The objective of the proposed research is to develop a feedforward neural network-based performance evaluator and controller, with the propose of quantifying and enhancing the robustness of dynamical systems subjected to bounded external disturbances and input constraints. The robustness of a dynamical systems is characterized by the smallest robust forward invariant set (RFIS), which represents the best performance under external disturbance, as well as the largest robust region of attraction (RRA), which defines the largest domain from which the system state could be driven to the internal smallest RFIS. Finding the RFIS and RRA of a high-dimensional dynamical system is a challenging task, while most of existing works in the literature only provide conservative estimations of these two performance metrics and suffer from the curse of dimensionality. Besides, this work also focuses on the controller design based on neural network, aiming to further improve the robustness by shrinking the smallest RFIS and enlarging the largest RRA for systems with input constraints. The contribution of this work is to use FNN to provide less conservative boundaries of both the smallest RFIS and the largest RRA, even when employing Lyapunov-based techniques. This methodology avoids simulating the system trajectories, which significantly reduces the computational burden for high-dimensional systems. The controller design also leverages the neural network-based method which provides a generalized framework for different dynamical systems. We will also design the performance enhancement controller for the proposed method in our lab-designed platform, the Georgia Tech Miniature Autonomous Blimp via inverted hovering in both simulator and actual blimp.