Adaptive observer design for a class of nonlinear fractional-order Lipschitz systems with unknown time-varying parameters

Abstract The confluence of nonlinearity, unavailable states, and unknown time-varying parameters poses profound estimation challenges in fractional-order dynamical systems. This paper presents a novel adaptive observer design for nonlinear fractional-order Lipschitz systems with unknown, slowly time-varying parameters. Drawing on recent advancements in fractional-order calculus, a rigorous stability analysis is conducted, deriving the updating law and formulating the observer’s viability and stability conditions in terms of linear matrix inequalities (LMIs) and linear matrix equalities (LMEs). The proposed observer ensures the stability of both state observation and parameter estimation errors, along with the asymptotic convergence of the observation error norm square mean value to zero. Empirical results from a case study on a fractional-order financial system validate the efficacy of the proposed observer, thereby advancing the field of states and parameters estimation theory for non-integer order nonlinear systems.