For multi-Global Navigation Satellite System (GNSS) constellations, the Geometric Dilution of Precision (GDOP) is an important parameter utilised for the selection of satellites. This paper has derived new formulae to describe the change of GDOP. The result shows that, for GNSS single point positioning solutions, if one more satellite belonging to the existing tracked multi-GNSS constellation used in the single point positioning solution is added, the GDOP always decreases with the number of the added satellites. On the other hand, when the constellation of the added satellite is not from the tracked existing constellations, the different numbers of the added satellites have different influences on the change of GDOP. Generally, adding one satellite from another constellation into the existing multi-GNSS constellations will increase the GDOP, but adding two satellites will decrease the GDOP compared with adding one from another constellation. Additionally, the GDOP also increases in the cases of adding two satellites from two different constellations into the tracked existing constellations.

Grid-connected converter with LCL-filter (LCL-GCC) plays an important role in renewable power generation systems. However, existing control techniques face several challenges in practice (e.g., difficulty in parameter design, slow dynamic response, and poor robustness under slowly time-varying filter parameter, dead-time of power devices and external disturbances). To solve these difficulties, a passivity-based fractional-order sliding-mode control (PBC-FOSMC) hybrid controller, which combines the merits of PBC and FOSMC, is proposed, where the inputs of the passive controller are the outputs of the FOSMC controller. First, a passive current controller is designed based on the Euler-Lagrange (EL) model established by damping injection according to the passivity-based control (PBC) theory, which can make the system automatically converge to meet the energy dissipation law. Secondly, a FOSMC controller is designed to further enhance the system robustness to counter disturbances and compensate the reference current accuracy of PBC. The introduced fractional order can successfully suppress the undesired chattering due to the switching behaviors in conventional SMC. Thirdly, the hybrid PBC-FOSMC controller is derived and system stability is analyzed. Finally, experimental results under 10-kW prototype validate the excellent performance of the proposed method in terms of robustness, dynamic performance, strong perturbation rejection ability, etc., and the desired control targets are achieved.

This paper develops a direct analytical framework for synchronizing and controlling fractional-order octonion-valued fuzzy bidirectional associative memory neural networks (FOOVFBAMNNs). Octonion algebra is neither commutative nor associative, which limits the application of standard analytical tools. To address this challenge, we first propose a generalized Cauchy–Schwarz inequality tailored to the octonionic domain, which operates directly without relying on system decomposition. This inequality lays the groundwork for a Lyapunov-based stability analysis that retains the system’s inherent geometric structure to avoid decomposition into real-valued components. Based on this framework, we derive concise 2-norm inequality criteria, which are sufficient to guarantee Mittag-Leffler synchronization of the proposed model. We also employ a Particle Swarm Optimization (PSO) algorithm to systematically optimize the flexible parameters in the generalized inequality, enhancing the practical performance of the synchronization scheme. To validate the effectiveness of the proposed method, we apply it to a multi-domain image restoration task. Numerical experiments verify the performance of our method. In terms of Peak Signal-to-Noise Ratio (PSNR), the octonion-valued network with PSO-tuned parameters achieves better results than its non-optimized counterpart as well as models constructed in complex or quaternion domains.

For single point positioning solutions with multi-Global Navigation Satellite System (GNSS) constellations, the Position Dilution Of Precision (PDOP) and Time Dilution Of Precision TDOP) are quality measures to specify the additional multiplicative affect of measurement error on positioning accuracy and the timing accuracy, respectively. Considering the dimension of the unknown vector (including the three-dimensional positioning information of the receiver as well as the receiver clock biases related to these single constellations), this paper theoretically derives new formulae to describe the change of PDOP and TDOP. In addition, the detailed expressions of the variable quantity of the PDOP and TDOP are also obtained. The results show that if the dimension of the unknown vector is invariant in case of removing one or more satellites belonging to the existing tracked multi-GNSS constellations, both the PDOP and TDOP increase. On the other hand, if removing satellites reduces the dimension of the unknown vector, the PDOP neither increases nor decreases. However, the change of the TDOP is different, and it becomes smaller.