Singular Perturbation Margin of INDI-Based Flight Control Systems With Delays

A linear singularly perturbed system (SPS) based on a linear approximation of incremental nonlinear dynamic inversion (INDI) is formulated for a general INDI-based flight control system. This system includes first-order actuator dynamics where its time constant is taken as the singular perturbation parameter. The stability of the SPS is studied, which gives an understanding of the sensitivity of INDI to delays. Two sources of delay are considered, one coming from the state derivative measurement and the other coming from the delayed input signal. It is shown in this article that the upper bound of the singular perturbation parameter serves as a robustness metric of the SPS to the two types of delays. Consequently, a relative singular perturbation margin (<inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\text{SPM}_{\text{R}}$</tex-math></inline-formula>) is proposed as a delay margin. The results are confirmed with two aircraft simulation cases for multiple-input multiple-output and single-input single-output systems. In the latter, the relationship between the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\text{SPM}_{\text{R}}$</tex-math></inline-formula> and the phase margin is analyzed.