Page 224 - Modern Control of DC-Based Power Systems

188 Modern Control of DC-Based Power Systems

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1 ð N
:G: 5 Tr G jωðÞGjωðÞ dω (5.210)
H
2 2π 2N
H
where G refers to Hermitian of G (complex conjugate transpose).

5.7.2.2 H N Norm
The H N spaces are set of all matrix values functions G where the H N
norm is bounded and analytic in open right-half plane. It is the set of all
stable and proper rational transfer functions referred to as RH N. The
H N norm is defined as follows:
:G: (5.211)
N :¼ sup σGðjωÞ
ω
The upper singular value represented by σ is determined by singular
value decomposition (SVD) of a matrix. The supremum operator denoted
by sup denotes the peak σ as the frequency ω is varied over the entire
range. Bisection algorithm is used to compute the infinity norm in an
efficient manner [69]. In both H 2 and H N theory, state-space formulation
is done by solving Algebraic Riccati Equations (ARE). Hence, we define
a Hamiltonian matrix (5.212) to denote the ARE of the form (5.213).

A R
(5.212)
2Q 2A
H :¼ H
H
A X 1 XA 1 XRX 1 Q 5 0 (5.213)
The stabilizing solution X of the Riccati equation is defined by the
Riccati operator (Ric) on the Hamiltonian H, which basically solves the
eigenvalue problem in Eq. (5.212).

(5.214)
X :¼ RicðHÞ

5.7.3 Linear Fractional Transformation
Consider the generalized plant P and the controller K as shown in
Fig. 5.53. The key idea behind norm based optimal control is to

w z
P
u v
K
Figure 5.53 Lower LFT structure.