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 for $n$ equations in $n+m$ variables. Systems with saddle Jacobi number 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.

翻译：我们考虑存在平坦输出且平坦输出为状态变量子集的平坦微分控制系统，并利用雅可比界对其进行研究。针对含有 n 个方程与 n+m 个变量的常微分系统，我们引入了鞍点雅可比界的概念。具有鞍点雅可比数的系统推广了链式系统和对角系统的多种形式，构成了允许状态变量子集作为平坦输出的最广泛系统类别，其平坦参数化可在无需对初始方程求导的情况下计算得出。我们研究了此类系统的表观奇异性与内禀平坦奇异性。以简化飞机模型为例，我们提供了新的平坦输出，并证明该模型除可能的失速状态外，在所有点处均具有平坦性。最后，数值仿真表明：采用这些平坦输出的反馈控制对扰动具有鲁棒性，且在使用更真实的气动模型时，还能补偿模型误差。