Page 293 - Modern Control of DC-Based Power Systems

252 Modern Control of DC-Based Power Systems

transforms the continuous-time signal in a sequence of points sepa-
rated by time intervals. Is it clear that the block represents the opera-
tion of the analog-to-digital converter.
• The block C represents the digital controller that calculates the control
output ukðÞ based on the samples xðkÞ and on the specific control algo-
rithm that uses the samples of the state. The control algorithm, usually
defined in the continuous-time, must be discretized as well in order to
be implemented in the digital controller.
• The block H represents the hold. It converts the digital control signal
sent by block C into a continuous-time signal that can be used in the
plant. The block holds the sample at time kT until the next sample is
available. The result is a piecewise constant continuous-time signal
uðtÞ, as described in Fig. 7.2. This represents the mathematical repre-
sentation of the digital-to-analog converter.
• The block Σ represents the plant, which maps the control input uðtÞ
to the state xðtÞ.
The calculation of the discrete-time system is different between the
linear time-invariant (LTI) system and the more generic nonlinear
system [1].
For the first case, the discretization follows a well-defined procedure
that is based on the exact calculation of the linear differential equation.
The LTI plant is described by the following equation:

_ x 5 fx; uÞ 5 Ax 1 Bu (7.3)
ð
The differential Eq. (7.3) is solved with the initial condition xðkTÞ at
time t 0ðÞ 5 kT and the input constant signal uðkTÞ.
The solution of Eq. (7.3) results in:
ð t
xtðÞ 5 e At2t 0 Þ xðt 0 Þ 1 e At2τÞ Bu τðÞdτ (7.4)
ð
ð
t 0
The continuous-time variables are substituted with their equivalent in
the discrete form:

8
xðtÞ 5 xððk 1 1ÞTÞ
>
>
>
>
xðt 0 Þ 5 xðkTÞ
<
t 5 ðk 1 1ÞT (7.5)
t 0 5 kT
>
>
>
>
:
uðτÞ 5 uðkTÞ