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By Efim N. Rosenwasser, Bernhard P. Lampe

During this booklet, the authors expand the parametric move functionality equipment, which include time-dependence, to the assumption of the parametric move matrix in a whole exposition of research and layout tools for multiple-input, multiple-output (MIMO) sampled-data platforms. Appendices overlaying uncomplicated mathematical formulae, MATLAB® toolboxes around out this self-contained consultant to multivariable keep an eye on platforms. The booklet will curiosity researchers in computerized regulate and to improvement engineers operating with complex regulate expertise.

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Example text

74), it is necessary and sufficient that their Smith-canonical forms coincide. 13 Polynomial Matrices of First Degree (Pencils) 39 3. 75) A(λ) = P A1 (λ)Q . e. the matrices are regular, then the matrices A(λ), B(λ) are only in that case equivalent, when they are strictly equivalent. e. 74) are anomalous, then the conditions for equivalence and strict equivalence do not coincide. 4. 74), following [51], we consider the n × n Jordan block ⎤ ⎡ a 1 0 ... 0 0 ⎢0 a 1 ... 0 0⎥ ⎥ ⎢ . ⎥ ⎢ ⎢ 0 0 a .. 76) Jn (a) = ⎢ .

Therefore, we will consider now only horizontal pairs. 67) are non-degenerated. If not supposed explicitly otherwise, we will always consider non-degenerated pairs. 2. 68) with polynomial matrices a1 (λ), b1 (λ). Then the matrix g(λ) is called a common left divisor of the pair (a(λ), b(λ)). The common left divisor g(λ) is named as a greatest common left divisor (GCLD) of the pair (a(λ), b(λ)), if for any left common divisor g1 (λ) g(λ) = g1 (λ)α(λ) with a polynomial matrix α(λ) is true. As known any two GCLD are rightequivalent [69].

Proof. Assume rank A(λ) = ρA , rank B(λ) = ρB , rank D(λ) = ρD with ρD > min{ρA , ρB } . 17, there exists a value λ = λ ˜ = ρA , rank A(λ) ˜ = ρB , rank B(λ) ˜ = ρD . rank D(λ) It is possible to apply the Sylvester inequalities to the number matrices ˜ = A(λ)B( ˜ ˜ , D(λ) λ) which gives ˜ ≤ min{rank A(λ) ˜ rank B(λ)} ˜ . 45). 43). 44) could be proved analogously because a corresponding inequality holds for constant matrices. 2. 46) and rank[A(λ)B(λ)] = rank A(λ) for rank B(λ) = . 47) Herein, rank A(λ) = can only be fulfilled for n ≥ , and rank B(λ) = only for m ≥ .

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