Asymptotic Stabilisation of Distributed Port-Hamiltonian Systems by Boundary Energy-Shaping Control

Abstract

This paper illustrates a general synthesis methodology of asymptotic stabilising, energy-based, boundary control laws, that is applicable to a large class of distributed portHamiltonian systems. Similarly to the finite dimensional case, the idea is to design a state feedback law able to perform the energy-shaping task, i.e. able to map the open-loop portHamiltonian system into a new one in the same form, but characterised by a new Hamiltonian with a unique and isolated minimum at the equilibrium. Asymptotic stability is then obtained via damping injection on the boundary, and is a consequence of the La Salle's Invariance Principle in infinite dimensions. The general theory is illustrated with the help of a simple concluding example, i.e. the boundary stabilisation of a transmission line with distributed dissipation.

Control by interconnection beyond the dissipation obstacle of finite and infinite dimensional port-Hamiltonian systems

“…Then, then same control obtained by relying on as energyshaping approach discussed e.g. in [26] can be obtained thanks to the control by interconnection strategy.…”

“…Due to the dissipative structure, (27) implies that there are no Casimir functions in closed-loop that allow to properly shape the Hamiltonian. On the other hand, it has been illustrated in [26] that there exists a boundary state-feedback law thanks to which it is possible to overcome the dissipation obstacle and obtain, in closed-loop, an energy function H(q, p) + C(q, p) with the desired stability properties. The function C satisfies (28), that now becomes ∂ ∂z δC δp (q, p) = 0 ∂ ∂z δC δq (q, p) + g δC δp (q, p) = 0 (36)…”

Twenty years of distributed port-Hamiltonian systems: a literature review

“…In this way the boundary control is not generated implicitly by means of Casimirs of the closed-loop system but directly as feedback control law. The methodology is applied to the whole class of (4.1) in Macchelli et al (2015a). In Macchelli (2015) the dissipation obstacle is overcome in a different way: by defining a new passive output and applying control by interconnection to the new input-output pair of the dpH system it was shown how to overcome the dissipation obstacle by means of Casimir generation for the new closed-loop-system.…”

Energy-based modeling and control of interactive aerial robots

“…In this way the boundary control is not generated implicitly by means of Casimirs of the closed-loop system but directly as feedback control law. The methodology is applied to the whole class of (3.4) in Macchelli et al [2015a]. In Macchelli [2015], the dissipation obstacle is overcome in a different way: by defining a new passive output and applying control by interconnection to the new input-output pair of the dpH system it was shown how to overcome the dissipation obstacle by means of Casimir generation for the new closed-loop-system.…”