Concurrent estimation and control of robotic systems remains an ongoing challenge, where controllers rely on data extracted from states/parameters riddled with uncertainties and noises. Framework suitability hinges on task complexity and computational constraints, demanding a balance between computational efficiency and mission-critical accuracy. This study leverages recent advancements in neuromorphic computing, particularly spiking neural networks (SNNs), for estimation and control applications. Our presented framework employs a recurrent network of leaky integrate-and-fire (LIF) neurons, mimicking a linear quadratic regulator (LQR) through a robust filtering strategy, a modified sliding innovation filter (MSIF). Benefiting from both the robustness of MSIF and the computational efficiency of SNN, our framework customizes SNN weight matrices to match the desired system model without requiring training. Additionally, the network employs a biologically plausible firing rule similar to predictive coding. In the presence of uncertainties, we compare the SNN-LQR-MSIF with non-spiking LQR-MSIF and the optimal linear quadratic Gaussian (LQG) strategy. Evaluation across a workbench linear problem and a satellite rendezvous maneuver, implementing the Clohessy-Wiltshire (CW) model in space robotics, demonstrates that the SNN-LQR-MSIF achieves acceptable performance in computational efficiency, robustness, and accuracy. This positions it as a promising solution for addressing dynamic systems' concurrent estimation and control challenges in dynamic systems.