Example State Space Model Transfer Function

In this example, we consider a second-order filter ($N = 2$) with two inputs ($p=2$) and two outputs ($q=2$):

$$\begin{eqnarray*} A &=& g\left[\begin{array}{rr} c & -s \\ [2pt] s & c \end{arra... ... \left[\begin{array}{cc} 0 & 0 \\ [2pt] 0 & 0 \end{array}\right] \end{eqnarray*}$$

so that

$$\begin{eqnarray*} \left[\begin{array}{c} x_1(n+1) \\ [2pt] x_2(n+1) \end{array}\... ...left[\begin{array}{c} x_1(n) \\ [2pt] x_2(n) \end{array}\right]. \end{eqnarray*}$$

From Eq. (E.5), the transfer function of this MIMO digital filter is then

$$\begin{eqnarray*} H(z) &=& C(zI-A)^{-1}B = (zI-A)^{-1} = \left[\begin{array}{cc}... ...cz^{-2}}{\displaystyle 1-2gcz^{-1}+g^2z^{-2}} \end{array}\right] \end{eqnarray*}$$

Note that when $g=1$, the state transition matrix $A$ is simply a 2D rotation matrix, rotating through the angle $\theta$ for which $c=\cos(\theta)$ and $s=\sin(\theta)$. For $g<1$, we have a type of normalized second-order resonator [51], and $g$ controls the ``damping'' of the resonator, while $\theta = 2\pi f_r/f_s$ controls the resonance frequency $f_r$. The resonator is ``normalized'' in the sense that the filter's state has a constant $L2$ norm (``preserves energy'') when $g=1$ and the input is zero:

$$\left\Vert\,{\underline{x}}(n+1)\,\right\Vert \isdef \sqrt{x_1^2(... ...x}}(n)\,\right\Vert \equiv \left\Vert\,{\underline{x}}(n)\,\right\Vert \protect$$ (E.6)

since a rotation does not change the $L2$ norm, as can be readily checked.

In this two-input, two-output digital filter, the input $u_1(n)$ drives state $x_1(n)$ while input $u_2(n)$ drives state $x_2(n)$. Similarly, output $y_1(n)$ is $x_1(n)$, while $y_2(n)$ is $x_2(n)$. The two-by-two transfer-function matrix $H(z)$ contains entries for each combination of input and output. Note that all component transfer functions have the same poles. This is a general property of physical linear systems driven and observed at arbitrary points: the resonant modes (poles) are always the same, but the zeros vary as the input or output location are changed. If a pole is not visible using a particular input/output pair, we say that the pole has been ``canceled'' by a zero associated with that input/output pair. In control-theory terms, the pole is ``uncontrollable'' from that input, or ``unobservable'' from that output, or both.