System Theory

Download Linear system theory : the state space approach by Lotfi A. Zadeh, Charles A. Desoer PDF

By Lotfi A. Zadeh, Charles A. Desoer

This landmark within the improvement of the nation area strategy used to be written by means of pioneers within the box. It mainly issues the technique's software to platforms defined via differential equations. difficulties of balance and controllability obtain specific consciousness, and connections among this strategy and classical innovations are highlighted. 1963 variation.

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Extra resources for Linear system theory : the state space approach

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The classes of structured matrices we consider in this paper are the following. Throughout the paper let A denote either the transpose AT or the conjugate transpose A∗ of a matrix A. Furthermore, let H = H ∈ Fn×n and −J T = J ∈ Fn×n be invertible. Then A ∈ Fn×n is called 1. H-selfadjoint, if F = C, = ∗, and A∗ H = HA; 2. H-symmetric, if F ∈ {R, C}, = T , and AT H = HA; 3. J-Hamiltonian, if F ∈ {R, C}, = T , and AT J = −JA. There is no need to consider H-skew-adjoint matrices A ∈ Cn,n satisfying A∗ H = −HA, because this case can be reduced to the case of H-selfadjoint matrices by considering iA instead.

Nm ), and hence, the algebraic multiplicity aU of A + U RU G at λ must be greater than or equal to a0 := nk+1 + · · · + nm . However, by hypothesis there exists one U0 so that A + U0 RU0 G has exactly the partial multiplicities (nk+1 , . . , nm ), so in particular it has the algebraic multiplicity aU0 = a0 . Therefore, by (2) the set Ω1 of all U ∈ Cn×k satisfying aU = a0 is generic and for all U ∈ Ω1 . Since (nk+1 , . . , nm ) is the only possible list of partial multiplicities that dominates (nk+1 , .

Am , where aj ≤ aj for j = 1, . . , m. Furthermore, let ε > 0 be such that the discs Dj := μ ∈ C |λj − μ| < ε2/n , j = 1, . . , m are pairwise disjoint. If for each j = 1, . . , m there exists a matrix Uj ∈ Cn×k with Uj < ε such that the matrix A + Uj RUj G has exactly (aj − aj ) simple eigenvalues in Dj different from λj , then generically (with respect to the entries of U if = T and with respect to the real and imaginary parts of the entries of U if = ∗) the eigenvalues of A + U RU G that are different from the eigenvalues of A are simple.

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