In aeroservoelastic (ASE)-stability analysis, one considers the coupling of the aerodynamic, inertial, structural, actuation, and control-system elements of the dynamics of an aircraft. The closed-loop interactions of these elements can introduce unexpected instabilities in flight if the analytical model used for synthesis and analysis is not accurate. Measures of allowable flight-condition variations, called "stability margins," should be computed to indicate the range of velocities and altitudes within which the aircraft can safely operate.

An approach known as the µ method was recently introduced for analyzing stability margins of open-loop flexible aircraft models. [The µ method was described in "Characterizing Worst-Case Flutter Margins From Flight Data" (DRC-97-03), NASA Tech Briefs, Vol. 21, No. 4 (April 1997), page 62.] This method is based on a formal mathematical concept of robustness that guarantees a level of modeling errors to which the aircraft is robustly stable. A realistic representation of errors can be formulated by describing differences between predicted responses and measured flight data. The structured singular value, µ, is used to compute a margin that is robust to these errors.

Figure 1. The Uncertainty Operators described in the text are incorporated into the mathematical model along with the nominal dynamical model of the aircraft, with a feedback-control gain matrix K, and with uncertainty-weighting operators Wadd and Win.

The µ method has now been extended to enable the evaluation of aeroservoelastic-stability margins of closed-loop, flexible aircraft models. For a given aircraft, uncertainty operators are introduced into the analysis to describe errors in the structural and aerodynamical models along with errors in the sensor and actuator models. The resulting stability margins are superior to such traditional measures as gain and phase margins, which cannot be easily interpreted as flight-condition information. Also, the extended µ method can be used to simultaneously compute closed-loop ASE stability margins and open-loop flutter stability margins.

Flight data are easily incorporated into the stability analysis in this method. Uncertainty operators are derived by model validation to ensure that the dynamics observed in the data are represented in a robust mathematical model. The stability-margin parameter, µ, is robust to the measured variations associated with the uncertainty operators. In this sense, the stability margins are worst-case margins with respect to the flight data.

Figure 2. Nominal and Robust Stability Margins — Γnom and Γrob, respectively — were calculated for closed-loop operation of the F/A-18 HARV equipped with a thrust-vectoring control subsystem.

In the extended µ method, an uncertainty description, as shown in Figure 1, is formulated for the mathematical model of a given aircraft. This description includes ΔA to account for errors in the modal parameters of the state matrix, Δin to account for multiplicative errors in the actuator models, and Δadd to account for remaining errors and unmodeled dynamics. An additional operator, δq is included to represent variations in flight condition and ensure the model is robust to all variations less than the stability margin. Magnitudes of the uncertainty operators are computed to account for errors observed between predicted responses of the model and measured flight data from accelerometers.

In an application of the foregoing methodology, ASE-stability margins were computed for the F/A-18 High Alpha Research Vehicle (HARV). There had been concern about the closed-loop stability margins of this aircraft operating with high angles of attack at high altitudes. The ASE stability margins are given in Figure 2 for the aircraft model at the extreme ranges of flight conditions in which the HARV operates. These margins are the biggest decreases in dynamic pressures that may be safely considered before an ASE instability can be encountered. The parameters Γnomare the stability margins computed without consideration of any modeling errors or uncertainties. These margins indicate that the nearest unstable flight condition for the nominal model is quite far from the flight envelope. The parameters Γrobare the stability margins computed with consideration for errors and uncertainties. These margins are considerably smaller than the nominal margins and indicate that the nearest instability may actually lie quite close to the flight envelope. In particular, the model at mach 0.3 and altitude of 30,000 ft (9.1 km) has very little robustness to the errors that are observed from the flight data.

This work was done by Martin Brenner of Dryden Flight Research Centerand Rick Lind of NRC. DRC-98-37

NASA Tech Briefs Magazine

This article first appeared in the September, 1998 issue of NASA Tech Briefs Magazine.

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