An Initial Value Problem on an Adaptive Fractional-Order Unscented Particle Filter for a Nonlinear Continuous-Time Fractional-Order System with Unknown Parameters

An adaptive unscented particle filter (UPF) with the initial value compensation (IVC) is proposed for a nonlinear continuous-time fractional-order system (FOS) with unknown parameters. The nonlinear continuous-time FOS is discretized by the Grünwald-Letnikov difference, and the corresponding difference equation is obtained. In addition, the augmented vector is proposed to establish the augmented equations with respect to the states, the initial values and the unknown parameters. The unscented transformation is used for each sampled particle, and a systematic resampling algorithm is used to optimize the particles. Besides, the proposed adaptive fractional-order UPF can simultaneously estimate the states and unknown parameters. Compared with the adaptive fractional-order UPF without IVC, the proposed adaptive fractional-order UPF with IVC has a higher accuracy for estimating states if the fractional-order of the nonlinear continuous-time FOS is relatively small in the interval (0,1). Finally, an experiment is used to verify the effectiveness of the proposed algorithm.