Discrete-Time Model

Let the system be described by the difference equation as

y(k) + a1y(k — 1) + ••• + any(k — n)

= b0u(k) + b1u(k — 1) + ••• + bmu(k — m) (2.3)

Using the time-shift operator [8], we obtain

q—1y(k) = y(k — 1)

A(q!) = 1 + aiq 1 +———– + n

B(q—1) = bo + biq~l H– h bmq~m

Thus, we have A(q—1)y(k) = B(q—1)u(k) and applying the Z-transformation we get the discrete TF (DTF/pulse TF or shift-form TF) for the discrete time-linear dynamic system:

b0 + b1Z 1 H——- H bmz m

1 + a1Z-1 +——- + a„z~"

Here, z is a complex variable, m, n (n > m) are the orders of the respective polyno­mials. The discrete Bode diagram can also be obtained. The roots of the denominator polynomial will give modes of the system in terms of frequency and damping ratio, which are important parameters to be ascertained from real flight data as first-cut results (Chapter 9). However, these are in the z domain and not in the continuous­time domain. For proper and formal interpretation, one should transform these into s-domain parameters. One major limitation of this approach is that there is no apparent correspondence between the numerical values of the coefficients of the DTF/SFTF and the corresponding CTTF of the real system. However, this corres­pondence is very closely obtained if the delta-operator form of the TF (DFTF) is employed for modeling and estimation as discussed in the sequel.

Example 2.4

Obtain the Bode diagram for the pulse TF [2] given as

Solution

We use the function dbode([3 4 0], [1 -1.2 0.45 -0.05], 1) to obtain/plot the Bode diagram for the given discrete-time system. Here, sampling time (interval) is chosen as 1. Change the sampling time to 0.1, 0.01 and obtain the Bode diagrams (Figure 2.8).

As the sampling time reduces, the frequency range increases.