Page 225 - Modern Control of DC-Based Power Systems

Control Approaches for Parallel Source Converter Systems 189

synthesize class of controller K which must satisfy conditions of internal
stability and secondly minimize a specific matrix norm of interest. The
controller input is the error signal v and its output is the control signal u,
whilst w and z are exogenous inputs and outputs respectively. The
input output relation could be mapped through the P matrix from
Eq. (5.215).

z w P 11 ðsÞ P 12 ðsÞ w
5 (5.215)
v 5 PsðÞ u P 21 ðsÞ P 22 ðsÞ u
In order to establish a direct mapping between w and z, one needs to
eliminate the second row of the Eq. (5.215). The steps involved to elimi-
nate u are as follows:
u 5 Kv.v 5 K 21 u (5.216)
v 5 P 21 w 1 P 22 u

K 21 u 5 P 21 w 1 P 22 u
ðK 21 2 P 22 Þu 5 P 21 w (5.217)
21
.u 5 ðK 21 2P 22 Þ P 21 w
21
u 5 KðI2P 22 KÞ P 21 w
Substituting the equation for u in the equation of z (first row), we get
the following result:
21
z 5 P 11 w 1 P 12 u 5 P 11 w 1 P 12 KðI2P 22 KÞ P 21 w (5.218)
21
.z5ðP 11 1P 12 KðI2P 22 KÞ P 21 Þw
The above rule is called as lower Linear Fractional Transformation
(LFT) of K with P. The lower LFT basically describes the transfer func-
tion T zw from exogenous inputs w to exogenous outputs z. By using the
MIMO rule, one can express the exogenous outputs in terms of exoge-
nous inputs to derive the generalized plant P and this is explained in the
next section.
z 5 F l ðP; KÞw (5.219)

21
F l P; Kð Þ 5 ðP 11 1P 12 KðI2P 22 KÞ P 21 Þ (5.220)

5.7.3.1 H 2 Optimal Control Problem
The H 2 optimal control statement is then to find a controller K which
minimizes the 2-Norm of the transfer function T zw .