Zonotopic Polynomial Set-Membership Fusion Estimation for Nonlinear Systems via Carleman Approximation: An Encoding-Decoding Scheme

In this paper, the zonotopic set-membership estimation (SME) problem is addressed for a class of nonlinear systems subjected to an encoding-decoding scheme, where the measurement information of each sensor is encoded and then transmitted to the remote fusion center. The objective of this paper is to develop a zonotope-based weighted measurement fusion method to fuse the received decoding signals, and to design a zonotopic SME algorithm that fully utilizes the higher-order partial derivatives of the nonlinear functions based on the fused signal. A zonotope-based weighted measurement fusion (WMF) method is proposed by means of the full rank decomposition technique, which enables the fusion of the decoding signals by solving a weighted least squares problem. To design the desired SME algorithm, the nonlinear system is first transformed into a linear time-varying system using the Carleman approximation technique. Then, based on the fused signal, a Kalman-type estimator is constructed and the zonotopes encompassing the prediction error and the estimation error are recursively calculated. The estimator parameter is obtained by minimizing the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$F$</tex-math></inline-formula>-radius of the zonotope enclosing the estimation error at each time instant. Furthermore, effect of the system smoothness level on the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$F$</tex-math></inline-formula>-radius is intensively analyzed. It is shown that incorporating the higher-order partial derivatives into the design of the SME algorithm enables the extraction of additional state constraints, and that the number of extractable constraints increases monotonically with the smoothness level. A method is subsequently proposed to leverage these constraints in order to improve the estimation accuracy. Moreover, the WMF method is proven to provide an equivalent estimation accuracy as compared to the most commonly used parallel fusion method while possessing lower computational complexity. Finally, two simulation experiments are conducted to demonstrate the efficacy and utility of the SME method.