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2.9 Advanced Stability Results 87

1. For any symmetric, positive-definite Q, there exists a symmetric, positive-
definite P solution of the Lyapunov equation

T
A P+PA=-Q
2. The matrices B and C satisfy

T
C=B P

The MKY Lemma gives conditions under which a transfer matrix has a
degree of robustness. Note that the conditions depend on both the input and
output matrices and thus a particular system may be SPR for a certain choice
of input/output pairs and not SPR for others. A modified version of the KY
lemma which relaxes the condition of controllability is given next.

LEMMA 2.9–2: Meyer-Kalman-Yakubovitch Given vector b, an
asymptotically stable A, a real vector v, scalars γ 0 and >0, and a positive-
definite matrix Q, then, there exist a vector q and a symmetric positive definite
P such that

1.

2.

if and only if

1. is small enough and,
-1
T
2. the transfer function γ/2+v (sI-A) b is SPR

In most of our applications, q=0. These lemmas find many applications in
nonlinear systems. It is usually possible to divide a nonlinear system into a
linear feed-forward subsystem, and a nonlinear, passive feedback. The
challenge is then to make the linear subsystem SPR so that the stability of the
combined system is guaranteed.

Copyright © 2004 by Marcel Dekker, Inc.