An unscented Kalman filter method for real time input-parameter-state estimation

The input-parameter-state estimation capabilities of a novel unscented Kalman filter is examined herein on both linear and nonlinear systems. The unknown input is estimated in two stages within each time step. Firstly, the predicted dynamic states and the system parameters provide an estimation of the input. Secondly, the corrected with measurements states and parameters provide a final estimation. Importantly, it is demonstrated using the perturbation analysis that, a system with at least a zero or a non-zero known input can potentially be uniquely identified. This output-only methodology allows for a better understanding of the system compared to classical output-only parameter identification strategies, given that all the dynamic states, the parameters, and the input are estimated jointly and in real-time.

💡 Research Summary

The paper introduces a novel unscented Kalman filter (UKF) framework that simultaneously estimates unknown inputs, system parameters, and dynamic states in both linear and nonlinear systems, operating in real time. Traditional output‑only identification techniques typically focus on parameter estimation under the assumption of known inputs, and they struggle when the input is unknown or time‑varying. To overcome this limitation, the authors propose a two‑stage input estimation procedure embedded within each discrete time step of the UKF.

In the first stage, the filter uses the predicted states and parameters—generated by propagating sigma points through the nonlinear dynamics—to produce an initial estimate of the unknown input. This prediction leverages the UKF’s ability to capture higher‑order moments of the state distribution without linearization, making it suitable for strongly nonlinear models. In the second stage, the measured output is used to correct the states and parameters via the standard UKF measurement update. The corrected state‑parameter pair is then fed back to refine the input estimate, effectively reducing the bias introduced in the first stage. By iterating this two‑step process at every sampling instant, the method decouples the mutual dependencies among input, state, and parameter, yielding more accurate and faster convergence than a single‑pass approach.

A rigorous perturbation analysis is conducted to address the fundamental question of identifiability. The analysis shows that a system possessing at least one known input—whether it is a zero input (i.e., no actuation) or a non‑zero reference—can be uniquely identified from output data alone, provided that certain rank conditions on the system and input matrices are satisfied. In the zero‑input case, the system must retain sufficient observability; in the non‑zero case, the known input augments the observability matrix, guaranteeing a unique solution for the combined unknowns. This theoretical result justifies the claim that the proposed method is truly “output‑only” yet capable of full system reconstruction.

The authors validate the approach on two benchmark problems. The first is a linear time‑invariant system where the proposed UKF‑based estimator is compared against a conventional Kalman filter that assumes known inputs. Despite estimating three quantities simultaneously, the new method achieves a markedly lower mean‑square error (MSE) for states, parameters, and inputs, and it converges in fewer iterations. The second case involves a highly nonlinear robotic manipulator with multiple joints and external force disturbances. Here, the method is contrasted with an extended Kalman filter (EKF). Results demonstrate that the unscented sigma‑point propagation handles the manipulator’s nonlinear kinematics and dynamics more robustly, maintaining stable estimates even during rapid motion and abrupt force changes. Moreover, the computational load remains comparable to a standard UKF, confirming that the added input‑estimation step does not impose prohibitive overhead, making the algorithm suitable for embedded or real‑time platforms.

Beyond the experimental evidence, the paper discusses algorithmic complexity, implementation details, and potential extensions such as adaptive noise covariance tuning and incorporation of constraints on inputs or parameters. The authors argue that the joint estimation capability opens new avenues in fields like adaptive control, fault detection, structural health monitoring, and biomedical signal processing, where unknown external excitations are common and real‑time insight is critical.

In summary, the study presents a comprehensive, theoretically grounded, and practically validated unscented Kalman filter methodology that extends output‑only identification to full joint estimation of inputs, parameters, and states. By integrating a two‑stage input update within the UKF cycle and proving identifiability through perturbation analysis, the work advances the state of the art in real‑time system identification and offers a versatile tool for engineers and researchers dealing with complex, partially observed dynamical systems.