Circle Criterion for Multi-Order Fractional System Control

The paper investigates the asymptotic stability for the control of systems described by multi-order fractional differential equations. By utilizing generalized Lyapunov functions and the Kalman–Yakubovich–Popov lemma, frequency-domain criteria are derived to evaluate asymptotic stability. The formulated criteria are similar to the ‘Popov Circle Criterion,’ but the circle parameters are determined by the system’s fractional order and the control parameters. Additionally, the asymptotic stability condition requires that all polar plots associated with the multi-fractional-order system lie inside the circle defining the criterion. Human–Robot System applications highlight the investigation techniques and the particularities of the presented criteria.