Nested Programming

$\displaystyle Y(Z) - c_0U(Z)$	$\textstyle =$	$\displaystyle \frac{(c_0+c_1Z^{-1}+\cdots +c_nZ^{-n})}{1+a_1Z^{-1}+\cdots +a_n^Z{-n}}U(Z)$
	$\textstyle =$	$\displaystyle \frac{\{(c_1-c_0a_1)Z^{-1}+(c_2-c_0a_2)^Z{-2}+\cdots +(c_n-c_0a_n)^Z{-n}\}}{1+a_1Z^{-1}+\cdots +a_nZ^{-n}}U(Z)$
	$\textstyle \equiv$	$\displaystyle X_1(Z)$	(4.25)

$\displaystyle \{(c_1-c_0a_1)Z^{-1}+(c_2-c_0a_2)Z^{-2}+\cdots +(c_n-c_0a_n)Z^{-n}\}U(Z)$
$\displaystyle = \{1+a_1Z^{-1}+\cdots +a_nZ^{-n}\}X_1(Z)$			(4.26)

$\displaystyle X_1(Z)$	$\textstyle =$	$\displaystyle \{-a_1Z^{-1}-a_2Z^{-2}-\cdots -a_nZ^{-n}\}X_1(Z)$
		$\displaystyle +\{(c_1-c_0a_1)Z^{-1}+(c_2-c_0a_2)Z^{-2} +\cdots +(c_n-c_0a_n)Z^{-n}\}U(Z)$
	$\textstyle =$	$\displaystyle \{-a_1X_1(Z)+(c_1-c_0a_1)U(Z)\}Z^{-1} +\{-a_2X_1(Z)+(c_2-c_0a_2)U(Z)\}Z^{-2}$
		$\displaystyle +\cdots +\{-a_nX_1(Z)+(c_n-c_0a_n)U(Z)\}Z^{-n}$	(4.27)

$\displaystyle ZX_1(Z)$	$\textstyle =$	$\displaystyle \{-a_1X_1(Z)+(c_1-c_0a_1)U(Z)\}+\{-a_2X_1(Z)+(c_2-c_0a_2)U(Z)\}Z^{-1}$
		$\displaystyle +\cdots +\{-a_nX_1(Z)+(c_n-c_0a_n)U(Z)\}Z^{-n+1}$
	$\textstyle \equiv$	$\displaystyle \{-a_1X_1+(c_1-c_0a_1)U(Z)\}+X_2(Z)$	(4.28)

$\begin{displaymath} x_1(k+1) = -a_1x_1(k)+x_2(k)+(c_1-c_0a_1)u(k) \end{displaymath}$

(4.29)

$\displaystyle ZX_2(Z)$	$\textstyle =$	$\displaystyle -a_2X_1(Z)+(c_2-c_0a_2)U(Z)+\{-a_3X_1(Z)+(c_3-c_0a_3)U(Z)\}Z^{-1}$
		$\displaystyle +\cdots +\{-a_nX_1(Z)+(c_n-c_0a_n)U(Z)\}Z^{-n+2}$
	$\textstyle \equiv$	$\displaystyle -a_2X_1(Z)+(c_2-c_0a_2)U(Z)+X_3(Z)$	(4.30)

$\begin{displaymath} x_2(k+1) = -a_2x_1(k)+x_3(k)+(c_2-c_0a_2)u(k) \end{displaymath}$

(4.31)

$\begin{displaymath} \left\{ \begin{array}{ll} x_{n-1}(k+1) = -a_{n-1}x_1(k)+x... ... x_n(k+1) = -a_nx_1(k)+(c_n-c_0a_n)u(k) \end{array} \right. \end{displaymath}$

(4.32)

$\begin{displaymath} \left[ \begin{array}{c} x_1(k+1) \\ x_2(k+1) \\ \vdot... ... c_{n-1}-a_{n-1}c_0 \\ c_n-a_nc_0 \end{array} \right]u(k) \end{displaymath}$

(4.33)

$\begin{displaymath} y(k) = \left[ \matrix{1 & 0 & \cdots & 0} \right] \left... ...x_2(k) \\ \vdots \\ x_n(k) \end{array} \right] + c_0u(k) \end{displaymath}$

(4.34)

**図 4.5:**
$\begin{figure}\begin{center} \psbox[scale=0.60]{eps/4-2-3.eps} \end{center} \end{figure}$

［例］ $\frac{1+2Z^{-1}+Z^{-2}}{1+5Z^{-1}+6Z^{-2}}$ をNested Programmingで表示する。

$\begin{displaymath} \begin{array}{ccc} a_0 = 1 & a_1 = 5 & a_2 = 6 \\ c_0 = 1 & c_1 = 2 & c_2 = 1 \end{array}\end{displaymath}$

$\begin{eqnarray*} c_1-a_1c_0 & = & -3 \\ c_2-a_2c_0 & = & -5 \end{eqnarray*}$

$\begin{displaymath} \left[ \begin{array}{c} x_1(k+1) \\ x_2(k+1) \end{array}... ... \left[ \begin{array}{c} -3 \\ -5 \end{array} \right]u(k) \end{displaymath}$

$\begin{displaymath} y(k) = \left[ \begin{array}{cc} 1 & 0 \end{array} \right]... ...gin{array}{c} x_1(k) \\ x_2(k) \end{array} \right] + u(k) \end{displaymath}$

**図 4.6:**
$\begin{figure}\begin{center} \psbox[scale=0.60]{eps/4-2-4.eps} \end{center} \end{figure}$