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8.23 Obtain a parallel realization for the transfer function matrix of Example 8-15. EXAMPLE 8.15 Determine the z-transfer function of a digitally controlled single-axis milling machine with a sampling period of 40 ms if its analog transfer function is given by² $G(s) = \text{diag}\left\{ \frac{3150}{(s+35)(s+150)}, \frac{1092}{(s+35)(s+30)} \right\}$ Find the poles and zeros of the transfer function. Solution Using the MATLAB command c2d, we obtain the transfer function matrix $G_{ZAS}(z) = \text{diag}\left\{ \frac{0.4075(z+0.1067)}{(z-0.2466)(z-0.2479 \times 10^{-2})}, \frac{0.38607(z+0.4182)}{(z-0.2466)(z-0.3012)} \right\}$ Because the transfer function is diagonal, the determinant is the product of its diagonal terms. The least common denominator of all nonzero minors is the denominator of the determinant of the transfer function $(z - 0.2479 \times 10^{-2})(z - 0.2466)^2(z - 0.3012)$ We therefore have poles at [0.2479 \times 10^{-2}, 0.2466, 0.2466, 0.3012]. The system is stable because all the poles are inside the unit circle.

          8.23 Obtain a parallel realization for the transfer function matrix of Example 8-15.
EXAMPLE 8.15
Determine the z-transfer function of a digitally controlled single-axis milling machine with
a sampling period of 40 ms if its analog transfer function is given by²
$G(s) = \text{diag}\left\{ \frac{3150}{(s+35)(s+150)}, \frac{1092}{(s+35)(s+30)} \right\}$
Find the poles and zeros of the transfer function.
Solution
Using the MATLAB command c2d, we obtain the transfer function matrix
$G_{ZAS}(z) = \text{diag}\left\{ \frac{0.4075(z+0.1067)}{(z-0.2466)(z-0.2479 \times 10^{-2})}, \frac{0.38607(z+0.4182)}{(z-0.2466)(z-0.3012)} \right\}$
Because the transfer function is diagonal, the determinant is the product of its diagonal
terms. The least common denominator of all nonzero minors is the denominator of the
determinant of the transfer function
$(z - 0.2479 \times 10^{-2})(z - 0.2466)^2(z - 0.3012)$
We therefore have poles at [0.2479 \times 10^{-2}, 0.2466, 0.2466, 0.3012]. The system is stable
because all the poles are inside the unit circle.

$8.23 Obtain a parallel realization for the transfer function matrix of Example 8-15. EXAMPLE 8.15 Determine the z-transfer function of a digitally controlled single-axis milling machine with a sampling period of 40 ms if its analog transfer function is given by² G(s) = diag{(3150)/((s+35)(s+150)), (1092)/((s+35)(s+30))} Find the poles and zeros of the transfer function. Solution Using the MATLAB command c2d, we obtain the transfer function matrix GZAS(z) = diag{(0.4075(z+0.1067))/((z-0.2466)(z-0.2479 × 10^-2)), (0.38607(z+0.4182))/((z-0.2466)(z-0.3012))} Because the transfer function is diagonal, the determinant is the product of its diagonal terms. The least common denominator of all nonzero minors is the denominator of the determinant of the transfer function (z - 0.2479 × 10^-2)(z - 0.2466)^2(z - 0.3012) We therefore have poles at [0.2479 ×10^-2, 0.2466, 0.2466, 0.3012]. The system is stable because all the poles are inside the unit circle.$

Added by Anthony W.

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