The Moore-Penrose Inverse of the First Markov Parameter

Abstract

In this chapter we employ the Moore-Penrose pseudoinverse of the first nonzero Markov parameter for the algebraic characterization of the system zeros , i.e., invariant, transmission and decoupling zeros. Strictly proper (D =0) and proper ($\mathbf{D \ne 0}$) systems are discussed separately. For a strictly proper system (2.1) as matrix-characterizing system zeros we take matrix K _ν A where K _ν A: = I–B (CA^ν B)⁺ CA^ν , CA^ν B stands for the first nonzero Markov parameter and “+” means the operation of taking the Moore-Penrose pseudoinverse. For a proper system (2.1) as matrix-describing system zeros we use matrix A – BD⁺C. The question of determining and interpreting the system zeros is based on the Kalman canonical decomposition theorem. As we shall see, the Kalman form of these matrices, with block partition consistent with the partition of the original system, discloses all decoupling zeros. Moreover, if the first nonzero Markov parameter has full rank and the system is nondegenerate, then these matrices characterize completely also invariant zeros.