Performing of stability analysis and control processes of complex order systems

Complex order mathematics, which generalizes the classical derivatives and integrals to degrees of complex numbers, can be considered as a generalization of fractional calculus. Studies on complex order systems are very limited and include many open problems. Because the physical meaning of these systems cannot be fully understood due to the imaginary part encountered in time responses and the inability to measure this part. To overcome this problem, the conjugate order derivative and integral operators are used. It has been observed that the real valued responses occur only when the complex order operators are paired with their conjugated orders. In this thesis, it is aimed to perform the stability analyzes and control processes of the complex conjugate order systems. The unit impulse and unit step responses of the complex order and the complex conjugate order systems are obtained and their characteristic features are investigated. The generalized modified Mikhailov (GD Mikhailov) stability method, which is a graphical method for the stability analysis of complex conjugate order systems, is adapted to these systems. The results obtained using the GD Mikhailov stability method are verified by comparing with different pole analysis based methods. The stabilization of complex conjugate order systems with PI, PID and fractional order PI controller has been performed with D-decomposition method. The robust stability of complex conjugate order systems with parameter uncertainty is investigated and it is shown that the Kharitonov theorem and the edge theorem cannot be used for such systems. The robust stability analysis of the family of complex conjugate order systems is performed using the zero exclusion principle supported by the visualization of the value sets, and also a method based on the GD Mikhailov stability criterion. Finally, the robust controller design is made for the complex conjugate order systems with parameter uncertainty. Keywords: Complex order systems, complex conjugate order systems, fractional order systems, Mikhailov stability criterion, stabilization, parameter uncertainty, value set, zero exclusion principle, robust control