Out of the three alternatives above, the bilinear transformation has the advantage of preserving the stability boundary (so that the stability boundary for the sampled model will be the inner part of the unit circle); see
Åström and Wittenmark (1997).

For all the cases the transformed model will in discrete-time be of the form

$$\begin{aligned} G_d(z) = \frac{B_d(z)}{A_d(z)} \; . \end{aligned}$$

(5.115)

The degree of the denominator polynomial

\(A_d(z)\) will always be

*n*. For the forward difference approximation the degree of

\(B_d(z)\) will be

\(\mathrm{deg}(B_c)\), which is smaller than

*n*. For the other two alternatives, the degree of

\(B_d(z)\) will be equal to

*n*, and then

\(G_d(z)\) will automatically have a direct term, that is

\(b_0 \ne 0\), cf. Sect.

5.2.

When the input behavior between the sampling instants is not known, also the standard zero-order-hold (ZOH) procedure can be regarded as a model approximation. (ZOH implies that the input is constant during the sampling intervals.) Write the effect of this sampling strategy as

$$\begin{aligned} G_d (z) = {\mathscr {S}}_\mathrm{ZOH} G_c(s) \; . \end{aligned}$$

(5.116)

If

\(G_c(s)\) is represented in state space form as

$$\begin{aligned} \begin{array}{rcl} \dot{x} &{} = &{} \mathbf {A}x + \mathbf {B}u \; ,\\ y &{} = &{} \mathbf {C}x \; , \end{array} \end{aligned}$$

(5.117)

then the ZOH sampled transfer function can be written as

$$\begin{aligned} G_d(z) = \mathbf {C}( z \mathbf {I}- \mathbf {F})^{-1} \mathbf {G}\; , \end{aligned}$$

(5.118)

where

$$\begin{aligned} \mathbf {F}= {\text {e}}^{\mathbf {A}h}, \ \ \ \mathbf {G}= \int _0 ^h {\text {e}}^{\mathbf {A}s} \mathbf {B}{\text {d}}s \; . \end{aligned}$$

(5.119)

One may alternatively write (

5.116) as

$$\begin{aligned} G_d(z) = \frac{z-1}{z} {\mathscr {Z}} {\mathscr {L}}^{-1} \frac{G_c(s)}{s} \; . \end{aligned}$$

(5.120)

This is to be interpreted as follows. Let

*g*(

*t*) be a continuous-time function, whose Laplace transform is

$$\begin{aligned} {\mathscr {L}} g(t) = \frac{G_c(s)}{s} \; . \end{aligned}$$

(5.121)

Then the

*z* transform of

*g*(

*t*), evaluated at the sampling instants,

\(t = 0, h, 2h, \dots \) is

$$\begin{aligned} {\mathscr {Z}} g(t) = \sum _{k=0}^{\infty } z^{-k} g(kh) \; . \end{aligned}$$

(5.122)

First-order-hold (FOH) sampling occurs when the input signal is interpolated linearly between the sampling points. This can be seen as a ZOH sampling of an integrator followed by the system

\(G_c(s)\). Due to the ZOH block, the output of the integrator will vary linearly with time. In order to get the correct dynamics, the presence of the integrator is compensated for after the ZOH sampling. Mathematically, the discrete-time system can in this case be computed according to

$$\begin{aligned} G_d (z) = \frac{z-1}{h} {\mathscr {S}}_\mathrm{ZOH} \left( \frac{1}{s} G_c(s) \right) \; . \end{aligned}$$

(5.123)

The transfer function in (

5.123) has normally

*equal* polynomial degrees in numerator and denominator. In contrast,

\(G_d(z) \) in (

5.118) has lower degree in the numerator than in the denominator.

A further possibility to perform time discretization is to assume that the input signal is band-limited. Details are developed in
Pintelon and Schoukens (2001).