[PPT] - Distribution of Eigenvalues of Linear Stochastic Systems S A DHIKARI PowerPoint Presentation

SLIDE 1

Distribution of Eigenvalues of Linear Stochastic Systems

S ADHIKARI† and R. S. LANGLEY‡

†Department of Aerospace Engineering, University of Bristol, Bristol, U.K.

Email: S.Adhikari@bristol.ac.uk URL: http://www.aer.bris.ac.uk/contact/academic/adhikari/home.html

‡Department of Engineering, University of Cambridge, Cambridge, UK

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SLIDE 2

Outline of the talk

Random eigenvalue problem
Perturbation Methods
Mean-centered perturbation method
α-centered perturbation method
Asymptotic analysis
PDF of the eigenvalues
Numerical Example
Conclusions & Open Problems

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Random eigenvalue problem

The random eigenvalue problem of undamped or proportionally damped linear systems:

K(x)φj = λjM(x)φj (1)

λj eigenvalues; φj eigenvectors; M(x) ∈ RN×N mass matrix and K(x) ∈ RN×N stiffness matrix.

x ∈ Rm is random parameter vector with pdf

p(x) = (2π)−m/2e−xTx/2

(2)

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The fundamental aim

To obtain the joint probability density function
f the eigenvalues and the eigenvectors.
If the matrix M−1K is GUE (Gaussian unitary

ensemble) or GOE (Gaussian orthogonal ensemble) an exact closed-form expression can be obtained for the joint pdf of the eigenvalues.

In general the system matrices for real

structures are not GUE or GOE.

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Mean-centered perturbation

Assume that M(0) = M0 and K(0) = K0 are ‘deterministic parts’. Deterministic eigenvalue problem: K0φj0 = λj0M0φj0. The eigenvalues λj(x) : Rm → R are non-linear functions of x. Expanding λj(x) by Taylor series about x = 0: λj(x) ≈ λj(0) + dT

λj(0)x + 1

2 xT Dλj(0)x (3) dλj(0) ∈ Rm: gradient vector, Dλj(0) ∈ Rm×m the

Hessian matrix of λj(x) evaluated at x = 0.

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α-centered perturbation

We are looking for a point x = α in the x-space such that the Taylor series expansion of λj(x) about this point λj(x) ≈ λj(α) + dT

λj(α) (x − α)

+ 1 2 (x − α)T Dλj(α) (x − α)

(4)

is optimal in some sense. The optimal point α is selected such that the mean or the first moment

f each eigenvalue is calculated most accurately.

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α-centered perturbation

The mean of λj(x) can be obtained as ¯ λj =

R

m λj(x)p(x) dx = (2π)−m/2

R

m e−h(x) dx

(5)

where h(x) = xT x/2 − ln λj(x)

(6)

Expand the function h(x) in a Taylor series about a point where h(x) attends its global minimum. By doing so the error in evaluating the integral (5) would be minimized.

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α-centered perturbation

Therefore, the optimal point can be obtained as ∂h(x) ∂xk = 0

r

xk = 1 λj(x) ∂λj(x) ∂xk , ∀k

(7)

Combining for all k we have dλj(α) = λj(α)α. Rearranging α = dλj(α)/λj(α)

(8)

This equation immediately gives a recipe for an iterative algorithm to obtain α.

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α-centered perturbation

Substituting dλj(α) in Eq. (4) λj(x) ≈ λj(α)

1 − |α|2

+ 1 2αT Dλj(α)α + αT λj(α)I − Dλj(α)

x + 1

2 xT Dλj(α)x (9)

This, like the mean-centered approach, also re- sults in a quadratic form in the random variable x.

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Eigenvalue statistics

Both approximations yield a quadratic form in Gaussian random variable of the form λj(x) ≈ cj + aT

j x + 1

2 xT Ajx (10)

The moment generating function: Mλj(s) = E

esλj(x)

≈ escj+ s2

2 aT j [I−sAj] −1aj

I − sAj

(11)

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Eigenvalue statistics

Cumulants: κr =

cj + 1

2Trace (Aj)

if r = 1,

r! 2 aT j Ar−2 j

aj + (r−1)!

2

Trace

Ar

j

if

r ≥ 2

(12)

Thus ¯ λj = κ1 = cj + 1 2Trace (Aj)

(13)

Var [λj] = κ2 = aT

j aj + 1

2Trace

A2

j

(14)

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Asymptotic analysis

We want to evaluate an integral of the following form: J =

R

m f(x)p(x) dx = (2π)−m/2

R

m e

h(x) dx

(15)

where

h(x) = ln f(x) − xT x/2

(16)

Assume f(x) : Rm → R is smooth and at least twice differentiable and h(x) reaches its global maximum at an unique point θ ∈ Rm.

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Asymptotic analysis

Therefore, at x = θ ∂ h(x) ∂xk = 0 or xk = ∂ ∂xk ln f(x), ∀k, or θ = ∂ ∂x ln f(θ).

(17)

Further assume that h(θ) is so large that

1
h(θ)

Dj( h(θ))

→ 0

for j > 2

(18)

Dj( h(θ)): jth order derivative of h(x) at x = θ.

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Asymptotic analysis

Under previous assumptions, using second-order Taylor series of h(x) the integral (12) can be evaluated asymptotically as J ≈ e

h(θ)

H(θ)

= f(θ)e

− θ

Tθ/2

H(θ)−1/2

(19)

H(θ) is the Hessian matrix of

h(x) at x = θ.

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Asymptotic analysis

An arbitrary rth order moment of the eigenvalues µ′

r =

R

m λr

j(x)p(x) dx,

r = 1, 2, 3 · · ·

(20)

Comparing this with Eq. (12) it is clear that f(x) = λr

j(x)

and

h(x) = r ln λj(x)− xT x/2 (21)

The optimal point θ can be obtained from (14) as θ = r dλj(θ)/λj(θ)

(22)

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Asymptotic analysis

The rth moment: µ′

r = λr j(θ)e− |θ|2

2

I + 1

rθθT − r λj(θ)

Dλj(θ)

−1/2

(23)

The mean of the eigenvalues (substitute r = 1): ¯ λj = λj(θ)e− |θ|2

2

I + θθT − Dλj(θ)/λj(θ)

−1/2

(24)

Central moments: E

(λj − ¯

λj)r = r

k=0

r

k

(−1)r−kµ′

k¯

λr−k

j

.

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Pdf of the eigenvalues

Theorem 1 λj(x) is distributed as a non-central χ2 random variable with noncentrality parameter δ2 and degrees-of-freedom m′ if and only if (a)

A2

j = Aj, (b) Trace (Aj) = m′ and (c)

aj = Ajaj, δ2 = cj = aT

j aj/4.

This implies that the the Hessian matrix Aj should be an idempotent matrix. In general this require- ment is not expected to be satisfied for eigenval- ues of real structural systems.

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Central χ2 approximation (Pearson’s)

Pdf of the jth eigenvalue pλj(u) ≈ 1

γpχ2

ν

u − η

γ
= (u −

η)ν/2−1e−(u−

η)/2 γ

(2 γ)ν/2Γ(ν/2)

(25)

where

η = −2κ22 + κ1κ3

κ3 , γ = κ3 4κ2 , and ν = 8κ23 κ32

(26)

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Non-central χ2 approximation

Pdf of the jth eigenvalue pλj(u) ≈ 1 γj pQj u − ηj γj

(27)

where pQj(u) =

e−(δj+u/2)um/2−1 2m/2

∞

r=0 (δu)r r! 2rΓ(m/2+r),

ηj = cj − 1

2aT j A−1 j aj, γj = Trace(Aj) 2m

, δ2

j = ρT j ρj and

ρj = A−1

j aj.

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Numerical example

Undamped two degree-of-system system: m1 = 1 Kg, m2 = 1.5 Kg, ¯ k1 = 1000 N/m, ¯ k2 = 1100 N/m and k3 = 100 N/m.

m

1

m

2 1 2

k

1

k

2

k

3

Only the stiffness parameters k1 and k2 are uncer- tain: ki = ¯ ki(1 + ǫixi), i = 1, 2. x = {x1, x2}T ∈ R2 and the ‘strength parameters’ ǫ1 = ǫ2 = 0.25.

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Numerical example

Following six methods are compared

1. Mean-centered first-order perturbation
2. Mean-centered second-order perturbation
3. α-centered first-order perturbation
4. α-centered second-order perturbation
5. Asymptotic method
6. Monte Carlo Simulation (10K samples) - can

be considered as benchmark.

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Numerical example

The percentage error: Errorith method = {µ′

k}ith method − {µ′ k}MCS

{µ′

k}MCS

× 100 i = 1, · · · 5.

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Numerical example

1 2 3 4 2 4 6 8 10 12 14 16 18 20 k−th order moment: E [λk

1]

Percentage error wrt MCS Mean−centered 1st−order Mean−centered 2nd−order α−centered 1st−order α−centered 2nd−order Asymptotic Method

Percentage error for the first four raw moments of the first eigenvalue

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Numerical example

1 2 3 4 −20 −18 −16 −14 −12 −10 −8 −6 −4 −2 k−th order moment: E [λk

2]

Percentage error wrt MCS Mean−centered 1st−order Mean−centered 2nd−order α−centered 1st−order α−centered 2nd−order Asymptotic Method

Percentage error for the first four raw moments of the second eigenvalue

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Numerical example

500 1000 1500 0.5 1 1.5 2 2.5 3 x 10−3 u pλ

1

(u) Mean−centered 1st−order Mean−centered 2nd−order α−centered 1st−order α−centered 2nd−order Asymptotic Method

Probability density function of the first eigenvalue

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Numerical example

400 600 800 1000 1200 1400 1600 1800 2000 2200 2400 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 x 10−3 u pλ

2

(u) Mean−centered 1st−order Mean−centered 2nd−order α−centered 1st−order α−centered 2nd−order Asymptotic Method

Probability density function of the second eigenvalue

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Conclusions

Two methods, namely (a) optimal point

expansion method, and (b) asymptotic moment method, are proposed.

The optimal point is obtained so that the

mean of the eigenvalues are estimated most accurately.

The asymptotic method assumes that the

eigenvalues are large compared to their 3rd

rder or higher derivatives.
Pdf of the eigenvalues are obtained in terms
f central and non-central χ2 densities.

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Open problems

Joint statistics (moments/pdf/cumulants) of

the eigenvalues with non-Gaussian system parameters.

Statistics of the difference and ratio of the

eigenvalues.

Statistics of a single eigenvector (for

GUE/GOE and general matrices).

Joint statistics of the eigenvectors.
Joint statistics of the eigenvalues and

eigenvectors.

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