State and disturbance simultaneous estimation for a class of nonlinear time-delay fractional-order systems

This article addresses the problem of estimating simultaneously the state and unknown disturbance of one-sided Lipschitz fractional-order systems with time-delay. The nominal models of nonlinearities are assumed to satisfy both the one-sided Lipschitz condition and the quadratically inner-bounded condition. Different from the state observer reported in the literature, which only dealt with one-sided Lipschitz integer-order time-delay systems or nonlinear fractional-order time-delay systems where the nonlinear function satisfying Lipschitz condition, the state observers in this article can be applied to a wide class of nonlinear time-delay systems (one-sided Lipschitz fractional-order time-delay systems and one-sided Lipschitz integer-order time-delay systems). We employ the Razumikhin stability theorem and a recent result on the Caputo fractional derivative of a quadratic function to derive a sufficient condition for the asymptotic stability of the observer error dynamic system. The stability condition is obtained in terms of linear matrix inequalities, which can be effectively solved using the MATLAB LMI Control Toolbox. Two examples are provided to show the effectiveness of the proposed design approach.