Frequency Stability Criteria for Multivariable Fractional Order Systems

Emerging technologies and cyber–physical systems have led to the development of complex mathematical models described by differential equations with multiple fractional orders. In this regard, this paper investigates the stability of control systems for this class of models, defined by state equations with multiple fractional orders ranging between 0 and 1. Matrix criteria and comparison principle for linear and nonlinear autonomous systems of different fractional orders are developed based on generalized Lyapunov functions for differential equations with multi-order fractional exponents. The results are extended to non-autonomous linear systems or systems with nonlinear components of different fractional orders. Application of the Yakubovich–Kalman–Popov lemma, adapted for this class of systems, allows us to obtain new stability criteria presented as frequency criteria and represented graphically by familiar frequency plots similar those of the Nyquist or Popov type. Numerical applications illustrate these results, such as models of complex human–machine systems described by state equations of multivariable fractional orders. An analysis of the advantages of the proposed methods compared to procedures and techniques used in other papers regarding the study of multi-order fractional exponent systems is presented. It is demonstrated that the proposed methods minimize the computational effort required for stability criteria.