Analytic Optimal Control for a Class of Driftless x-Flat Systems
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arXiv
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| Main Authors: | , , , |
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| Format: | Preprint |
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2026
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| _version_ | 1866917385420668928 |
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| author | Buchinger, Raphael Hartl, Georg Ecker, Lukas Schöberl, Markus |
| author_facet | Buchinger, Raphael Hartl, Georg Ecker, Lukas Schöberl, Markus |
| contents | This paper studies optimal trajectory-tracking for driftless, x-flat nonlinear systems with three states and two inputs. The tracking problem is formulated in Bolza form with a quadratic cost of the tracking error and its derivative. Applying Pontryagin's maximum principle yields a mixed regular-singular optimal control problem. By exploiting geometric properties and a specific relation between the weighting matrices, a closed-form expression for the costate and an explicit feedback law for both inputs is derived. Thereby, the numerical solution of a two-point boundary-value problem is avoided. The singular input leads to a bang-singular-bang optimal control structure, while on the singular arc, the tracking error dynamics reduces to a linear dynamics of order two. The approach is illustrated for the kinematic model of a steerable axle, demonstrating accurate trajectory-tracking. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_02249 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Analytic Optimal Control for a Class of Driftless x-Flat Systems Buchinger, Raphael Hartl, Georg Ecker, Lukas Schöberl, Markus Optimization and Control Dynamical Systems This paper studies optimal trajectory-tracking for driftless, x-flat nonlinear systems with three states and two inputs. The tracking problem is formulated in Bolza form with a quadratic cost of the tracking error and its derivative. Applying Pontryagin's maximum principle yields a mixed regular-singular optimal control problem. By exploiting geometric properties and a specific relation between the weighting matrices, a closed-form expression for the costate and an explicit feedback law for both inputs is derived. Thereby, the numerical solution of a two-point boundary-value problem is avoided. The singular input leads to a bang-singular-bang optimal control structure, while on the singular arc, the tracking error dynamics reduces to a linear dynamics of order two. The approach is illustrated for the kinematic model of a steerable axle, demonstrating accurate trajectory-tracking. |
| title | Analytic Optimal Control for a Class of Driftless x-Flat Systems |
| topic | Optimization and Control Dynamical Systems |
| url | https://arxiv.org/abs/2604.02249 |