Sigma-Point Kalman Filter with Nonlinear Unknown Input Estimation via Optimization and Data-Driven Approach for Dynamic Systems

Most works on joint state and unknown input (UI) estimation require the assumption that the UIs are linear; this is potentially restrictive as it does not hold in many intelligent autonomous systems. To overcome this restriction and circumvent the need to linearize the system, we propose a derivative-free UI sigma-point Kalman filter (SPKF-nUI), where the SPKF is interconnected with a general nonlinear UI estimator that can be implemented via nonlinear optimization and data-driven approaches. The nonlinear UI estimator uses the posterior state estimate, which is less susceptible to state prediction error. In addition, we introduce a joint sigma-point transformation scheme to incorporate both the state and UI uncertainties in the estimation of SPKF-nUI. An in-depth stochastic stability analysis proves that the proposed SPKF-nUI yields exponentially converging estimation error bounds under reasonable assumptions. Finally, two case studies are carried out on a simulation-based rigid robot and a physical soft robot, i.e., the robots made of soft materials with complex dynamics, to validate the effectiveness of the proposed filter on nonlinear dynamic systems. Our results demonstrate that the proposed SPKF-nUI achieves the lowest state and UI estimation errors when compared to the existing nonlinear state-UI filters.