Dirac structures on Hilbert spaces and boundary control of distributed port-Hamiltonian systems

Abstract

Aim of this paper is to show how the Dirac structure properties can be exploited in the development of energy-based boundary control laws for distributed port-Hamiltonian systems. Usually, stabilization of non-zero equilibria has been achieved by looking at, or generating, a set of structural invariants, namely Casimir functions, in closed-loop. Since this approach fails when an infinite amount of energy is required at the equilibrium (dissipation obstacle), this paper illustrates a novel approach that enlarges the class of stabilizing controllers. The starting point is the parametrization of the dynamics provided by the image representation of the Dirac structure, that is able to show the effects of the boundary inputs on the state evolution. In this way, energy-balancing and control by state-modulated source methodologies are extended to the distributed parameter scenario, and a geometric interpretation of these control techniques is provided. The theoretical results are discussed with the help of a simple but illustrative example, i.e. a transmission line with an RLC load in both serial and parallel configurations. In the latter case, energy-balancing controllers are not able to stabilize non-zero equilibria because of the dissipation obstacle. The problem is solved thanks to a (boundary) state-modulated source.

Introduction

This paper deals with the energy-based boundary control of distributed port-Hamiltonian systems  [1], [2]. In recents works, see e.g.  [3], [4], [5], [6], [7], [8], this task has been accomplished by looking at, or generating, a set of Casimir functions in the closed-loop system that robustly (i.e., independently from the Hamiltonian function) relates the state of the infinite dimensional port-Hamiltonian system with the state of the controller. The controller is a finite dimensional port-Hamiltonian system which is interconnected to the boundary of the distributed parameter system. The shape of the closed-loop energy function is changed by choosing the Hamiltonian of the controller e.g. to introduce a minimum in a desired configuration. This procedure is the generalization of the control by interconnection via Casimir generation developed for finite dimensional systems  [9], [10]. The result is an energy-balancing passivity-based controller that is not able to deal with equilibria that require an infinite amount of supplied energy in steady state, i.e. with the “dissipation obstacle”.

The limits of the energy–Casimir method are intrinsic, and due to the fact that Casimir functions are invariants that do not depend on the particular Hamiltonian, i.e. they are completely determined by the Dirac structure of the system  [9], [6]. Main advantage, however, is that it provides a constructive way to develop the feedback law, and to choose the closed-loop Hamiltonian with desired stability properties. Moreover, this approach is able to provide a control action without explicitly dealing with the trajectories of open and closed-loop systems. On the other hand, this is also the main problem of the method, since the controller is developed somehow independently from the “real” evolution of the state, which is in fact a “consequence” of a specific Hamiltonian.

The class of controllers can be enlarged beyond the dissipation obstacle by focusing on the trajectories that correspond to a particular Hamiltonian, rather than on the geometric structure (i.e., the Dirac structure), of the system only. Then, the regulator is developed to map the open-loop trajectories into the trajectories of a target system with (at least) a different Hamiltonian and, clearly, characterized by the desired stability properties. This is the same concept adopted for finite dimensional in case of stabilization with state-modulated sources  [10], or with the more general IDA-PBC control technique  [11].

With this in mind, it is clear that in case of boundary control of distributed port-Hamiltonian systems, the first issue is to understand the effects of the inputs on the state evolution. A possible solution is to “map” the inputs into the system dynamics, as discussed in [12]. On the other hand, this approach does not take advantage from the geometric structure associated to a port-Hamiltonian system, i.e. its Dirac structure, that is present also in the distributed parameter case. In finite dimensions in fact, the Dirac structure provides an elegant way to parametrize the system dynamics once inputs, resistive structure, and Hamiltonian have been fixed  [13]. Furthermore, as proposed in  [14] for implicit port-Hamiltonian systems (i.e., port-Hamiltonian systems described by DAEs), the control synthesis can take advantage from this parametrization: energy-balancing control, and control by state-modulated source are applied to this class of systems, and the solution explicitly determined on the basis of geometric and energetic properties.

Implicit port-Hamiltonian systems appear, for example, from the spatial discretization of distributed port-Hamiltonian systems [15], and they have been already proven to be a “bridge” between lumped and distributed parameter systems as far as the stabilization via the energy–Casimir method is concerned  [16], [17], [18]. In this paper, the approach discussed in  [14] in the implicit and finite dimensional case is extended to the distributed parameter scenario, and conditions for the existence of energy-balancing passivity-based boundary controllers are determined. Furthermore, the dissipation obstacle is tackled by formulating the control by state-modulated source in the distributed parameter case.

The starting point is the definition of Dirac structures on Hilbert spaces proposed in  [19], and in particular their kernel and image representations  [20]. Even if Dirac structures for distributed port-Hamiltonian systems (also called, Stokes–Dirac structures) do not require the space of power variables to be a Hilbert space, but just a space of e.g.  $C^{\infty }$ functions [1], their definition and description in terms of linear operators on Hilbert spaces have been adopted to heavily rely on the theory of boundary control systems in port-Hamiltonian form proposed in [21].

The general methodology is applied to a simple but illustrative example, i.e. a transmission line with RLC load in series and parallel configurations (see also  [22], [23] for some preliminary results). In the second case, energy-balance passivity-based control and energy–Casimir methods fail due to the dissipation obstacle. However, it is shown that asymptotic stability of non-zero equilibria can be achieved thanks to boundary state-modulated source. Stability in the sense of  [24] is achieved in both the cases by properly shaping the Hamiltonian function, while asymptotic stability is obtained via damping injection and proved via La Salle’s invariance principle  [25].

The paper is organized as follows. In Section  2, a brief background on Dirac structures on Hilbert spaces, and on the class of distributed port-Hamiltonian systems under investigation is given. Boundary energy-balancing control and control via state modulated sources are discussed in Sections  3 Boundary energy-balancing control, 4 Boundary control via state-modulated source, respectively. The illustrative examples are reported in Section  5, while conclusions are in Section  6.