On some properties of the Lyapunov equation for damped systems

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1 On some properties of the Lyapuno equation for damped systems Ninosla Truhar, Uniersity of Osijek, Department of Mathematics, 31 Osijek, Croatia 1 ntruhar@mathos.hr Krešimir Veselić Lehrgebiet Mathematische Physik, Fernuniersität, 5884 Hagen, Germany kresimir.eselic@fernuni-hagen.de Abstract We consider a damped linear ibrational system whose dampers depend linearly on the iscosity parameter. We show that the trace of the corresponding Lyapuno solution can be represented as a rational function of whose poles are the eigenalues of a certain skew symmetric matrix. This makes it possible to derie an asymptotic expansion of the solution in the neighborhood of zero (small damping). 1 Introduction We consider a damped linear ibrational system Mẍ + Cẋ + Kx = (1.1) the matrices M, C, K (mass, damping, stiffness) are symmetric, M, K are positie definite and C is positie semidefinite. If internal damping is neglected C has often small rank as it describes a few dampers built in to calm down dangerous ibrations. Often C has the form C = C is a ariable iscosity and C describes the geometry of a damper. C will hae rank 1,2 or 3 according to whether the damper can exhibit linear, planar or spatial displacements. 1 A part of this work was written while the author was isiting researcher on the Lehrgebiet Mathematische Physik, Fernuniersität, Hagen 1

2 An example is the so-called n-mass oscillator or oscillator ladder (Fig. 1) M = diag(m 1,m 2,...,m n ) k 1 k 1 k 1 k 1 + k 2 k 2 K = , k n 2 k n 2 + k n 1 k n 1 k n 1 k n 1 + k n C = e 1 e T 1 + (e 3 e 2 )(e 3 e 2 ) T. (1.2) Figure 1: The n-mass oscillator with two dampers Here m i > are the masses, k i > the spring constants or stiffnesses, e i is the i-th canonical basis ector, and is the iscosity of the damper applied on the i-th mass. After the substitution y 1 = ΩΦ 1 x, y 2 = Φ 1 ẋ, (1.3) KΦ = MΦΩ 2, Φ T MΦ = I (1.4) Ω = diag(ω 1,...,ω n ), ω 1 <... < ω n (1.5) is the eigenreduction of the symmetric positie definite matrix pair K,M, the system (1.1) goes oer into y1 ẏ = Ay, y =, (1.6) Then E = 1 2 y 2 Ω A =, D = Φ T CΦ. (1.7) Ω D (ẋt Mẋ + x T Kx ) dt = 2 y 2 dt = e A t y 2 dt, (1.8)

3 y is the initial data. Thus, soles the Lyapuno equation E E(y ) = y T Xy, X = e AT t e At dt (1.9) A T X + XA = I. (1.1) Our penalty function is obtained by aeraging E oer all initial data with the equal energy, that is, we form Ē = y T Xy dσ y =1 dσ is a probability measure on the unit sphere in R 2n. Since by the map X y T Xy dσ y =1 is gien a linear functional on the space of the symmetric matrices, by Riesz theorem there exists a symmetric matrix Z such that X y T Xy dσ = Tr(ZX), for all symmetric matrices X. y =1 Let y R 2n be arbitrary. Set X = yy T. Then y T Xy dσ = Tr(ZX) = Tr(Zyy T ) = Tr(y T Zy), y =1 hence Z is always positie semi-definite. For any gien measure there is a unique positie semidefinite matrix Z such that Ē = Tr(ZX). (1.11) For the measure σ generated by the Lebesgue measure (i.e. the usual surface measure) on R 2n, we obtain Z = 1 2nI. For the conenience of the reader, we gie a sketch of the proof: Recall, Z ij = y i y j σ(dy). S One can easily see using Minkowski formula (see [2]) that Z ij = y i y j σ(dy) = 1 2ε lim y i y j σ(dy), ε S d(y,s) ε 3

4 here S denotes the unit sphere in R 2n and d(y,s) is a corresponding distance. Obiously, Z ij = for i j and Z ii = Z jj, for i,j {1,2,...,2n}. Since 1 Z 11 + Z Z 2n2n = lim ε 2ε ol ( y R 2n : d(y,s) ε ) = 1, it follows Z = 1 2nI. The more details about the structure of the matrix Z one can find in [4]. We hae shown that y T Xy dσ = min is equialent to y =1 Tr(ZX) = min. (1.12) Z is a symmetric positie semidefinite matrix which may be normalized to hae a unit trace. If one is interested in damping a certain part of the spectrum of the matrix A (which is ery important in applications) then the matrix Z will hae a special structure. For example, let σ = σ 1 σ 2 σ 1 σ 2, σ 1 is a measure on the frequency subspace determined by ω ω max ω s generated by Lebesgue measure, that is σ 1 is a measure on the frequency subspace which corresponds to the eigenfrequencies (defined by (1.5)) ω 1,...ω s and σ 2 is Dirac measure on the complement. Then we obtain that the corresponding matrix Z has the form I s Z s = Z = 1 2s I s, (1.13) I s is the identity matrix of the dimension s which is defined by ω max = ω s. Here ω max = ω s is critical frequency with the property that the eigenfrequencies from (1.5) greater then ω s are not dangerous. Hence, we damp first s eigenfrequencies. The construction of Z from (1.13) is a generalization of the simplest case Z = 1 2n I. In [7] a simple solution of the problem (1.11) has been presented for A from (1.7) and rank(c) = 1. In particular, Tr(ZX()) = const + a + b, a,b >, (1.14) which made it possible to find the minimum by a simple formula explicitly. The case rank(c) > 1 seems to be essentially more difficult to handle. The main result of this paper is the explicit formula: X() = Ψ 1 Ψ Ψ 1 + s i=1 λ i (λ i Φ i Υ i ) λ 2 i + 2. (1.15) 4

5 Ψ 1, Ψ, Ψ 1, Φ i and Υ i are m m matrices and λ i are eigenalues of the pencil (A, D) A and D are matrices which correspond to the linear operators X A X + XA, X DX + XD, respectiely. Thus, technically, we hae turned the iscosity into the spectral parameter. Further, we hae obtained a simple formula for the Ψ 1 in (1.15). This matrix is responsible for the behaior of the solution X() in the neighborhood of zero (small damping): D Ψ 1 =, D D = (diag (D)) 1. We will use the following notation: matrices written in the simple Roman fonts, M, D or K for example will hae n 2 entries. Matrices written in the mathematical bold fonts, A, B will hae m 2 entries, m = 2n (that is A, B are matrices defined on the 2n-dimensional phase space). Finally, matrices written in the Blackboard bold fonts A, or D will hae more than m 2 entries. 2 The main result As we hae said in the Introduction, our aim is to obtain the solution X of the Lyapuno equation A() T X + XA() = I, (2.1) Z is defined by (1.13) and I is m m identity matrix. From (1.7) it follows that A() can be written as Ω 1 Ω 2 A() A D, A =, Ω i =... Ωn [ ] ωi ω i (2.2) and D = D D T,... d 11 d d 1r... D = d 21 d d 2r d n1 d n2... d nr, 5

6 d ij are entries of the matrix D = P, C = L C L T C, L C and P is the perfect shuffling permutation. Now, we proceed with soling equation (2.1). As it is well known, Lyapuno equation (2.1) is equialent to ([3, Theorem ]) ( I (A D) T + (A D) T I ) ec(x) = ec(i), (2.3) L T denotes the Kronecker product of L and T, and ec(i) is the ector formed by stacking the columns of I into one long ector. Further, we will need the following two m 2 m 2 matrices defined by A = I A T + A T I, D = I D D T + D D T I. (2.4) It is easy to show that D = D F D T F, D F = [ I D D I ]. (2.5) Now, using (2.5) and (2.4) it follows that solution ec(x) of equation (2.3) can be written as ec(x) = ( A D F D T F) 1 ec(i). (2.6) Obiously, there exists a unitary matrix U such that [ U T A U =, (2.7) Â] A is the skew-symmetric matrix corresponds with linear operator defined in (2.4) and  is a non-singular block diagonal matrix defined by µi  = diag(ξ 1,...,Ξ m2 ) Ξ i = (2.8) µ i ±iµ i, i = 1,...,m 2 are non-zero eigenalues of matrix A, that is, µ i = (±ω i ) (±ω j ) for i,j = 1,...,n (see [3, Corollary ]). Note that m 2 = (m 2 m)/2. Set D1 = U T D F, D 2 D F is defined in (2.5). Now, ([ ) A D F D T D1 D F = U Â] T 1 D 1 D T 2 D 2 D T 1 D 2 D T U T. (2.9) 2 Taking [ ] I G = D 2 D T 1 (D 1 D T 1 ) 1 I (2.1) 6

7 we obtain Note that we can write A D F D T F = U G 1 [ D1 D T 1 ] G T U T, (2.11) à à =  D 2 (I D T 1 (D 1 D T 1 ) 1 D 1 )D T 2. (2.12) D 2 (I D T 1 (D 1 D T 1 ) 1 D 1 )D T 2 = FF T. Further, we hae to find the inerse of à =  FF T. This is obtained by using the Sherman-Morrison-Woodbury formula ([1, (2.1.4), pg.51.]), that is, ( ) 1 ( FF T ) 1 =  1 +  1 F I F T  1 F F T  1. (2.13) The inerse of I F T  1 Λ = diag F remains to be found. Let ] λ2,,..., λ 2 ([ λ1 λ 1 [ λs λ s ]), ±iλ 1, ±iλ 2,..., ±iλ s are non-anishing finite eigenalues of the problem (A λd)ec(y) =. Since F T  1 F is skew-symmetric, then there exists an orthogonal matrix U S of order 2(r 1)m such that U T S F T  1 FU S = Γ = Λ 1. Using (2.11), (2.13), (2.14) and (2.6) it follows, (2.14) Γ ec(x) = U G T [ 1 2 ] GU T ec(i), (2.15) 1 = (D 1D T 1 ) 1, 2 =  1 +  1 FU S Since Γ is block diagonal we hae ( [ (I Γ) 1 1 λ 2 = diag 1 λ 1 λ λ 1 λ 2 1 Using (2.16) and (2.15) it follows ( V 1 s ec(x) = V V 1 + i=1 7 ],..., I (I Γ) 1 Us T F T  1. 1 λ 2 s + 2 (2.16) ) λ 2 s λ s λ s λ 2. s (2.17) ) λ i (λ i W i Z i ) λ 2 i + ec(i), (2.18) 2

8 matrices V 1, V, V 1, W i and Z i are constructed using (2.1), (2.16) and (2.15). By reshaping ectors in (2.18) back into m m matrices we obtain the solution of equation (2.1) X = Ψ 1 Ψ Ψ 1 + s i=1 λ i (λ i Φ i Υ i ) λ 2 i + 2. (2.19) If one is interested in deriing an optimal damping, then according to (1.11) one has to minimize the function f() = tr(zx()), This gies tr(zx()) = ec(z) T ec(x) = ec(z) T ( A D F D T F) 1ec(I). Tr(ZX) = X 1 X X 1 + s i=1 X 1 = ec(z) T V 1 ec(i), λ i (λ i X i Y i ) λ 2 i + 2, (2.2) X = ec(z) T V ec(i), (2.21) X 1 = ec(z) T V 1 ec(i), and X i = ec(z) T W i ec(i), Y i = ec(z) T Z i ec(i). (2.22) Note that the function Tr(ZX) from (2.2) is a generalization of the function Tr(X) defined in (1.14). Remark 2.1 In the case when Z is a diagonal matrix, the function Tr(ZX) from (2.2) has the following simpler form: Tr(ZX) = X 1 X 1 + s i=1 λ 2 i X i λ 2 i + 2. Finally we derie an explicit formula for the matrix Ψ 1. Let U be that part of U corresponding to the null-space of A. From (2.18) it follows that ec(ψ 1 ) = V 1 ec(i) U (D 1 D T 1 ) 1 U T ec(i). (2.23) It is easily seen that i-th column of the matrix U can be written as ec(o i ), O i is a block-diagonal matrix and O 2i 1 = diag(,...,,o ii,,...,) O 2i = diag(,...,,ôii,,...,) 8

9 O ii and Ôii are orthogonal solutions of Ω i O ij + O ij Ω j = i,j = 1,...,n, that is O ii = 1 2, and Ôii = 1. 2 Let A,B = Tr(A T B) be the usual Frobenius scalar product. Then, we can write the (p,q)-th element of U T D F D T F U as (U ) T (:,p) D F D T F(U ) (:,q) = D pq O p + O p D pq,o q p,q = 1,...,m, D pq = (D pq ) and D pq = D (p,:) D T. (q,:) The orthonormality property O p,o q = δ pq, implies ( U T D F D T F U ) 1 = diag(1/(d)22,1/(d) 22,...,1/(D) mm,1/(d) mm ). Using the fact that U U T ec(i) = ec(i), from (2.23) it follows that ( ) Ψ 1 = diag,,...,,. (2.24) D 22 D 22 D mm D mm After applying a perfect shuffle permutation we hae D Ψ 1 =, (2.25) D D = (diag (D)) 1. The explicitness of the obtained formulas is attractie for possible numerical computation. Our first attempts to perform this task did not succeed due to unexpected complexity problems. We will come back to this issue in our future research. References [1] G.H. Golub and Ch.F. an Loan, Matrix computations, J. Hopkins Uniersity Press, Baltimore,

10 [2] Herbert Federer, Geometric Measure Theory. Springer-Verlag, New York Inc., New York, [3] P. Lancaster and M. Tismenetsky, The Theory of Matrices, Academic Press, New York [4] I. Nakić, Optimal damping of ibrational systems, Ph. D. thesis Fernuniersität, Hagen, 22. [5] N. Truhar, An efficient algorithm for damper optimization for linear ibrating systems using Lyapuno equation, J. Comput. Appl. Math (24), [6] K. Veselić, K. Brabender and K. Delinić, Passie control of linear systems, in: Applied Mathematics and Computation, (M. Rogina et al. Eds.) Dept. of Math. Uni. Zagreb, 21, [7] K. Veselić, On linear ibrational systems with one dimensional damping II, Integral Eq. Operator Th., 13(199),