By Henry M. Paynter

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By setting as interconnection u˜ = u¯ and y˜ = y¯ , we by construction have that ∂H ∂x lim H0 (x) = lim H (x) = ∞ ⇒ x → ∞ t→∞ t→∞ which proves instability of the coupled system having a state diverging. The previous proof is simple and reasonably straightforward, but the theorem’s implications are far reaching. First of all, the theorem is general and nonlinear. This means that, if a controlled robot is not passive, it is possible to construct an environment, maybe by a second controlled robot, which would be passive and if connected to the original robot would result in an unstable system.

We proved that the resulting interconnection is again a Dirac structure with interconnection losses. Consider a network of N routers, each router being modeled as a port-Hamiltonian system. We have the corresponding Dirac structures Di , i = 1, . . , N . Assume that D1 is a source sending packets, at a constant rate η(t), to the destination D N . Further, assume that each Di , i = 2, . . , N − 1 also receive traffic from other sources. 7, by Li j , i = j, i = 1 : N − 1, j = 2 : N . , L = i=N , j=N −1 Li j .

H. Ramírez, Y. Le Gorrec, A. Macchelli, H. Zwart, Exponential stabilization of boundary controlled port-Hamiltonian systems with dynamic feedback. IEEE Trans. Autom. Control 59(10), 2849–2855 (2014) 23. H. Rodriguez, A. van der Schaft, R. Ortega, On Stabilization of Nonlinear Distributed Parameter Port-Controlled Hamiltonian Systems via Energy Shaping, in Proceedings of the 40th IEEE Conference on Decision and Control (CDC 2001), vol. 1, 2001, pp. 131–136 24. M. Schöberl, A. Siuka, On Casimir functionals for infinite-dimensional port-Hamiltonian control systems.

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Analysis and design engineering systems by Henry M. Paynter

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