An algebraic method to determine the common divisor, poles and transmission zeros of matrix transfer functions

Abstract

A purely algebraic method which uses the matrix Routh algor-ithm and its reverse process of the algorithm is presented to decompose a mat.ri x transfer function into a pair of right co-prime polynomial matrices or left co-pr-ime polynomial matrices. The poles and transmission zeros of the matrix transfer function arc determined from a. pair of relatively prime polynomial matrices. Also, tho common divisor of two matrix polynomials CRn be obtained by using the matrix Routh algorithm and the matrix Routh array.

Approximate Linear Regulator and Kalman Filter

“…and then minimize the mean-square error, T- (1)(2)(3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15)(16)(17)(18)(19)(20) where a T(t) represents any linear combination of the state variables.…”

“…Note that the differential equation in (1)(2)(3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15)(16)) is defined only if we accept the notion of continuous-time white noise. In discrete-time systems, white noise is well defined,and the problem does not arise.…”

“…Once the designer has specified Q, H and R, representing different weightings in (1)(2)(3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13), the optimal closed-loop system will be (1-15a) y(t) -Cx(t) (1-15b)…”

“…Given the continuous-time system matrix A and input maitrix B, the approximated discrete-time system matrix D) and input matrix E. can bj oj be calculated by using (2)(3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14) and (2-15). The roundoff errors between the exact mode and the approximated mode increase as J jAj T increases.…”

“…One popular method is to truncate the infinite series, i.e., (2)(3)(4)(5)(6)(7)(8)(9)(10)(11)(12) Keeping the first (j+l) terms in the series of (2)(3)(4)(5)(6)(7)(8)(9)(10)(11)(12) and approximating the other terms of the series by a geometric series with a weighting factor (i/(jn.j!)) for the term (AT) 3~ rather than l/(j+n)l (=1/(J+n) (J+n-l)...(j+l).j!)…”