Abstract
As described in the previous Chaps. 3 and 4, (cyclo-)passive systems are defined by the existence of a storage function (nonnegative in case of passivity) satisfying the dissipation inequality with respect to the supply rate \(s(u,y)=u^Ty\). In contrast, port-Hamiltonian systems, the topic of the current chapter are endowed with the property of (cyclo-)passivity as a consequence of their system formulation. In fact, port-Hamiltonian systems arise from first principles physical modeling. They are defined in terms of a Hamiltonian function together with two geometric structures (corresponding, respectively, to power-conserving interconnection and energy dissipation), which are such that the Hamiltonian function automatically satisfies the dissipation inequality.
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Notes
- 1.
As before, \(\frac{\partial H}{\partial x}(x)\) denotes the column vector of partial derivatives of H.
- 2.
However, asymptotic feedback stabilization using discontinuous or time-varying feedback may still be possible.
- 3.
Note that the function \(\widehat{H}_{\bar{x}}\) admits the following geometric interpretation. Consider the surface in \(\mathbbm {R}^{n+1}\) defined by H, and the tangent plane at the point \((\bar{x}, H(\bar{x}) \in \mathbbm {R}^n\) to this surface. Then \(\widehat{H}_{\bar{x}}(x)\) is the vertical distance above the point \(x \in \mathbbm {R}^n\) from this tangent plane to the surface.
- 4.
The matrix J in the Hamiltonian refers to the inertia of the generators; not to be confused with the Poisson structure.
- 5.
Since \(K(x) + K^T(x) \ge 0\) the condition (6.103) is automatically satisfied in case K does not depend on x.
- 6.
We may also allow F and E to be \(l' \times l\) matrices with \(l' \ge l\), and satisfying (6.121). This is called a relaxed kernel representation.
- 7.
The Whitney sum of two vector bundles with the same base space is defined as the vector bundle whose fiber above each element of this common base space is the product of the fibers of each individual vector bundle.
- 8.
The minus sign is inserted in order to have a consistent power flow convention.
- 9.
Note that \(y_c\) serves as an input to (6.169) and \(u_c\) as an output, contrary to the intuitive use of a PID-controller, where \(u_c\) equals the output of the plant system and \(-y_c\) is the input applied to the plant system. This is of course caused by the fact that the D-action involves a differentiation.
- 10.
Since otherwise the same analysis can be performed on each connected component of the graph.
- 11.
If the mapping \(q \mapsto \frac{\partial H}{\partial q}(q,0)\) is surjective, then there exists for every \(\bar{f}_b\) such a \(\bar{q}\) if and only if \({{\mathrm{im\,}}}E \subset {{\mathrm{im\,}}}D_s\).
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van der Schaft, A. (2017). Port-Hamiltonian 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_6
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DOI: https://doi.org/10.1007/978-3-319-49992-5_6
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