Port-Hamiltonian formulation and symplectic discretization of plate models Part I: Mindlin model for thick plates

The port-Hamiltonian formulation is a powerful method for modeling and interconnecting systems of different natures. In this paper, the port-Hamiltonian formulation in tensorial form of a thick plate described by the Mindlin–Reissner model is presented. Boundary control and observation are taken into account. Thanks to tensorial calculus, it can be seen that the Mindlin plate model mimics the interconnection structure of its one-dimensional counterpart, i.e. the Timoshenko beam. The Partitioned Finite Element Method (PFEM) is then extended to both the vectorial and tensorial formulations in order to obtain a suitable, i.e. structure-preserving, finite-dimensional port-Hamiltonian system (pHs), which preserves the structure and properties of the original distributed parameter system. Mixed boundary conditions are finally handled by introducing some algebraic constraints. Numerical examples are finally presented to validate this approach

Exponential stability of a class of boundary control systems

We study a class of partial differential equations (with variable coefficients) on a one dimensional spatial domain with control and observation at the boundary. For this class of systems we provide simple tools to check exponential stability. This class is general enough to include models of flexible structures, traveling waves, heat exchangers, and bioreactors among others. The result is based on the use of a generating function (the energy for physical systems) and an inequality condition at the boundary. Furthermore, based on the port Hamiltonian approach, we give a constructive method to reduce this inequality to a simple matrix inequality

On the port-Hamiltonian representation of systems described by partial differential equations

We consider infinite dimensional port-Hamiltonian systems. Based on a power balance relation we introduce the port-Hamiltonian system representation where we pay attention to two different scenarios, namely the non-differential operator case and the differential operator case regarding the structural mapping, the dissipation mapping and the in/output mapping. In contrast to the well-known representation on the basis of the underlying Stokes-Dirac structure our approach is not necessarily based on using energy-variables which leads to a different port-Hamiltonian representation of the analyzed partial differential equations.Comment: A definitive version has been published in ifac-papersonline.ne

Port Hamiltonian formulation of infinite dimensional systems I. Modeling

In this paper, some new results concerning the modeling of distributed parameter systems in port Hamiltonian form are presented. The classical finite dimensional port Hamiltonian formulation of a dynamical system is generalized in order to cope with the distributed parameter and multivariable case. The resulting class of infinite dimensional systems is quite general, thus allowing the description of several physical phenomena, such as heat conduction, piezoelectricity and elasticity. Furthermore, classical PDEs can be rewritten within this framework. The key point is the generalization of the notion of finite dimensional Dirac structure in order to deal with an infinite dimensional space of power variables

Structure Preserving Spatial Discretization of a 1-D Piezoelectric Timoshenko Beam

In this paper we show how to spatially discretize a distributed model of a piezoelectric beam representing the dynamics of an inflatable space reflector in port-Hamiltonian (pH) form. This model can then be used to design a controller for the shape of the inflatable structure. Inflatable structures have very nice properties, suitable for aerospace applications, e.g., inflatable space reflectors. With this technology we can build inflatable reflectors which are about 100 times bigger than solid ones. But to be useful for telescopes we have to achieve the desired surface accuracy by actively controlling the surface of the inflatable. The starting point of the control design is modeling for control. In this paper we choose lumped pH modeling since these models offer a clear structure for control design. To be able to design a finite dimensional controller for the infinite dimensional system we need a finite dimensional approximation of the infinite dimensional system which inherits all the structural properties of the infinite dimensional system, e.g., passivity. To achieve this goal first divide the one-dimensional (1-D) Timoshenko beam with piezoelectric actuation into several finite elements. Next we discretize the dynamics of the beam on the finite element in a structure preserving way. These finite elements are then interconnected in a physical motivated way. The interconnected system is then a finite dimensional approximation of the beam dynamics in the pH framework. Hence, it has inherited all the physical properties of the infinite dimensional system. To show the validity of the finite dimensional system we will present simulation results. In future work we will also focus on two-dimensional (2-D) models.

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

The port-Hamiltonian (pH) theory for distributed parameter systems has developed greatly in the past two decades. The theory has been successfully extended from finite-dimensional to infinite-dimensional systems through a lot of research efforts. This article collects the different research studies carried out for distributed pH systems. We classify over a hundred and fifty studies based on different research focuses ranging from modeling, discretization, control and theoretical foundations. This literature review highlights the wide applicability of the pH systems theory to complex systems with multi-physical domains using the same tools and language. We also supplement this article with a bibliographical database including all papers reviewed in this paper classified in their respective groups

Boundary port Hamiltonian control of a class of nanotweezers.

International audienceBoundary controlled-port Hamiltonian systems have proven to be of great use for the analysis and control of a large class of systems described by partial differential equations. The use of semi-group theory, combined with the underlying physics of Hamiltonian systems permits to prove existence, well-possessedness and stability of solutions using constructive techniques. On other hand, the differential geometric representation of these systems has lead to finite dimension approximation methods that conserves physical properties such as the interconnection structure and the energy. These results are applied to the modelling and control of a class of nanotweezers used for DNA-manipulation. The Nanotweezer may be modelled as a flexible beam interconnected with a finite dimensional dynamical system representing the manipulated object. A boundary controlled-port Hamiltonian model for the ensemble and an exponentially stabilizing controller are proposed. A geometric approximation scheme is used to reduce the infinite dimensional system and numerical simulations of the closed-loop system presented