Flat singularities of chained systems, illustrated with an aircraft model

TL;DR

This work extends nonlinear control theory by introducing ō-systems, a broad class of block-structured differential systems with flat outputs lying among state variables. It defines the saddle Jacobi number and shows that when the saddle bound is zero and a related determinant is nonvanishing, the system is flat, enabling lazy flat parametrizations that avoid differentiating the original equations. The authors develop algorithmic criteria to identify ō-systems and test regularity, then apply the framework to a simplified aircraft model using GNA-based aerodynamics, revealing multiple valid flat-output sets and stall-related singularities. Numerical simulations in Python and Maple demonstrate robust feedback designs that compensate model errors and disturbances, illustrating the practical viability of flat-based control for complex, real‑world systems. Overall, the paper provides a cohesive theory and practical toolkit for exploiting flatness in block-triangular systems, with a concrete aircraft application that showcases new flat outputs and regularity insights under nonideal flight conditions.

Abstract

We consider flat differential control systems for which there exist flat outputs that are part of the state variables and study them using Jacobi bound. We introduce a notion of saddle Jacobi bound for an ordinary differential system of $n$ equations in $n+m$ variables. Systems with saddle Jacobi number equal to $0$ generalize various notions of chained and diagonal systems and form the widest class of systems admitting subsets of state variables as flat output, for which flat parametrization may be computed without differentiating the initial equations. We investigate apparent and intrinsic flat singularities of such systems. As an illustration, we consider the case of a simplified aircraft model, providing new flat outputs and showing that it is flat at all points except possibly in stalling conditions. Finally, we present numerical simulations showing that a feedback using those flat outputs is robust to perturbations and can also compensate model errors, when using a more realistic aerodynamic model.

Figures (8)

• Figure 1: a) Earth frame and b) body frame.
• Figure 2: a) Wind frame and b) From wind to body frame.
• Figure 3: Range of values for the GNA model
• Figure 4: F-4: values of $V$ and $F$ depending on $\alpha$. a) & b) full generic nonlinear aerodynamic model; c) & d) simplified model
• Figure 5: The values of the GTM trajectory, one engine: the aircraft converges toward the reference trajectory. a) Trajectory, $3$D view; b)--e) Histories of $x,y,z,\beta$

Theorems & Definitions (111)

• Definition 1
• Example 2
• Example 3
• Example 4
• Definition 5
• Example 6
• Definition 7
• Example 8
• Remark 9
• Example 10