arXiv subjects

Georg Hartl

Publications and source records attributed to Georg Hartl.

7 recordsLinked to original sources

arXiv2026

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

math.OC↗

arXiv2026

A Structurally Flat Triangular Form for Three-Input Systems

We present a broadly applicable structurally flat triangular form for x-flat control-affine systems with three inputs. Building on recent results for the derivative structure of flat outputs, we define the triangular form together with regularity conditions that guarantee structural flatness, and derive necessary and sufficient conditions for a system with a given x-flat output to be static feedback equivalent to this form. Further, we present sufficient conditions under which general x-flat three-input systems can be rendered static feedback equivalent to the proposed triangular form after a finite number of input prolongations.

math.DS↗

arXiv2026

On the Linearization of Flat Multi-Input Systems via Prolongations

We examine when differentially flat nonlinear control systems with more than two inputs can be rendered static feedback linearizable by a minimal number of prolongations of suitably chosen inputs after applying a static input transformation. We derive sufficient conditions that guarantee such prolongations yield a static feedback linearizable system. For $(x,u)$-flat two-input systems, prior work established precise links between the relative degrees, the highest derivative orders occurring in the flat parameterization, and the minimal dimension of a linearizing dynamic extension, leading to necessary and sufficient criteria for flatness of systems that become static feedback linearizable after at most two prolongations of such suitably chosen inputs. Building on the structure of the time derivatives of a flat output, this work extends this analysis to systems with three inputs.

math.DS↗

arXiv2025

A Flat Triangular Structure Based on a Multi-Chained Form

Determining whether a nonlinear multi-input system is differentially flat remains challenging. One way to obtain computationally tractable sufficient conditions is to give complete characterizations of flat normal forms. We introduce a structurally flat triangular form for control-affine systems with at least three inputs that is based on a multi-chained form. For two specific instances of this structure, we provide complete geometric characterizations, i.e., necessary and sufficient conditions under which a control-affine system is static-feedback equivalent to the respective triangular form. These characterizations yield sufficient conditions for differential flatness and, in turn, constructive procedures for computing flat outputs.

math.DS↗

arXiv2025

On Triangular Forms for x-Flat Control-Affine Systems With Two Inputs

This paper examines a broadly applicable triangular normal form for x-flat control-affine systems with two inputs. First, we show that this triangular form encompasses a wide range of established normal forms. Next, we prove that any x-flat system can be transformed into this triangular structure after a finite number of prolongations of each input. Finally, we introduce a refined algorithm for identifying candidates for x-flat outputs. Through illustrative examples, we demonstrate the usefulness of our results. In particular, we show that the refined algorithm exceeds the capabilities of existing methods for computing flat outputs based on triangular forms.

math.DS↗

arXiv2024

On the Exact Linearization of Minimally Underactuated Configuration Flat Lagrangian Systems in Generalized State Representation

In this paper, we examine the exact linearization of configuration flat Lagrangian control systems in generalized state representation with p degrees of freedom and p-1 control inputs by quasi-static feedback of its generalized state. We formally introduce generalized Lagrangian control systems, which are obtained when configuration variables are considered as inputs instead of forces. This work presents all possible lengths of integrator chains achieved by an exact linearization with a quasi-static feedback law of the generalized state that allows for rest-to-rest transitions. We show that such feedback laws can be systematically derived without using Brunovsk\'y states.

math.DS↗

arXiv2023

Exact Linearization of Minimally Underactuated Configuration Flat Lagrangian Control Systems by Quasi-Static Feedback of Classical States

We study the exact linearization of configuration flat Lagrangian control systems with p degrees of freedom and p-1 inputs by quasi-static feedback of classical states. First, we present a detailed analysis of the structure of the parameterization of the system variables by the flat output. Based on that, we systematically construct a linearizing quasi-static feedback law of the classical state such that the closed-loop system shows the behavior of decoupled integrator chains. Our approach shows that the construction of a generalized Brunovsky state can be completely circumvented. Furthermore, we present a method for determining the lengths of the integrator chains achieved by quasi-static feedback laws that allow for rest-to-rest transitions.

math.DS↗