Abstract
In this chapter we elaborate on the characterization of finite \(L_2\)-gain for state space systems, continuing on the general theory of dissipative systems.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsNotes
- 1.
Or assuming the existence of coordinates in which g(x) is constant, which is, under the assumption that \({\mathrm {rank}}g(x)=m\), equivalent to the Lie brackets of the vector fields \(g_1, \ldots , g_m\) defined by the columns of g(x) to be zero [233].
- 2.
The typical situation being that each row in the \(\mathcal {A}\) matrix contains only one 1. Multiple occurrence of ones in a row is allowed but will imply an equality constraint on the corresponding outputs, leading to algebraic constraints between the state variables.
- 3.
The subsequent argumentation directly extends to the case of non-differentiable storage functions, replacing the differential dissipation inequalities by their integral counterparts.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Copyright information
© 2017 Springer International Publishing AG
About this chapter
Cite this chapter
van der Schaft, A. (2017). \(L_2\)-Gain and the Small-Gain Theorem. In: L2-Gain and Passivity Techniques in Nonlinear Control. Communications and Control Engineering. Springer, Cham. https://doi.org/10.1007/978-3-319-49992-5_8
Download citation
DOI: https://doi.org/10.1007/978-3-319-49992-5_8
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-49991-8
Online ISBN: 978-3-319-49992-5
eBook Packages: EngineeringEngineering (R0)