Page 191 - Modern Control of DC-Based Power Systems

Control Approaches for Parallel Source Converter Systems 155

The system would be stable if there were no modeling errors or dis-
turbances are present and the virtual disturbance estimate is perfect. This
would mean that the set-point trajectory generator alone could output
the control signal which drives the system to its equilibrium point.
Since in real applications this is not the case, an additional LQR con-
troller is added for optimal state feedback. The goal of this control strat-
egy is the minimization of a cost function that includes the states as well
as the control signal. By assuming that the controller will operate for a
longer period than the transient time of the optimal gains [50], the LQG
controller is supposed to minimize the cost function for an infinite time
interval.
The steady-state solution of the following cost function (J i Þ has to be
found:

ð N
2 2
J i 5 E Q V 2V nom Þ 1 R d2d e Þ dt (5.119)
y
ð
ð
0
Eq. (5.119) can be transformed into the general cost function form in
[33]:
ð N
T T
J 5 E ð x2x e Þ Qx 2 x e Þ 1 d2d e Þ R d 2 d e Þ dt (5.120)
ð
ð
ð
0
T
Where Q 5 C QC is a symmetric, positive-semidefinite weighting
matrix and R becomes a positive weighting scalar.
By using the feedback gain matrix of the LQR on the state estimates
of the Kalman filter a state feedback which minimizes the optimization
problem can be obtained:
21
T
d 52 R B Px 52 K ^x (5.121)
P is found from the steady-state solution of the Riccati differential
equation which satisfies the algebraic Riccati equation:

21 T
T
PA 1 A P 2 PBR B P 1 Q 5 0 (5.122)

5.4.2.2 Augmented Local Kalman Filter
The design will be performed in two consecutive steps. First, the distur-
bance I d in the system is simply included as noise rather than as a state
and the Kalman filter is introduced to estimate the inductor current of
the buck converter. Second, this Kalman filter is augmented to include