Estimation of the Unitary Interactor Matrices

Abstract

The idea of the time delay term can be easily generalized to the multivariate case in terms of the impulse response coefficient or the Markov parameter matrices. In the multivariate case, the notion of a delay corresponds to the fewest number of impulse response coefficients or Markov parameter matrices whose linear combination is nonsingular. This means that a set of inputs, acting via this specific linear combination of Markov parameter matrices, can have a desired effect on the output. This linear combination of impulse response matrices can be expressed in a polynomial matrix form. The determinant of this polynomial matrix has, as its roots, the infinite zeros of the discrete time multivariate system. Simple examples to illustrate these concepts are considered in Shah et al. (1987). The knowledge of the interactor matrix is an important prerequisite to high performance control strategies such as minimum variance control. However a knowledge of the delay or the interactor matrix is, until recently, tantamount to the knowledge of the entire process transfer function matrix. As per the above definition, it should appear that relatively simple tests can be performed to determine if a linear combination of the first few Markov or impulse response matrices is singular or not. This is precisely the purpose of this chapter in which we propose the use of a singular value decomposition (SVD) based procedure to determine if a linear combination of a set of matrices has full rank. Note that the determination of a rank of a matrix is a non-trivial computational problem. SVD-based techniques are useful in determining the rank of a given matrix. The proposed procedure allows us to compute the time delay matrix with minimum effort using closed- or open-loop data with dither excitation and its subsequent use in multivariate control loop performance assessment or control law design.