Page 294 - Modern Control of DC-Based Power Systems

Hardware In the Loop Implementation and Challenges 253

The discrete form of the LTI plant is obtained as:

xk 1 1ÞTÞ 5 e AT xkTÞ
ð
ð
ð
1 Ð ð k11ÞT Ak11ðð ÞT2τÞ (7.6)
e
kT dτBu kTÞ
ð
5 e AT xkTÞ 1 Ð T Aτ
0 e dτBuðkTÞ
ð
Therefore, the Eq. (7.6) allows the adoption of the discrete-time sys-
tem description:
(7.7)
ð
xk 1 1Þ 5 FxðkÞ 1 GuðkÞ
The procedure for obtaining the discrete system can be performed in
Matlab by means of the command c2d [2].
In the case of nonlinear system Σ, the exact discrete model is usually
impossible to find, given that the nonlinear differential equation is almost
impossible to solve. Therefore, the discrete nonlinear model can be
obtained only by applying an approximation. Among the different meth-
ods, which have different degree of approximation, the simpler and more
popular method is the Euler approximation [1]. This is based on the fact
that the time derivative of a variable x can be written as:

dx
_ x 5 5 lim xt 1 ΔTÞ 2 xðtÞ xt 1 TÞ 2 xðtÞ (7.8)
ð
ð
dt ΔT-0 ΔT T
The application of (7.8) in the differential Eq. (7.1) results in:
(7.9)
ð
xk 1 1Þ xkðÞ 1 TUfx kðÞ; ukðÞÞ
ð
The Euler approximation is a general method that can be applied also
in the case of a linear system. A general rule is that the time constant T
must be small enough not to bring the system to instability.
The Euler approximation can be applied to the nonlinear controllers
previously described, which means applying (7.9) to the control output
calculated in continuous time.
In the case of using the LQR or the LQG control strategies, the stan-
dard control theory describes the procedure for obtaining the equivalent
discrete form of the two controllers, based on the calculation of the dis-
crete form of the Riccati Equation [3]. Matlab integrates these function-
alities with the functions lqrd [4] and kalmd [5] and lqg [6], if the system is
already in the discrete form.