Introduction –Structured Linear Dynamical Systems– Linear system (standard) Linear system (standard) Time �→ Frequency E ˙ x ( t ) = Ax ( t ) + Bu ( t ) , G ( s ) = C ( s E − A ) − 1 B . y ( t ) = Cx ( t ) . Second-order system Second-order system Time �→ Frequency M ¨ x ( t ) + D ˙ x ( t ) + Kx ( t ) = Bu ( t ) , G ( s ) = C ( s 2 M + s D + K ) − 1 B . y ( t ) = Cx ( t ) . Delay system Delay system Time �→ Frequency E ˙ x ( t ) = Ax ( t ) + A τ x ( t − τ ) + Bu ( t ) , s E − A − A τ e − sτ � − 1 B . � G ( s ) = C y ( t ) = Cx ( t ) . Integro system Integro system � t E ˙ x ( t ) = Ax ( t ) + A τ x ( τ ) dτ + Bu ( t ) , Time �→ Frequency � − 1 � s E − A − 1 G ( s ) = C s A τ B . 0 y ( t ) = Cx ( t ) . Igor Pontes Duff, pontes@mpi-magdeburg.mpg.de Automatic Generation of Minimal and Reduced Models for Structured and Nonlinear Parametric Dynamical Systems 6/34
Problem Formulation Problem Formulation Approximate the transfer function of an n -dimensional system, H ( s ) = C ( s ) K ( s ) − 1 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
Problem Formulation Problem Formulation Approximate the transfer function of an n -dimensional system, H ( s ) = C ( s ) K ( s ) − 1 B ( s ) , by the transfer function of a system K ( s ) − 1 ˆ H ( s ) = ˆ C ( s ) ˆ 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
Problem Formulation Problem Formulation Approximate the transfer function of an n -dimensional system, H ( s ) = C ( s ) K ( s ) − 1 B ( s ) , K ( s ) B ( s ) by the transfer function of a system C ( s ) K ( s ) − 1 ˆ H ( s ) = ˆ C ( s ) ˆ ˆ B ( s ) , ⇓ of order r ≪ n , such that � H ( s ) − ˆ H ( s ) � < tolerance ∀ s. ˆ ˆ K ( s ) B ( s ) � H − ˆ = ⇒ Optimization problem: min H � . ˆ C ( s ) order ( ˆ H ) ≤ r Igor Pontes Duff, pontes@mpi-magdeburg.mpg.de Automatic Generation of Minimal and Reduced Models for Structured and Nonlinear Parametric Dynamical Systems 7/34
Problem Formulation –Structure Preservation– Structured system C ( s ) = � k i =1 α i ( s ) C i ∈ R q × n , K ( s ) = � l i =1 β i ( s ) A i ∈ R n × n , H ( s ) = C ( s ) K ( s ) − 1 B ( s ) , B ( s ) = � m i =1 γ i ( s ) B i ∈ R n × m , α i ( s ) , β i ( s ) and γ i ( 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
Problem Formulation –Structure Preservation– Structured system C ( s ) = � k i =1 α i ( s ) C i ∈ R q × n , K ( s ) = � l i =1 β i ( s ) A i ∈ R n × n , H ( s ) = C ( s ) K ( s ) − 1 B ( s ) , B ( s ) = � m i =1 γ i ( s ) B i ∈ R n × m , α i ( s ) , β i ( s ) and γ i ( s ) are meromorphic functions (dynamical structures). Structured reduced system ˆ i =1 α i ( s ) ˆ C ( s ) = � k C i ∈ R q × r , ˆ i =1 β i ( s ) ˆ K ( s ) = � g A i ∈ R r × r , K ( s ) − 1 ˆ H ( s ) = ˆ ˆ C ( s ) ˆ B ( s ) , ˆ i =1 γ i ( s ) ˆ B ( s ) = � m B i ∈ R r × 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
Construction of reduced-order systems Petrov-Galerkin-type projection For given projection matrices V , W ∈ R n × 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
Construction of reduced-order systems Petrov-Galerkin-type projection For given projection matrices V , W ∈ R n × r , leading to ˆ C ( s ) = α 1 ( s ) C 1 V + α 2 ( s ) C 2 V + · · · + α k ( s ) C k V , = α 1 ( s ) ˆ C 1 + α 2 ( s ) ˆ C 2 + · · · + α ( s ) ˆ C k , K ( s ) = β 1 ( s ) W T A 1 V + · · · + β g ( s ) W T A g V , ˆ = + β 1 ( s ) ˆ A 1 + · · · + β g ( s ) ˆ A g , B ( s ) = γ 1 ( s ) W T B 1 + γ 2 ( s ) W T B 2 + · · · + γ m ( s ) W T B m , ˆ = γ 1 ( s ) ˆ B 1 + γ 2 ( s ) ˆ B 2 + · · · + γ m ( s ) ˆ B m , Igor Pontes Duff, pontes@mpi-magdeburg.mpg.de Automatic Generation of Minimal and Reduced Models for Structured and Nonlinear Parametric Dynamical Systems 9/34
Construction of reduced-order systems Petrov-Galerkin-type projection For given projection matrices V , W ∈ R n × r , leading to ˆ C ( s ) = α 1 ( s ) C 1 V + α 2 ( s ) C 2 V + · · · + α k ( s ) C k V , = α 1 ( s ) ˆ C 1 + α 2 ( s ) ˆ C 2 + · · · + α ( s ) ˆ C k , K ( s ) = β 1 ( s ) W T A 1 V + · · · + β g ( s ) W T A g V , ˆ = + β 1 ( s ) ˆ A 1 + · · · + β g ( s ) ˆ A g , B ( s ) = γ 1 ( s ) W T B 1 + γ 2 ( s ) W T B 2 + · · · + γ m ( s ) W T B m , ˆ = γ 1 ( s ) ˆ B 1 + γ 2 ( s ) ˆ B 2 + · · · + γ m ( s ) ˆ B m , � 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
Some Literature Common existing approaches 1. Interpolating reduced-order systems [ Beattie/Gugercin ’09 ] H ( σ i ) = ˆ H ( σ i ) , for i = 1 , . . . , 2 r – How to choose σ i ?? Igor Pontes Duff, pontes@mpi-magdeburg.mpg.de Automatic Generation of Minimal and Reduced Models for Structured and Nonlinear Parametric Dynamical Systems 10/34
Some Literature Common existing approaches 1. Interpolating reduced-order systems [ Beattie/Gugercin ’09 ] H ( σ i ) = ˆ H ( σ i ) , for i = 1 , . . . , 2 r – How to choose σ i ?? 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
Some Literature Common existing approaches 1. Interpolating reduced-order systems [ Beattie/Gugercin ’09 ] H ( σ i ) = ˆ H ( σ i ) , for i = 1 , . . . , 2 r – How to choose σ i ?? 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
Some Literature Common existing approaches 1. Interpolating reduced-order systems [ Beattie/Gugercin ’09 ] H ( σ i ) = ˆ H ( σ i ) , for i = 1 , . . . , 2 r – How to choose σ i ?? 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
First-order systems –Interpolation-based MOR– An n -dimensional linear system � Transfer function ⇒ x ( t ) ˙ = Ax ( t ) + Bu ( t ) , Σ := y ( t ) = Cx ( t ) . H ( s ) := C ( s I − A ) − 1 B . Igor Pontes Duff, pontes@mpi-magdeburg.mpg.de Automatic Generation of Minimal and Reduced Models for Structured and Nonlinear Parametric Dynamical Systems 11/34
First-order systems –Interpolation-based MOR– An n -dimensional linear system � Transfer function ⇒ x ( t ) ˙ = Ax ( t ) + Bu ( t ) , Σ := y ( t ) = Cx ( t ) . H ( s ) := C ( s I − A ) − 1 B . Theorem (simplified) [ Villemagne/Skelton 1987,Grimme 1997 ] If � � ( σ 1 I − A ) − 1 B , . . . , ( σ r I − A ) − 1 B range ( V ) ⊇ span , � ( µ 1 I − A ) − 1 C , . . . , ( µ r I − A ) − T C T � range ( W ) ⊇ span , Igor Pontes Duff, pontes@mpi-magdeburg.mpg.de Automatic Generation of Minimal and Reduced Models for Structured and Nonlinear Parametric Dynamical Systems 11/34
First-order systems –Interpolation-based MOR– An n -dimensional linear system � Transfer function ⇒ x ( t ) ˙ = Ax ( t ) + Bu ( t ) , Σ := y ( t ) = Cx ( t ) . H ( s ) := C ( s I − A ) − 1 B . Theorem (simplified) [ Villemagne/Skelton 1987,Grimme 1997 ] If � � ( σ 1 I − A ) − 1 B , . . . , ( σ r I − A ) − 1 B range ( V ) ⊇ span , � ( µ 1 I − A ) − 1 C , . . . , ( µ r I − A ) − T C T � range ( W ) ⊇ span , then H ( s ) = ˆ H ( s ) , s ∈ { σ 1 , σ r , µ 1 . . . , µ r } . Igor Pontes Duff, pontes@mpi-magdeburg.mpg.de Automatic Generation of Minimal and Reduced Models for Structured and Nonlinear Parametric Dynamical Systems 11/34
First-order systems –Reachable and observable subspaces– An n -dimensional linear system � t � ⇒ e Aσ Bu ( t − σ ) dσ Input to state map: x ( t ) = x ( t ) ˙ = Ax ( t ) + Bu ( t ) , Σ := 0 y ( t ) = Cx ( t ) . State to output map: y ( t ) = C e A t x 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
First-order systems –Reachable and observable subspaces– An n -dimensional linear system � t � ⇒ e Aσ Bu ( t − σ ) dσ Input to state map: x ( t ) = x ( t ) ˙ = Ax ( t ) + Bu ( t ) , Σ := 0 y ( t ) = Cx ( t ) . State to output map: y ( t ) = C e A t x 0 . Reachable and observable subspaces The reachable subspace R and the observable subspace O are the smallest subspaces of C n such that e A T t C T ∈ O for every t ≥ 0 . e A t B ∈ R and Igor Pontes Duff, pontes@mpi-magdeburg.mpg.de Automatic Generation of Minimal and Reduced Models for Structured and Nonlinear Parametric Dynamical Systems 12/34
First-order systems –Reachable and observable subspaces– An n -dimensional linear system � t � ⇒ e Aσ Bu ( t − σ ) dσ Input to state map: x ( t ) = x ( t ) ˙ = Ax ( t ) + Bu ( t ) , Σ := 0 y ( t ) = Cx ( t ) . State to output map: y ( t ) = C e A t x 0 . Reachable and observable subspaces The reachable subspace R and the observable subspace O are the smallest subspaces of C n such that e A T t C T ∈ O for every t ≥ 0 . e A t B ∈ R and or, in the frequency domain, ( s I − A ) − T C T ∈ O for every s ∈ i R . ( s I − A ) − 1 B ∈ R and Igor Pontes Duff, pontes@mpi-magdeburg.mpg.de Automatic Generation of Minimal and Reduced Models for Structured and Nonlinear Parametric Dynamical Systems 12/34
First-order systems –Reachable and observable subspaces– An n -dimensional linear system � t � ⇒ e Aσ Bu ( t − σ ) dσ Input to state map: x ( t ) = x ( t ) ˙ = Ax ( t ) + Bu ( t ) , Σ := 0 y ( t ) = Cx ( t ) . State to output map: y ( t ) = C e A t x 0 . Reachable and observable subspaces The reachable subspace R and the observable subspace O are the smallest subspaces of C n such that e A T t C T ∈ O for every t ≥ 0 . e A t B ∈ R and or, in the frequency domain, ( s I − A ) − T C T ∈ O for every s ∈ i R . ( s I − A ) − 1 B ∈ R and 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
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
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): �� B A n − 1 B �� A 2 B R = range , and AB . . . �� ( A n − 1 ) T C T �� A T C T ( A 2 ) T C T C T O = range . . . , Igor Pontes Duff, pontes@mpi-magdeburg.mpg.de Automatic Generation of Minimal and Reduced Models for Structured and Nonlinear Parametric Dynamical Systems 13/34
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): �� B A n − 1 B �� A 2 B R = range , and AB . . . �� ( A n − 1 ) T C T �� A T C T ( A 2 ) T C T C T O = range . . . , Rational Krylov subspaces [ Anderson/Antoulas 90’ ] �� ( σ 1 I − A ) − 1 B ( σ n I − A ) − 1 B �� ( σ 2 I − A ) − 1 B R = range ( R ) = range . . . , and �� ( σ n I − A ) − T C T �� ( σ 1 I − A ) − T C T ( σ 2 I − A ) − T C T O = range ( O ) = range . . . Igor Pontes Duff, pontes@mpi-magdeburg.mpg.de Automatic Generation of Minimal and Reduced Models for Structured and Nonlinear Parametric Dynamical Systems 13/34
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): �� B A n − 1 B �� A 2 B R = range , and AB . . . �� ( A n − 1 ) T C T �� A T C T ( A 2 ) T C T C T O = range . . . , Rational Krylov subspaces [ Anderson/Antoulas 90’ ] �� ( σ 1 I − A ) − 1 B ( σ n I − A ) − 1 B �� ( σ 2 I − A ) − 1 B R = range ( R ) = range . . . , and �� ( σ n I − A ) − T C T �� ( σ 1 I − A ) − T C T ( σ 2 I − A ) − T C T O = range ( O ) = range . . . 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
First-order systems –Minimal realization– � ( σ 1 I − A ) − 1 B ( σ n I − A ) − 1 B � ( σ 2 I − A ) − 1 B R = . . . , and � ( σ n I − A ) − T C T � ( σ 1 I − A ) − T C T ( σ 2 I − A ) − T C T O = . . . Minimal order [ Anderson/Antoulas 90’ ] � order of the minimal realization obtained by �� �� O T R O T AR rank = 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
First-order systems –Minimal realization– � ( σ 1 I − A ) − 1 B ( σ n I − A ) − 1 B � ( σ 2 I − A ) − 1 B R = . . . , and � ( σ n I − A ) − T C T � ( σ 1 I − A ) − T C T ( σ 2 I − A ) − T C T O = . . . Minimal order [ Anderson/Antoulas 90’ ] � order of the minimal realization obtained by �� �� O T R O T AR rank = removing unreachable and unobservable states Construction of minimal or reduced-order approximation e.g., [ Mayo/Antoulas ’07 ] Matrices O T R and O T 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
An Illustrative Example A demo example H ( s ) = C ( s E − A ) − 1 B , − 1 − 1 − 1 1 0 0 1 1 C T = , , , , 0 1 0 0 − 2 − 1 2 0 E = A = B = 0 0 1 0 0 − 3 1 0 Igor Pontes Duff, pontes@mpi-magdeburg.mpg.de Automatic Generation of Minimal and Reduced Models for Structured and Nonlinear Parametric Dynamical Systems 15/34
An Illustrative Example A demo example H ( s ) = C ( s E − A ) − 1 B , 1 0 0 − 1 − 1 1 1 − 1 C T = , , , , 0 1 0 0 − 2 − 1 2 0 E = A = B = 0 0 1 0 0 − 3 1 0 Decay of singular values 10 5 Singular values 10 − 10 10 − 25 2 4 6 k Igor Pontes Duff, pontes@mpi-magdeburg.mpg.de Automatic Generation of Minimal and Reduced Models for Structured and Nonlinear Parametric Dynamical Systems 15/34
An Illustrative Example A demo example H ( s ) = C ( s E − A ) − 1 B , − 1 − 1 − 1 1 0 0 1 1 C T = , , , , 0 1 0 0 − 2 − 1 2 0 E = A = B = 0 0 1 0 0 − 3 1 0 Construction of a minimal system � ˆ � H ( s ) − ˆ � H ( s ) � , n = 3 H ( s ) � , r = 2 H ( s ) � 10 0 10 − 16 10 − 1 10 − 18 10 − 2 10 − 2 10 − 1 10 − 2 10 − 1 10 0 10 1 10 2 10 0 10 1 10 2 Igor Pontes Duff, pontes@mpi-magdeburg.mpg.de Automatic Generation of Minimal and Reduced Models for Structured and Nonlinear Parametric Dynamical Systems 15/34
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 ) − 1 B ( 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
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 ) − 1 B ( s ) , . Theorem (simplified) [ Beattie/Gugercin ’09 ] If � � K ( σ 1 ) − 1 B ( σ 1 ) , . . . , K ( σ r ) − 1 B ( σ r ) range ( V ) ⊇ span , K ( µ 1 ) − T C ( µ 1 ) T , . . . , K ( µ r ) − T C ( µ r ) T � � range ( W ) ⊇ span , Igor Pontes Duff, pontes@mpi-magdeburg.mpg.de Automatic Generation of Minimal and Reduced Models for Structured and Nonlinear Parametric Dynamical Systems 16/34
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 ) − 1 B ( s ) , . Theorem (simplified) [ Beattie/Gugercin ’09 ] If � � K ( σ 1 ) − 1 B ( σ 1 ) , . . . , K ( σ r ) − 1 B ( σ r ) range ( V ) ⊇ span , K ( µ 1 ) − T C ( µ 1 ) T , . . . , K ( µ r ) − T C ( µ r ) T � � range ( W ) ⊇ span , then H ( s ) = ˆ H ( s ) , s ∈ { σ 1 , σ r , µ 1 . . . , µ r } . Igor Pontes Duff, pontes@mpi-magdeburg.mpg.de Automatic Generation of Minimal and Reduced Models for Structured and Nonlinear Parametric Dynamical Systems 16/34
Structured Transfer Function –Reachability and observability– An n -dimensional linear system Transfer function ⇒ Input to state map: X ( s ) = K ( s ) − 1 B ( s ) U ( s ) State to output map: Y ( s ) = C ( s ) K ( s ) − 1 X ( s ) . H ( s ) = C ( s ) K ( s ) − 1 B ( 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
Structured Transfer Function –Reachability and observability– An n -dimensional linear system Transfer function ⇒ Input to state map: X ( s ) = K ( s ) − 1 B ( s ) U ( s ) State to output map: Y ( s ) = C ( s ) K ( s ) − 1 X ( s ) . H ( s ) = C ( s ) K ( s ) − 1 B ( s ) , . Reachable and observable subspaces for structured systems The reachable subspace R and the observable subspace O are the smallest subspaces of C n such that K ( s ) − T C ( s ) T ∈ O for every s ∈ i R . K ( s ) − 1 B ( s ) ∈ R and Igor Pontes Duff, pontes@mpi-magdeburg.mpg.de Automatic Generation of Minimal and Reduced Models for Structured and Nonlinear Parametric Dynamical Systems 17/34
Structured Transfer Function –Reachability and observability– An n -dimensional linear system Transfer function ⇒ Input to state map: X ( s ) = K ( s ) − 1 B ( s ) U ( s ) State to output map: Y ( s ) = C ( s ) K ( s ) − 1 X ( s ) . H ( s ) = C ( s ) K ( s ) − 1 B ( s ) , . Reachable and observable subspaces for structured systems The reachable subspace R and the observable subspace O are the smallest subspaces of C n such that K ( s ) − T C ( s ) T ∈ O for every s ∈ i R . K ( s ) − 1 B ( s ) ∈ R and 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
Structured Transfer Function –Reachability and observability– An n -dimensional linear system Transfer function ⇒ Input to state map: X ( s ) = K ( s ) − 1 B ( s ) U ( s ) State to output map: Y ( s ) = C ( s ) K ( s ) − 1 X ( s ) . H ( s ) = C ( s ) K ( s ) − 1 B ( s ) , . Reachable and observable subspaces for structured systems The reachable subspace R and the observable subspace O are the smallest subspaces of C n such that K ( s ) − T C ( s ) T ∈ O for every s ∈ i R . K ( s ) − 1 B ( s ) ∈ R and 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
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 ) − 1 B ( s ) . Characterization Controllable and Observable Subspaces (simplified) [ Benner/Goyal/P. ’19 ] Let � K ( σ 1 ) − 1 B ( σ 1 ) K ( σ g ) − 1 B ( σ g ) � K ( σ 2 ) − 1 B ( σ 2 ) R = . . . , � K ( σ 1 ) − T C ( σ 1 ) T K ( σ 2 ) − T C ( σ 2 ) T K ( σ g ) − T C ( σ g ) T � 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
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 ) − 1 B ( s ) . Characterization Controllable and Observable Subspaces (simplified) [ Benner/Goyal/P. ’19 ] Let � K ( σ 1 ) − 1 B ( σ 1 ) K ( σ g ) − 1 B ( σ g ) � K ( σ 2 ) − 1 B ( σ 2 ) R = . . . , � K ( σ 1 ) − T C ( σ 1 ) T K ( σ 2 ) − T C ( σ 2 ) T K ( σ g ) − T C ( σ g ) T � O = . . . . 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
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 ) − 1 B ( s ) . Characterization Controllable and Observable Subspaces (simplified) [ Benner/Goyal/P. ’19 ] Let � K ( σ 1 ) − 1 B ( σ 1 ) K ( σ g ) − 1 B ( σ g ) � K ( σ 2 ) − 1 B ( σ 2 ) R = . . . , � K ( σ 1 ) − T C ( σ 1 ) T K ( σ 2 ) − T C ( σ 2 ) T K ( σ g ) − T C ( σ g ) T � O = . . . . 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
Structured Transfer Function –Minimal realization– Structured transfer function Consider an n-dimensional linear system, whose structure transfer function is given by l � H ( s ) := C ( s ) K ( s ) − 1 B ( s ) , with K ( s ) = β i ( s ) A i , i =1 and let � � K ( σ 1 ) − 1 B ( σ 1 ) K ( σ 2 ) − 1 B ( σ 2 ) K ( σ N ) − 1 B ( σ N ) R = . . . , � K ( σ 1 ) − T C ( σ 1 ) T K ( σ 2 ) − T C ( σ 2 ) T K ( σ N ) − T C ( σ N ) T � O = . . . . 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
Structured Transfer Function –Minimal realization– Structured transfer function Consider an n-dimensional linear system, whose structure transfer function is given by l � H ( s ) := C ( s ) K ( s ) − 1 B ( s ) , with K ( s ) = β i ( s ) A i , i =1 and let � � K ( σ 1 ) − 1 B ( σ 1 ) K ( σ 2 ) − 1 B ( σ 2 ) K ( σ N ) − 1 B ( σ N ) R = . . . , � K ( σ 1 ) − T C ( σ 1 ) T K ( σ 2 ) − T C ( σ 2 ) T K ( σ N ) − T C ( σ N ) T � O = . . . . such that R = range ( R ) and O = range ( O ) . Minimal order (simplified) [ Benner/Goyal/P. ’19 ] � order of the minimal realization obtained by �� �� O T A 1 R O T A l R rank . . . = 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
Structured Transfer Function –Towards model reduction– We propose a method enabling to identify simultaneously the states that are unreachable and unobservable. Assume O T A 1 R . �� �� O T A 1 R O T A l R . rank = rank = r, . . . . O T A l 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
Structured Transfer Function –Towards model reduction– We propose a method enabling to identify simultaneously the states that are unreachable and unobservable. Assume O T A 1 R . �� �� O T A 1 R O T A l R . rank = rank = r, . . . . O T A l R with range ( R ) = R and range ( O ) = O . Then, we consider the compact SVDs O T A 1 R . � O T A 1 R O T A l R � = W 1 Σ l ˜ = ˜ V T W Σ r V T . . . . and 1 . . O T A l R Let W := OW 1 and V := RV 1 be two projection matrices and let us consider the lower-order K ( s ) − 1 ˆ realization ˆ C ( s ) 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
Structured Transfer Function –Towards model reduction– Petrov-Galerkin projections: W := OW 1 and V := RV 1 Theorem [ Benner/Goyal/P. ’19 ] K ( s ) − 1 ˆ The lower-order system ˆ C ( s ) ˆ B ( s ) of order r , obtained by Petrov-Galerkin projection with V and W , realizes the original transfer function, i.e., K ( s ) − 1 ˆ C ( s ) ˆ ˆ B ( s ) = C ( s ) K ( s ) − 1 B ( s ) for every s ∈ i R . Igor Pontes Duff, pontes@mpi-magdeburg.mpg.de Automatic Generation of Minimal and Reduced Models for Structured and Nonlinear Parametric Dynamical Systems 21/34
Structured Transfer Function –Towards model reduction– Petrov-Galerkin projections: W := OW 1 and V := RV 1 Theorem [ Benner/Goyal/P. ’19 ] K ( s ) − 1 ˆ The lower-order system ˆ C ( s ) ˆ B ( s ) of order r , obtained by Petrov-Galerkin projection with V and W , realizes the original transfer function, i.e., K ( s ) − 1 ˆ C ( s ) ˆ ˆ B ( s ) = C ( s ) K ( s ) − 1 B ( s ) for every s ∈ i R . 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
Structured Transfer Function –Towards model reduction– Petrov-Galerkin projections: W := OW 1 and V := RV 1 Theorem [ Benner/Goyal/P. ’19 ] K ( s ) − 1 ˆ The lower-order system ˆ C ( s ) ˆ B ( s ) of order r , obtained by Petrov-Galerkin projection with V and W , realizes the original transfer function, i.e., K ( s ) − 1 ˆ C ( s ) ˆ ˆ B ( s ) = C ( s ) K ( s ) − 1 B ( s ) for every s ∈ i R . 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
Structured Transfer Function –Towards model reduction– Petrov-Galerkin projections: W := OW 1 and V := RV 1 Theorem [ Benner/Goyal/P. ’19 ] K ( s ) − 1 ˆ The lower-order system ˆ C ( s ) ˆ B ( s ) of order r , obtained by Petrov-Galerkin projection with V and W , realizes the original transfer function, i.e., K ( s ) − 1 ˆ C ( s ) ˆ ˆ B ( s ) = C ( s ) K ( s ) − 1 B ( s ) for every s ∈ i R . 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
Algorithm to Construct Structured ROMs Algorithm: Dominant Reachable and Observable Projection ( DROP ) 1. Take σ i , µ i , 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
Algorithm to Construct Structured ROMs Algorithm: Dominant Reachable and Observable Projection ( DROP ) 1. Take σ i , µ i , i = 1 , . . . , N . � � K ( σ 1 ) − 1 B ( σ 1 ) , . . . , K ( σ N ) − 1 B ( σ N ) 2. Compute R = . � K ( µ 1 ) − T C ( µ 1 ) T , . . . , K ( µ N ) − T C ( µ N ) T � 3. Compute O = . Igor Pontes Duff, pontes@mpi-magdeburg.mpg.de Automatic Generation of Minimal and Reduced Models for Structured and Nonlinear Parametric Dynamical Systems 22/34
Algorithm to Construct Structured ROMs Algorithm: Dominant Reachable and Observable Projection ( DROP ) 1. Take σ i , µ i , i = 1 , . . . , N . � � K ( σ 1 ) − 1 B ( σ 1 ) , . . . , K ( σ N ) − 1 B ( σ N ) 2. Compute R = . � K ( µ 1 ) − T C ( µ 1 ) T , . . . , K ( µ N ) − T C ( µ N ) T � 3. Compute O = . 4. Determine L ( i ) = O T A i 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
Algorithm to Construct Structured ROMs Algorithm: Dominant Reachable and Observable Projection ( DROP ) 1. Take σ i , µ i , i = 1 , . . . , N . � � K ( σ 1 ) − 1 B ( σ 1 ) , . . . , K ( σ N ) − 1 B ( σ N ) 2. Compute R = . � K ( µ 1 ) − T C ( µ 1 ) T , . . . , K ( µ N ) − T C ( µ N ) T � 3. Compute O = . 4. Determine L ( i ) = O T A i R . 5. Compute singular value decomposition: L (1) . L (1) , . . . , L ( l ) �� � Y 1 , Σ 1 , X 1 � = svd �� � Y 2 , Σ 2 , X 2 � = svd . , . . 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
Algorithm to Construct Structured ROMs Algorithm: Dominant Reachable and Observable Projection ( DROP ) 1. Take σ i , µ i , i = 1 , . . . , N . � � K ( σ 1 ) − 1 B ( σ 1 ) , . . . , K ( σ N ) − 1 B ( σ N ) 2. Compute R = . � K ( µ 1 ) − T C ( µ 1 ) T , . . . , K ( µ N ) − T C ( µ N ) T � 3. Compute O = . 4. Determine L ( i ) = O T A i R . 5. Compute singular value decomposition: L (1) . L (1) , . . . , L ( l ) �� � Y 1 , Σ 1 , X 1 � = svd �� � Y 2 , Σ 2 , X 2 � = svd . , . . L ( l ) 6. Determine projection matrices: V := RX 2 (: , 1 : r ) , W := OY 1 (: , 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
Algorithm to Construct Structured ROMs Algorithm: Dominant Reachable and Observable Projection ( DROP ) 1. Take σ i , µ i , i = 1 , . . . , N . � � K ( σ 1 ) − 1 B ( σ 1 ) , . . . , K ( σ N ) − 1 B ( σ N ) 2. Compute R = . � K ( µ 1 ) − T C ( µ 1 ) T , . . . , K ( µ N ) − T C ( µ N ) T � 3. Compute O = . 4. Determine L ( i ) = O T A i R . 5. Compute singular value decomposition: L (1) . L (1) , . . . , L ( l ) �� � Y 1 , Σ 1 , X 1 � = svd �� � Y 2 , Σ 2 , X 2 � = svd . , . . L ( l ) 6. Determine projection matrices: V := RX 2 (: , 1 : r ) , W := OY 1 (: , 1 : r ) . 7. Determine reduced-order system K ( s ) = W T K ( s ) V , B ( s ) = W T B ( s ) , ˆ ˆ 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
Algorithm to Construct Structured ROMs Can be easily parallarized Algorithm: Dominant Reachable and Observable Projection ( DROP ) not need to solve all shifted systems 1. Take σ i , µ i , i = 1 , . . . , N . Make use of Low-rank solvers. � � K ( σ 1 ) − 1 B ( σ 1 ) , . . . , K ( σ N ) − 1 B ( σ N ) 2. Compute R = . � K ( µ 1 ) − T C ( µ 1 ) T , . . . , K ( µ N ) − T C ( µ N ) T � 3. Compute O = . 4. Determine L ( i ) = O T A i R . 5. Compute singular value decomposition: L (1) . L (1) , . . . , L ( l ) �� � Y 1 , Σ 1 , X 1 � = svd �� � Y 2 , Σ 2 , X 2 � = svd . , . . L ( l ) 6. Determine projection matrices: V := RX 2 (: , 1 : r ) , W := OY 1 (: , 1 : r ) . 7. Determine reduced-order system K ( s ) = W T K ( s ) V , B ( s ) = W T B ( s ) , ˆ ˆ 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
Algorithm to Construct Structured ROMs Can be easily parallarized Algorithm: Dominant Reachable and Observable Projection ( DROP ) not need to solve all shifted systems 1. Take σ i , µ i , i = 1 , . . . , N . Make use of Low-rank solvers. � � K ( σ 1 ) − 1 B ( σ 1 ) , . . . , K ( σ N ) − 1 B ( σ N ) 2. Compute R = . � K ( µ 1 ) − T C ( µ 1 ) T , . . . , K ( µ N ) − T C ( µ N ) T � 3. Compute O = . 4. Determine L ( i ) = O T A i R . 5. Compute singular value decomposition: L (1) . L (1) , . . . , L ( l ) �� � Y 1 , Σ 1 , X 1 � = svd �� � Y 2 , Σ 2 , X 2 � = svd . , . . L ( l ) Efficient variant of SVDs can be 6. Determine projection matrices: V := RX 2 (: , 1 : r ) , W := OY 1 (: , 1 : r ) . applied including randomized SVD. 7. Determine reduced-order system K ( s ) = W T K ( s ) V , B ( s ) = W T B ( s ) , ˆ ˆ 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
Low Rank Solvers for Sylvester Equations If we take enough points ( σ i ) , the matrix � � K ( σ 1 ) − 1 B ( σ 1 ) K ( σ N ) − 1 B ( σ N ) R = . . . , encodes the C n 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
Low Rank Solvers for Sylvester Equations If we take enough points ( σ i ) , the matrix � � K ( σ 1 ) − 1 B ( σ 1 ) K ( σ N ) − 1 B ( σ N ) R = . . . , encodes the C n reachable subspace. Notice that R solves l m � � A i RM i = B i b i , i =1 i =1 where M i = diag ( β i ( σ 1 ) , . . . , β i ( σ N )) and b i = [ γ i ( σ 1 ) , . . . , γ i ( σ N )] . Igor Pontes Duff, pontes@mpi-magdeburg.mpg.de Automatic Generation of Minimal and Reduced Models for Structured and Nonlinear Parametric Dynamical Systems 23/34
Low Rank Solvers for Sylvester Equations If we take enough points ( σ i ) , the matrix � � K ( σ 1 ) − 1 B ( σ 1 ) K ( σ N ) − 1 B ( σ N ) R = . . . , encodes the C n reachable subspace. Notice that R solves l m � � A i RM i = B i b i , i =1 i =1 where M i = diag ( β i ( σ 1 ) , . . . , β i ( σ N )) and b i = [ γ i ( σ 1 ) , . . . , γ i ( σ N )] . 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
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
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 ) = M 1 + d M 2 ∈ R n , D ( d, θ ) = θ ( D 1 + d D 2 ) , Figure: Semantic gyroscope diagram. K ( d ) = T 1 + 1 d T 2 + d T 3 . Parameters and frequency range: 10 − 5 , 10 − 7 � θ ∈ � , d ∈ � 1 , 2 � f ∈ � 0 . 025 , 40 � . Order of the system: 17 , 913 . Igor Pontes Duff, pontes@mpi-magdeburg.mpg.de Automatic Generation of Minimal and Reduced Models for Structured and Nonlinear Parametric Dynamical Systems 24/34
Parametric extension –Problem formulation– Parametric Problem Formulation Approximate the transfer function of an n -dimensional system, K ( s, p ) B ( s, p ) H ( s, p ) = C ( s, p ) K ( s, p ) − 1 B ( s, p ) , by the transfer function of a system C ( s, p ) K ( s, p ) − 1 ˆ H ( s, p ) = ˆ ˆ C ( s, p ) ˆ B ( s, p ) , ⇓ of order r ≪ n , such that ˆ ˆ K ( s, p ) B ( s, p ) � H ( s, p ) − ˆ H ( s, p ) � < tolerance ∀ s and ∀ 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
Parametric extension –Parametric transfer function– Parametric structured linear system C ( s, p ) = � k i =1 α i ( s, p ) C i ∈ R q × n , K ( s, p ) = � l i =1 β i ( s, p ) A i ∈ R n × n , H ( s ) = C ( s, p ) K ( s, p ) − 1 B ( s, p ) , B ( s, p ) = � m i =1 γ i ( s, p ) B i ∈ R n × 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
Parametric extension –Parametric transfer function– Parametric structured linear system C ( s, p ) = � k i =1 α i ( s, p ) C i ∈ R q × n , K ( s, p ) = � l i =1 β i ( s, p ) A i ∈ R n × n , H ( s ) = C ( s, p ) K ( s, p ) − 1 B ( s, p ) , B ( s, p ) = � m i =1 γ i ( s, p ) B i ∈ R n × m , Reachable and observable subspaces for parametric structured systems The reachable subspace R and the observable subspace O are the smallest subspaces of C n such that K ( s, p ) − T C ( s, p ) T ∈ O for every s ∈ i R and p ∈ Ω . K ( s, p ) − 1 B ( s, p ) ∈ R and Igor Pontes Duff, pontes@mpi-magdeburg.mpg.de Automatic Generation of Minimal and Reduced Models for Structured and Nonlinear Parametric Dynamical Systems 26/34
Parametric extension –Parametric transfer function– Parametric structured linear system C ( s, p ) = � k i =1 α i ( s, p ) C i ∈ R q × n , K ( s, p ) = � l i =1 β i ( s, p ) A i ∈ R n × n , H ( s ) = C ( s, p ) K ( s, p ) − 1 B ( s, p ) , B ( s, p ) = � m i =1 γ i ( s, p ) B i ∈ R n × m , Reachable and observable subspaces for parametric structured systems The reachable subspace R and the observable subspace O are the smallest subspaces of C n such that K ( s, p ) − T C ( s, p ) T ∈ O for every s ∈ i R and p ∈ Ω . K ( s, p ) − 1 B ( s, p ) ∈ R and � � K ( σ 1 , p 1 ) − 1 B ( σ 1 , p 1 ) K ( σ 2 . p 2 ) − 1 B ( σ 2 , p 2 ) K ( σ g , p g ) − 1 B ( σ g , p g ) R = . . . , � K ( σ 1 , p 1 ) − T C ( σ 1 , p 1 ) T K ( σ 2 , p 2 ) − T C ( σ 2 , p 2 ) T K ( σ g , p g ) − T C ( σ g , p g ) T � O = . . . . 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
Parametric extension –Minimal and reduced order model– Structured transfer function Consider an n-dimensional linear system, whose structure transfer function is given by l � H ( s, p ) := C ( s, p ) K ( s, p ) − 1 B ( s, p ) , K ( s, p ) = with β i ( s, p ) A i , i =1 Igor Pontes Duff, pontes@mpi-magdeburg.mpg.de Automatic Generation of Minimal and Reduced Models for Structured and Nonlinear Parametric Dynamical Systems 27/34
Parametric extension –Minimal and reduced order model– Structured transfer function Consider an n-dimensional linear system, whose structure transfer function is given by l � H ( s, p ) := C ( s, p ) K ( s, p ) − 1 B ( s, p ) , K ( s, p ) = with β i ( s, p ) A i , i =1 Minimal order (simplified) [ Benner/Goyal/P. ’19 ] � order of the minimal realization obtained by �� �� O T A 1 R O T A l R rank = . . . 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
Parametric extension –Minimal and reduced order model– Structured transfer function Consider an n-dimensional linear system, whose structure transfer function is given by l � H ( s, p ) := C ( s, p ) K ( s, p ) − 1 B ( s, p ) , K ( s, p ) = with β i ( s, p ) A i , i =1 Minimal order (simplified) [ Benner/Goyal/P. ’19 ] � order of the minimal realization obtained by �� �� O T A 1 R O T A l R rank = . . . 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
Numerical Examples –Delay demo example– A time-delay demo system s I − A 1 − A 2 e − s � − 1 B � H ( s ) = C T − 1 0 0 1 1 1 1 1 , , , A 1 = 0 − 1 0 A 2 = 0 0 0 B = 0 C = 1 , 0 0 − 1 0 0 0 0 0 Igor Pontes Duff, pontes@mpi-magdeburg.mpg.de Automatic Generation of Minimal and Reduced Models for Structured and Nonlinear Parametric Dynamical Systems 28/34
Numerical Examples –Delay demo example– A time-delay demo system s I − A 1 − A 2 e − s � − 1 B � H ( s ) = C T − 1 0 0 1 1 1 1 1 , , , A 1 = 0 − 1 0 A 2 = 0 0 0 B = 0 C = 1 , 0 0 − 1 0 0 0 0 0 Let us construct, for σ i = [1 , 2 , 3 , 4 , 5 , 6] , K ( σ 1 ) − 1 B K ( σ 6 ) − 1 B R = � � . . . , K ( σ 1 ) − T C T K ( σ 6 ) − T C T � � O = . . . . Igor Pontes Duff, pontes@mpi-magdeburg.mpg.de Automatic Generation of Minimal and Reduced Models for Structured and Nonlinear Parametric Dynamical Systems 28/34
Numerical Examples –Delay demo example– A time-delay demo system s I − A 1 − A 2 e − s � − 1 B � H ( s ) = C T − 1 0 0 1 1 1 1 1 , , , A 1 = 0 − 1 0 A 2 = 0 0 0 B = 0 C = 1 , 0 0 − 1 0 0 0 0 0 Let us construct, for σ i = [1 , 2 , 3 , 4 , 5 , 6] , K ( σ 1 ) − 1 B K ( σ 6 ) − 1 B R = � � . . . , K ( σ 1 ) − T C T K ( σ 6 ) − T C T � � O = . . . . � nonreachable rank ( R ) = 2 , rank ( O ) = 1 . � 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
Numerical Examples –Delay demo example– A time-delay demo system s I − A 1 − A 2 e − s � − 1 B � H ( s ) = C T − 1 0 0 1 1 1 1 1 , , , A 1 = 0 − 1 0 A 2 = 0 0 0 B = 0 C = 1 , 0 0 − 1 0 0 0 0 0 Let us construct, for σ i = [1 , 2 , 3 , 4 , 5 , 6] , Then, using DROP , we get the projection matrices K ( σ 1 ) − 1 B K ( σ 6 ) − 1 B R = � � . . . , K ( σ 1 ) − T C T K ( σ 6 ) − T C T � � O = . . . . V = RX (: , 1) and W = OY (: , 1) . � nonreachable rank ( R ) = 2 , rank ( O ) = 1 . � nonobservable O T R O T A 1 R O T A 2 R �� �� rank = 1 . (minimal realization order) Igor Pontes Duff, pontes@mpi-magdeburg.mpg.de Automatic Generation of Minimal and Reduced Models for Structured and Nonlinear Parametric Dynamical Systems 28/34
Numerical Examples –Delay demo example– A time-delay demo system s I − A 1 − A 2 e − s � − 1 B � H ( s ) = C T − 1 0 0 1 1 1 1 1 , , , A 1 = 0 − 1 0 A 2 = 0 0 0 B = 0 C = 1 , 0 0 − 1 0 0 0 0 0 Let us construct, for σ i = [1 , 2 , 3 , 4 , 5 , 6] , Then, using DROP , we get the projection matrices K ( σ 1 ) − 1 B K ( σ 6 ) − 1 B R = � � . . . , K ( σ 1 ) − T C T K ( σ 6 ) − T C T � � O = . . . . V = RX (: , 1) and W = OY (: , 1) . The ˆ H obtained using V and W satisfies � nonreachable rank ( R ) = 2 , rank ( O ) = 1 . � nonobservable H ( s ) = ˆ H ( s ) , ∀ s. O T R O T A 1 R O T A 2 R �� �� rank = 1 . (minimal realization order) Igor Pontes Duff, pontes@mpi-magdeburg.mpg.de Automatic Generation of Minimal and Reduced Models for Structured and Nonlinear Parametric Dynamical Systems 28/34
Numerical Examples –Delay demo example– A time-delay demo system s I − A 1 − A 2 e − s � − 1 B � H ( s ) = C T − 1 0 0 1 1 1 1 1 , , , 0 − 1 0 0 0 0 0 1 A 1 = A 2 = B = C = , 0 0 − 1 0 0 0 0 0 Decay of singular values Let us construct, for σ i = [1 , 2 , 3 , 4 , 5 , 6] , Then, using DROP , we get the projection matrices 10 5 K ( σ 1 ) − 1 B K ( σ 6 ) − 1 B � Singular values � R = . . . , K ( σ 1 ) − T C T K ( σ 6 ) − T C T � � O = . . . . V = RX (: , 1) and W = OY (: , 1) . 10 − 10 The ˆ H obtained using V and W satisfies � nonreachable � rank ( R ) = 2 , rank ( O ) = 1 . nonobservable H ( s ) = ˆ 10 − 25 H ( s ) , ∀ s. O T R O T A 1 R O T A 2 R �� �� rank = 1 . (minimal 2 4 6 realization order) k Igor Pontes Duff, pontes@mpi-magdeburg.mpg.de Automatic Generation of Minimal and Reduced Models for Structured and Nonlinear Parametric Dynamical Systems 28/34
Numerical Examples –Delay demo example– A time-delay demo system s I − A 1 − A 2 e − s � − 1 B � H ( s ) = C T − 1 0 0 1 1 1 1 1 , , , A 1 = 0 − 1 0 A 2 = 0 0 0 B = 0 C = 1 , 0 0 − 1 0 0 0 0 0 Construction of a minimal system � ˆ � H ( s ) − ˆ � H ( s ) � ( n = 3) H ( s ) � ( r = 1) H ( s ) � 10 1 10 − 15 10 0 10 − 1 10 − 17 10 − 2 10 − 1 10 0 10 1 10 − 2 10 − 1 10 0 10 1 10 2 Igor Pontes Duff, pontes@mpi-magdeburg.mpg.de Automatic Generation of Minimal and Reduced Models for Structured and Nonlinear Parametric Dynamical Systems 28/34
Numerical Examples –Parametric demo example– A parametric demo dynamical system H ( s, p ) = C ( s I − A 1 − p A 2 ) − 1 B , − 2 0 0 0 1 0 1 1 C T = , , , , A 1 = 0 − 1 0 A 2 = − 1 0 0 B = 0 1 0 0 − 2 1 0 0 1 0 Igor Pontes Duff, pontes@mpi-magdeburg.mpg.de Automatic Generation of Minimal and Reduced Models for Structured and Nonlinear Parametric Dynamical Systems 29/34
Numerical Examples –Parametric demo example– A parametric demo dynamical system H ( s, p ) = C ( s I − A 1 − p A 2 ) − 1 B , − 2 0 0 0 1 0 1 1 C T = , , , , A 1 = 0 − 1 0 A 2 = − 1 0 0 B = 0 1 0 0 − 2 1 0 0 1 0 For l = 20 points ( σ i , p i ) , let � K ( σ 1 , p 1 ) − 1 B K ( σ l , p l ) − 1 B � R = . . . , � K ( σ l , p l ) − T C T � K ( σ 1 , p 1 ) − T C T O = . . . . Igor Pontes Duff, pontes@mpi-magdeburg.mpg.de Automatic Generation of Minimal and Reduced Models for Structured and Nonlinear Parametric Dynamical Systems 29/34
Numerical Examples –Parametric demo example– A parametric demo dynamical system H ( s, p ) = C ( s I − A 1 − p A 2 ) − 1 B , − 2 0 0 0 1 0 1 1 C T = , , , , A 1 = 0 − 1 0 A 2 = − 1 0 0 B = 0 1 0 0 − 2 1 0 0 1 0 For l = 20 points ( σ i , p i ) , let � K ( σ 1 , p 1 ) − 1 B K ( σ l , p l ) − 1 B � R = . . . , � K ( σ l , p l ) − T C T � K ( σ 1 , p 1 ) − T C T O = . . . . �� O T R O T A 1 R O T A 2 R �� rank = 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
Numerical Examples –Parametric demo example– A parametric demo dynamical system H ( s, p ) = C ( s I − A 1 − p A 2 ) − 1 B , − 2 0 0 0 1 0 1 1 C T = , , , , A 1 = 0 − 1 0 A 2 = − 1 0 0 B = 0 1 0 0 − 2 1 0 0 1 0 Decay of Singular values For l = 20 points ( σ i , p i ) , let 10 5 � K ( σ 1 , p 1 ) − 1 B K ( σ l , p l ) − 1 B � R = . . . , � K ( σ l , p l ) − T C T � K ( σ 1 , p 1 ) − T C T O = . . . . 10 − 10 �� O T R O T A 1 R O T A 2 R �� rank = 2 . Compute projectors V and W and ˆ H ( s, p ) . 10 − 25 0 5 10 15 20 Then, H ( s, p ) = ˆ H ( s, p ) . Igor Pontes Duff, pontes@mpi-magdeburg.mpg.de Automatic Generation of Minimal and Reduced Models for Structured and Nonlinear Parametric Dynamical Systems 29/34
Numerical Examples –Parametric demo example– A parametric demo dynamical system H ( s, p ) = C ( s I − A 1 − p A 2 ) − 1 B , − 2 0 0 0 1 0 1 1 C T = , , , , A 1 = 0 − 1 0 A 2 = − 1 0 0 B = 0 1 0 0 − 2 1 0 0 1 0 Absolute error For l = 20 points ( σ i , p i ) , let 10 − 14 � K ( σ 1 , p 1 ) − 1 B K ( σ l , p l ) − 1 B � R = . . . , � K ( σ l , p l ) − T C T � K ( σ 1 , p 1 ) − T C T O = . . . . 10 − 18 �� O T R O T A 1 R O T A 2 R �� rank = 2 . Compute projectors V and W and ˆ 10 − 22 H ( s, p ) . 10 − 3 10 − 2 10 − 1 10 0 10 1 10 2 Then, H ( s, p ) = ˆ H ( s, p ) . Igor Pontes Duff, pontes@mpi-magdeburg.mpg.de Automatic Generation of Minimal and Reduced Models for Structured and Nonlinear Parametric Dynamical Systems 29/34
Numerical Examples –Time-delay example– Delay example [ Beattie/Gugercin ’09 ] E ˙ x ( t ) = Ax ( t ) + A τ x ( t − τ ) + Bu ( t ) , H ( s ) = C ( s E − A 1 − A τ e − sτ ) B y ( t ) = Cx ( t ) . 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
Numerical Examples –Time-delay example– Delay example [ Beattie/Gugercin ’09 ] E ˙ x ( t ) = Ax ( t ) + A τ x ( t − τ ) + Bu ( t ) , H ( s ) = C ( s E − A 1 − A τ e − sτ ) B y ( t ) = Cx ( t ) . 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 Balanced truncation [ Breiten ’16 ] DROP 10 0 10 − 7 10 − 14 10 − 20 12 25 50 75 100 Figure: Delay example: relative decay of the singular values using the proposed method and structured balanced Igor Pontes Duff, pontes@mpi-magdeburg.mpg.de Automatic Generation of Minimal and Reduced Models for Structured and Nonlinear Parametric Dynamical Systems 30/34 truncation.
Numerical Examples –Time-delay example– Delay example [ Beattie/Gugercin ’09 ] E ˙ x ( t ) = Ax ( t ) + A τ x ( t − τ ) + Bu ( t ) , H ( s ) = C ( s E − A 1 − A τ e − sτ ) B y ( t ) = Cx ( t ) . 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 ] 100 10 − 2 10 − 2 10 − 7 10 − 4 10 − 12 10 − 2 10 − 1 100 101 102 103 104 10 − 2 10 − 1 100 101 102 103 104 freq ( s ) 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
Numerical Examples –Fractional derivative example– Fractional Maxwell equations. [ Feng/Benner ’08 ] � − 1 � s 2 I − 1 H ( s ) = s B T √ s D + A B , � � Full order model n = 29 , 295 . Frequency range is F := 4 e 9 , 8 e 9 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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