**1.2 Background and motivation**

Control theory significantly progressed during the last two decades of the 20th century. *Linear control theory* (Zhou et al., 1996) was developed for systems whose states are limited to a neighborhood around stable points. The theory was extended to include particular classes of distributed parameter systems and nonlinear systems (Khalil, 2001; Isidori, 1995). However, dispite this progress, simpler and more intuitive methods like *PID controls* (Brogliato et al., 2006) are still in the mainstream of practical control designs. One reason for this trend is that advanced methods do not always remarkably produce significant improvements to the performance of controlled systems despite their theoretical complexity; rather, they are prone to modeling errors. The other reason is that simple methods are understandable and adjustable online, although the resulting performance is not exactly optimal.

On the other hand, actual controlled systems can be regarded as distributed parameter systems from a macroscopic viewpoint, e.g., as elastic continuums, and as discrete nonlinear

**1.4 Construction of this chapter**

distributed controls.

from this decomposition.

certain constitutive relations.

conservation law,

**2.1 Ideal magnetohydrodynamical equations**

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

Hamiltonian Representation of Magnetohydrodynamics for Boundary Energy Controls 169

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

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

Magnetohydrodynamics (MHD) is a discipline involving modeling magnetically confined plasmas (Wesson, 2004; Pironti and Walker, 2005; Ariola and Pironti, 2008). The ideal MHD system is a coupled system consisting of a single fluid and an electromagnetic field with

The fluid is described by the two equations in three dimensions. The first is the mass

*<sup>∂</sup><sup>t</sup>* <sup>+</sup> ∇ · (*ρv*) = 0, (1)

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

**2. Port-Hamiltonian systems and passivity-based controls for MHD**

*∂ρ*

MHD (Wesson, 2004) in the sense of dynamical systems and boundary controls.

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 and dissipation of energy in port-Hamiltonian systems.
