterfly Gyroscope-

Parametric Butterfly Gyroscope

[morwiki, Modified Gyroscope]

M(d)¨ x(t) + D(d, θ) ˙ x(t) + K(θ) = Bu(t), y(t) = Cx(t), where M(d) = M1 + dM2, D(d, θ) = θ (D1 + dD2) , K(d) = T1 + 1 dT2 + dT3. Parameters and frequency range: θ ∈

  • 10−5, 10−7

, d ∈

  • 1, 2
  • freq ∈
  • 0.025, 40
  • .

Order of the system: n = 17, 913. Figure: Semantic gyroscope diagram.

Igor Pontes Duff, pontes@mpi-magdeburg.mpg.de Automatic Generation of Minimal and Reduced Models for Structured and Nonlinear Parametric Dynamical Systems 32/34

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SLIDE 104

Numerical Examples

–Parametric Butterfly Gyroscope-

Parametric Butterfly Gyroscope

50 80 150 250 100 10−7 10−15 10−20

Figure: Gyro example: relative decay of the singular values obtained using the proposed method.

We take 500 points for frequency s in the logarithmic way and the same number of random points for parameter p = d, θT in the considered range.

Igor Pontes Duff, pontes@mpi-magdeburg.mpg.de Automatic Generation of Minimal and Reduced Models for Structured and Nonlinear Parametric Dynamical Systems 33/34

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SLIDE 105

Numerical Examples

–Parametric Butterfly Gyroscope-

Parametric Butterfly Gyroscope

  • Orig. sys. (n = 17,913)

DROP (r = 80) Method in [Feng, Antoulas, Benner 17’] (r = 210) 0.5 1 1.5 10−7 10−5 Freq (s) 0.5 1 1.5 10−7 10−5 10−3 Freq (s) 0.5 1 1.5 10−8 10−5 10−2 Freq (s) 0.5 1 1.5 10−5 10−3 Freq (s)

Figure: p(1) :

  • 1.00, 10−7

, p(2) :

  • 1.33, 4.64 · 10−7

, p(3) :

  • 1.67, 2.15 · 10−6

, p(4) :

  • 2.00, 10−5

.

Igor Pontes Duff, pontes@mpi-magdeburg.mpg.de Automatic Generation of Minimal and Reduced Models for Structured and Nonlinear Parametric Dynamical Systems 33/34

slide-106
SLIDE 106

Numerical Examples

–Parametric Butterfly Gyroscope-

Parametric Butterfly Gyroscope

  • Orig. sys. (n = 17,913)

DROP (r = 80) Method in [Feng, Antoulas, Benner 17’] (r = 210) 0.5 1 1.5 10−15 10−11 10−7 Freq (s) 0.5 1 1.5 10−15 10−11 10−7 Freq (s) 0.5 1 1.5 10−15 10−11 10−7 Freq (s) 0.5 1 1.5 10−15 10−10 10−5 Freq (s)

Figure: p(1) :

  • 1.00, 10−7

, p(2) :

  • 1.33, 4.64 · 10−7

, p(3) :

  • 1.67, 2.15 · 10−6

, p(4) :

  • 2.00, 10−5

.

Igor Pontes Duff, pontes@mpi-magdeburg.mpg.de Automatic Generation of Minimal and Reduced Models for Structured and Nonlinear Parametric Dynamical Systems 33/34

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SLIDE 107

Outlook

Contributions of the talk

Minimal realization and reduced-order modeling for structured linear systems. DROP (Dominant Reachable and Observable Projection) algorithm. Computational aspects in a large-scale setting (low-rank factors, randomized SVDs). Extend results to parametric systems. Application to benchmarks examples.

Igor Pontes Duff, pontes@mpi-magdeburg.mpg.de Automatic Generation of Minimal and Reduced Models for Structured and Nonlinear Parametric Dynamical Systems 34/34

slide-108
SLIDE 108

Outlook

Contributions of the talk

Minimal realization and reduced-order modeling for structured linear systems. DROP (Dominant Reachable and Observable Projection) algorithm. Computational aspects in a large-scale setting (low-rank factors, randomized SVDs). Extend results to parametric systems. Application to benchmarks examples.

Open questions and future work

Errors estimators and parameter choice. Stability of reduced-order systems. Reference: Benner, P., Goyal, P., & Pontes Duff, I. (2019). Identification of Dominant Subspaces for Linear Structured Parametric Systems and Model Reduction. arXiv preprint arXiv:1910.13945.

Igor Pontes Duff, pontes@mpi-magdeburg.mpg.de Automatic Generation of Minimal and Reduced Models for Structured and Nonlinear Parametric Dynamical Systems 34/34