Page 223 - Modern Control of DC-Based Power Systems

Control Approaches for Parallel Source Converter Systems 187

major breakthrough in the field happened with the introduction of state-
space solutions to H 2 and H N problem. An elegant formulation of the
theory through the works of Doyle [70] enabled the application of the
technique in a more practical sense. The H N controller found its first avi-
ation application with the VSTOL (vertical short takeoff and landing) air-
craft model through the work of Hyde in 1995 [70].
The H N controller does not make assumptions on the statistics. The
basic philosophy of the H N approach is to perform minimization of a
cost function for the worst-case disturbance input and hence the H N
approach is a mini-max optimization problem where the approach is pes-
simistic. These generalities that the H N assumes about nature of the noise
makes it more robust towards noise and uncertainties. This is of supreme
importance for futuristic MVDC grids where one deals not only with fast
dynamics but also with the presence of noise of unknown statistics. Apart
from the influence of noise, the nonlinearities of power converters are
also an important issue especially with the trend of increase in switching
frequency of converters. The H N approach allows modeling of paramet-
ric uncertainties and frequency-dependent uncertainties arising due to
small signal approximations. The application of the H N control problem
to the stabilization of DC link in power electronic converters is investi-
gated in [71].

5.7.2 Preliminaries

Firstly, we define a few mathematical preliminaries before going into
norm optimization based control synthesis. A notation for state-space to
transfer function transformation is defined as follows:

A B 21
:¼ CsI2AÞ B 1 D (5.209)
C D
GsðÞ :¼ ð
A few definitions highlighting the important matrix norms and the
notation of Riccati and Hamiltonian operators are described in a stan-
dardized way which will enable the reader to understand detailed theory
from other literature.

5.7.2.1 H 2 Norm
The H 2 spaces are set of all matrix values functions G where the H 2
norm is bounded and analytic in open right-half plane. The H 2 norm is
defined as follows: