By Jerzy Tokarzewski
This e-book offers a country house method of the research of zeros of MIMO LTI discrete-time platforms, utilizing the Moore-Penrose pseudoinverse and singular price decomposition of the 1st nonzero Markov parameter of a method. The ebook starts with definition of invariant zeros and is going so far as a common characterization of output-zeroing inputs and the corresponding options, particular formulation for maximal output-nulling invariant subspaces and for the 0 dynamics.
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Additional resources for Finite Zeros in Discrete Time Control Systems
Example text
4). 8, a clear dynamical interpretation in the context of the output-zeroing problem. In nondegenerate systems we can identify the notion of Smith zeros with the notion of invariant zeros. 3. 1) is nondegenerate, then for a given complex number λ the following statements are equivalent: (i) λ is an invariant zero of the system; λ is a Smith zero of the system; (ii) (iii) λ is a root of the zero polynomial. Proof. 1 (ii). The statements (ii) and (iii) are equivalent by virtue of the definition of Smith zeros (see Chap.
A number λ ∈ C is an invariant zero of the system if and only if det P (λ) = 0 , where zI − A − b . , g (z) ≡ 0 ). 3. 1) for which the matrix has full column rank. Prove the followD ing statements: a) λ ∈ C is an invariant zero of the system if and only if det P (λ) = 0 ; b) the system is degenerate if and only if det P (z) ≡ 0 (or, equivalently, det G (z) ≡ 0 ). x o Hint. a) Let det P (λ) = 0 . 4). Suppose that in this vector is x o = 0 . , . This g = 0 g = is D 0 g nonzero.
1). 4) if and only if the input (i) ~ (k ) = g for k = 0 u k λ g for k = 1,2,... and the initial condition x o ≠ 0 yield ~ y (k ) = 0 for all k ∈ N . Moreover, in the triple under considerations is g ≠ 0 , and the solution ~ x (k ) correo ~ sponding to x and u (k ) takes the form (ii) x o for k = 0 ~ . x (k ) = k o for k = 1,2,... λ x 32 2 Zeros and the Output-Zeroing Problem Proof. 7, the input sequence of the form (i) applied to the system (treated as a complex one) at the initial condition ~ x (0) = x o yields a solution of the state equation of the form (ii) and the identically zero output sequence.