Model reduction of aeroelastic systems in the Loewner framework

Kurzfassung

Model order reduction (MOR) techniques have found many applications in the field of aeroelasticity, which considers the interaction between the motion of a structure and the aerodynamic flow around it. After spatial discretization of the governing nonlinear partial differential equations (PDEs) together with appropriate boundary conditions a set of nonlinear ordinary differential equations (ODEs) containing typically millions of degrees of freedom is obtained. The specific models available to solve the set of nonlinear ODEs describing the unsteady aerodynamic flow are typically formulated in the frequency domain. The Loewner framework is applied here in order to obtain a complete reduced aeroelastic model for a transport aircraft flying in the transonic regime. The model includes the effect of control surfaces in order to alleviate the loads experienced by the airframe when encountering atmospheric disturbances. A residualization of the non-proper part of the aerodynamic transfer function results in the appearance of the time derivative of the control surface command. The minimal order model size produced by the Loewner approach allows for the design of an optimal controller, for which a specific method that explicitly considers the time derivative of the control signal has been developed. Applications showing the combination of the Loewner framework together with the optimal design controller for loads alleviation are presented. In order to analyze the applicability of recent extensions of the Loewner framework for nonlinear systems to weak nonlinear aerodynamic phenomena, the quasi one-dimensional Euler equations describing the unsteady compressible flow in a nozzle are considered next. Due to the specific formulation of the boundary conditions the system cannot be formulated as having a bilinear dependency on the input. The analytical expression for the second order Volterra kernel is derived, showing that the functional series expansion of second order is able to capture weak nonlinear behaviour exhibited by the unsteady compressible flow, motivating the application of the Loewner framework for the description of weak nonlinear aerodynamic phenomena.