Transfer Functions

As developed in Book II [452], a discrete-time transfer function is the z transform of the impulse response of a linear, time-invariant (LTI) system. In a physical modeling context, we must specify the input and output signals we mean for each transfer function to be associated with the LTI model. For example, if the system is a simple mass sliding on a surface, the input signal could be an external applied force, and the output could be the velocity of the mass in the direction of the applied force. In systems containing many masses and other elements, there are many possible different input and output signals. It is worth emphasizing that a system can be reduced to a set of transfer functions only in the LTI case, or when the physical system is at least nearly linear and only slowly time-varying (compared with its impulse-response duration).

As we saw in the previous section, the state-space formulation nicely organizes all possible input and output signals in a linear system. Specifically, for inputs, each input signal is multiplied by a ``$B$ vector'' (the corresponding column of the $B$ matrix) and added to the state vector; that is, each input signal may be arbitrarily scaled and added to any state variable. Similarly, each state variable may be arbitrarily scaled and added to each output signal via the row of the $C$ matrix corresponding to that output signal.

Using the closed-form sum of a matrix geometric series (again as detailed in Book II), we may easily calculate the transfer function of the state-space model of Eq.(1.8) above as the z transform of the impulse response given in Eq.(1.10) above:

$$H(z) \eqsp D + C \left(zI - A\right)^{-1}B \protect$$ (2.11)

Note that if there are $p$ inputs and $q$ outputs, $H(z)$ is a $p\times q$ transfer-function matrix (or ``matrix transfer function'').

In the force-driven-mass example of the previous section, defining the input signal as the driving force $f_n$ and the output signal as the mass velocity $v_n$ , we have $p=q=1$ , $B=[0,T/m]^T$ , $C=[0,1]$ , and $D=0$ , so that the force-to-velocity transfer function is given by

$$\begin{eqnarray*} H(z) &=& D + C \left(zI - A\right)^{-1}B\\ [5pt] &=&\begin{array}{r}\left[\begin{array}{cc} 0 & 1 \end{array}\right]\\ [2pt]{}\end{array}\left[\begin{array}{cc} z-1 & -T \\ [2pt] 0 & z-1 \end{array}\right]^{-1}\left[\begin{array}{c} 0 \\ [2pt] T/m \end{array}\right]\\ [5pt] &=&\frac{1}{(z-1)^2}\begin{array}{r}\left[\begin{array}{cc} 0 & 1 \end{array}\right]\\ [2pt]{}\end{array}\left[\begin{array}{cc} z-1 & T \\ [2pt] 0 & z-1 \end{array}\right]\left[\begin{array}{c} 0 \\ [2pt] T/m \end{array}\right]\\ [5pt] &=&\frac{T}{m}\frac{z-1}{(z-1)^2} \eqsp \zbox {\frac{T}{m}\frac{z^{-1}}{1-z^{-1}}.} \end{eqnarray*}$$

Thus, the force-to-velocity transfer function is a one-pole filter with its pole at $z=1$ (an integrator). The unit-sample delay in the numerator guards against delay-free loops when this element (a mass) is combined with other elements to build larger filter structures.

Similarly, the force-to-position transfer function is a two-pole filter:

$$\begin{eqnarray*} H(z) &=&\frac{1}{(z-1)^2}\begin{array}{r}\left[\begin{array}{cc} 1 & 0 \end{array}\right]\\ [2pt]{}\end{array}\left[\begin{array}{cc} z-1 & T \\ [2pt] 0 & z-1 \end{array}\right]\left[\begin{array}{c} 0 \\ [2pt] T/m \end{array}\right]\\ [5pt] &=&\frac{T^2/m}{(z-1)^2} \eqsp \frac{T^2}{m}\,\frac{z^{-2}}{1-2z^{-1}+z^{-2}} \end{eqnarray*}$$

Now we have two poles on the unit circle at $z=1$ , and the impulse response of this filter is a ramp, as already discovered from the previous impulse-response calculation.

Once we have transfer-function coefficients, we can realize any of a large number of digital filter types, as detailed in Book II [452, Chapter 9].