Page 104 -

2.9 Advanced Stability Results 85

1. All elements of T(s) have no poles in the region Re(s)>0,
2. Any poles of T(s) on the jw axis are simple with positive-definite residues,
and
3. The matrix He[T(s)] is positive semidefinite for Re(s)>0.

EXAMPLE 2.9–2: PR Matrices
Consider the matrix

This matrix is PR as can be checked.

DEFINITION 2.9–3 An m×m matrix T(s) of proper real rational functions
which is not identically zero is strictly-positive-real or SPR if

1. All elements of T(s) have no poles in the region Re(s) 0, and
2. The matrix He[T(s)] is positive definite for Re(s)>0.

EXAMPLE 2.9–3: SPR Matrices

Consider the matrix

This matrix is SPR as can be checked.

Lure’s Problem

Consider the following feedback interconnected system

(2.9.3)

Copyright © 2004 by Marcel Dekker, Inc.