Distributed Cascaded Spline-Based Adaptive Graph Filters for Nonlinear Systems: Design and Performance Analysis

The distributed nonlinear adaptive graph filter (DNAGF) is developed with the single nonlinear graph filter model (NGFM) to handle streaming datasets. However, the current DNAGFs tend to underperform when predicting unknown nonlinear dynamic systems. This suboptimal performance is due to their reliance on a single NGFM and the network's limited computational burden. To address these issues, two novel cascaded DNAGFs considering the spline interpolation method, i.e. a distributed Wiener spline adaptive graph filter (DWSAGF) and distributed Hammerstein spline adaptive graph filter (DHSAGF), are proposed to improve the capacity for nonlineaprediction in this paper. By utilizing piecewise low-order nonlinear spline functions, the proposed DWSAGF and DHSAGF can adapt locally to improve the fitting of the predicted nonlinear system to the unknown one. In DWSAGF and DHSAGF, the cascaded architectures containing linear and nonlinear subsystems are employed, which are more flexible than the single NGFM. Particularly, since DHSAGF has a memory through the constructed matrix ${\boldsymbol {\bar{U}}}_{m}^{\bar{t}}(r)$, it generates higher performance than DWSAGF for complex or time-varying nonlinear systems. In addition, the detailed performance analyses regarding DWSAGF and DHSAGF in the mean and mean-square senses are presented. Simulations are exhibited to validate the theoretical analysis and to show the performance superiorities of DWSAGF and DHSAGF.