Control-oriented Modeling of an Air-breathing Hypersonic Vehicle
Design and development of future high speed aircraft require the use of high fidelity modeling tools early on in the design phase to study and analyze complex aeroelastic, thermoelastic, and aerothermal interactions. Due to their high lift-to-drag ratio and low operational costs, air-breathing hypersonic vehicles have spurred some interest in the field of high speed aircraft design over the last few decades. Modeling aeroelastic effects for such an aircraft is challenging due to its tightly integrated airframe and propulsion system that leads to significant deflections in the thrust vector caused by flexing of the airframe under extreme aerodynamic and thermal loads. These changes in the orientation of the thrust vector in turn introduce low frequency oscillations in the flight path angle, which make control system design a challenging task. Inclusion of such effects in the vehicle dynamics model to develop accurate control laws is an important part of control-oriented modeling.
The air-breathing hypersonic vehicle considered here is modeled as a thin-walled structure, where deformations due to axial, bending, shear, and torsion are modeled using the six independent displacements of a rigid cross section. The modeling framework also accounts for the effect of non-structural mass as shown above. Free vibration mode shapes for the first 40 modes are computed using a novel scheme previously developed that is equipped to accurately estimate high-frequency mode shapes. These mode shapes are broadly classified into axial-transverse and lateral-torsional modes due to a natural modal decoupling due to the aircraft geometry.
The six displacements of the rigid cross-section are used to represent deformations at any point on the aircraft (shown below left), while the dimensions of the cross section change across the length of the aircraft (represented by x_1).
The figure (left) presents an overview of the AHV modeling framework, which relies on the free vibration mode shapes generated and adopts an aerodynamic model that uses oblique shock theory for surfaces experiencing compression, and Prandtl-Meyer expansion theory for
surfaces adjacent to expanding hypersonic flow. The structural deformations influence the aerodynamic forces, which are heavily coupled with the scramjet propulsion system (as shown below).
where virtual work due to internal forces equals virtual work due to external forces.
The aeropropulsive forces (external), control forces (external), structural generalized forces (internal), and inertial forces (internal) are used to compose the nonlinear equations of motion for the aircraft using principle of virtual work,
The nonlinear equations of motion derived (above left) are linearized about the equilibrium configuration to obtain linearized equations of motion (above right). Optimal control using linear quadratic regulator (LQR) moves unstable eigen values to stability (as shown below).