Abstract
In this chapter, we will exploit the port-Hamiltonian structure for control, going beyond passivity. We will mainly concentrate on the problem of set-point stabilization. Section 7.1 focusses on control by interconnection, by attaching a controller port-Hamiltonian system to the plant port-Hamiltonian system. Section 7.2 takes a different perspective by emphasizing direct shaping of the Hamiltonian and the structure matrices by state feedback. Other control opportunities will be indicated in Sect. 7.3; see also the Notes at the end of this chapter.
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Notes
- 1.
“Local” or “global”; we leave this open for flexibility of exposition.
- 2.
Indeed, suppose we have r Casimirs \(C_i(x,\xi ), i=1, \ldots ,r\), where the partial Jacobian matrix \(\frac{\partial C}{\partial \xi }(x,\xi )\) of the map \(C: \mathcal {X}\times \mathcal {X}_c \rightarrow \mathbbm {R}^r\) with components \(C_i\) has full rank r. Then by an application of the Implicit Function theorem the level sets \(C_1(x,\xi )=c_1, \ldots , C_r(x,\xi )=c_r\) for constants \(c_1,\ldots ,c_r\), can be equivalently described by level sets of functions of the form (7.14).
- 3.
Note that in this case the extended resistive structure matrix \(\begin{bmatrix} R(x)&P(x) \\ P^T(x)&S(x) \end{bmatrix}\) is a minimal rank extension of R(x), since its rank is equal to the rank of R(x) (\(=n\)).
- 4.
In view of (7.46) this means that the input vector fields \(g_j\) are “Leibniz vector field” with respect to the Leibniz structure \(J(x)-R(x)\) and potential functions \(-F_j\). Note that the same assumption was made in (6.106) in the context of generating Lyapunov functions for port-Hamiltonian systems driven by constant inputs.
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van der Schaft, A. (2017). Control of 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_7
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DOI: https://doi.org/10.1007/978-3-319-49992-5_7
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