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Intro H∞ Damping Examples Concl
Model reduction of large scale second-ordersystems with modal damping
Christopher Beattie and Serkan GugercinVirginia Tech
Research supported under NSF DMS0505971 and DMS0513542
Sixth International Workshopon Accurate Solution of Eigenvalue Problems
Beattie Model reduction of second-order systems w/ modal damping
Intro H∞ Damping Examples Concl Structural Dynamics Structured Model Reduction
Structural Dynamics
Second-order dynamical system:
Mx(t) + Gx(t) + Kx(t) = b u(t),y(t) = cT x(t)
symmetric positive semidefinite M, G, K ∈ Rn×n
models distributed mass, damping, and stiffnessb, c ∈ R
n models spatial distribution of input and output.Mass (M) and stiffness (K) distributions are well-modeledbut damping distribution (G) is often heuristic.Need “input-output” map u 7→ y.
Frequency domain: y(ω) = H(ıω)u(ω)
Transfer function: H(s) = cT(Ms2 + Gs + K)−1bBut n can be too large for efficient capture of i/o map u 7→ y
Beattie Model reduction of second-order systems w/ modal damping
Intro H∞ Damping Examples Concl Structural Dynamics Structured Model Reduction
Structural Dynamics
Second-order dynamical system:
Mx(t) + Gx(t) + Kx(t) = b u(t),y(t) = cT x(t)
symmetric positive semidefinite M, G, K ∈ Rn×n
models distributed mass, damping, and stiffnessb, c ∈ R
n models spatial distribution of input and output.Mass (M) and stiffness (K) distributions are well-modeledbut damping distribution (G) is often heuristic.Need “input-output” map u 7→ y.
Frequency domain: y(ω) = H(ıω)u(ω)
Transfer function: H(s) = cT(Ms2 + Gs + K)−1bBut n can be too large for efficient capture of i/o map u 7→ y
Beattie Model reduction of second-order systems w/ modal damping
Intro H∞ Damping Examples Concl Structural Dynamics Structured Model Reduction
Structured Model Reduction
Construct a simpler input-output map u 7→ y that maintainsthe character of the mediating system.Generate, for some r � n, an r degree of freedom“condensed structure”
Mrxr(t) + Grxr(t) + Krxr(t) = br u(t),yr(t) = cT
r xr(t)
symmetric positive semidefinite Mr, Gr, Kr ∈ Rr×r
(condensed property matrices) and br, cr ∈ Rr.
Can maintain structure with a Ritz approximation onto asubspace Wr = Ran(W) with WTW = Ir and
Mr = WTMW, Gr =WTGW, Kr = WTKW,
br = WTb, and cTr = cT W
Choose Wr so yr ≈ y over a wide range of u(t).
Beattie Model reduction of second-order systems w/ modal damping
Intro H∞ Damping Examples Concl Structural Dynamics Structured Model Reduction
Structured Model Reduction
Construct a simpler input-output map u 7→ y that maintainsthe character of the mediating system.Generate, for some r � n, an r degree of freedom“condensed structure”
Mrxr(t) + Grxr(t) + Krxr(t) = br u(t),yr(t) = cT
r xr(t)
symmetric positive semidefinite Mr, Gr, Kr ∈ Rr×r
(condensed property matrices) and br, cr ∈ Rr.
Can maintain structure with a Ritz approximation onto asubspace Wr = Ran(W) with WTW = Ir and
Mr = WTMW, Gr =WTGW, Kr = WTKW,
br = WTb, and cTr = cT W
Choose Wr so yr ≈ y over a wide range of u(t).
Beattie Model reduction of second-order systems w/ modal damping
Intro H∞ Damping Examples Concl Structural Dynamics Structured Model Reduction
Structured Model Reduction
Construct a simpler input-output map u 7→ y that maintainsthe character of the mediating system.Generate, for some r � n, an r degree of freedom“condensed structure”
Mrxr(t) + Grxr(t) + Krxr(t) = br u(t),yr(t) = cT
r xr(t)
symmetric positive semidefinite Mr, Gr, Kr ∈ Rr×r
(condensed property matrices) and br, cr ∈ Rr.
Can maintain structure with a Ritz approximation onto asubspace Wr = Ran(W) with WTW = Ir and
Mr = WTMW, Gr =WTGW, Kr = WTKW,
br = WTb, and cTr = cT W
Choose Wr so yr ≈ y over a wide range of u(t).
Beattie Model reduction of second-order systems w/ modal damping
Intro H∞ Damping Examples Concl PerfMeas Interpolation RatKry
H∞ Performance Measure
‖yr − y‖L2 ≤ (small number) ‖u‖L2 ;Choose Wr to make (small number) small
Frequency domain:
Full response: y(ω) = H(ıω)u(ω)
Reduced order response: yr(ω) = Hr(ıω)u(ω)
with transfer functions:
H(s) = cT(Ms2 + Gs + K)−1b
Hr(s) = cTr (Mrs2 + Grs + Kr)
−1br
Uniformly small L2 error guaranteed by:
‖yr − y‖L2 ≤(
maxω∈R
|Hr(ıω) − H(ıω)|)
︸ ︷︷ ︸
small H∞ error
‖u‖L2
Beattie Model reduction of second-order systems w/ modal damping
Intro H∞ Damping Examples Concl PerfMeas Interpolation RatKry
Approximation by Interpolation
Want a reduced order model leading to small H∞ error:maxω∈R |Hr(ıω) − H(ıω)|.
Suppose H(opt)r is a stable transfer function producing the
optimal H∞ error
maxω∈R
|H(opt)r (ıω) − H(ıω)| ≤ max
ω∈R
|Hr(ıω) − H(ıω)|
and Hr is a stable transfer function that interpolates H at2r + 1 points in the RHP.Then
minω∈R
|Hr(ıω) − H(ıω)| ≤ maxω∈R
|H(opt)r (ıω) − H(ıω)|
Beattie Model reduction of second-order systems w/ modal damping
Intro H∞ Damping Examples Concl PerfMeas Interpolation RatKry
Approximation by Interpolation
Want a reduced order model leading to small H∞ error:maxω∈R |Hr(ıω) − H(ıω)|.
Suppose H(opt)r is a stable transfer function producing the
optimal H∞ error
maxω∈R
|H(opt)r (ıω) − H(ıω)| ≤ max
ω∈R
|Hr(ıω) − H(ıω)|
and Hr is a stable transfer function that interpolates H at2r + 1 points in the RHP.Then
minω∈R
|Hr(ıω) − H(ıω)| ≤ maxω∈R
|H(opt)r (ıω) − H(ıω)|
Beattie Model reduction of second-order systems w/ modal damping
Intro H∞ Damping Examples Concl PerfMeas Interpolation RatKry
Approximation by Interpolation
Want a reduced order model leading to small H∞ error:maxω∈R |Hr(ıω) − H(ıω)|.
Suppose H(opt)r is a stable transfer function producing the
optimal H∞ error
maxω∈R
|H(opt)r (ıω) − H(ıω)| ≤ max
ω∈R
|Hr(ıω) − H(ıω)|
and Hr is a stable transfer function that interpolates H at2r + 1 points in the RHP.Then
minω∈R
|Hr(ıω) − H(ıω)| ≤ maxω∈R
|H(opt)r (ıω) − H(ıω)|
Beattie Model reduction of second-order systems w/ modal damping
Intro H∞ Damping Examples Concl PerfMeas Interpolation RatKry
Approximation by Interpolation
Related to ”near-circularity” in error of best uniform rationalapproximation on unit disk (Trefethen, 1981)complex analog of de la Valee-Poussin theorem.Hr(s) is close to the best H∞ approx to H(s) when
minω∈R
|Hr(ıω) − H(ıω)| ≈ maxω∈R
|Hr(ıω) − H(ıω)|
(that is, |H(ω) − Hr(ω)| ≈ constant)and
Hr(s) interpolates H(s) on at least 2r + 1 points in RHP
Best uniform approximations are hard to calculate.Interpolants are (comparatively) easy to calculate.
Beattie Model reduction of second-order systems w/ modal damping
Intro H∞ Damping Examples Concl PerfMeas Interpolation RatKry
Approximation by Interpolation
Related to ”near-circularity” in error of best uniform rationalapproximation on unit disk (Trefethen, 1981)complex analog of de la Valee-Poussin theorem.Hr(s) is close to the best H∞ approx to H(s) when
minω∈R
|Hr(ıω) − H(ıω)| ≈ maxω∈R
|Hr(ıω) − H(ıω)|
(that is, |H(ω) − Hr(ω)| ≈ constant)and
Hr(s) interpolates H(s) on at least 2r + 1 points in RHP
Best uniform approximations are hard to calculate.Interpolants are (comparatively) easy to calculate.
Beattie Model reduction of second-order systems w/ modal damping
Intro H∞ Damping Examples Concl PerfMeas Interpolation RatKry
Approximation by Interpolation
Related to ”near-circularity” in error of best uniform rationalapproximation on unit disk (Trefethen, 1981)complex analog of de la Valee-Poussin theorem.Hr(s) is close to the best H∞ approx to H(s) when
minω∈R
|Hr(ıω) − H(ıω)| ≈ maxω∈R
|Hr(ıω) − H(ıω)|
(that is, |H(ω) − Hr(ω)| ≈ constant)and
Hr(s) interpolates H(s) on at least 2r + 1 points in RHP
Best uniform approximations are hard to calculate.Interpolants are (comparatively) easy to calculate.
Beattie Model reduction of second-order systems w/ modal damping
Intro H∞ Damping Examples Concl PerfMeas Interpolation RatKry
How to choose interpolation points ?
|H(ω) − Hr(ω)| ≈ constant ?Φ(s) = log |H(s) − Hr(s)|· has positive singularities at system eigenvalues.· has negative singularities at interpolation points.· is harmonic everywhere else
Φ is a potential function - electrostatic analogyLocate interpolation points (negative point charges) tobalance equipotentials of log |H(s) − Hr(s)| (makeslog |H(s) − Hr(s)| nearly constant along the imaginary axis)Interpolate at points that mirror singularities across theimaginary axis (but there are too many !)So mirror “equivalent charges” instead; e.g., Ritz values.Mirrored Ritz values are optimal choice for H2 minimizationas well (Gugercin/Antoulas/B [2004])
Beattie Model reduction of second-order systems w/ modal damping
Intro H∞ Damping Examples Concl PerfMeas Interpolation RatKry
How to choose interpolation points ?
|H(ω) − Hr(ω)| ≈ constant ?Φ(s) = log |H(s) − Hr(s)|· has positive singularities at system eigenvalues.· has negative singularities at interpolation points.· is harmonic everywhere else
Φ is a potential function - electrostatic analogyLocate interpolation points (negative point charges) tobalance equipotentials of log |H(s) − Hr(s)| (makeslog |H(s) − Hr(s)| nearly constant along the imaginary axis)Interpolate at points that mirror singularities across theimaginary axis (but there are too many !)So mirror “equivalent charges” instead; e.g., Ritz values.Mirrored Ritz values are optimal choice for H2 minimizationas well (Gugercin/Antoulas/B [2004])
Beattie Model reduction of second-order systems w/ modal damping
Intro H∞ Damping Examples Concl PerfMeas Interpolation RatKry
How to choose interpolation points ?
|H(ω) − Hr(ω)| ≈ constant ?Φ(s) = log |H(s) − Hr(s)|· has positive singularities at system eigenvalues.· has negative singularities at interpolation points.· is harmonic everywhere else
Φ is a potential function - electrostatic analogyLocate interpolation points (negative point charges) tobalance equipotentials of log |H(s) − Hr(s)| (makeslog |H(s) − Hr(s)| nearly constant along the imaginary axis)Interpolate at points that mirror singularities across theimaginary axis (but there are too many !)So mirror “equivalent charges” instead; e.g., Ritz values.Mirrored Ritz values are optimal choice for H2 minimizationas well (Gugercin/Antoulas/B [2004])
Beattie Model reduction of second-order systems w/ modal damping
Intro H∞ Damping Examples Concl PerfMeas Interpolation RatKry
How to choose interpolation points ?
|H(ω) − Hr(ω)| ≈ constant ?Φ(s) = log |H(s) − Hr(s)|· has positive singularities at system eigenvalues.· has negative singularities at interpolation points.· is harmonic everywhere else
Φ is a potential function - electrostatic analogyLocate interpolation points (negative point charges) tobalance equipotentials of log |H(s) − Hr(s)| (makeslog |H(s) − Hr(s)| nearly constant along the imaginary axis)Interpolate at points that mirror singularities across theimaginary axis (but there are too many !)So mirror “equivalent charges” instead; e.g., Ritz values.Mirrored Ritz values are optimal choice for H2 minimizationas well (Gugercin/Antoulas/B [2004])
Beattie Model reduction of second-order systems w/ modal damping
Intro H∞ Damping Examples Concl PerfMeas Interpolation RatKry
Rational Krylov-based model reduction
Examine the pointwise error: H(s) − Hr(s)
Define Kσ = Mσ2 + Gσ + K
H(σ) − Hr(σ) = cT [K−1
σ − Wr(Mrs2 + Grs + Kr)−1WT
r]
b
So K−1σ b ∈ Wr implies H(σ) = Hr(σ).
Matching H′(σ) = H′r(σ) and higher moments can be done
similarly.
Beattie Model reduction of second-order systems w/ modal damping
Intro H∞ Damping Examples Concl PerfMeas Interpolation RatKry
Rational Krylov-based model reduction
Examine the pointwise error: H(s) − Hr(s)
Define Kσ = Mσ2 + Gσ + K
H(σ) − Hr(σ) = cT [K−1
σ − Wr(Mrs2 + Grs + Kr)−1WT
r]
b
= ct[I − Wr(Mrs2 + Grs + Kr)−1WT
r Kσ]K−1σ b
(factor out K−1σ )
So K−1σ b ∈ Wr implies H(σ) = Hr(σ).
Matching H′(σ) = H′r(σ) and higher moments can be done
similarly.
Beattie Model reduction of second-order systems w/ modal damping
Intro H∞ Damping Examples Concl PerfMeas Interpolation RatKry
Rational Krylov-based model reduction
Examine the pointwise error: H(s) − Hr(s)
Define Kσ = Mσ2 + Gσ + K
H(σ) − Hr(σ) = cT [K−1
σ − Wr(Mrs2 + Grs + Kr)−1WT
r]
b
= cT [I − Wr(Mrs2 + Grs + Kr)−1WT
r Kσ︸ ︷︷ ︸
projection onto Wr∆= Pr(σ)
]K−1σ b
So K−1σ b ∈ Wr implies H(σ) = Hr(σ).
Matching H′(σ) = H′r(σ) and higher moments can be done
similarly.
Beattie Model reduction of second-order systems w/ modal damping
Intro H∞ Damping Examples Concl PerfMeas Interpolation RatKry
Rational Krylov-based model reduction
Examine the pointwise error: H(s) − Hr(s)
Define Kσ = Mσ2 + Gσ + K
H(σ) − Hr(σ) = cT [K−1
σ − Wr(Mrs2 + Grs + Kr)−1WT
r]
b
= cT [I − Wr(Mrs2 + Grs + Kr)−1WT
r Kσ︸ ︷︷ ︸
projection onto Wr∆= Pr(σ)
]K−1σ b
= cT [I − Pr(σ)]K−1σ b
So K−1σ b ∈ Wr implies H(σ) = Hr(σ).
Matching H′(σ) = H′r(σ) and higher moments can be done
similarly.
Beattie Model reduction of second-order systems w/ modal damping
Intro H∞ Damping Examples Concl PerfMeas Interpolation RatKry
Rational Krylov-based model reduction
Examine the pointwise error: H(s) − Hr(s)
Define Kσ = Mσ2 + Gσ + K
H(σ) − Hr(σ) = cT [K−1
σ − Wr(Mrs2 + Grs + Kr)−1WT
r]
b
= cT [I − Wr(Mrs2 + Grs + Kr)−1WT
r Kσ︸ ︷︷ ︸
projection onto Wr∆= Pr(σ)
]K−1σ b
= cT [I − Pr(σ)]K−1σ b
= 0 if K−1σ b ∈ Wr
So K−1σ b ∈ Wr implies H(σ) = Hr(σ).
Matching H′(σ) = H′r(σ) and higher moments can be done
similarly.
Beattie Model reduction of second-order systems w/ modal damping
Intro H∞ Damping Examples Concl PerfMeas Interpolation RatKry
Rational Krylov-based model reduction
Examine the pointwise error: H(s) − Hr(s)
Define Kσ = Mσ2 + Gσ + K
H(σ) − Hr(σ) = cT [K−1
σ − Wr(Mrs2 + Grs + Kr)−1WT
r]
b
= cT [I − Wr(Mrs2 + Grs + Kr)−1WT
r Kσ︸ ︷︷ ︸
projection onto Wr∆= Pr(σ)
]K−1σ b
= cT [I − Pr(σ)]K−1σ b
= 0 if K−1σ b ∈ Wr
So K−1σ b ∈ Wr implies H(σ) = Hr(σ).
Matching H′(σ) = H′r(σ) and higher moments can be done
similarly.
Beattie Model reduction of second-order systems w/ modal damping
Intro H∞ Damping Examples Concl PerfMeas Interpolation RatKry
Rational Krylov-based model reduction
K−1σ b ∈ Wr implies H(σ) = Hr(σ)
In particular, if for {σ1, σ2, . . . , σr, } ⊂ C,
span{K−1σ1
b, K−1σ2
b, . . . , K−1σr
b} = Wr
then H(σ) = Hr(σ) for σ = σ1, σ2, . . . , σr.Wr is not a rational Krylov subspace in the usual senseexcept in special circumstances.(Note there is no useful commutation property in general:KσiKσj 6= KσjKσi , KσiM−1Kσj 6= KσjM−1Kσi , orKσiK−1Kσj 6= KσjK−1Kσi)Bai[2003] discovered these interpolation conditions andcalled the associated spaces “second-order Krylovsubspaces”
Beattie Model reduction of second-order systems w/ modal damping
Intro H∞ Damping Examples Concl PerfMeas Interpolation RatKry
Rational Krylov-based model reduction
K−1σ b ∈ Wr implies H(σ) = Hr(σ)
In particular, if for {σ1, σ2, . . . , σr, } ⊂ C,
span{K−1σ1
b, K−1σ2
b, . . . , K−1σr
b} = Wr
then H(σ) = Hr(σ) for σ = σ1, σ2, . . . , σr.Wr is not a rational Krylov subspace in the usual senseexcept in special circumstances.(Note there is no useful commutation property in general:KσiKσj 6= KσjKσi , KσiM−1Kσj 6= KσjM−1Kσi , orKσiK−1Kσj 6= KσjK−1Kσi)Bai[2003] discovered these interpolation conditions andcalled the associated spaces “second-order Krylovsubspaces”
Beattie Model reduction of second-order systems w/ modal damping
Intro H∞ Damping Examples Concl PerfMeas Interpolation RatKry
Rational Krylov-based model reduction
K−1σ b ∈ Wr implies H(σ) = Hr(σ)
In particular, if for {σ1, σ2, . . . , σr, } ⊂ C,
span{K−1σ1
b, K−1σ2
b, . . . , K−1σr
b} = Wr
then H(σ) = Hr(σ) for σ = σ1, σ2, . . . , σr.Wr is not a rational Krylov subspace in the usual senseexcept in special circumstances.(Note there is no useful commutation property in general:KσiKσj 6= KσjKσi , KσiM−1Kσj 6= KσjM−1Kσi , orKσiK−1Kσj 6= KσjK−1Kσi)Bai[2003] discovered these interpolation conditions andcalled the associated spaces “second-order Krylovsubspaces”
Beattie Model reduction of second-order systems w/ modal damping
Intro H∞ Damping Examples Concl Modal ShftSel PropDmp
Second-order systems with modal damping
Assume that KσiM−1Kσj = Kσj M−1Kσi independent of {σ`}.Then M, G, and K can be simultaneously diagonalized:
XTMX = I, XTKX = diag(ω2i ), XTGX = diag(γ2
i )
Simple heuristic model of damping: γ2i = 2ξiωi
Modal damping can be characterized by a function g(z)that is real analytic on R+ that interpolates the values
g(ω2i ) = γ2
i and then G = Mg(M−1K)
More generally G = Mg1(M−1K)P1 + Mg2(M−1K)P2where P1 and P2 are complementary spectral projectors.Common choices:
g(z) = α + βz with α, β > 0 (“proportional damping”)g(z) = 2ξ
√z with ξ > 0 (“fractional damping”)
Beattie Model reduction of second-order systems w/ modal damping
Intro H∞ Damping Examples Concl Modal ShftSel PropDmp
Second-order systems with modal damping
Assume that KσiM−1Kσj = Kσj M−1Kσi independent of {σ`}.Then M, G, and K can be simultaneously diagonalized:
XTMX = I, XTKX = diag(ω2i ), XTGX = diag(γ2
i )
Simple heuristic model of damping: γ2i = 2ξiωi
Modal damping can be characterized by a function g(z)that is real analytic on R+ that interpolates the values
g(ω2i ) = γ2
i and then G = Mg(M−1K)
More generally G = Mg1(M−1K)P1 + Mg2(M−1K)P2where P1 and P2 are complementary spectral projectors.Common choices:
g(z) = α + βz with α, β > 0 (“proportional damping”)g(z) = 2ξ
√z with ξ > 0 (“fractional damping”)
Beattie Model reduction of second-order systems w/ modal damping
Intro H∞ Damping Examples Concl Modal ShftSel PropDmp
Second-order systems with modal damping
Assume that KσiM−1Kσj = Kσj M−1Kσi independent of {σ`}.Then M, G, and K can be simultaneously diagonalized:
XTMX = I, XTKX = diag(ω2i ), XTGX = diag(γ2
i )
Simple heuristic model of damping: γ2i = 2ξiωi
Modal damping can be characterized by a function g(z)that is real analytic on R+ that interpolates the values
g(ω2i ) = γ2
i and then G = Mg(M−1K)
More generally G = Mg1(M−1K)P1 + Mg2(M−1K)P2where P1 and P2 are complementary spectral projectors.Common choices:
g(z) = α + βz with α, β > 0 (“proportional damping”)g(z) = 2ξ
√z with ξ > 0 (“fractional damping”)
Beattie Model reduction of second-order systems w/ modal damping
Intro H∞ Damping Examples Concl Modal ShftSel PropDmp
Second-order systems with modal damping
The damped eigenvalue λ(ω) associated with ω satisfiesλ2 + λ g(ω2) + ω2 = 0.
Damped eigenvalues are constrained to curves in LHPRe(λ) ≤ g(ω2)
2 ; |λ| = ω
-6 -4 -2 2 4 6
-3
-2
-1
1
2
3
Beattie Model reduction of second-order systems w/ modal damping
Intro H∞ Damping Examples Concl Modal ShftSel PropDmp
Shift Selection for Modal Damping
Damped eigenvalues are constrained to lie on curves inLHP (change in stiffness or mass properties only changeseigenvalue distribution on the curve).Effective shift strategies (mirroring eigenvalue distribution)could be constructed on the basis of g(z).
-6 -4 -2 2 4 6
-3
-2
-1
1
2
3
Beattie Model reduction of second-order systems w/ modal damping
Intro H∞ Damping Examples Concl Modal ShftSel PropDmp
Shift Selection for Modal Damping
Damped eigenvalues are constrained to lie on curves inLHP (change in stiffness or mass properties only changeseigenvalue distribution on the curve).Effective shift strategies (mirroring eigenvalue distribution)could be constructed on the basis of g(z).
(largely independent of mass and stiffness)
-6 -4 -2 2 4 6
-3
-2
-1
1
2
3
Beattie Model reduction of second-order systems w/ modal damping
Intro H∞ Damping Examples Concl Modal ShftSel PropDmp
Shift Selection for Modal Damping
Damped eigenvalues are constrained to lie on curves inLHP (change in stiffness or mass properties only changeseigenvalue distribution on the curve).Effective shift strategies (mirroring eigenvalue distribution)could be constructed on the basis of g(z).
Replace w/ equivalent charges. (Balyage)
-6 -4 -2 2 4 6
-3
-2
-1
1
2
3
Beattie Model reduction of second-order systems w/ modal damping
Intro H∞ Damping Examples Concl Modal ShftSel PropDmp
Shift Selection for Modal Damping
Interesting special case: proportional damping
Mx + (αM + βK)x + Kx = b u(t).
All damped eigenvalues (system poles) are on circle withcenter:− 1
β, radius:
√1−αββ
and on ray (∞, − 1β].
Distribution depends on undamped natural frequencies,but usual elastic vibration models lead to distributions thatare close to “equilibrium condenser distributions”
Beattie Model reduction of second-order systems w/ modal damping
Intro H∞ Damping Examples Concl Modal ShftSel PropDmp
Shift Selection for Modal Damping
Interesting special case: proportional damping
Mx + (αM + βK)x + Kx = b u(t).
All damped eigenvalues (system poles) are on circle withcenter:− 1
β, radius:
√1−αββ
and on ray (∞, − 1β].
Distribution depends on undamped natural frequencies,but usual elastic vibration models lead to distributions thatare close to “equilibrium condenser distributions”
-6 -4 -2 2 4 6
-3
-2
-1
1
2
3
Beattie Model reduction of second-order systems w/ modal damping
Intro H∞ Damping Examples Concl Modal ShftSel PropDmp
Shift Selection for Modal Damping
Interesting special case: proportional damping
Mx + (αM + βK)x + Kx = b u(t).
All damped eigenvalues (system poles) are on circle withcenter:− 1
β, radius:
√1−αββ
and on ray (∞, − 1β].
Distribution depends on undamped natural frequencies,but usual elastic vibration models lead to distributions thatare close to “equilibrium condenser distributions”
-6 -4 -2 2 4 6
-3
-2
-1
1
2
3
Beattie Model reduction of second-order systems w/ modal damping
Intro H∞ Damping Examples Concl Modal ShftSel PropDmp
Shift Selection for Proportional Damping
Only ONE shift is necessary - replace aggregate ofinterpolation points (negative charge distribution) with
single shift (an equivalent lumped charge) at σ∗ =
√α
β.
Optimal choice for condenser distribution of system poles;pretty good choice for most K and M.Wr is a true rational Krylov space in this case.
Beattie Model reduction of second-order systems w/ modal damping
Intro H∞ Damping Examples Concl Modal ShftSel PropDmp
Shift Selection for Proportional Damping
Only ONE shift is necessary - replace aggregate ofinterpolation points (negative charge distribution) with
single shift (an equivalent lumped charge) at σ∗ =
√α
β.
Optimal choice for condenser distribution of system poles;pretty good choice for most K and M.Wr is a true rational Krylov space in this case.
Beattie Model reduction of second-order systems w/ modal damping
Intro H∞ Damping Examples Concl Modal ShftSel PropDmp
Shift Selection for Proportional Damping
Only ONE shift is necessary - replace aggregate ofinterpolation points (negative charge distribution) with
single shift (an equivalent lumped charge) at σ∗ =
√α
β.
Optimal choice for condenser distribution of system poles;pretty good choice for most K and M.Wr is a true rational Krylov space in this case.
Beattie Model reduction of second-order systems w/ modal damping
Intro H∞ Damping Examples Concl Condenser Beam1 Beam2
Exact Condenser Distribution• Pick α, β ∈ (0, 1)
K =α
β
2
6
6
6
6
6
6
6
6
6
6
6
6
4
2−√
1−αβ√
1−αβ−1 0 . . .
−1 2√1−αβ
0. . . 0
.
.
. 2√1−αβ
−1
0 −1 2−√
1−αβ√
1−αβ
3
7
7
7
7
7
7
7
7
7
7
7
7
5
M =
2
6
6
6
6
6
6
6
6
6
6
6
6
4
2+√
1−αβ√
1−αβ1 0 . . .
1 2√1−αβ
0. . . 0
.
.
. 2√1−αβ
1
0 1 2+√
1−αβ√
1−αβ
3
7
7
7
7
7
7
7
7
7
7
7
7
5
.
• G = α M + β KSingle shift is exactly equivalentto mirrored eigenvalue distribution !!
Beattie Model reduction of second-order systems w/ modal damping
Intro H∞ Damping Examples Concl Condenser Beam1 Beam2
• Reduction from n = 2000 to r = 30 using a single shift• α = β = 0.05, b = c = [ 1 0 0 · · · 0 ]T .
−40 −35 −30 −25 −20 −15 −10 −5 0−20
−15
−10
−5
0
5
10
15
20
Real
Ima
g
Pole locations for exact condenser distribution
10−3
10−2
10−1
100
101
102
10−2
10−1
100
101
H∞ error vs interpolation point
H∞ e
rro
r
σ
• σ∗ =√
αβ
= 1 is the optimal shift.
Beattie Model reduction of second-order systems w/ modal damping
Intro H∞ Damping Examples Concl Condenser Beam1 Beam2
A 1-D Beam Model
• n = 2000. α = 1/10, β = 1/500, b = e1, c = e200.
−1000 −900 −800 −700 −600 −500 −400 −300 −200 −100 0−500
−400
−300
−200
−100
0
100
200
300
400
500
Imag
Real
Distributiion of Observed and Condenser Poles
Observed Poles
Condenser Poles
2 4 6 8 10 12 14
100
101
σ
||
G(s
) −
Gr(s
) ||
∞
H∞ error vs interpolation point
r=5
r=10
r=15
• σ∗ =√
αβ
= 7.0711: Very close to being optimal.
Beattie Model reduction of second-order systems w/ modal damping
Intro H∞ Damping Examples Concl Condenser Beam1 Beam2
Another 1-D Beam Model
• n = 200 and α = β = 1/300, b = c = e1.• Compare with balanced truncation and other shift selections• Balanced reduction done on ‘linearized’ system to r(1) = 40(r = 20).
10−3 10−2 10−1 100 101 102 10310−7
10−6
10−5
10−4
10−3
freq (rad/sec)
||
H(
jw )
|| 2
Amplitude bode plots for the beam model
HBT
H(s)
Hσ*
10−3 10−2 10−1 100 101 102 10310−10
10−9
10−8
10−7
10−6
10−5
10−4
freq (rad/sec)
||
Herr
or(
jw )
|| 2
Amplitude bode plots of the error systems for the beam model
H − Hσ*
H − HBT
Beattie Model reduction of second-order systems w/ modal damping
Intro H∞ Damping Examples Concl Condenser Beam1 Beam2
10−3 10−2 10−1 100 101 102 103
10−6
10−5
10−4
freq (rad/sec)
||
H(
jw )
|| 2
Amplitude bode plots for the beam model
σ = 1
σi, i=1:40
0 10 20 30 40 50 60 70 80 90 100−5.6
−5.4
−5.2
−5
−4.8
−4.6
−4.4
−4.2
−4
−3.8
−3.6
lo
g10 |
| H
er |
| Hin
f
r
Beam with n=200, α = β = 1/300
σ = 1
σ = 5
σ = 10
Observed Convergence rate: 0.9428 Expected Convergence rate: 0.9867
• Convergence rates are very close to what is predicted by“condenser capacity” estimates.
Beattie Model reduction of second-order systems w/ modal damping
Intro H∞ Damping Examples Concl
Conclusions
Considered strategies for structure-maintaining modelreduction of large scale second-order systems.
Mx + Gx + Kx = b u(t)
y(t) = cTx(t)
The use of modal damping models G = Mg(M−1K) canfacilitate model reduction through well chosen interpolationpoints.Proportional damping provides special computationaladvantages: an equivalent single shift (lumped charge) toaggregate interpolation points produced by mirroring poles.This single shift is exactly optimal for a class of mass andstiffness matrices (condenser distribution)Close to optimal in general.Future work? Extend to other damping models.
Beattie Model reduction of second-order systems w/ modal damping