**7. References**


30 Will-be-set-by-IN-TECH

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Notwithstanding, as we have already mentioned in the end of Sec. 5.1, the possibility for the onset of a Kelvin–Helmholtz instability of kink waves running along soft X-ray coronal jets should not be excluded – at high enough flow speeds, which in principal are reachable, one can expect a dramatic change in the waves' behaviour associated with an emerging instability,

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quiet corona."

**7. References**


**0**

**7**

<sup>1</sup>*RIKEN*

*Japan*

<sup>2</sup>*Nagoya University*

**Hamiltonian Representation of**

Gou Nishida1 and Noboru Sakamoto2

**Energy Controls**

**Magnetohydrodynamics for Boundary**

This chapter shows that basic boundary control strategies for *magnetohydrodynamics* (MHD) can be derived from a formal system representation, called a *port-Hamiltonian system* (Van der Schaft and Maschke, 2002). The port-Hamiltonian formulation clarifies collocated input/output pairs used for stabilizing and assigning a global stable point. The controls called *passivity-based controls* (Arimoto, 1996; Ortega et al., 1998; Van der Schaft, 2000; Duindam et al., 2009) are simple and robust to disturbances. Moreover, port-Hamiltonian systems can be connected while keeping their consistency with respect to energy flows. Finally, we show that port-Hamiltonian systems can be used for boundary controls. In the future, this theory might be specialized, for instance, in order to control disruptions of Tokamak plasmas (Wesson, 2004; Pironti and Walker, 2005; Ariola and Pironti, 2008). This chapter emphasizes the versatility of

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

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

adjustable online, although the resulting performance is not exactly optimal.

**1. Introduction**

**1.1 Brief summary of this chapter**

control system representations.

**1.2 Background and motivation**

