Introduction Polynomial Matrix Eigenproblem Spectral Symmetry Cayley Transformations Structured Linearization Conclusions On A Quadratic Eigenproblem Arising In The Analysis of Delay Equations Heike Faßbender AG Numerik Institut Computational Mathematics TU Braunschweig Joint work with E. Jarlebring, N. & D.S. Mackey Heike Faßbender On A Quadratic Eigenproblem Arising In The Analysis of Delay Equ
Introduction Polynomial Matrix Eigenproblem TDS Spectral Symmetry Critical System Cayley Transformations Quadratic eigenproblem Structured Linearization Conclusions Outline Time Delay System Polynomial Eigenvalue Problem Spectral Symmetry Structured Linearization Conclusions Heike Faßbender On A Quadratic Eigenproblem Arising In The Analysis of Delay Equ
Introduction Polynomial Matrix Eigenproblem TDS Spectral Symmetry Critical System Cayley Transformations Quadratic eigenproblem Structured Linearization Conclusions Time Delay Systems (TDS) m � x ( t ) = A 0 x ( t ) + ˙ A k x ( t − h k ) , t > 0 (Σ) k =1 x ( t ) = ϕ ( t ) , t ∈ [ − h m , 0] with 0 < h 1 < . . . < h m and A k ∈ R n × n . Heike Faßbender On A Quadratic Eigenproblem Arising In The Analysis of Delay Equ
Introduction Polynomial Matrix Eigenproblem TDS Spectral Symmetry Critical System Cayley Transformations Quadratic eigenproblem Structured Linearization Conclusions Time Delay Systems (TDS) m � x ( t ) = A 0 x ( t ) + ˙ A k x ( t − h k ) , t > 0 (Σ) k =1 x ( t ) = ϕ ( t ) , t ∈ [ − h m , 0] with 0 < h 1 < . . . < h m and A k ∈ R n × n . Definition Eigenvalue s and eigenvector v � = 0: � � m � A k e − h k s M ( s ) v := − sI n + A 0 + v = 0 k =1 Heike Faßbender On A Quadratic Eigenproblem Arising In The Analysis of Delay Equ
Introduction Polynomial Matrix Eigenproblem TDS Spectral Symmetry Critical System Cayley Transformations Quadratic eigenproblem Structured Linearization Conclusions Time Delay Systems (TDS) m � x ( t ) = A 0 x ( t ) + ˙ A k x ( t − h k ) , t > 0 (Σ) k =1 x ( t ) = ϕ ( t ) , t ∈ [ − h m , 0] with 0 < h 1 < . . . < h m and A k ∈ R n × n . Definition Eigenvalue s and eigenvector v � = 0: � � m � A k e − h k s M ( s ) v := − sI n + A 0 + v = 0 k =1 spectrum σ (Σ): set of all eigenvalues Heike Faßbender On A Quadratic Eigenproblem Arising In The Analysis of Delay Equ
Introduction Polynomial Matrix Eigenproblem TDS Spectral Symmetry Critical System Cayley Transformations Quadratic eigenproblem Structured Linearization Conclusions Time Delay Systems (TDS) m � x ( t ) = A 0 x ( t ) + ˙ A k x ( t − h k ) , t > 0 (Σ) k =1 x ( t ) = ϕ ( t ) , t ∈ [ − h m , 0] with 0 < h 1 < . . . < h m and A k ∈ R n × n . Definition Eigenvalue s and eigenvector v � = 0: � � m � A k e − h k s M ( s ) v := − sI n + A 0 + v = 0 k =1 spectrum σ (Σ): set of all eigenvalues stable: σ (Σ) ⊂ C − Heike Faßbender On A Quadratic Eigenproblem Arising In The Analysis of Delay Equ
Introduction Polynomial Matrix Eigenproblem TDS Spectral Symmetry Critical System Cayley Transformations Quadratic eigenproblem Structured Linearization Conclusions Critical System Problem For what h 1 ,. . . ,h m is there an ω s.t M ( ıω ) v = 0 . Heike Faßbender On A Quadratic Eigenproblem Arising In The Analysis of Delay Equ
Introduction Polynomial Matrix Eigenproblem TDS Spectral Symmetry Critical System Cayley Transformations Quadratic eigenproblem Structured Linearization Conclusions Critical System Problem For what h 1 ,. . . ,h m is there an ω s.t M ( ıω ) v = 0 . Definition Σ is called critical iff σ (Σ) ∩ ı R � = ∅ . Heike Faßbender On A Quadratic Eigenproblem Arising In The Analysis of Delay Equ
Introduction Polynomial Matrix Eigenproblem TDS Spectral Symmetry Critical System Cayley Transformations Quadratic eigenproblem Structured Linearization Conclusions Example (Jarlebring 2005) Two delay problem: ˙ x ( t ) = − x ( t − h 1 ) − 2 x ( t − h 2 ) Heike Faßbender On A Quadratic Eigenproblem Arising In The Analysis of Delay Equ
Introduction Polynomial Matrix Eigenproblem TDS Spectral Symmetry Critical System Cayley Transformations Quadratic eigenproblem Structured Linearization Conclusions Example (Jarlebring 2005) Two delay problem: ˙ x ( t ) = − x ( t − h 1 ) − 2 x ( t − h 2 ) Critical curves: 10 9 8 7 6 h 2 5 4 3 2 1 0 0 1 2 3 4 5 6 7 8 9 10 h 1 Heike Faßbender On A Quadratic Eigenproblem Arising In The Analysis of Delay Equ
Introduction Polynomial Matrix Eigenproblem TDS Spectral Symmetry Critical System Cayley Transformations Quadratic eigenproblem Structured Linearization Conclusions Critical System Problem For what h 1 ,. . . ,h m is there an ω s.t M ( ıω ) v = 0 . Heike Faßbender On A Quadratic Eigenproblem Arising In The Analysis of Delay Equ
Introduction Polynomial Matrix Eigenproblem TDS Spectral Symmetry Critical System Cayley Transformations Quadratic eigenproblem Structured Linearization Conclusions Critical System Problem For what h 1 ,. . . ,h m is there an ω s.t M ( ıω ) v = 0 . Hale & Huang 1993: Scalar two delays: Geometric classification Chen & Gu & Nett 1995: Commensurate delays Louisell 2001: Single delay, neutral, moderate size Sipahi & Olgac 2003 : Small systems, few delays: Form determinant + Routh table + Rekasius Substitution. Heike Faßbender On A Quadratic Eigenproblem Arising In The Analysis of Delay Equ
Introduction Polynomial Matrix Eigenproblem TDS Spectral Symmetry Critical System Cayley Transformations Quadratic eigenproblem Structured Linearization Conclusions Given free parameters ϕ k , k = 1 , . . . , m − 1. Theorem (Jarlebring 2005) The point in delay space ( h 1 , . . . , h m ) is critical iff ϕ k + 2 p π h k = , k = 1 , . . . , m − 1 ω Arg s + 2 q π h m = ω � � m − 1 � � � I ⊗ A k e − ıϕ k + e ıϕ k A k ⊗ I s 2 I ⊗ A m + s + A m ⊗ I u = 0 , k =0 where s = e ıω , u = vec vv ∗ = v ⊗ ¯ v and � � m − 1 � A k e − ıϕ k + A m s ω = − ı v ∗ A 0 + v . k =1 Heike Faßbender On A Quadratic Eigenproblem Arising In The Analysis of Delay Equ
Introduction Polynomial Matrix Eigenproblem TDS Spectral Symmetry Critical System Cayley Transformations Quadratic eigenproblem Structured Linearization Conclusions Quadratic eigenproblem � � m − 1 � � � I ⊗ A k e − ıϕ k + e ıϕ k A k ⊗ I s 2 ( I ⊗ A m ) + s + ( A m ⊗ I ) u = 0 , k =0 Heike Faßbender On A Quadratic Eigenproblem Arising In The Analysis of Delay Equ
Introduction Polynomial Matrix Eigenproblem TDS Spectral Symmetry Critical System Cayley Transformations Quadratic eigenproblem Structured Linearization Conclusions Quadratic eigenproblem G � �� � M K � �� � � �� � � � m − 1 � � � I ⊗ A k e − ıϕ k + e ıϕ k A k ⊗ I s 2 ( I ⊗ A m ) + s + ( A m ⊗ I ) u = 0 , k =0 Heike Faßbender On A Quadratic Eigenproblem Arising In The Analysis of Delay Equ
Introduction Polynomial Matrix Eigenproblem TDS Spectral Symmetry Critical System Cayley Transformations Quadratic eigenproblem Structured Linearization Conclusions Quadratic eigenproblem G � �� � M K � �� � � �� � � � m − 1 � � � I ⊗ A k e − ıϕ k + e ıϕ k A k ⊗ I s 2 ( I ⊗ A m ) + s + ( A m ⊗ I ) u = 0 , k =0 Quadratic Eigenvalue Problem R n 2 × n 2 M ∈ C n 2 × n 2 G ∈ R n 2 × n 2 K ∈ Heike Faßbender On A Quadratic Eigenproblem Arising In The Analysis of Delay Equ
Introduction Polynomial Matrix Eigenproblem TDS Spectral Symmetry Critical System Cayley Transformations Quadratic eigenproblem Structured Linearization Conclusions Theorem (Horn, Johnson) There exists an involutary permutation matrix P ∈ R n 2 × n 2 such that B ⊗ C = P ( C ⊗ B ) P for all B , C ∈ R n × n . Heike Faßbender On A Quadratic Eigenproblem Arising In The Analysis of Delay Equ
Introduction Polynomial Matrix Eigenproblem TDS Spectral Symmetry Critical System Cayley Transformations Quadratic eigenproblem Structured Linearization Conclusions Theorem (Horn, Johnson) There exists an involutary permutation matrix P ∈ R n 2 × n 2 such that B ⊗ C = P ( C ⊗ B ) P for all B , C ∈ R n × n . In particular, n � E ij ⊗ E T ij = [ E T ij ] n P = i , j =1 , i , j =1 where E ij ∈ R n × n has entry 1 in position i , j and all other entries are zero. Heike Faßbender On A Quadratic Eigenproblem Arising In The Analysis of Delay Equ
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