Minimization of the energy of a vibrational system . . . . . - - PowerPoint PPT Presentation

Motivation and introduction Optimality criteria Results Related results

. . . . . . .

Minimization of the energy of a vibrational system

Ivica Nakić

Department of Mathematics Faculty of Natural Sciences and Mathematics University of Zagreb

Second Najman Conference on Spectral Problems for Operators and Matrices, 2009

Ivica Nakić Minimization of a vibrational system

Motivation and introduction Optimality criteria Results Related results

. Outline

. . .

1

Motivation and introduction . . .

2

Optimality criteria . . .

3

Results . . .

4

Related results

Ivica Nakić Minimization of a vibrational system

Motivation and introduction Optimality criteria Results Related results

. A class of vibrational systems

We consider a damped vibrational system described by the differential equation M ¨ x + C ˙ x + Kx = 0, x(0) = x0, ˙ x(0) = ˙ x0, where M, C, and K (called mass, damping and stiffness matrix, respectively) are real, symmetric matrices of order n with M, K positive definite, and C positive semi–definite matrices. Our aim is to optimize this vibrational system in the sense of finding an optimal damping matrix C such that the energy of the system is minimal.

Ivica Nakić Minimization of a vibrational system

Motivation and introduction Optimality criteria Results Related results

. Linearization

The choice of linearization is such that the underlying phase space has energy norm. ˙ y = Ay y(0) = y0 with the solution y(t) = eAty0. Here y1 = L∗

1x, y2 = L∗ 2 ˙

x, K = L1L∗

1, M = L2L∗ 2, y = ( y1 y2 ),

y0 = ( L∗

1x0

L∗

2 ˙

x0 ) , and A = ( L1L−∗

2

−L−1

2 L1

−L2−1CL−∗

2

) .

Ivica Nakić Minimization of a vibrational system

Motivation and introduction Optimality criteria Results Related results

. Linearization

A = ( L1L−∗

2

−L−1

2 L1

−L2−1CL−∗

2

) . The underlying norm is energy-norm: ∥y(t)∥2 = ˙ x(t)∗M ˙ x(t) + x(t)∗Kx(t) = 2E(t). This linearization is closest to normality among all linearizations of the form y1 = W1x, y2 = W2 ˙ x, W1, W2 nonsingular (in the sense that ∥A∥2

F − ∑ |λi(A)|2 is minimal).

Ivica Nakić Minimization of a vibrational system

Motivation and introduction Optimality criteria Results Related results

. Linearization

A = ( L1L−∗

2

−L−1

2 L1

−L2−1CL−∗

2

) . The underlying norm is energy-norm: ∥y(t)∥2 = ˙ x(t)∗M ˙ x(t) + x(t)∗Kx(t) = 2E(t). This linearization is closest to normality among all linearizations of the form y1 = W1x, y2 = W2 ˙ x, W1, W2 nonsingular (in the sense that ∥A∥2

F − ∑ |λi(A)|2 is minimal).

Ivica Nakić Minimization of a vibrational system

Motivation and introduction Optimality criteria Results Related results

. Convenient form

Let L−1

2 L1 = U2ΩU∗ 1 be SVD–decomposition of the matrix L−1 2 L1,

with Ω = diag(ω1, . . . , ωn) > 0. We can assume ω1 ≤ ω2 ≤ · · · ≤ ωn. Set U = (

U1 0 0 U2

) . Then

• A = U∗AU =

[ 0 Ω −Ω − ˆ C ] , where ˆ C = U∗

2 L−1 2 CL−∗ 2 U2 is positive semi–definite.

If we denote F = L−∗

2 U2, then F ∗MF = I, F ∗KF = Ω. Thus we

have obtained a particularly convenient, the so–called modal representation of the problem. In the following we will work in the basis in which matrix A has this form.

Ivica Nakić Minimization of a vibrational system

Motivation and introduction Optimality criteria Results Related results

. The choice of optimality criterion

A number of optimality criteria in use: spectral abscissa criterion ˆ C → s(A) := max

k

ℜλk → min minimization of the number of oscillations ˆ C → max

k

ℜλk |λk| → min minimization of the total energy of the system ˆ C → ∫ ∞ E(t)dt → min We will concentrate on the last criterion.

Ivica Nakić Minimization of a vibrational system

Motivation and introduction Optimality criteria Results Related results

. The choice of optimality criterion

A number of optimality criteria in use: spectral abscissa criterion ˆ C → s(A) := max

k

ℜλk → min minimization of the number of oscillations ˆ C → max

k

ℜλk |λk| → min minimization of the total energy of the system ˆ C → ∫ ∞ E(t)dt → min We will concentrate on the last criterion.

Ivica Nakić Minimization of a vibrational system

Motivation and introduction Optimality criteria Results Related results

. The choice of optimality criterion

A number of optimality criteria in use: spectral abscissa criterion ˆ C → s(A) := max

k

ℜλk → min minimization of the number of oscillations ˆ C → max

k

ℜλk |λk| → min minimization of the total energy of the system ˆ C → ∫ ∞ E(t)dt → min We will concentrate on the last criterion.

Ivica Nakić Minimization of a vibrational system

Motivation and introduction Optimality criteria Results Related results

. The choice of optimality criterion

A number of optimality criteria in use: spectral abscissa criterion ˆ C → s(A) := max

k

ℜλk → min minimization of the number of oscillations ˆ C → max

k

ℜλk |λk| → min minimization of the total energy of the system ˆ C → ∫ ∞ E(t)dt → min We will concentrate on the last criterion.

Ivica Nakić Minimization of a vibrational system

Motivation and introduction Optimality criteria Results Related results

. The choice of optimality criterion

A number of optimality criteria in use: spectral abscissa criterion ˆ C → s(A) := max

k

ℜλk → min minimization of the number of oscillations ˆ C → max

k

ℜλk |λk| → min minimization of the total energy of the system ˆ C → ∫ ∞ E(t)dt → min We will concentrate on the last criterion.

Ivica Nakić Minimization of a vibrational system

Motivation and introduction Optimality criteria Results Related results

. Minimal energy criterion

We want to get rid of the dependence on the initial state. (At least) three viable approaches: take maximum over all initial states max

∥y0∥=1

∫ ∞ E(t; y0)dt take average over all initial states ∫

∥y0∥=1

∫ ∞ E(t; y0)dt dσ take ”smoothed” maximum over all initial states max

∥y0∥=1

∫

{z0:∥z0∥=1,∥z0−y0∥≤δ}

∫ ∞ E(t; z0)dt dσ

Ivica Nakić Minimization of a vibrational system

Motivation and introduction Optimality criteria Results Related results

. Minimal energy criterion

We want to get rid of the dependence on the initial state. (At least) three viable approaches: take maximum over all initial states max

∥y0∥=1

∫ ∞ E(t; y0)dt take average over all initial states ∫

∥y0∥=1

∫ ∞ E(t; y0)dt dσ take ”smoothed” maximum over all initial states max

∥y0∥=1

∫

{z0:∥z0∥=1,∥z0−y0∥≤δ}

∫ ∞ E(t; z0)dt dσ

Ivica Nakić Minimization of a vibrational system

Motivation and introduction Optimality criteria Results Related results

. Minimal energy criterion

We want to get rid of the dependence on the initial state. (At least) three viable approaches: take maximum over all initial states max

∥y0∥=1

∫ ∞ E(t; y0)dt take average over all initial states ∫

∥y0∥=1

∫ ∞ E(t; y0)dt dσ take ”smoothed” maximum over all initial states max

∥y0∥=1

∫

{z0:∥z0∥=1,∥z0−y0∥≤δ}

∫ ∞ E(t; z0)dt dσ

Ivica Nakić Minimization of a vibrational system

Motivation and introduction Optimality criteria Results Related results

. Lyapunov equation

It’s easy to see that ∫ ∞ E(t; y0)dt = 1 2y∗

0Xy0,

where X is the solution of the Lyapunov equation A∗X + XA = −I. Hence maximum over all initial states ֌ 1

2∥X∥

average over all initial states ֌ trace(XZ) ”smoothed” maximum over all initial states ֌ α∥X∥ + β trace(XZ)

Ivica Nakić Minimization of a vibrational system

Motivation and introduction Optimality criteria Results Related results

. Lyapunov equation

It’s easy to see that ∫ ∞ E(t; y0)dt = 1 2y∗

0Xy0,

where X is the solution of the Lyapunov equation A∗X + XA = −I. Hence maximum over all initial states ֌ 1

2∥X∥

average over all initial states ֌ trace(XZ) ”smoothed” maximum over all initial states ֌ α∥X∥ + β trace(XZ)

Ivica Nakić Minimization of a vibrational system

Motivation and introduction Optimality criteria Results Related results

. Lyapunov equation

It’s easy to see that ∫ ∞ E(t; y0)dt = 1 2y∗

0Xy0,

where X is the solution of the Lyapunov equation A∗X + XA = −I. Hence maximum over all initial states ֌ 1

2∥X∥

average over all initial states ֌ trace(XZ) ”smoothed” maximum over all initial states ֌ α∥X∥ + β trace(XZ)

Ivica Nakić Minimization of a vibrational system

Motivation and introduction Optimality criteria Results Related results

. Lyapunov equation

It’s easy to see that ∫ ∞ E(t; y0)dt = 1 2y∗

0Xy0,

where X is the solution of the Lyapunov equation A∗X + XA = −I. Hence maximum over all initial states ֌ 1

2∥X∥

average over all initial states ֌ trace(XZ) ”smoothed” maximum over all initial states ֌ α∥X∥ + β trace(XZ)

Ivica Nakić Minimization of a vibrational system

Motivation and introduction Optimality criteria Results Related results

. Average case

The matrix Z is explicitely computable ... in some cases. In the Lebesgue measure case Z =

1 2nI.

What is the optimal matrix ˆ C among all positive semi–definite matrices for which the corresponding system is stable? . Theorem (Lebesgue measure case) . . . . . . . . The optimal ˆ C is 2Ω. In terms of the original matrices: C = 2M1/2(M −1/2KM −1/2)1/2M1/2 .

Ivica Nakić Minimization of a vibrational system

Motivation and introduction Optimality criteria Results Related results

. Average case

The matrix Z is explicitely computable ... in some cases. In the Lebesgue measure case Z =

1 2nI.

What is the optimal matrix ˆ C among all positive semi–definite matrices for which the corresponding system is stable? . Theorem (Lebesgue measure case) . . . . . . . . The optimal ˆ C is 2Ω. In terms of the original matrices: C = 2M1/2(M −1/2KM −1/2)1/2M1/2 .

Ivica Nakić Minimization of a vibrational system

Motivation and introduction Optimality criteria Results Related results

. Average case

The matrix Z is explicitely computable ... in some cases. In the Lebesgue measure case Z =

1 2nI.

What is the optimal matrix ˆ C among all positive semi–definite matrices for which the corresponding system is stable? . Theorem (Lebesgue measure case) . . . . . . . . The optimal ˆ C is 2Ω. In terms of the original matrices: C = 2M1/2(M −1/2KM −1/2)1/2M1/2 .

Ivica Nakić Minimization of a vibrational system

Motivation and introduction Optimality criteria Results Related results

. Average case

The matrix Z is explicitely computable ... in some cases. In the Lebesgue measure case Z =

1 2nI.

What is the optimal matrix ˆ C among all positive semi–definite matrices for which the corresponding system is stable? . Theorem (Lebesgue measure case) . . . . . . . . The optimal ˆ C is 2Ω. In terms of the original matrices: C = 2M1/2(M −1/2KM −1/2)1/2M1/2 .

Ivica Nakić Minimization of a vibrational system

Motivation and introduction Optimality criteria Results Related results

. Average case

The matrix Z is explicitely computable ... in some cases. In the Lebesgue measure case Z =

1 2nI.

What is the optimal matrix ˆ C among all positive semi–definite matrices for which the corresponding system is stable? . Theorem (Lebesgue measure case) . . . . . . . . The optimal ˆ C is 2Ω. In terms of the original matrices: C = 2M1/2(M −1/2KM −1/2)1/2M1/2 .

Ivica Nakić Minimization of a vibrational system

Motivation and introduction Optimality criteria Results Related results

. Outline of the proof

Define f(C) = trace(X(C)). Then f′(C) = 0 if and only if B∗X(C)Y (C)B = 0, where B = [ 0 I ]∗, and Y (C) is the solution of the dual Lyapunov equation AY + Y A∗ = −I. X solution of Lyapunov equation ⇔ Y = JXJ is the solution of the dual Lyapunov equation, where J = [ I

0 −I

] (A is J–symmetric). From these ingredients one obtains that 2Ω is the unique stationary point of the function f. And since f goes to infinity when A(C) approaches the boundary of the set of stable matrices, we have the result.

Ivica Nakić Minimization of a vibrational system

Motivation and introduction Optimality criteria Results Related results

. Outline of the proof

Define f(C) = trace(X(C)). Then f′(C) = 0 if and only if B∗X(C)Y (C)B = 0, where B = [ 0 I ]∗, and Y (C) is the solution of the dual Lyapunov equation AY + Y A∗ = −I. X solution of Lyapunov equation ⇔ Y = JXJ is the solution of the dual Lyapunov equation, where J = [ I

0 −I

] (A is J–symmetric). From these ingredients one obtains that 2Ω is the unique stationary point of the function f. And since f goes to infinity when A(C) approaches the boundary of the set of stable matrices, we have the result.

Ivica Nakić Minimization of a vibrational system

Motivation and introduction Optimality criteria Results Related results

. Outline of the proof

Define f(C) = trace(X(C)). Then f′(C) = 0 if and only if B∗X(C)Y (C)B = 0, where B = [ 0 I ]∗, and Y (C) is the solution of the dual Lyapunov equation AY + Y A∗ = −I. X solution of Lyapunov equation ⇔ Y = JXJ is the solution of the dual Lyapunov equation, where J = [ I

0 −I

] (A is J–symmetric). From these ingredients one obtains that 2Ω is the unique stationary point of the function f. And since f goes to infinity when A(C) approaches the boundary of the set of stable matrices, we have the result.

Ivica Nakić Minimization of a vibrational system

Motivation and introduction Optimality criteria Results Related results

. Outline of the proof

Define f(C) = trace(X(C)). Then f′(C) = 0 if and only if B∗X(C)Y (C)B = 0, where B = [ 0 I ]∗, and Y (C) is the solution of the dual Lyapunov equation AY + Y A∗ = −I. X solution of Lyapunov equation ⇔ Y = JXJ is the solution of the dual Lyapunov equation, where J = [ I

0 −I

] (A is J–symmetric). From these ingredients one obtains that 2Ω is the unique stationary point of the function f. And since f goes to infinity when A(C) approaches the boundary of the set of stable matrices, we have the result.

Ivica Nakić Minimization of a vibrational system

Motivation and introduction Optimality criteria Results Related results

. General case

We assume that the measure σ is Gaussian measure with zero mean and covariance matrix ˆ Σ = diag(Σ, Σ) where F ∗ΣF is diagonal (F diagonalizes M and K). We assume that we work in the basis generated by F. Then the corresponding Z has the form Z = diag( ˜ Z, ˜ Z) where ˜ Z = diag(α1, . . . , αn), αi ≥ 0. Further assume α1, . . . , αs > 0, as+1 = · · · = αn = 0. Then . Theorem . . . . . . . . The set of optimal matrices ˆ C is Cmin = {[2Ωs H ] : H ≥ 0 } , where Ωs = diag(ω1, . . . , ωs).

Ivica Nakić Minimization of a vibrational system

Motivation and introduction Optimality criteria Results Related results

. General case

We assume that the measure σ is Gaussian measure with zero mean and covariance matrix ˆ Σ = diag(Σ, Σ) where F ∗ΣF is diagonal (F diagonalizes M and K). We assume that we work in the basis generated by F. Then the corresponding Z has the form Z = diag( ˜ Z, ˜ Z) where ˜ Z = diag(α1, . . . , αn), αi ≥ 0. Further assume α1, . . . , αs > 0, as+1 = · · · = αn = 0. Then . Theorem . . . . . . . . The set of optimal matrices ˆ C is Cmin = {[2Ωs H ] : H ≥ 0 } , where Ωs = diag(ω1, . . . , ωs).

Ivica Nakić Minimization of a vibrational system

Motivation and introduction Optimality criteria Results Related results

. Outline of proof

Again we use use dual Lyapunov equation, this time AY + Y A∗ = −Z. (♣) Then trace(XZ) = trace(Y ) (we use J–symmetry of A and the fact that Z and J commute). Let Zi = diag( ˜ Zi, ˜ Zi), where ˜ Zi is diagonal matrix with all entries zero except the i–th which is αi. Let Yi be the solution of the Lyapunov equation AY + Y A∗ = −Zi. (♢) Then the solution of (♣) is Y = Y1 + · · · Ys. Then we show min{trace(Y ) : Y solves (♢) , A(C) stable } ≥ 2αi ωi . Final step: if ˆ C is such that trace(X(C)Z) = 2 ∑s

i=1 αi ωi , then

C ∈ Cmin.

Ivica Nakić Minimization of a vibrational system

Motivation and introduction Optimality criteria Results Related results

. Outline of proof

Again we use use dual Lyapunov equation, this time AY + Y A∗ = −Z. (♣) Then trace(XZ) = trace(Y ) (we use J–symmetry of A and the fact that Z and J commute). Let Zi = diag( ˜ Zi, ˜ Zi), where ˜ Zi is diagonal matrix with all entries zero except the i–th which is αi. Let Yi be the solution of the Lyapunov equation AY + Y A∗ = −Zi. (♢) Then the solution of (♣) is Y = Y1 + · · · Ys. Then we show min{trace(Y ) : Y solves (♢) , A(C) stable } ≥ 2αi ωi . Final step: if ˆ C is such that trace(X(C)Z) = 2 ∑s

i=1 αi ωi , then

C ∈ Cmin.

Ivica Nakić Minimization of a vibrational system

Motivation and introduction Optimality criteria Results Related results

. Outline of proof

Again we use use dual Lyapunov equation, this time AY + Y A∗ = −Z. (♣) Then trace(XZ) = trace(Y ) (we use J–symmetry of A and the fact that Z and J commute). Let Zi = diag( ˜ Zi, ˜ Zi), where ˜ Zi is diagonal matrix with all entries zero except the i–th which is αi. Let Yi be the solution of the Lyapunov equation AY + Y A∗ = −Zi. (♢) Then the solution of (♣) is Y = Y1 + · · · Ys. Then we show min{trace(Y ) : Y solves (♢) , A(C) stable } ≥ 2αi ωi . Final step: if ˆ C is such that trace(X(C)Z) = 2 ∑s

i=1 αi ωi , then

C ∈ Cmin.

Ivica Nakić Minimization of a vibrational system

Motivation and introduction Optimality criteria Results Related results

. Outline of proof

Again we use use dual Lyapunov equation, this time AY + Y A∗ = −Z. (♣) Then trace(XZ) = trace(Y ) (we use J–symmetry of A and the fact that Z and J commute). Let Zi = diag( ˜ Zi, ˜ Zi), where ˜ Zi is diagonal matrix with all entries zero except the i–th which is αi. Let Yi be the solution of the Lyapunov equation AY + Y A∗ = −Zi. (♢) Then the solution of (♣) is Y = Y1 + · · · Ys. Then we show min{trace(Y ) : Y solves (♢) , A(C) stable } ≥ 2αi ωi . Final step: if ˆ C is such that trace(X(C)Z) = 2 ∑s

i=1 αi ωi , then

C ∈ Cmin.

Ivica Nakić Minimization of a vibrational system

Motivation and introduction Optimality criteria Results Related results

. Related results

Spectral abscissa case

. Theorem (Freitas, Lancaster) . . . . . . . . On the set of symmetric matrices, s(A(C)) attains its minimum, which is equal to ( det K det M )2n . The minimizer is unique, if and only if the matrices K and M are proportional.

Ivica Nakić Minimization of a vibrational system

Motivation and introduction Optimality criteria Results Related results

. Related results

Maximal energy case: . Theorem (Cox) . . . . . . . . Among all friction damping matrices C = 2αM, α > 0 the optimal parameter is α = ω1 √√

5−1 2

. Infinite–dimensional average energy case: . . . . . . . The set of optimal operators has the same form, but we need additional assumption C ≥ δM (internal damping).

Ivica Nakić Minimization of a vibrational system

Motivation and introduction Optimality criteria Results Related results

. Related results

Maximal energy case: . Theorem (Cox) . . . . . . . . Among all friction damping matrices C = 2αM, α > 0 the optimal parameter is α = ω1 √√

5−1 2

. Infinite–dimensional average energy case: . . . . . . . The set of optimal operators has the same form, but we need additional assumption C ≥ δM (internal damping).

Ivica Nakić Minimization of a vibrational system

Motivation and introduction Optimality criteria Results Related results

. Related results

Maximal energy case: . Theorem (Cox) . . . . . . . . Among all friction damping matrices C = 2αM, α > 0 the optimal parameter is α = ω1 √√

5−1 2

. Infinite–dimensional average energy case: . . . . . . . The set of optimal operators has the same form, but we need additional assumption C ≥ δM (internal damping).

Ivica Nakić Minimization of a vibrational system

Motivation and introduction Optimality criteria Results Related results

. References

• S. J. Cox

Designing for optimal energy absortion, I: Lumped parameter systems. ASME J. Vibration and Acoustics. 120 (2) (1998), 339–-345.

• S. J. Cox, I.N., A. Rittmann, K. Veselić

Lyapunov optimization of a damped system. Systems Control Lett.. 53 (3-4) (2004), 187–-194.

• P. Freitas, P. Lancaster

Optimal decay of energy for a system of linear oscillators. SIAM J. Matrix Anal. Appl. 21 (1999), no. 1, 195–-208.

• I. N.

Minimization of the trace of the solution of Lyapunov equation connected with damped vibrational systems. Systems Control Lett.. to appear.

Ivica Nakić Minimization of a vibrational system

Motivation and introduction Optimality criteria Results Related results

. References

• S. J. Cox

Designing for optimal energy absortion, I: Lumped parameter systems. ASME J. Vibration and Acoustics. 120 (2) (1998), 339–-345.

• S. J. Cox, I.N., A. Rittmann, K. Veselić

Lyapunov optimization of a damped system. Systems Control Lett.. 53 (3-4) (2004), 187–-194.

• P. Freitas, P. Lancaster

Optimal decay of energy for a system of linear oscillators. SIAM J. Matrix Anal. Appl. 21 (1999), no. 1, 195–-208.

• I. N.

Minimization of the trace of the solution of Lyapunov equation connected with damped vibrational systems. Systems Control Lett.. to appear.

Ivica Nakić Minimization of a vibrational system

Motivation and introduction Optimality criteria Results Related results

. References

• S. J. Cox

Designing for optimal energy absortion, I: Lumped parameter systems. ASME J. Vibration and Acoustics. 120 (2) (1998), 339–-345.

• S. J. Cox, I.N., A. Rittmann, K. Veselić

Lyapunov optimization of a damped system. Systems Control Lett.. 53 (3-4) (2004), 187–-194.

• P. Freitas, P. Lancaster

Optimal decay of energy for a system of linear oscillators. SIAM J. Matrix Anal. Appl. 21 (1999), no. 1, 195–-208.

• I. N.

Minimization of the trace of the solution of Lyapunov equation connected with damped vibrational systems. Systems Control Lett.. to appear.

Ivica Nakić Minimization of a vibrational system

Motivation and introduction Optimality criteria Results Related results

. References

• S. J. Cox

Designing for optimal energy absortion, I: Lumped parameter systems. ASME J. Vibration and Acoustics. 120 (2) (1998), 339–-345.

• S. J. Cox, I.N., A. Rittmann, K. Veselić

Lyapunov optimization of a damped system. Systems Control Lett.. 53 (3-4) (2004), 187–-194.

• P. Freitas, P. Lancaster

Optimal decay of energy for a system of linear oscillators. SIAM J. Matrix Anal. Appl. 21 (1999), no. 1, 195–-208.

• I. N.

Minimization of the trace of the solution of Lyapunov equation connected with damped vibrational systems. Systems Control Lett.. to appear.

Ivica Nakić Minimization of a vibrational system