On Non-Commensurate Fractional-Order System Control

The control systems for models described by non-commensurate fractional-order differential equations are based on their transformation into large commensurate-order systems, which impose difficulties in determining control laws. In this context, in this paper, a new control method for this class of systems is proposed. The results obtained are based on Lyapunov methods for differential equations with fractional exponents and on the application of the Yakubovich–Kalman–Popov lemma adapted for this class of systems. The stability criterion is presented as a frequency criterion and represented graphically by familiar frequency plots similar to those of the Nyquist or Popov type. If the parameters that define the model can be defined in a closed domain, the frequency criterion can be interpreted as “Popov’s circle criterion”. The two numerical applications present two important cases. The first studies the stability criterion in the case where the viscosity coefficients determine non-commensurate fractional-order exponents in the dynamic model of the system. The second example studies the complex problem of the human–machine system in which the human model imposes dynamics determined by non-commensurate fractional-order systems. The proposed investigation methods allow for a reduction in computational effort by several orders of magnitude for non-commensurate fractional-order systems, eliminate stability conditions that use matrix-based criteria for large-scale systems, and introduce standard frequency-domain criteria.