Eigenvalue placement for regular matrix pencils with rank one perturbations

Transcription

1 Eigenvalue placement for regular matrix pencils with rank one perturbations Hannes Gernandt (joint work with Carsten Trunk) TU Ilmenau Annual meeting of GAMM and DMV Braunschweig 2016

2 Model of an electrical network A model for a LC-circuit with simplified transistor behavior is given by the differential-algebraic equation d dt 0 c p g m x(t) = x(t) R p 0 } {{ 0 L p } } {{ 0 0 } =:E =:A the solutions can be described with the eigenvectors and eigenvalues of a matrix pencil A(s) = se A, s C. Problems: E is singular, A non-hermitian

3 Problem statement Given: Network described by the pencil se A, E, A C n n Task: Improve the properties of the network by moving the eigenvalues into a certain region G C How to move: Add step-by-step new elements to network. Example element: A capacity can be described by the pencil C ij (s) := sc ij (e i e j )(e i e j ) T, c ij > 0. New circuit: s(e + c ij (e i e j )(e i e j ) T ) A Difficulty: Number of elements types is restricted, therefore the class of perturbations is restricted. Today: Which eigenvalue sets can be obtained under arbitrary rank one perturbations?

4 Eigenvalues and regularity For fixed λ C observe that A(λ) = λe A is a matrix, so we define the spectrum of A as σ(a) := {λ C 0 is eig.val. of λe A}, σ(a) := {λ C 0 is eig.val. of λe A} { }, A(s) = se A regular : det(se A) 0 for E invertible and for E singular. For A(s) = se A regular the set σ(a) \ { } is the zero set of det(se A). For A(s) = se A singular we have σ(a) = C

5 Eigenvalues and regularity For fixed λ C observe that A(λ) = λe A is a matrix, so we define the spectrum of A as σ(a) := {λ C 0 is eig.val. of λe A}, σ(a) := {λ C 0 is eig.val. of λe A} { }, A(s) = se A regular : det(se A) 0 for E invertible and for E singular. For A(s) = se A regular the set σ(a) \ { } is the zero set of det(se A). For A(s) = se A singular we have σ(a) = C

6 The rank of matrix pencils The rank of A(s) = se A is rk A := max λ C rk (λe A). A is regular if and only if rk A = n Proposition The pencil P(s) = sf G with F, G C n n has rank one if and only if there exists u, v, w C n with w 0 and u 0 or v 0 such that P(s) = (su + v)w T or P(s) = w(su T + v T ).

7 Jordan chains and root subspaces For λ σ(a) \ { } we call {g 0, g 1,..., g m 1 } C n a Jordan chain of length m at λ iff g 0 0 and (A λe)g 0 = 0, (A λe)g 1 = Eg 0,..., (A λe)g m 1 = Eg m 2. We call {g 0, g 1,..., g m 1 } C n a Jordan chain of length m at iff g 0 0 and Eg 0 = 0, Eg 1 = Ag 0,..., Eg m 1 = Ag m 2. Define the (kth) root subspace of A at λ as L k λ (A) := {g i C n g i {g 0,..., g k 1 } is JC of A at λ} L λ (A) := {g C n g i {g 0,..., g m 1 } is JC of A at λ}

8 Jordan chains and root subspaces For λ σ(a) \ { } we call {g 0, g 1,..., g m 1 } C n a Jordan chain of length m at λ iff g 0 0 and (A λe)g 0 = 0, (A λe)g 1 = Eg 0,..., (A λe)g m 1 = Eg m 2. We call {g 0, g 1,..., g m 1 } C n a Jordan chain of length m at iff g 0 0 and Eg 0 = 0, Eg 1 = Ag 0,..., Eg m 1 = Ag m 2. Define the (kth) root subspace of A at λ as L k λ (A) := {g i C n g i {g 0,..., g k 1 } is JC of A at λ} L λ (A) := {g C n g i {g 0,..., g m 1 } is JC of A at λ}

9 Rank one perturbation of root subspaces Lemma 2.1 in Dopico, Moro, De Terán 08 implies the following perturbation result. Theorem (G., Trunk 16; Dopico, Moro, De Terán 08) For A regular and P rank one such that A + P is regular, we have for all λ C and all k N \ {0} dim ker(a + P)(λ) dim ker A(λ) 1, Lk+1 λ (A + P) dim dim Lk+1 λ (A) (A + P) L k λ (A) 1, L k λ dim L k λ (A + P) dim Lk λ (A) k. Similar results by Savchenko 03 for matrices and Behrndt, Leben, Martinez Peria, Trunk 15 for operators

10 Rank one perturbations of root subspaces Denote by m(λ) the length of the longest JC at λ σ(a) and define M(A) := λ σ(a) m(λ) Proposition (G, Trunk 16) Let A be regular and P be of rank one such that A + P is regular, then the following estimates hold: dim L λ (A) m(λ) dim L λ (A + P), dim L λ (A + P) dim L λ (A) + M(A) m(λ), 0 dim L µ (A + P) M(A), µ C\σ(A) for λ σ(a), for λ σ(a),

11 Eigenvalue placement for regular pencils Given regular A(s) = se A with σ(a) = {λ 1,..., λ m } and multiplicities m(λ i ) Given pairwise distinct µ 1,..., µ l C with multiplicities m i > 0 Construct rank one P with σ(a + P) = {µ 1,..., µ l } with multiplicities m i at µ i? Previous Results: Golub 73 for E = I n, A symmetric, Krupnik 92 for E = I n (only inverse problem), Elhay, Golub, Ram 03 for E > 0 and A symmetric

12 Eigenvalue placement for regular pencils Given regular A(s) = se A with σ(a) = {λ 1,..., λ m } and multiplicities m(λ i ) Given pairwise distinct µ 1,..., µ l C with multiplicities m i > 0 Construct rank one P with σ(a + P) = {µ 1,..., µ l } with multiplicities m i at µ i? Previous Results: Golub 73 for E = I n, A symmetric, Krupnik 92 for E = I n (only inverse problem), Elhay, Golub, Ram 03 for E > 0 and A symmetric

13 Eigenvalue placement for regular pencils Given regular A(s) = se A with σ(a) = {λ 1,..., λ m } and multiplicities m(λ i ) Given pairwise distinct µ 1,..., µ l C with multiplicities m i > 0 Construct rank one P with σ(a + P) = {µ 1,..., µ l } with multiplicities m i at µ i? Previous Results: Golub 73 for E = I n, A symmetric, Krupnik 92 for E = I n (only inverse problem), Elhay, Golub, Ram 03 for E > 0 and A symmetric

14 Eigenvalue placement for regular pencils Theorem (G, Trunk 16) Given pairwise distinct µ 1,..., µ l C, with multiplicities m i N \ {0} such that l i=1 m i = M(A), then we find such that P(s) = (αs β)uv T, α, β C u, v C n σ(a + P) = {µ 1,..., µ l } {λ σ(a) dim ker A(λ) 2}. and dim L λ (A + P) is equal to dim L λ (A) m(λ) + m i, for λ = µ i σ(a), dim L λ (A) m(λ), for λ σ(a) \ {µ 1,..., µ l }, m i, for λ = µ i / σ(a), 0, for λ / σ(a) {µ 1,..., µ l }.

15 Remarks on the placement There is only one Jordan chain at each λ σ(a + P) \ σ(a) Similar result for E, A R n n and {µ 1,..., µ l } symmetric w.r.t. the real line, then we can find α, β R and u, v R n. Corollary For A C n n with dim ker(λ A) 1 and µ 1,..., µ l C with multiplicities m i N \ {0} satisfying l i=1 m i = n there exist u, v C n such that σ(a + uv T ) = {µ 1,..., µ l } and { dim L λ (A + uv T m i, if λ = µ i, ) = 0, if λ / {µ 1,..., µ l }.

16 Application: DAEs with feedback Given a single input DAE with state feedback u(t) = f T x(t), f C n E d dt x(t) = Ax(t) + b f T x(t), x(0) = x 0. Solutions are given by the eigenvalues and Jordan chains of (A + P)(s) := se (A + bf T ), P(s) = bf T

17 Application: DAEs with feedback Given a single input DAE with state feedback u(t) = f T x(t), f C n E d dt x(t) = Ax(t) + b f T x(t), x(0) = x 0. Solutions are given by the eigenvalues and Jordan chains of (A + P)(s) := se (A + bf T ), P(s) = bf T Let s compare the rank one perturbations P(s) = (αs β)uv T vs. P(s) = bf T. Here b C n is fixed and this affects the placement.

18 Application: DAEs with feedback Given a single input DAE with state feedback u(t) = f T x(t), f C n E d dt x(t) = Ax(t) + b f T x(t), x(0) = x 0. Solutions are given by the eigenvalues and Jordan chains of (A + P)(s) := se (A + bf T ), P(s) = bf T Let s compare the rank one perturbations P(s) = (αs β)uv T vs. P(s) = bf T. Here b C n is fixed and this affects the placement.

19 Application: DAEs with feedback For simplicity, assume that A(s) = se A, E, A C n n satisfies the Hautus condition rk [λe A, b] = n, for all λ C. (1) This implies dim ker(λe A) 1 for λ C hence M(A) = n. Theorem (Feedback placeability) Let se A be regular satisfying (1) with E singular such that se (A + bf T ) is regular. Given µ 1 = and µ 2,..., µ l C and multiplicities m i N \ {0} with l i=1 m i = M(A) there exists an f C n such that P(s) = bf T satisfies σ(a + P) = {, µ 2,..., µ l } {λ σ(a) : dim ker A(λ) 2}

20 References J. Behrndt, L.Leben, F. Martinez Peria, C. Trunk: The effect of finite rank perturbations on Jordan chains of linear operators, Linear Algebra Appl. 458 (2015), pp T. Berger, A. Ilchmann, S. Trenn: The quasi-weierstraß form for regular matrix pencils, Linear Algebra Appl. 436 (2012), pp F.M. Dopico, J. Moro, F. De Teran: Low rank perturbation of Weierstrass structure, SIAM J. Matrix Anal. Appl. 30 (2008), pp H. Gernandt, C. Trunk: Eigenvalue placement for regular matrix pencils with rank one perturbations, to appear at SIMAX, see also arxiv: M. Krupnik: Changing the spectrum of an operator by perturbation, Linear Algebra Appl. 167 (1992), pp