Analysing dynamic deep stall recovery using a nonlinear frequency approach

Abstract

Based on bifurcation theory, nonlinear frequency response analysis is a recent development in the field of flight dynamics studies. Here, we consider how this method can be used to inform us on how to devise the control input such that the system transitions from an undesirable equilibrium solution—an aircraft deep stall solution in our case—to a desirable solution. We show that it is still possible to induce a large-amplitude oscillation via harmonic forcing of the pitch control device and escape the otherwise unrecoverable deep stall, despite very little control power available in such a high angle-of-attack flight condition. The forcing frequencies that excite these resonances are reflected as asymptotically unstable solutions using bifurcation analysis and Floquet theory. Due to the softening behaviour observed in the frequency response, these unstable (divergent) solutions have slightly lower frequencies than the value predicted using linear analysis. Subharmonic resonances are also detected, which are reflected in the time-domain unforced responses. These nonlinear phenomena show strong dependency on the forcing/perturbation amplitude and result in complex dynamics that can impede recovery if the existing procedures are followed. The proposed method is shown to be a useful tool for nonlinear flight dynamics analysis as well as to complement the rather thin literature on deep stall analysis—a topic of relevance for recent research on unconventional landing techniques in unmanned aerial vehicles. A full description of the aircraft model used, the unstable F-16 fighter jet, is provided in the appendix.

Derivation of control inputs for deep stall recovery using nonlinear frequency analysis

“…Therefore, it can be inferred that past studies that only considered the short-period dynamics and ignored variations in V and θ [7][8][9] cannot accurately reflect the aircraft dynamics in this high-α regime, making unsuitable for analysing dynamic deep stall recovery. It is also worth noting that no subharmonic resonance (additional peaks at low frequencies) is detected in the GTT frequency response, unlike in the unstable fighter jet example [11].…”

“…Recent developments in the field saw the use of a harmonic forcing function to generate a 'nonlinear Bode plot', which facilitates assessments of the non-stationary nonlinear elements like sub/super-harmonic resonances [16] and actuator rate limiting [17] while also providing information on resonance and stability -none of which is available in linear-based frequency analysis. In an earlier study on the frequency-domain dynamics of an unstable fighter aircraft model at high angles of attack, we discovered that a simple harmonic elevator input can lead to nonlinear resonances with diverging amplitudes [11]. This phenomenon destabilises the statically stable deep stall trim point, which allows the pilot to do the pitch rocking manoeuvre in an open-loop manner with a higher chance of recovery.…”

“…The high workload can be attributed to the fact that upon entering a deep stall, there is a limited window to observe the aircraft's transient motion and figure out an input frequency before the motion damps out [3]. Furthermore, when using a mathematical model to determine a suitable pilot response, the nonlinearities in the aerodynamic data and the equations of motion will also affect the frequency characteristics: Depending on how the aircraft entered the deep stall, the ensuing oscillation may have different and varying frequencies, therefore making it more challenging to observe the motion and provide an appropriate forcing term [11]. Relying on visual cues to devise an escape manoeuvre can therefore be hit or miss, especially when the psychological and physical stresses on the pilot are considered.…”

“…This phenomenon destabilises the statically stable deep stall trim point, which allows the pilot to do the pitch rocking manoeuvre in an open-loop manner with a higher chance of recovery. The deep stall observed in [11] is caused by the aft centre-of-gravity position -commonly seen in high performance fighter jets -so the feasibility of the pitch rocking method on a T-tail aircraft has not been discussed.…”

Vision‐assisted deep stall landing for a fixed‐wing UAV

“…Heinrich (2019) analyzed vertical speed, horizontal speed, angle of attack, lift and drag through open‐loop deep stall flight tests. Nguyen et al (2021) proposed a nonlinear frequency domain approach for deep stall recovery. Precision deep stall landing was performed by nonlinear model predictive control (NMPC) (Mathisen et al, 2015, 2021).…”