Post on 25-May-2020
Quadratification for second order model reduction
Christian Schroder
Research center Matheon, TU Berlin
AbsolventenseminarBerlin, October 27th 2016
Work in progress!
Introduction
Introduction
Background on MOR for 1st and 2nd order
Linearization and quadratification
Counter arguments
The EndC. Schroder (Matheon and TU Berlin) Quadratification for 2nd order model reduction Abs Sem, Oct 2016 2 / 15
Introduction
Second order model reductionI linear time invariant (LTI) dynamical system of second order
x = A0x + A1x + B0u + B1u, (x(0) = x0, x(0) = x1)
y = C0x + C1x + Du
I x(t): state ∈ Rn, u(t): input ∈ Rm, y(t): output ∈ Rp
I arises in mechanical systemsI often n� m, p (semi discretization of PDE)⇒ simulation and control are expensive
I wanted: reduced order model of state dimension r � n
xr = Ar 0xr + Ar 1xr + Br 0u + Br 1u
yr = Cr 0xr + Cr 1xr + Dru (notation: subr= reduced to size r)
I such that y ≈ yr , i.e., ‖G − Gr‖ is small, usually H2- or H∞-norm
G (s) = (C0 + sC1)(s2I − sA1 − A0)−1(B0 + sB1) + D
I here: use linearization and quadratification
C. Schroder (Matheon and TU Berlin) Quadratification for 2nd order model reduction Abs Sem, Oct 2016 2 / 15
Introduction
Second order model reductionI linear time invariant (LTI) dynamical system of second order
x = A0x + A1x + B0u + B1u, (x(0) = x0, x(0) = x1)
y = C0x + C1x + Du
I x(t): state ∈ Rn, u(t): input ∈ Rm, y(t): output ∈ Rp
I arises in mechanical systemsI often n� m, p (semi discretization of PDE)⇒ simulation and control are expensive
I wanted: reduced order model of state dimension r � n
xr = Ar 0xr + Ar 1xr + Br 0u + Br 1u
yr = Cr 0xr + Cr 1xr + Dru (notation: subr= reduced to size r)
I such that y ≈ yr
, i.e., ‖G − Gr‖ is small, usually H2- or H∞-norm
G (s) = (C0 + sC1)(s2I − sA1 − A0)−1(B0 + sB1) + D
I here: use linearization and quadratification
C. Schroder (Matheon and TU Berlin) Quadratification for 2nd order model reduction Abs Sem, Oct 2016 2 / 15
Introduction
Second order model reductionI linear time invariant (LTI) dynamical system of second order
x = A0x + A1x + B0u + B1u, (x(0) = x0, x(0) = x1)
y = C0x + C1x + Du
I x(t): state ∈ Rn, u(t): input ∈ Rm, y(t): output ∈ Rp
I arises in mechanical systemsI often n� m, p (semi discretization of PDE)⇒ simulation and control are expensive
I wanted: reduced order model of state dimension r � n
xr = Ar 0xr + Ar 1xr + Br 0u + Br 1u
yr = Cr 0xr + Cr 1xr + Dru (notation: subr= reduced to size r)
I such that y ≈ yr , i.e., ‖G − Gr‖ is small, usually H2- or H∞-norm
G (s) = (C0 + sC1)(s2I − sA1 − A0)−1(B0 + sB1) + D
I here: use linearization and quadratification
C. Schroder (Matheon and TU Berlin) Quadratification for 2nd order model reduction Abs Sem, Oct 2016 2 / 15
Background on MOR for 1st and 2nd order
Introduction
Background on MOR for 1st and 2nd order
Linearization and quadratification
Counter arguments
The EndC. Schroder (Matheon and TU Berlin) Quadratification for 2nd order model reduction Abs Sem, Oct 2016 3 / 15
Background on MOR for 1st and 2nd order
1st order MOR
χ = Aχ+ Bu χr = Arχr + Bruy = Cχ+Du yr = Crχr +Dru
G (s) = C(sI −A)−1B +D (curly = 1st order)
I numerous good/trusted/well-understood methods existsI most are projection methods: Vr ∈ Rn,r ,Wr ∈ Rr ,n,WrVr = Ir
Ar = WrAVr , Br = WrB, Cr = CVr , Dr = DI e.g. partial realization, imW T
r = Kr (AT , cT ), Vr = similarmatches 2r + 1 moments at ∞
I interpolation, imW Tr = span{(σi I −A)−T cT}ri=1 ⇒ G (σi )
2r=
i = 1Gr (σi )
I IRKA, if {σ1, . . . , σr} = eig(−Ar ), then Gr is H2-optimalI balanced truncation, Wr ,Vr from balancing transformation,
sequence of ROMs with H∞error bound
C. Schroder (Matheon and TU Berlin) Quadratification for 2nd order model reduction Abs Sem, Oct 2016 3 / 15
Background on MOR for 1st and 2nd order
1st order MOR
χ = Aχ+ Bu χr = Arχr + Bruy = Cχ+Du yr = Crχr +Dru
G (s) = C(sI −A)−1B +D (curly = 1st order)
I numerous good/trusted/well-understood methods existsI most are projection methods: Vr ∈ Rn,r ,Wr ∈ Rr ,n,WrVr = Ir
Ar = WrAVr , Br = WrB, Cr = CVr , Dr = DI e.g.
partial realization, imW Tr = Kr (AT , cT ), Vr = similar
matches 2r + 1 moments at ∞I interpolation, imW T
r = span{(σi I −A)−T cT}ri=1 ⇒ G (σi )2r=
i = 1Gr (σi )
I IRKA, if {σ1, . . . , σr} = eig(−Ar ), then Gr is H2-optimalI balanced truncation, Wr ,Vr from balancing transformation,
sequence of ROMs with H∞error bound
C. Schroder (Matheon and TU Berlin) Quadratification for 2nd order model reduction Abs Sem, Oct 2016 3 / 15
Background on MOR for 1st and 2nd order
1st order MOR
χ = Aχ+ Bu χr = Arχr + Bruy = Cχ+Du yr = Crχr +Dru
G (s) = C(sI −A)−1B +D (curly = 1st order)
I numerous good/trusted/well-understood methods existsI most are projection methods: Vr ∈ Rn,r ,Wr ∈ Rr ,n,WrVr = Ir
Ar = WrAVr , Br = WrB, Cr = CVr , Dr = DI e.g. partial realization, imW T
r = Kr (AT , cT ), Vr = similarmatches 2r + 1 moments at ∞
I interpolation, imW Tr = span{(σi I −A)−T cT}ri=1 ⇒ G (σi )
2r=
i = 1Gr (σi )
I IRKA, if {σ1, . . . , σr} = eig(−Ar ), then Gr is H2-optimalI balanced truncation, Wr ,Vr from balancing transformation,
sequence of ROMs with H∞error bound
C. Schroder (Matheon and TU Berlin) Quadratification for 2nd order model reduction Abs Sem, Oct 2016 3 / 15
Background on MOR for 1st and 2nd order
1st order MOR
χ = Aχ+ Bu χr = Arχr + Bruy = Cχ+Du yr = Crχr +Dru
G (s) = C(sI −A)−1B +D (curly = 1st order)
I numerous good/trusted/well-understood methods existsI most are projection methods: Vr ∈ Rn,r ,Wr ∈ Rr ,n,WrVr = Ir
Ar = WrAVr , Br = WrB, Cr = CVr , Dr = DI e.g. partial realization, imW T
r = Kr (AT , cT ), Vr = similarmatches 2r + 1 moments at ∞
I interpolation, imW Tr = span{(σi I −A)−T cT}ri=1 ⇒ G (σi )
2r=
i = 1Gr (σi )
I IRKA, if {σ1, . . . , σr} = eig(−Ar ), then Gr is H2-optimalI balanced truncation, Wr ,Vr from balancing transformation,
sequence of ROMs with H∞error bound
C. Schroder (Matheon and TU Berlin) Quadratification for 2nd order model reduction Abs Sem, Oct 2016 3 / 15
Background on MOR for 1st and 2nd order
1st order MOR
χ = Aχ+ Bu χr = Arχr + Bruy = Cχ+Du yr = Crχr +Dru
G (s) = C(sI −A)−1B +D (curly = 1st order)
I numerous good/trusted/well-understood methods existsI most are projection methods: Vr ∈ Rn,r ,Wr ∈ Rr ,n,WrVr = Ir
Ar = WrAVr , Br = WrB, Cr = CVr , Dr = DI e.g. partial realization, imW T
r = Kr (AT , cT ), Vr = similarmatches 2r + 1 moments at ∞
I interpolation, imW Tr = span{(σi I −A)−T cT}ri=1 ⇒ G (σi )
2r=
i = 1Gr (σi )
I IRKA, if {σ1, . . . , σr} = eig(−Ar ), then Gr is H2-optimal
I balanced truncation, Wr ,Vr from balancing transformation,sequence of ROMs with H∞error bound
C. Schroder (Matheon and TU Berlin) Quadratification for 2nd order model reduction Abs Sem, Oct 2016 3 / 15
Background on MOR for 1st and 2nd order
1st order MOR
χ = Aχ+ Bu χr = Arχr + Bruy = Cχ+Du yr = Crχr +Dru
G (s) = C(sI −A)−1B +D (curly = 1st order)
I numerous good/trusted/well-understood methods existsI most are projection methods: Vr ∈ Rn,r ,Wr ∈ Rr ,n,WrVr = Ir
Ar = WrAVr , Br = WrB, Cr = CVr , Dr = DI e.g. partial realization, imW T
r = Kr (AT , cT ), Vr = similarmatches 2r + 1 moments at ∞
I interpolation, imW Tr = span{(σi I −A)−T cT}ri=1 ⇒ G (σi )
2r=
i = 1Gr (σi )
I IRKA, if {σ1, . . . , σr} = eig(−Ar ), then Gr is H2-optimalI balanced truncation, Wr ,Vr from balancing transformation,
sequence of ROMs with H∞error boundC. Schroder (Matheon and TU Berlin) Quadratification for 2nd order model reduction Abs Sem, Oct 2016 3 / 15
Background on MOR for 1st and 2nd order
Projection methods for 2nd order
Vr ∈ Rn,r ,Wr ∈ Rr ,n,WrVr = Ir
Ai r = WrAiVr , Bi r = WrBi , Ci r = CiVr , Dr = D
I most methods have been adapted, work often well. . .
I but have some sort of problem
I partial realization, SOAR (second order Arnoldi method), but nolucky breakdown
I interpolation, imW Tr = span{(σ2
i I − σiA1 − A0)−T (c0 + σic1)T}ri=1
I 2r interp. points, but no optimal points [Beattie and Benner, 2014]
I balanced truncation, choice of positions or velocities [Stykel and Reis]
The other standard approach for 2nd order systems is. . .
C. Schroder (Matheon and TU Berlin) Quadratification for 2nd order model reduction Abs Sem, Oct 2016 4 / 15
Background on MOR for 1st and 2nd order
Projection methods for 2nd order
Vr ∈ Rn,r ,Wr ∈ Rr ,n,WrVr = Ir
Ai r = WrAiVr , Bi r = WrBi , Ci r = CiVr , Dr = D
I most methods have been adapted, work often well. . .
I but have some sort of problem
I partial realization, SOAR (second order Arnoldi method), but nolucky breakdown
I interpolation, imW Tr = span{(σ2
i I − σiA1 − A0)−T (c0 + σic1)T}ri=1
I 2r interp. points, but no optimal points [Beattie and Benner, 2014]
I balanced truncation, choice of positions or velocities [Stykel and Reis]
The other standard approach for 2nd order systems is. . .
C. Schroder (Matheon and TU Berlin) Quadratification for 2nd order model reduction Abs Sem, Oct 2016 4 / 15
Background on MOR for 1st and 2nd order
Projection methods for 2nd order
Vr ∈ Rn,r ,Wr ∈ Rr ,n,WrVr = Ir
Ai r = WrAiVr , Bi r = WrBi , Ci r = CiVr , Dr = D
I most methods have been adapted, work often well. . .
I but have some sort of problem
I partial realization, SOAR (second order Arnoldi method), but nolucky breakdown
I interpolation, imW Tr = span{(σ2
i I − σiA1 − A0)−T (c0 + σic1)T}ri=1
I 2r interp. points, but no optimal points [Beattie and Benner, 2014]
I balanced truncation, choice of positions or velocities [Stykel and Reis]
The other standard approach for 2nd order systems is. . .
C. Schroder (Matheon and TU Berlin) Quadratification for 2nd order model reduction Abs Sem, Oct 2016 4 / 15
Background on MOR for 1st and 2nd order
Projection methods for 2nd order
Vr ∈ Rn,r ,Wr ∈ Rr ,n,WrVr = Ir
Ai r = WrAiVr , Bi r = WrBi , Ci r = CiVr , Dr = D
I most methods have been adapted, work often well. . .
I but have some sort of problem
I partial realization, SOAR (second order Arnoldi method), but nolucky breakdown
I interpolation, imW Tr = span{(σ2
i I − σiA1 − A0)−T (c0 + σic1)T}ri=1
I 2r interp. points, but no optimal points [Beattie and Benner, 2014]
I balanced truncation, choice of positions or velocities [Stykel and Reis]
The other standard approach for 2nd order systems is. . .
C. Schroder (Matheon and TU Berlin) Quadratification for 2nd order model reduction Abs Sem, Oct 2016 4 / 15
Background on MOR for 1st and 2nd order
Projection methods for 2nd order
Vr ∈ Rn,r ,Wr ∈ Rr ,n,WrVr = Ir
Ai r = WrAiVr , Bi r = WrBi , Ci r = CiVr , Dr = D
I most methods have been adapted, work often well. . .
I but have some sort of problem
I partial realization, SOAR (second order Arnoldi method), but nolucky breakdown
I interpolation, imW Tr = span{(σ2
i I − σiA1 − A0)−T (c0 + σic1)T}ri=1
I 2r interp. points, but no optimal points [Beattie and Benner, 2014]
I balanced truncation, choice of positions or velocities [Stykel and Reis]
The other standard approach for 2nd order systems is. . .
C. Schroder (Matheon and TU Berlin) Quadratification for 2nd order model reduction Abs Sem, Oct 2016 4 / 15
Linearization and quadratification
Introduction
Background on MOR for 1st and 2nd order
Linearization and quadratification
Counter arguments
The EndC. Schroder (Matheon and TU Berlin) Quadratification for 2nd order model reduction Abs Sem, Oct 2016 5 / 15
Linearization and quadratification
Linearizationthe 2nd order system
x = A0x + A1x + B0u + B1u
y = C0x + C1x + Du
can be linearized (= reformulated in 1st order) to[xz
]=
[0 IA0 A1
] [xz
]+
[B1
B0 + A1B1
]u
y =[C0 C1
] [xz
]+ (D + C1B1)u
I χ = Aχ+ Bu, y = Cχ+Du with A in companion form
I transfer functions G coincide
I (A0,A1,B0,B1,C0,C1,D)linearization�
quadratification(A,B, C,D)
C. Schroder (Matheon and TU Berlin) Quadratification for 2nd order model reduction Abs Sem, Oct 2016 5 / 15
Linearization and quadratification
Linearizationthe 2nd order system
x = A0x + A1x + B0u + B1u
y = C0x + C1x + Du
can be linearized (= reformulated in 1st order) to[xz
]=
[0 IA0 A1
] [xz
]+
[B1
B0 + A1B1
]u
y =[C0 C1
] [xz
]+ (D + C1B1)u
I χ = Aχ+ Bu, y = Cχ+Du with A in companion form
I transfer functions G coincide
I (A0,A1,B0,B1,C0,C1,D)linearization�
quadratification(A,B, C,D)
C. Schroder (Matheon and TU Berlin) Quadratification for 2nd order model reduction Abs Sem, Oct 2016 5 / 15
Linearization and quadratification
Linearizationthe 2nd order system
x = A0x + A1x + B0u + B1u
y = C0x + C1x + Du
can be linearized (= reformulated in 1st order) to[xz
]=
[0 IA0 A1
] [xz
]+
[B1
B0 + A1B1
]u
y =[C0 C1
] [xz
]+ (D + C1B1)u
I χ = Aχ+ Bu, y = Cχ+Du with A in companion form
I transfer functions G coincide
I (A0,A1,B0,B1,C0,C1,D)linearization�
quadratification(A,B, C,D)
C. Schroder (Matheon and TU Berlin) Quadratification for 2nd order model reduction Abs Sem, Oct 2016 5 / 15
Linearization and quadratification
Algorithm
1. given (A0,A1,B0,B1,C0,C1,D) with state dim n
2. linearize, state dim 2n
A =
[0 IA0 A1
], B =
[B1
B0 + A1B1
], C =
[C0 C1
], D = D+C1B1
3. reduce to state dimension 2r
A2r = W2rAV2r , B2r = W2rB, C2r = CV2r , D2r = D
4. quadratify to state dim r , i.e., partition and read off blocks
A2r =
[0 IAr 0 Ar 1
],B2r =
[Br 1
Br 0 + Ar 1Br 1
], C2r =
[Cr 0 Cr 1
],D2r =Dr+Cr 1Br 1
I but A2r is not in companion form!
. . . or is it?
I replace W = [W1,W2]→[W1 00 W2
],V =
[V1
V2
]→[V1 00 V2
]this often works, but doubles the dimension and changes Gr→ bad!
C. Schroder (Matheon and TU Berlin) Quadratification for 2nd order model reduction Abs Sem, Oct 2016 6 / 15
Linearization and quadratification
Algorithm
1. given (A0,A1,B0,B1,C0,C1,D) with state dim n
2. linearize, state dim 2n
A =
[0 IA0 A1
], B =
[B1
B0 + A1B1
], C =
[C0 C1
], D = D+C1B1
3. reduce to state dimension 2r
A2r = W2rAV2r , B2r = W2rB, C2r = CV2r , D2r = D
4. quadratify to state dim r , i.e., partition and read off blocks
A2r =
[0 IAr 0 Ar 1
],B2r =
[Br 1
Br 0 + Ar 1Br 1
], C2r =
[Cr 0 Cr 1
],D2r =Dr+Cr 1Br 1
I but A2r is not in companion form!
. . . or is it?
I replace W = [W1,W2]→[W1 00 W2
],V =
[V1
V2
]→[V1 00 V2
]this often works, but doubles the dimension and changes Gr→ bad!
C. Schroder (Matheon and TU Berlin) Quadratification for 2nd order model reduction Abs Sem, Oct 2016 6 / 15
Linearization and quadratification
Algorithm
1. given (A0,A1,B0,B1,C0,C1,D) with state dim n
2. linearize, state dim 2n
A =
[0 IA0 A1
], B =
[B1
B0 + A1B1
], C =
[C0 C1
], D = D+C1B1
3. reduce to state dimension 2r
A2r = W2rAV2r , B2r = W2rB, C2r = CV2r , D2r = D
4. quadratify to state dim r , i.e., partition and read off blocks
A2r =
[0 IAr 0 Ar 1
],B2r =
[Br 1
Br 0 + Ar 1Br 1
], C2r =
[Cr 0 Cr 1
],D2r =Dr+Cr 1Br 1
I but A2r is not in companion form!
. . . or is it?
I replace W = [W1,W2]→[W1 00 W2
],V =
[V1
V2
]→[V1 00 V2
]this often works, but doubles the dimension and changes Gr
→ bad!
C. Schroder (Matheon and TU Berlin) Quadratification for 2nd order model reduction Abs Sem, Oct 2016 6 / 15
Linearization and quadratification
Algorithm
1. given (A0,A1,B0,B1,C0,C1,D) with state dim n
2. linearize, state dim 2n
A =
[0 IA0 A1
], B =
[B1
B0 + A1B1
], C =
[C0 C1
], D = D+C1B1
3. reduce to state dimension 2r
A2r = W2rAV2r , B2r = W2rB, C2r = CV2r , D2r = D
4. quadratify to state dim r , i.e., partition and read off blocks
A2r =
[0 IAr 0 Ar 1
],B2r =
[Br 1
Br 0 + Ar 1Br 1
], C2r =
[Cr 0 Cr 1
],D2r =Dr+Cr 1Br 1
I but A2r is not in companion form!
. . . or is it?
I replace W = [W1,W2]→[W1 00 W2
],V =
[V1
V2
]→[V1 00 V2
]this often works, but doubles the dimension and changes Gr→ bad!
C. Schroder (Matheon and TU Berlin) Quadratification for 2nd order model reduction Abs Sem, Oct 2016 6 / 15
Linearization and quadratification
Algorithm
1. given (A0,A1,B0,B1,C0,C1,D) with state dim n
2. linearize, state dim 2n
A =
[0 IA0 A1
], B =
[B1
B0 + A1B1
], C =
[C0 C1
], D = D+C1B1
3. reduce to state dimension 2r
A2r = W2rAV2r , B2r = W2rB, C2r = CV2r , D2r = D
4. quadratify to state dim r , i.e., partition and read off blocks
A2r =
[0 IAr 0 Ar 1
],B2r =
[Br 1
Br 0 + Ar 1Br 1
], C2r =
[Cr 0 Cr 1
],D2r =Dr+Cr 1Br 1
I but A2r is not in companion form! . . . or is it?
I replace W = [W1,W2]→[W1 00 W2
],V =
[V1
V2
]→[V1 00 V2
]this often works, but doubles the dimension and changes Gr→ bad!
C. Schroder (Matheon and TU Berlin) Quadratification for 2nd order model reduction Abs Sem, Oct 2016 6 / 15
Linearization and quadratification
When is Ar in companion form?
I Theorem For A ∈ Rn,n and V2r ,WT2r ∈ Rn,2r with W2rV2r = I let
A2r = W2rAV2r . Then
W2r =
[Wup
WupA
]⇒ A2r =
[0 Ir∗ ∗
].
I partial realization: imW T
→ quadratifying Arnoldi method [Li et al., 2012]
I interpolation (SISO only)span{(A− σ1I )
−T cT , (A− σ2I )−T cT} = span{w ,ATw} with
w = (A− σ1I )−T (A− σ2I )
−T cT
I does not hold for MIMO systems
I does not hold for balanced truncation
I what to do then?
C. Schroder (Matheon and TU Berlin) Quadratification for 2nd order model reduction Abs Sem, Oct 2016 7 / 15
Linearization and quadratification
When is Ar in companion form?
I Theorem For A ∈ Rn,n and V2r ,WT2r ∈ Rn,2r with W2rV2r = I let
A2r = W2rAV2r . Then
W2r =
[Wup
WupA
]⇒ A2r =
[0 Ir∗ ∗
].
I partial realization: imW T = span[cT ,AT cT ,A2T cT ,A3T cT , . . .]
→ quadratifying Arnoldi method [Li et al., 2012]
I interpolation (SISO only)span{(A− σ1I )
−T cT , (A− σ2I )−T cT} = span{w ,ATw} with
w = (A− σ1I )−T (A− σ2I )
−T cT
I does not hold for MIMO systems
I does not hold for balanced truncation
I what to do then?
C. Schroder (Matheon and TU Berlin) Quadratification for 2nd order model reduction Abs Sem, Oct 2016 7 / 15
Linearization and quadratification
When is Ar in companion form?
I Theorem For A ∈ Rn,n and V2r ,WT2r ∈ Rn,2r with W2rV2r = I let
A2r = W2rAV2r . Then
W2r =
[Wup
WupA
]⇒ A2r =
[0 Ir∗ ∗
].
I partial realization: imW T = span[cT ,A2T cT , ..,AT cT ,A3T cT , ..]→ quadratifying Arnoldi method [Li et al., 2012]
I interpolation (SISO only)span{(A− σ1I )
−T cT , (A− σ2I )−T cT} = span{w ,ATw} with
w = (A− σ1I )−T (A− σ2I )
−T cT
I does not hold for MIMO systems
I does not hold for balanced truncation
I what to do then?
C. Schroder (Matheon and TU Berlin) Quadratification for 2nd order model reduction Abs Sem, Oct 2016 7 / 15
Linearization and quadratification
When is Ar in companion form?
I Theorem For A ∈ Rn,n and V2r ,WT2r ∈ Rn,2r with W2rV2r = I let
A2r = W2rAV2r . Then
W2r =
[Wup
WupA
]⇒ A2r =
[0 Ir∗ ∗
].
I partial realization: imW T = span[cT ,A2T cT , ..,AT cT ,A3T cT , ..]→ quadratifying Arnoldi method [Li et al., 2012]
I interpolation (SISO only)span{(A− σ1I )
−T cT , (A− σ2I )−T cT} = span{w ,ATw} with
w = (A− σ1I )−T (A− σ2I )
−T cT
I does not hold for MIMO systems
I does not hold for balanced truncation
I what to do then?
C. Schroder (Matheon and TU Berlin) Quadratification for 2nd order model reduction Abs Sem, Oct 2016 7 / 15
Linearization and quadratification
When is Ar in companion form?
I Theorem For A ∈ Rn,n and V2r ,WT2r ∈ Rn,2r with W2rV2r = I let
A2r = W2rAV2r . Then
W2r =
[Wup
WupA
]⇒ A2r =
[0 Ir∗ ∗
].
I partial realization: imW T = span[cT ,A2T cT , ..,AT cT ,A3T cT , ..]→ quadratifying Arnoldi method [Li et al., 2012]
I interpolation (SISO only)span{(A− σ1I )
−T cT , (A− σ2I )−T cT} = span{w ,ATw} with
w = (A− σ1I )−T (A− σ2I )
−T cT
I does not hold for MIMO systems
I does not hold for balanced truncation
I what to do then?
C. Schroder (Matheon and TU Berlin) Quadratification for 2nd order model reduction Abs Sem, Oct 2016 7 / 15
Linearization and quadratification
Post processing
I Theorem For A2r ∈ R2r ,2r and T ∈ R2r ,2r invertible we have
T =
[Tup
TupA2r
]⇔ TA2rT
−1 =
[0 Ir∗ ∗
].
I Is there such an invertible T? e.g. A2r = 0, I
[De Teran et al., 2015]No, iff A2r has an eigenvalue of geometric multiplicity > r .Yes, for almost all Tup, otherwise.
I How to get T?I proof is constructiveI run quadratifying Arnoldi method to the endI random Tup
I T means update: V ← VT−1,W ← TW
I so, A2r can be brought to companion form
C. Schroder (Matheon and TU Berlin) Quadratification for 2nd order model reduction Abs Sem, Oct 2016 8 / 15
Linearization and quadratification
Post processing
I Theorem For A2r ∈ R2r ,2r and T ∈ R2r ,2r invertible we have
T =
[Tup
TupA2r
]⇔ TA2rT
−1 =
[0 Ir∗ ∗
].
I Is there such an invertible T? [De Teran et al., 2015]No, iff A2r has an eigenvalue of geometric multiplicity > r .Yes, for almost all Tup, otherwise.
I How to get T?
I proof is constructiveI run quadratifying Arnoldi method to the endI random Tup
I T means update: V ← VT−1,W ← TW
I so, A2r can be brought to companion form
C. Schroder (Matheon and TU Berlin) Quadratification for 2nd order model reduction Abs Sem, Oct 2016 8 / 15
Linearization and quadratification
Post processing
I Theorem For A2r ∈ R2r ,2r and T ∈ R2r ,2r invertible we have
T =
[Tup
TupA2r
]⇔ TA2rT
−1 =
[0 Ir∗ ∗
].
I Is there such an invertible T? [De Teran et al., 2015]No, iff A2r has an eigenvalue of geometric multiplicity > r .Yes, for almost all Tup, otherwise.
I How to get T?I proof is constructive
I run quadratifying Arnoldi method to the endI random Tup
I T means update: V ← VT−1,W ← TW
I so, A2r can be brought to companion form
C. Schroder (Matheon and TU Berlin) Quadratification for 2nd order model reduction Abs Sem, Oct 2016 8 / 15
Linearization and quadratification
Post processing
I Theorem For A2r ∈ R2r ,2r and T ∈ R2r ,2r invertible we have
T =
[Tup
TupA2r
]⇔ TA2rT
−1 =
[0 Ir∗ ∗
].
I Is there such an invertible T? [De Teran et al., 2015]No, iff A2r has an eigenvalue of geometric multiplicity > r .Yes, for almost all Tup, otherwise.
I How to get T?I proof is constructiveI run quadratifying Arnoldi method to the end
I random Tup
I T means update: V ← VT−1,W ← TW
I so, A2r can be brought to companion form
C. Schroder (Matheon and TU Berlin) Quadratification for 2nd order model reduction Abs Sem, Oct 2016 8 / 15
Linearization and quadratification
Post processing
I Theorem For A2r ∈ R2r ,2r and T ∈ R2r ,2r invertible we have
T =
[Tup
TupA2r
]⇔ TA2rT
−1 =
[0 Ir∗ ∗
].
I Is there such an invertible T? [De Teran et al., 2015]No, iff A2r has an eigenvalue of geometric multiplicity > r .Yes, for almost all Tup, otherwise.
I How to get T?I proof is constructiveI run quadratifying Arnoldi method to the endI random Tup
I T means update: V ← VT−1,W ← TW
I so, A2r can be brought to companion form
C. Schroder (Matheon and TU Berlin) Quadratification for 2nd order model reduction Abs Sem, Oct 2016 8 / 15
Linearization and quadratification
Post processing
I Theorem For A2r ∈ R2r ,2r and T ∈ R2r ,2r invertible we have
T =
[Tup
TupA2r
]⇔ TA2rT
−1 =
[0 Ir∗ ∗
].
I Is there such an invertible T? [De Teran et al., 2015]No, iff A2r has an eigenvalue of geometric multiplicity > r .Yes, for almost all Tup, otherwise.
I How to get T?I proof is constructiveI run quadratifying Arnoldi method to the endI random Tup
I T means update: V ← VT−1,W ← TW
I so, A2r can be brought to companion form
C. Schroder (Matheon and TU Berlin) Quadratification for 2nd order model reduction Abs Sem, Oct 2016 8 / 15
Linearization and quadratification
Remarks
about the linearize-reduce-quadratizize method:
I (Ar 0,Ar 1,Br 0,Br 1,Cr 0,Cr 1,Dr ) inherits any properties of Gr
I linearization + partial realization: match 4r + 1 moments at ∞I linearization + IRKA: H2 optimal; interpolates 4r conditions
I linearization + BT: H∞ bound
I this is no projection method, in particular
X
{= 0
small rank6⇒ Xr
{= 0
small rank,X ∈ {Ai r ,Bi r ,Ci r ,Dr}
I D changes (difference to projection approach)
I initial conditions can be dealt with
I A does not have to be in companion form
C. Schroder (Matheon and TU Berlin) Quadratification for 2nd order model reduction Abs Sem, Oct 2016 9 / 15
Counter arguments
Introduction
Background on MOR for 1st and 2nd order
Linearization and quadratification
Counter arguments
The EndC. Schroder (Matheon and TU Berlin) Quadratification for 2nd order model reduction Abs Sem, Oct 2016 10 / 15
Counter arguments
“Only a dead B1 is a good B1”B1 6= 0 is non standard
x = A0x + A1x + B0u + 0u ⇒ xr = Ar 0xr + Ar 1xr + Br 0u + Br 1u
y = C0x + C1x + Du yr = Cr 0xr + Cr 1xr + Dru
I “My u is not differentiable!”→ rewrite state eqn. to
ddt z = d
dt (xr − Br 1u) = Ar 0xr + Ar 1xr + Br 0u
I “My code can’t handle B1!”→ reformulate system as
x1 = A0x1 + A1x1 + B0u
x2 = A0x2 + A1x2 + B1u
y = C0x1 + C1x1 + C1A0x2 + (C0 + C1A1)x2 + (D + C1B1)u
I “I still don’t like it!”→ in random algo. choose Tup s.t. TupB2r = 0Lemma [Meyer and Srinivasan, 1996]Such Tup exists ⇔ rank[B2r ,A2rB2r ] = 2 rank B2r .
C. Schroder (Matheon and TU Berlin) Quadratification for 2nd order model reduction Abs Sem, Oct 2016 10 / 15
Counter arguments
“Only a dead B1 is a good B1”B1 6= 0 is non standard
x = A0x + A1x + B0u + 0u ⇒ xr = Ar 0xr + Ar 1xr + Br 0u + Br 1u
y = C0x + C1x + Du yr = Cr 0xr + Cr 1xr + Dru
I “My u is not differentiable!”
→ rewrite state eqn. to
ddt z = d
dt (xr − Br 1u) = Ar 0xr + Ar 1xr + Br 0u
I “My code can’t handle B1!”→ reformulate system as
x1 = A0x1 + A1x1 + B0u
x2 = A0x2 + A1x2 + B1u
y = C0x1 + C1x1 + C1A0x2 + (C0 + C1A1)x2 + (D + C1B1)u
I “I still don’t like it!”→ in random algo. choose Tup s.t. TupB2r = 0Lemma [Meyer and Srinivasan, 1996]Such Tup exists ⇔ rank[B2r ,A2rB2r ] = 2 rank B2r .
C. Schroder (Matheon and TU Berlin) Quadratification for 2nd order model reduction Abs Sem, Oct 2016 10 / 15
Counter arguments
“Only a dead B1 is a good B1”B1 6= 0 is non standard
x = A0x + A1x + B0u + 0u ⇒ xr = Ar 0xr + Ar 1xr + Br 0u + Br 1u
y = C0x + C1x + Du yr = Cr 0xr + Cr 1xr + Dru
I “My u is not differentiable!”→ rewrite state eqn. to
ddt z = d
dt (xr − Br 1u) = Ar 0xr + Ar 1xr + Br 0u
I “My code can’t handle B1!”→ reformulate system as
x1 = A0x1 + A1x1 + B0u
x2 = A0x2 + A1x2 + B1u
y = C0x1 + C1x1 + C1A0x2 + (C0 + C1A1)x2 + (D + C1B1)u
I “I still don’t like it!”→ in random algo. choose Tup s.t. TupB2r = 0Lemma [Meyer and Srinivasan, 1996]Such Tup exists ⇔ rank[B2r ,A2rB2r ] = 2 rank B2r .
C. Schroder (Matheon and TU Berlin) Quadratification for 2nd order model reduction Abs Sem, Oct 2016 10 / 15
Counter arguments
“Only a dead B1 is a good B1”B1 6= 0 is non standard
x = A0x + A1x + B0u + 0u ⇒ xr = Ar 0xr + Ar 1xr + Br 0u + Br 1u
y = C0x + C1x + Du yr = Cr 0xr + Cr 1xr + Dru
I “My u is not differentiable!”→ rewrite state eqn. to
ddt z = d
dt (xr − Br 1u) = Ar 0xr + Ar 1xr + Br 0u
I “My code can’t handle B1!”→ reformulate system as
x1 = A0x1 + A1x1 + B0u
x2 = A0x2 + A1x2 + B1u
y = C0x1 + C1x1 + C1A0x2 + (C0 + C1A1)x2 + (D + C1B1)u
I “I still don’t like it!”→ in random algo. choose Tup s.t. TupB2r = 0Lemma [Meyer and Srinivasan, 1996]Such Tup exists ⇔ rank[B2r ,A2rB2r ] = 2 rank B2r .
C. Schroder (Matheon and TU Berlin) Quadratification for 2nd order model reduction Abs Sem, Oct 2016 10 / 15
Counter arguments
“Only a dead B1 is a good B1”B1 6= 0 is non standard
x = A0x + A1x + B0u + 0u ⇒ xr = Ar 0xr + Ar 1xr + Br 0u + Br 1u
y = C0x + C1x + Du yr = Cr 0xr + Cr 1xr + Dru
I “My u is not differentiable!”→ rewrite state eqn. to
ddt z = d
dt (xr − Br 1u) = Ar 0xr + Ar 1xr + Br 0u
I “My code can’t handle B1!”
→ reformulate system as
x1 = A0x1 + A1x1 + B0u
x2 = A0x2 + A1x2 + B1u
y = C0x1 + C1x1 + C1A0x2 + (C0 + C1A1)x2 + (D + C1B1)u
I “I still don’t like it!”→ in random algo. choose Tup s.t. TupB2r = 0Lemma [Meyer and Srinivasan, 1996]Such Tup exists ⇔ rank[B2r ,A2rB2r ] = 2 rank B2r .
C. Schroder (Matheon and TU Berlin) Quadratification for 2nd order model reduction Abs Sem, Oct 2016 10 / 15
Counter arguments
“Only a dead B1 is a good B1”B1 6= 0 is non standard
x = A0x + A1x + B0u + 0u ⇒ xr = Ar 0xr + Ar 1xr + Br 0u + Br 1u
y = C0x + C1x + Du yr = Cr 0xr + Cr 1xr + Dru
I “My u is not differentiable!”→ rewrite state eqn. to
ddt z = d
dt (xr − Br 1u) = Ar 0xr + Ar 1xr + Br 0u
I “My code can’t handle B1!”→ reformulate system as
x1 = A0x1 + A1x1 + B0u
x2 = A0x2 + A1x2 + B1u
y = C0x1 + C1x1 + C1A0x2 + (C0 + C1A1)x2 + (D + C1B1)u
I “I still don’t like it!”→ in random algo. choose Tup s.t. TupB2r = 0Lemma [Meyer and Srinivasan, 1996]Such Tup exists ⇔ rank[B2r ,A2rB2r ] = 2 rank B2r .
C. Schroder (Matheon and TU Berlin) Quadratification for 2nd order model reduction Abs Sem, Oct 2016 10 / 15
Counter arguments
“Only a dead B1 is a good B1”B1 6= 0 is non standard
x = A0x + A1x + B0u + 0u ⇒ xr = Ar 0xr + Ar 1xr + Br 0u + Br 1u
y = C0x + C1x + Du yr = Cr 0xr + Cr 1xr + Dru
I “My u is not differentiable!”→ rewrite state eqn. to
ddt z = d
dt (xr − Br 1u) = Ar 0xr + Ar 1xr + Br 0u
I “My code can’t handle B1!”→ reformulate system as
x1 = A0x1 + A1x1 + B0u
x2 = A0x2 + A1x2 + B1u
y = C0x1 + C1x1 + C1A0x2 + (C0 + C1A1)x2 + (D + C1B1)u
I “I still don’t like it!”
→ in random algo. choose Tup s.t. TupB2r = 0Lemma [Meyer and Srinivasan, 1996]Such Tup exists ⇔ rank[B2r ,A2rB2r ] = 2 rank B2r .
C. Schroder (Matheon and TU Berlin) Quadratification for 2nd order model reduction Abs Sem, Oct 2016 10 / 15
Counter arguments
“Only a dead B1 is a good B1”B1 6= 0 is non standard
x = A0x + A1x + B0u + 0u ⇒ xr = Ar 0xr + Ar 1xr + Br 0u + Br 1u
y = C0x + C1x + Du yr = Cr 0xr + Cr 1xr + Dru
I “My u is not differentiable!”→ rewrite state eqn. to
ddt z = d
dt (xr − Br 1u) = Ar 0xr + Ar 1xr + Br 0u
I “My code can’t handle B1!”→ reformulate system as
x1 = A0x1 + A1x1 + B0u
x2 = A0x2 + A1x2 + B1u
y = C0x1 + C1x1 + C1A0x2 + (C0 + C1A1)x2 + (D + C1B1)u
I “I still don’t like it!”→ in random algo. choose Tup s.t. TupB2r = 0Lemma [Meyer and Srinivasan, 1996]Such Tup exists ⇔ rank[B2r ,A2rB2r ] = 2 rank B2r .
C. Schroder (Matheon and TU Berlin) Quadratification for 2nd order model reduction Abs Sem, Oct 2016 10 / 15
Counter arguments
important directions, twice as large
I “What are the important directions?”I for projection we have x(t) ≈ Vxr (t) and x(t) ≈ V xr (t)I the columns of V are the important directionsI with linearization: similar relation
x(t) ≈ V1
[xr (t)
xr (t)− Br 1u(t)
], x(t) ≈ [V2,B1]
xr (t)xr (t)− Br 1u(t)
u(t)
,V =
[V1
V2
]I So, different important directions for x and x
I “The linearization is twice as large!”→ operations with companion form matrices are cheaper
C. Schroder (Matheon and TU Berlin) Quadratification for 2nd order model reduction Abs Sem, Oct 2016 11 / 15
Counter arguments
important directions, twice as large
I “What are the important directions?”I for projection we have x(t) ≈ Vxr (t) and x(t) ≈ V xr (t)I the columns of V are the important directionsI with linearization: similar relation
x(t) ≈ V1
[xr (t)
xr (t)− Br 1u(t)
], x(t) ≈ [V2,B1]
xr (t)xr (t)− Br 1u(t)
u(t)
,V =
[V1
V2
]I So, different important directions for x and x
I “The linearization is twice as large!”→ operations with companion form matrices are cheaper
C. Schroder (Matheon and TU Berlin) Quadratification for 2nd order model reduction Abs Sem, Oct 2016 11 / 15
The End
Introduction
Background on MOR for 1st and 2nd order
Linearization and quadratification
Counter arguments
The EndC. Schroder (Matheon and TU Berlin) Quadratification for 2nd order model reduction Abs Sem, Oct 2016 12 / 15
The End
Conclusions
I discussed the linearize-reduce-quadratizize approach to 2nd ordermodel reduction
I A2r needs to be in companion formsometimes it is naturally,otherwise can be forced (almost always)
I enabled ROMs with H2 optimality or with H∞ error bound
I not a projection method
I generalization to DAE and higher order straight forward for the mostpart
I open: structure preservation
Thanks for your attention!
C. Schroder (Matheon and TU Berlin) Quadratification for 2nd order model reduction Abs Sem, Oct 2016 12 / 15
The End
GeneralizationsI DAE: linearization needs im B1 ⊂ im A2; for quadratification:[
Wup
Wlow
](E ,A)
[V1,V2
]=
([I 0∗ ∗
],
[0 I∗ ∗
])⇔[WupEWupA
] [V1,V2
]= I
I higher order...x = A0x + A1x + A2x + B0u + B1u + B2u
y = C0x + C1x + C2x + D0u + D1u
can be linearized (= reformulated in 1st order) toxzv
=
0 I 00 0 IA0 A1 A2
xzv
+
B2
B1 + A2B2
B0 + (A1 + A22)B2 + A2B1
u
y =[C0 C1 C2
] xzv
+ (D0 + C1B2 + C2B1 + C2A2B2)u + (D1 + C2B2)u
T =
Tup
TupA3r
TupA23r
C. Schroder (Matheon and TU Berlin) Quadratification for 2nd order model reduction Abs Sem, Oct 2016 13 / 15
The End
GeneralizationsI DAE: linearization needs im B1 ⊂ im A2; for quadratification:[
Wup
Wlow
](E ,A)
[V1,V2
]=
([I 0∗ ∗
],
[0 I∗ ∗
])⇔[WupEWupA
] [V1,V2
]= I
I higher order...x = A0x + A1x + A2x + B0u + B1u + B2u
y = C0x + C1x + C2x + D0u + D1u
can be linearized (= reformulated in 1st order) toxzv
=
0 I 00 0 IA0 A1 A2
xzv
+
B2
B1 + A2B2
B0 + (A1 + A22)B2 + A2B1
u
y =[C0 C1 C2
] xzv
+ (D0 + C1B2 + C2B1 + C2A2B2)u + (D1 + C2B2)u
T =
Tup
TupA3r
TupA23r
C. Schroder (Matheon and TU Berlin) Quadratification for 2nd order model reduction Abs Sem, Oct 2016 13 / 15
The End
References
Beattie, C. and Benner, P. (2014).
H2-optimality conditions for structured dynamicalsystems.preprint MPIMD 14/18, Max-Planck-Institut furDynamik komplexer technischer Systeme Magdeburg.
De Teran, F., Dopico, F. M., and Van Dooren, P.
(2015).Matrix polynomials with completely prescribedeigenstructure.SIAM J. Matrix Anal. Appl., 36(1):302–328.
De Teran, F., Dopico, F. M., and Van Dooren, P.
(2016).Constructing strong `-ifications from dual minimalbases.Linear Algebra Appl., 495:344–372.
Lawrence, P. W., Van Barel, M., and Van Dooren, P.
(2016).Backward error analysis of polynomial eigenvalueproblems solved by linearization.SIAM J. Matrix Anal. Appl., 37(1):123–144.
Li, Y.-T., Bai, Z., Lin, W.-W., and Su, Y. (2012).
A structured quasi-Arnoldi procedure for model orderreduction of second-order systems.Linear Algebra Appl., 436(8):2780–2794.
Meyer, D. G. and Srinivasan, S. (1996).
Balancing and model reduction for second-order formlinear systems.IEEE Trans. Automat. Control, 41(11):1632–1644.
Tern, F. D., Dopico, F. M., and Mackey, D. S. (2014).
Spectral equivalence of matrix polynomials and theindex sum theorem.Linear Algebra and its Applications, 459:264 – 333.
Tisseur, F. and Zaballa, I. (2013).
Triangularizing quadratic matrix polynomials.SIAM J. Matrix Anal. Appl., 34(2):312–337.
Wyatt, S. (2012).
Issues in Interpolatory Model Reduction: InexactSolves, Second-order Systems and DAEs.Phd thesis, Virginia Tech, Blacksburg, VA.
C. Schroder (Matheon and TU Berlin) Quadratification for 2nd order model reduction Abs Sem, Oct 2016 14 / 15
The End
papers to cite
I DeTeranDopicaMackeyVandooren, quadratization foreigenvalues[Tern et al., 2014, De Teran et al., 2015,De Teran et al., 2016]
I VanDooren, backward error, lineariztaion[Lawrence et al., 2016]
I quadratifying arnoldi[Li et al., 2012]
I conditions for 2nd order formulation (ref byserkan)[Meyer and Srinivasan, 1996, no pdf]
I 2nd order Schur (Tisseur, Zaballa, ...)[Tisseur and Zaballa, 2013]
I PhDthesis of serkans student[Wyatt, 2012]
I Benner, conditions for 2nd order optimal interpolationpoints[Beattie and Benner, 2014]
C. Schroder (Matheon and TU Berlin) Quadratification for 2nd order model reduction Abs Sem, Oct 2016 15 / 15