Page 239 - Modern Control of DC-Based Power Systems

Control Approaches for Parallel Source Converter Systems 203

Eq. (5.246) describes the dynamics of the control loop during the slid-
ing motion. It has to be highlighted that the state variable in this case is
given by the algebraic Eq. (5.245) and the equation is therefore omitted.
The differential Eq. (5.246) are no longer dependent on the parameters
of the controlled system. This means that the control loop (5.246) is
robust against parameter fluctuations of the controlled system. It is also
noteworthy that the system order has decreased by one degree to order
n 2 1 and the coefficients r t of the switching surface (5.244) form the
coefficients of the characteristic polynomial of the linear dynamics (5.246)
in the case of the sliding state.

5.8.3 Proof of Robustness
The main advantage of SMCs is their robustness against variation of the
control loop parameters ΔA or external disturbances d(t), if these appear
in the system description:

_ x 5 A 1 ΔAÞx 1 bu 1 d (5.247)
ð
This means that the dynamics of the closed loop during the sliding
motion depend as in Eq. (5.246), only from the parameters of the switch-
ing surface (5.242). The dynamics are independent of ΔA and d(t). The
robustness therefore applies if both of the following conditions are met
[23]:
1. There exists a vector p, such that ΔA 5 bp T
2. There exists an αðtÞ, so that d tðÞ 5 bαðtÞ.
For example, if the system is cab written in the controllable canonical
T
form, the condition of ΔA 5 bp can be met if only the coefficients a i
vary.
2 3 2 3
0 1 0 ? 0 0
0 0 1 ? 0 0
6 7 6 7
6 7 6 7
_ x 5 6 ^ ^ ^ & ^ 7 x 1 ^ u (5.248)
6 7
6 7 6 7
0 0 0 ? 1 0
4 5 4 5
? 1
2a 0 2a 1 2a 2 2a n21
5.8.4 Application to DC DC Converters
In this section we apply the SMC to the buck converter supplying a resis-
tive load (Fig. 5.65).