Abstract
In this chapter we focus on passive systems as an outstanding subclass of dissipative systems, firmly rooted in the mathematical modeling of physical systems.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
With thanks to Anders Rantzer for a useful conversation.
- 2.
In fact, in Sect. 9.4 we will see how this can be extended to general systems \(\Sigma \).
- 3.
This can be extended to \(u=-Dy +v,\) with D a matrix satisfying \(D+D^T>0\).
- 4.
Note that this storage function does not have an interpretation in terms of physical energy. It is instead a function that is directly related to the geometry of the dynamics (4.48) on \(S^3\), integrating \(\omega \).
- 5.
In fact, balancedness of a communication or flow Laplacian matrix implies that all connected components are strongly connected; cf. [70].
- 6.
Note that the secant function is given as \(\sec \, \phi = \frac{1}{\cos \, \phi }\).
- 7.
Not to be confused with the Laplacian matrix of the previous section; too many mathematicians with a name starting with “L.”
- 8.
A simple proof runs as follows (with thanks to J.W. Polderman and I.M.Y. Mareels). Take for simplicity \(n=1\). Then, since \({d\over dt} e^2(t)=2e(t)\dot{e}(t),~ e^2(t_2)-e^2(t_1) = 2\int _{t_1}^{t_2} e(t)\dot{e}(t) dt ~\le ~ \int _{t_1}^{t_2}[e^2(t)+\dot{e}^2(t)]dt \rightarrow 0\) for \(t_1,t_2 \rightarrow \infty \). Thus for any sequence of time instants \(t_1,t_2, \ldots , t_k, \ldots \) with \(t_k \rightarrow \infty \) for \(k \rightarrow \infty \) the sequence \(e^2(t_i)\) is a Cauchy sequence, implying that \(e^2(t_i)\) and thus \(e^2(t)\) converges to some finite value for \(t_i,t \rightarrow \infty \), which has to be zero since \(e\in L_2(\mathbbm {R})\).
- 9.
Note that in this case the subscript \({}{\bar{x}}\) does not refer to differentiation.
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). Passive State Space Systems. 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_4
Download citation
DOI: https://doi.org/10.1007/978-3-319-49992-5_4
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-49991-8
Online ISBN: 978-3-319-49992-5
eBook Packages: EngineeringEngineering (R0)