Robust D--stability analysis of fractional order interval systems of commensurate and incommensurate orders

This study focuses on the robust $\mathcal {D}$D-stability analysis of fractional order interval systems (FOISes). The concept of interval means that the coefficients of the systems transfer functions are uncertain parameters that each adopts a value in a real interval. Initially, some new bounds on the poles of the FOISes are produced so that they reduce the computational burden in the case of the ${\theta }$theta-stability. Then, the concept of the value set is extended to analyse the robust $\mathcal {D}$D-stability of the FOISes, and a new necessary and sufficient condition is presented. The value set of the FOISes is obtained analytically, and based on it an auxiliary function is introduced to check the condition. The obtained results are applicable to systems of both commensurate and incommensurate orders. Moreover, it is perceived that if a family of a FOIS is of commensurate order, then the robust ${\theta }$theta-stability can be checked by checking the ${\theta }$theta-stability of a finite number of family members, i.e. a generalisation of Kharitonov's theorem. Finally, the presented theorems are applied to the control system of a space tether system.