Page 161 - Modern Control of DC-Based Power Systems

Control Approaches for Parallel Source Converter Systems 125

During operation the system is influenced by a dynamic disturbance
M i ðtÞ:

(5.37)
dM i ðtÞ
dt 5 f mi ðxÞ
The first step of the procedure is defining a macrovariable (ψ)asa
function of the state variables:
(5.38)
ψ 5 ψ xðÞ
The characteristic of the macrovariable can be selected in accordance
to the control specifications which can be either regulating or stabilizing
the output of the system. The definition of the macrovariable defines
how the system behaves once it reaches the manifold. For each input
channel a macrovariable is needed.
It has been shown in [19], that (5.39) is a solution of the optimum
problem whose objective function J is given in (5.40).
_
Tψ 1 ψ 5 0 (5.39)
ð 1 t
_
J 5 Lðt; ψ; ψÞ (5.40)
t 0
where {t 0 , t 1 } are the initial and final time, and L is defined as:

Lt; ψ; ψÞ 5 ψK Kψ 1 ψ ψ (5.41)
T
T
ð
T is a design parameter that specifies the time it takes the macrovari-
able to converge to zero or evolve into the manifold, which can also be
viewed as the speed at which the system variable reaches the manifold.
Substituting (5.36) and (5.38) into (5.39) leads to:
@ψ @ψ
_
T ψ 1 ψ 5 T f ðx; uÞ 1 ψ 5 0 (5.42)
@x @x
By defining an appropriate macrovariable and choosing the parameter
T, the control output can be derived from (5.42).
There is no unique way of constructing the manifold. The manner in
which the manifold is constructed depends on the type of problem (regu-
lation or tracking) and determines the quality of the performance of the
controller [23].