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A local theory of inputoutput stability of dynamical systems文档
A Local TheoryofInput/Output Stability of
Dynamical Systems
by H O R A C I O MARQUEZ~
Department of Enyineering, Royal Roads Military College, F M O Victoria,
British Columbia, VOS 1BO, Canada
ABSTRACT: Thepurpose of this paper is topresent a local theory of input/output stability OJ
dynamical systems. More precisely, the intention is to obtain local open loop conditionsfor
local closed loop stability offeedback interconnections. It will be shown that, by setting the
problem in the context of extended binormed spaces it ispossible to obtain local versions of
the so-called small 9ain theorem and circle criterionfor stability of nonlinear systems. The
results clearly determine the extent of input~output stability, i.e. the set of input.[hnctions./br
which stability is 9uaranteed.
L Introduction
In this paper we are concerned with stability of dynamical systems. Roughly
speaking, there are two conceptually different approaches to stability theory: (i)
Lyapunov theory, and (ii) the input/output approach. The former studies the
stability of an equilibrium state of the autonomous, or unforced system, while the
latter regards systems as mappinys between inputs and outputs. The Lyapunov
theory was initiated by A. M. Lyapunov in the 1890s and it is summarized in several
textbooks [see for example (1-3)], while the input/output theory was initiated in
the 1960s by G. Zames (4, 5), and I. Sandberg (6, 7). See also references (8, 9) for
thorough expositions of the input/output theory. Intrinsic in the philosophical
differences between these two approaches is the fact that, while the Lyapunov
theory is es
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