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# Formulation of a State Equation Including Fractional-Order State-Vectors

[+] Author Affiliations
Masaharu Kuroda

National Institute of Advanced Industrial Science and Technology, Tsukuba, Japan

Paper No. DETC2007-35273, pp. 285-294; 10 pages
doi:10.1115/DETC2007-35273
From:
• ASME 2007 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference
• Volume 5: 6th International Conference on Multibody Systems, Nonlinear Dynamics, and Control, Parts A, B, and C
• Las Vegas, Nevada, USA, September 4–7, 2007
• Conference Sponsors: Design Engineering Division and Computers and Information in Engineering Division
• ISBN: 0-7918-4806-X | eISBN: 0-7918-3806-4

## abstract

In recent years, applications of fractional calculus have flourished in various science and engineering fields. Particularly in engineering, control engineering appears to be expanding aggressively in its applications. Exemplary are the CRONE controller and the PIλ Dμ controller, which is categorizable into applications of fractional calculus in classical control theory. A state equation can be called the foundation of modern control theory. However, the relationship between fractional derivatives and the state equation has not been examined sufficiently. Consequently, a systematic procedure referred to by every researcher on the fractional-calculus side or control-theory side has not yet been established. For this study, therefore, involvement of fractional-order derivatives into a state equation is demonstrated here for ready comprehension by researchers. First, the procedures are explained generally; then the technique to incorporate the fractional-order state-vector into a conventional state equation is given as an example of the applications. The state-space representation in this study is useful not only for modeling a controlled system with fractional dynamics, but also for design and implementation of a controller to control fractional-order states. After we complete installation of the basic parts, we can apply the benefits of modern control theory, including robust control theories such as H-infinity and μ-analysis and synthesis in their integrities, to this fractional-order state-equation.

Topics: Equations

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