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Analytic Optimal Control for a Class of Driftless x-Flat Systems

Raphael Buchinger, Georg Hartl, Lukas Ecker, Markus Schöberl

Abstract

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.

Analytic Optimal Control for a Class of Driftless x-Flat Systems

Abstract

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.

Paper Structure

This paper contains 9 sections, 2 theorems, 48 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Notation and Preliminaries
  3. Differential Geometry
  4. Optimal Control Problem
  5. Differential Flatness
  6. Problem Statement
  7. Main results
  8. Example
  9. Conclusion

Key Result

Theorem D.1

Consider the system eq:system together with the OCP eq:opt_problem, with the cost functional eq:cost_functional and the associated terminal and running costs eq:terminal_cost and eq:running_cost. The components of the costate $\lambda \in \mathcal{T}^*(\mathcal{M})$ on the singular arc are given by with $Q_{ij} = \bar{Q}_{il}M^{lm}\bar{Q}_{mj}$. $\blacktriangleleft$$\blacktriangleleft$

Figures (2)

  • Figure F1: Trajectories for three different initial conditions: $x_{0,1}=(0,-1,\pi/3)^T$, $x_{0,2}=(3,-1,-\pi/4)^T$, $x_{0,3}=(1,-\tfrac{1}{2},\pi/3)^T$. The solid line denotes the desired trajectory $y_d(t)=(2\cos(2\pi t/T),\,\sin(\pi t/T))^T$ with $T=5$. The input bounds are set to $u^1_{\max}=u^2_{\max}=10$.
  • Figure F2: Time evolution of $f^2$ and $h^2=f^1/f^2$ for the trajectories given in fig:trajectory. The plot shows that $f^2$ remains strictly positive in a neighborhood of the singular manifold, while $h^2$ stays in the interior of the admissible set $\mathcal{U}$, i.e., $|h^2|<u^2_{\max}$.

Theorems & Definitions (4)

  • Theorem D.1
  • proof
  • Theorem D.2
  • proof : Proof of th:opt_solution