# Rapid Parametric Modeling and Robust Analysis for the Hypersonic Ascent Based on Gap Metrics
## 1 Introduction  
Reusable hypersonic flight vehicles for future aerospace access will dramatically improve economy and efficiency [1]. However, such aerospace vehicles must pass through the large flight envelope from subsonic flight to hypersonic, leading to unexpected dynamic features compared with other traditional vehicles. Hence, researchers on hypersonic aerospace vehicles need to adopt a multidisciplinary integrated strategy concerning the propulsion system, aerodynamics, control, and structural dynamics [2]. The parametric modeling approach offers a favorable way for the aerospace vehicle to implement the multidisciplinary integrated design [3]. In particular, an analytical aeropropulsive/aeroelastic aerospace vehicle model was established [4], and the model involves a two-dimensional aerodynamic structure, a one-dimensional SCRAMjet-type propulsion system, and an elastic airframe [5]. Alternatively, a nonlinear longitudinal dynamical model was derived from first principles, capturing inertial coupling effects between the propulsion system, aerodynamics, and structural dynamics [6; 7]. Furthermore, a control-oriented model of the aerospace vehicle was obtained in [8] by replacing complex force and moment equations with curve-fitted approximations, and this model allows an understanding of the applicability of control-oriented techniques. In addition, a 3D simulation model of the aerospace vehicle was built [9], and intrinsic flexibility was considered with a one-dimensional Euler-Bernoulli beam representation. This model was subjected to stability analysis and control-based design. However, some problems still need to be addressed to enhance the overall performance, including the integrated design of the trajectory and vehicle shape, control-centric trade study, and multidisciplinary design optimization. Additionally, the control-centric trade studies were conducted based on a parametric model, and this work explores the parameterization types, parameter ranges, and objectives available for the analysis and optimization of the aerospace vehicle [10]. Moreover, amultidisciplinary optimization method was proposed [11], and the optimal vehicle configuration and flight trajectory are determined to accomplish the mission requirements. The parametric model provides a suitable way to change a vehicle optimized for a single design point to be more effective along a trajectory [12]. In summary, the parametric model is a fundamental issue for further studies. Furthermore, the model provides an effective tool to construct the mutual relations between the propulsion, aerodynamics, control, and structural dynamics so that the optimal trade results for the aerospace vehicle design can be easily obtained in the conceptual stage. Some control-relevant issues are revealed for hypersonic flight based on these proposed models [13], including the coupled features of the plant and trajectory [14]. In addition, hypersonic flight suffers from control system constraints, which limits the performance for trajectory tracking [15]. Moreover, the robustness of flight control is particularly significant on account of the uncertainty of the model and environment. Many robust control design methods have been developed [16; 17; 18]. Their works focused on the development of a nonlinear adaptive robust control method to deal with model uncertainty. However, few methods have been proposed for quantifying the robustness of the flight control system. A robust analysis method based on a gap metric is proposed in this paper. The benefit of the proposed method is the quantitative analysis ability to identify the boundary of the robustness, whereas the drawback is the restriction from the nonlinear system. One has to formulate a series of linearized systems to approximate the nonlinearity. The structure of the paper is organized as follows. Section 1 introduces the model of aerospace vehicles and deduces the polynomial fitting model of longitudinal dynamics for the following control design.