Zeros and Kalman canonical form of MIMO LTI systems

The problem of determining system zeros (transmission+decoupling) of a linear system S(A, B, C, D) employing the canonical decomposition theorem is discussed. The zeros are characterized as eigenvalues of an appropriate real matrix of order of the state matrix. When the system is transformed to the Kalman canonical form that matrix takes an upper-triangular block form consistent with the Kalman structure of the state matrix. The submatrices of the diagonal describe successively output decoupling, transmission, input-output decoupling and input decoupling zeros. A suitable procedure for the computation of the zeros is proposed. As is shown, each system S(A, B, C, D) may be diagonally decoupled by state feedback and squaring down. The presented approach is based on the Moore-Penrose pseudoinverse of the first nonzero Markov parameter.