Design and Analysis of Control Systems by Humberto Stein Shiromoto

Author:Humberto Stein Shiromoto
Language: eng
Format: epub
Publisher: Springer International Publishing, Cham

(3.6)

holds, for every . Also, there exist functions and such that the condition

(3.7a)

implies

(3.7b)

Interconnecting systems (3.1) and (3.5) yields the system

(3.8)

Using vectorial notation, system (3.8) is denoted as . A solution to (3.8) starting at and evaluated at time t is denoted as Y(t, y). The set of inequalities (3.4) and (3.7) are rewritten, respectively, as

(3.9a)

and

(3.9b)

The ISS-Lyapunov gains are defined as the functions and .

A sufficient condition to ensure that the equilibrium of the origin is globally asymptotically stable for system (3.8) is provided by the small-gain theorem. This result is roughly stated here, the interested reader may consult [17, Theorem 3.1] for further details.

If the ISS-Lyapunov gains and are such that the small-gain condition

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