Formulation of A State Equation Including Fractional-Order State Vectors

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 their applications. Exemplary are the CRONE controller and the $PIλDμ$ controller, which is categorizable into applications of fractional calculus in classical control theory. A state equation can be called as 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 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 a 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 to model a controlled system with fractional dynamics but also for design and implementation of a controller to control fractional-order states. After introducing the basic parts, the benefits of modern control theory including robust control theories, such as $H∞$ and $μ$-analysis and synthesis in their integrities, can be applied to this fractional-order state equation.