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Hardware In the Loop Implementation and Challenges 251

7.3 DISCRETIZATION

The result of each time-step of the real time simulator can be used
by the digital controller to calculate the control output. These measure-
ments can be, generally, analog or digital signals. In both cases, the digital
controller samples the measurement signals and calculates the control out-
put that is sent to the digital simulator, forming the HiL setup.
Considering a continuous-time system in the form:

(7.1)
_ x 5 fx t ðÞ; ut ðÞð Þ
where the input u is mapped into the state x by means of Σ:u-x, which
represents the plant.
With the definition of the time sample T, the continuous-time system
can originate the equivalent discete-time system, by using the samples
obtained every T seconds. The resulting system has the following
form [1]:
(7.2)
ð
ð
xk 1 1Þ 5 fx kðÞ; ukðÞÞ
The discretization process is described in Fig. 7.2, consisting of the
blocks H; Σ; S; C.
• The block S, representing the sampler, produces the discrete output
sequence of samples xkTÞ 5 xðkÞ, obtained from the input x: The
ð
sampling is generally performed with regular intervals, determined by
the sampling period T. As it can be observed from Fig. 7.2, the block S

u(kT) u(t) x(t) x(kT)

k t t k
H Σ S

C

Figure 7.2 The discretization process.