**1.4 Construction of this chapter**

2 Magnetohydrodynamics

systems from a microscopic viewpoint, e.g., as molecular dynamics systems. Moreover, their stable points are not always unique and vary according to the environment. Multi-physics and multi-scaling models are becoming increasingly significant in science and engineering because of rapid advances in computational devices and micromachining technology. However, such complexities have tended to be ignored in system modeling of conventional control designs, because controllers have to be simple enough to be integrated with other mechanisms and be quickly adjustable. Moreover, numerical analyses using more detailed models can be executed off-line by trial and error and in circumstance where there are no physical size limitations on the computational devices. Hence, it would be desirable to have a new framework of simple control designs like PID controls, but for complex systems. The port-Hamiltonian system, which is introduced in this chapter, is one of the most promising frameworks for this purpose. This chapter addresses the issue of how to derive simple and versatile controls for partial differential equations (PDEs), especially, those of MHD, from considerations about the storage

Port-Hamiltonian systems are a framework for passivity-based controls. *Passivity* (Van der Schaft, 2000) is a property by which the energy supplied from the outside of systems through input/output variables can be expressed as a function of the stored energy. The storage function is equivalent to a Hamiltonian in dynamical systems. The collocated input/output variable pairs, called *port variables*, are defined systematically in terms of port-Hamiltonian systems, and they are used as controls and for making observations. Passivity-based controls consist of *shaping Hamiltonians* and *damping assignments*. The Hamiltonians of these systems can be changed by "connecting" them to other port-Hamiltonian systems by means of the port variables. The Hamiltonian of controlled systems is equal to the sum of those of the original system and controllers. Thus, if we can design such a changed Hamiltonian beforehand, the connections give the Hamiltonian of the original system "shaping". Such connected port-Hamiltonian systems with a shaped Hamiltonian can be stabilized to the minimum of

The energy preserving properties of port-Hamiltonian systems can be described in terms of a *Dirac structure* (Van der Schaft, 2000; Courant, 1990), which is the generalization of symplectic and Poisson structures (Arnold, 1989). Dirac structures enable us to model complex systems as port-Hamiltonian representations, e.g., distributed parameter systems with nonlinearity (Van der Schaft and Maschke, 2002), systems with higher order derivatives (Le Gorrec et al., 2005; Nishida, 2004), thermodynamical systems (Eberard et al., 2007), discretized distributed systems (Golo et al., 2004; Voss and Scherpen, 2011) and their coupled systems. This chapter mainly uses the port-Hamiltonian representation of PDEs for boundary controls based on passivity, i.e., the DPH system. The boundary integrability of DPH systems is derived from a *Stokes-Dirac structure*(Van der Schaft and Maschke, 2002), which is an extended Dirac structure in the sense of Stokes theorem. Because of this boundary integrability, the change in the internal energy of DPH systems is equal to the energy supplied through port variables defined on the boundary of the system domain. Hence, passivity-based controls for distributed

the storage function by adding dissipating elements to the port variables.

parameter systems can be considered to be boundary energy controls.

and dissipation of energy in port-Hamiltonian systems.

**1.3 History of topic and relevant research**

In Section 2, we derive the geometric formulation of MHD defined by using differential forms (Flanders, 1963; Morita, 2001). After that, we rewrite the model in terms of DPH systems. The modeling procedure is systematically determined by a given Hamiltonian. Next, we explain passivity-based controls that can be applied to the DPH system of MHD, and their energy flows by means of the bond graph (Karnopp et al., 2006). Finally, we show that the boundary power balance equation of the DPH system is the extended energy principle of MHD (Wesson, 2004) in the sense of dynamical systems and boundary controls.

In Section 3, we extend the DPH model of MHD to include non-Hamiltonian subsystems corresponding to external force terms in Euler-Lagrange equations. Actual controlled systems represented by MHD might be affected by model perturbations, e.g., disturbances or other controllers, or model improvements. Such variations cannot always be modeled in terms of Hamiltonian systems. Some systems of PDEs can be decomposed into a Hamiltonian subsystem, which we call an *exact subsystem*, and a non-Hamiltonian subsystem, which we call a *dual-exact subsystem* (Nishida et al., 2007a). Through this decomposition, a PDE system can be described as a coupled system consisting of a port-Hamiltonian subsystem determined by a *pseudo potential* and other subsystems representing, e.g., external forces, dissipations and distributed controls.

In Section 4, we derive a boundary observer for detecting symmetry breaking (Nishida et al., 2009) from the DPH system of conservation laws associated with MHD. For example, Hamiltonian systems can be regarded as the conservation law with a symmetry that is the invariance of energy with respect to the time evolution. If a symmetry is broken, the associated conservation law becomes invalid. Symmetry breaking can be detected by checking whether quantities are conserved with the boundary port variables of the DPH system. Furthermore, we present a basic strategy for detecting the topological transitions of the domain of DPH systems. The formulation using differential forms defined on Riemannian manifolds can describe systems affected by such transitions. We use a general decomposition of differential forms on Riemannian manifolds and of vector fields on three-dimensional Riemannian manifolds and derive the boundary controls for creating a desired topological energy flow from this decomposition.

The last section is devoted to a brief introduction of future work on this topic.
