Flat singularities of chained systems, illustrated with an aircraft model

We investigate apparent and intrinsic singularities of block diagonal systems. These systems admit a partition of the set of state functions $\Xi=\bigcup_{i=1}^{r}\Xi_{i}$, such that derivatives of a state variable in $\Xi_{i}$ only depend on state variables in $\Xi_{1}$, ..., $\Xi_{i+1}$. They include various notions of "chained systems". Such systems are flat in the generic case, which means that their solutions may be parametrized by a finite set of differentially independent functions, called flat outputs and a finite number of their derivatives, provided that some Jacobian determinant does not vanish. Using theoretical results related to Jacobi's bound, flatness conditions can be reduced to the non vanishing of Jacobi's truncated determinant. An algorithm is provided to test if a system is block diagonal. We illustrate such systems with a study of a simplified aircraft model. We exhibit new sets of flat outputs, provide explicit regularity conditions for them and interpret some flat singularities as stalling conditions. This simplified model remains flat using some alternative controls such as differential thrust in case of rudder failure, or when one or all engines are lost. We conclude this work with numerical simulation showing that a feed-back using those fkat outputs is robust to perturbations and can also compensate model errors, when using a more realistic aerodynamic model.