Well-Posedness and Stability of Infinite-Dimensional Linear Port-Hamiltonian Systems with Nonlinear Boundary Feedback

Abstract

Boundary feedback stabilisation of linear port-Hamiltonian systems on an interval is considered. Generation and stability results already known for linear feedback are extended to nonlinear dissipative feedback, both to static feedback control and dynamic control via an (exponentially stabilising) nonlinear controller. A design method for nonlinear controllers of linear port-Hamiltonian systems is introduced. As a special case the Euler-Bernoulli beam is considered.

Stabilization of port-Hamiltonian systems by nonlinear boundary control in the presence of disturbances

“…What we are interested in here is the stabilization of such a system S by means of dynamic boundary control, that is, by coupling the system to a dynamic controller S c that acts on the system only via the boundary points a, b of the spatial domain (a, b). Since realistic controllers often exhibit nonlinear behavior (due to nonquadratic potential energy or nonlinear damping terms, for instance), we want to work with nonlinear controllers -just like [1], [27], [59], [42]. Since, moreover, realistic controllers are typically affected by external disturbances, we -unlike [1], [27], [59], [42] -also want to incorporate such actuator disturbances which corrupt the output of the controller before it is fed back into the system.…”

“…Since realistic controllers often exhibit nonlinear behavior (due to nonquadratic potential energy or nonlinear damping terms, for instance), we want to work with nonlinear controllers -just like [1], [27], [59], [42]. Since, moreover, realistic controllers are typically affected by external disturbances, we -unlike [1], [27], [59], [42] -also want to incorporate such actuator disturbances which corrupt the output of the controller before it is fed back into the system. Coupling such a controller to the system S by standard feedback interconnection In this paper, we establish the input-to-state stability for the closed-loop systemS w.r.t.…”

“…In [54], [1], [27], [59], [42], no disturbances are considered at all, that is, the situation depicted above is considered in the special case where d = 0. Stabilization of port-Hamiltonian systems of various degrees of generality is achieved by means of linear dynamic boundary controllers in [54] and by means of nonlinear dynamic boundary controllers in [1], [27], [59], [42]. In [51], sensor -instead of actuator -disturbances are considered, that is, disturbances do occur in [51] but they corrupt the input of the controller instead of its output.…”

Well-posedness of infinite-dimensional linear systems with nonlinear feedback

“…The paper (Augner, 2016) provides conditions on a nonlinear boundary feedback interconnected with a linear port-Hamiltonian system to determine a nonlinear contraction semigroup. Even if those nonlinearities comprise some classes of locally Lipschitz continuous functions, wellposedness in the sense of (Tucsnak and Weiss, 2014) is not addressed for the closed-loop system.…”

“…In this work, we introduce a more general class of closedloop well-posed systems composed of a well-posed linear Email addresses: anthony.hastir@unamur.be (Anthony Hastir), f.califano@utwente.nl (Federico Califano), h.j.zwart@utwente.nl, h.j.zwart@tue.nl (Hans Zwart) infinite-dimensional system whose input to output map is coercive for small times interconnected with static and monotone nonlinear feedback, which includes the class of locally Lipschitz continuous functions considered in (Augner, 2016). This paper is organized as follows.…”