Development of Generalized Aeroservoelastic Reduced Order Models (2009)


The pursuit of increasingly advanced vehicles creates a demand for a strong capability for generating vehicle math models. This demand is well exhibited through the example of high speed transport aircraft design. Significant study conducted during the NASA High Speed Research (HSR) program identified the importance of performing active control for flutter suppression, gust load alleviation, and ride quality enhancement. These requirements apply across the transonic flight regime, making the problem significantly more challenging. Ongoing research is aimed at developing and testing active control laws for this class of vehicle. To enable design of these control laws, state space representations of the vehicle’s aeroservoelastic behavior must be created. Methods for generating reduced order state space models for aeroservoelastic control law design are therefore in high demand.

Existing methods for generating state space models can be categorized into test based methods and analysis based methods. Test based methods are capable of calculating state space models directly from transfer function data collected in test. While the direct use of test data offers high accuracy, this data is not available during the vehicle design process and cannot be easily recomputed for changes in configuration. Methods based on analysis data have been developed for varying levels of analysis fidelity. Systems using potential flow have generally been based on the Doublet-Lattice Method as seen in numerous analytical tools and applications. This approach is can be extended to use corrected unsteady aerodynamic data to generate models that are linearized about high fidelity data points. More recently, generation of highly accurate ROMs has been performed directly from Euler or Navier-Stokes CFD data. Conversion of ASE data to state space models can be accomplished by approximating the unsteady aerodynamic content using Rational Function Approximation (RFA) or by matching the aeroelastic behavior directly using the P-Transform technique. The RFA approach has the drawback of generating a high-order model, and model reduction is often employed to create a lower order ROM. Neither the PTransform approach nor the reduced models from the RFA approach retain the physical consistency of the model states. Specifically, the meaning of a given term in the state space matricies is not consistent for calculations at various flight conditions. As such, interpolation of the state space model among flight conditions is not valid.

Using a conventional approach such as Roger’s RFA initially generates a high order state space model. In order to convert this model into a suitable form for control law synthesis and evaluation, the model order must be reduced. A powerful method for performing this reduction is conversion of the state space model into a balanced realization followed by truncation. A balanced realization is a means of ordering the modes of a model based on the modes that are the most observable and the most controllable. This ordering is accomplished by the calculation of the controllability and observability Gramians. Once the modes are ordered in this way order reduction is accomplished  by truncating the modes which are the least observable and controllable.

Development of the Generalized Reduced Order Model generation (GROM) system fills a need for an approach that can be queried across multiple flight conditions. The current implementation of this method is based on Doublet-Lattice unsteady aerodynamic approach to which correction factors can be applied. Conversion of aerodynamic data to state space form is accomplished though the use of Roger’s Rational Function Approximation. The significant divergence from previous approaches lies in the reduction of the high-order state space model. In this approach, the controllability and observability Gramians are calculated across a range of flight conditions. These quantities are then summed resulting in an indication of the  significance of a given mode to a range of flight conditions. Using these summations, the model is consistently converted into a realization that is balanced in a generalized sense. Due to the fact that the same transformation is applied to the state space models for each flight condition, the models maintain consistency. This consistency has been enforced through the structure inherent in assembling the full order state space model and preserved through the use of a common transformation matrix. It is in this form that the insignificant states are truncated leaving a family of reduced order models that can be interpolated or curve fit.