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.
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Data availability
Simulation data for Maple implementations are available with the Maple packages.
Code availability
The most recent implementation in Maple for aircraft motion planning, including generalized flatness, is available at http://www.lix.polytechnique.fr/~ollivier/GFLAT/.
Notes
Allowing change of independent variable, i.e., in control change of time, so that Monge problem is more precisely to test orbital flatness (Fliess et al. 1999)
From the Greek “nothing”, or “zero” for Iamblichus, and , “saddle”.
Or Lie-Bäcklund transform.
In the case of a system \(X'=AX\), the characteristic polynomial of A is the determinant of \(M-\lambda \textrm{Id}\).
The general case is more complicated, already for time-varying linear systems (see Ritt 1935). Then, there exists an analog of Smith normal form, due to Jacobson (1937), but no suitable notion of divisors, as factorization in \(\mathbb {R}(t)[\textrm{d}/\textrm{d}t]\) is not unique. Indeed, \((\textrm{d}/\textrm{d}t)^{2}\) is equal to \((\textrm{d}/\textrm{d}t+1/(x+\alpha ))(\textrm{d}/\textrm{d}t-1/(x+\alpha ))\), for any \(\alpha \in \mathbb {R}\) (Chyzak et al. 2022). One must also notice that \(\textrm{diag}(\textrm{d}/\textrm{d}t, (\textrm{d}/\textrm{d}t)(\textrm{d}/\textrm{d}t+1))(x_{1}x_{2})^{\textrm{t}}\) is a Smith normal form with \(\mathbb {R}[\textrm{d}/\textrm{d}t]\) as the base ring, but not a Jacobson normal form with base ring \(\mathbb {R}(t)[\textrm{d}/\textrm{d}t]\), as then the quotient module may be generated by a single element \(x_{1}+tx_{2}\).
This convention, introduced by Ritt (see Ollivier 2023, § 4 for details), is known as the strong bound. The convention \(\textrm{ord}_{x_{j}}P_{i}=0\) is the weak bound.
Here, \(\sharp A\) denotes the number of elements of a set A.
Jacobi named it determinans mancum sive determinans mutilatum, because only the terms \(\partial P_{i}/\partial x_{j}^{(a_{i,j})}\), such that \(a_{i,j}=\alpha _{i}+\beta _{j}\) appear in it.
The algebraic ideal is a proper subset of the differential ideal.
Considering the system as a system in the variables of Y only, and the remaining variables as parametric variables, we reduce to a system of differential dimension 0 that is indeed block triangular, according to the definition in (Ollivier 2023, 4.3).
We denote for brevity sets of rows or columns by the sets of corresponding indices.
Wing span a and mean aerodynamic chord b are, respectively, denoted by b and c in Grauer and Morelli (2014).
More precisely, such angles are known as Tait–Bryan angles.
The angles are not physical angles, but rather measures related to some physical angles and that are calibrated by the aircraft producer.
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Acknowledgements
The authors thank Jean Lévine for advices and inspiration.
Funding
F. Ollivier thanks the French ANR project ANR-22-CE48-0016 NODE (numeric-symbolic resolution of differential equations) and the ANR project ANR-22-CE48-0008 OCCAM (theory and practice of differential elimination).
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Y. J. Kaminski is responsible for Python implementations and F. Ollivier for Maple implementations. Y.J. Kaminski and F. Ollivier have contributed to the study conception and realization or to the writing and typesetting process.
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Communicated by Nadhir Messai.
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Kaminski, Y., Ollivier, F. Flat singularities of chained systems, illustrated with an aircraft model. Comp. Appl. Math. 43, 135 (2024). https://doi.org/10.1007/s40314-024-02605-w
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DOI: https://doi.org/10.1007/s40314-024-02605-w
Keywords
- Differentially flat systems
- Flat singularities
- Flat outputs
- Aircraft aerodynamics models
- Gravity-free flight
- Engine failure
- Rudder jam
- Differential thrust
- Forward slip landing
- Jacobi’s bound
- Hungarian method