arXiv subjects

Markus Schöberl

Publications and source records attributed to Markus Schöberl.

At least 19 recordsLinked to original sources

arXiv2026

Flatness-Based Controller Design for Backward-Flat Systems

This paper addresses the design of flatness-based tracking controllers for backward-flat nonlinear discrete-time systems. First, we show that backward-flat discrete-time systems can be constructively obtained from differentially flat continuous-time systems by applying an implicit Euler discretization to a structurally flat triangular representation. Subsequently, two approaches for the exact linearization of backward-flat systems are considered. Besides a dynamic feedback based on prelongations, we show that a linearizing feedback involving lower-order backward-shifts of the flat output can be employed without explicitly implementing the corresponding dynamic extension. This allows the order of the resulting tracking error dynamics to be reduced while requiring only stored values of past system trajectories. Based on the resulting linear input-output representation, flatness-based tracking control laws are derived. The proposed discretization and controller design are illustrated for a 2D gantry crane.

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arXiv2026

On the Exact Linearization of Multi-Input Discrete-Time Flat Systems

In this contribution, we address the exact linearization of forward- and backward-flat discrete-time nonlinear systems with an arbitrary number of inputs. Building on structural properties of the flat parameterization, we show that forward-flat systems can be rendered static feedback linearizable by prolongations of suitably transformed inputs. Analogously, for backward-flat systems, we derive a constructive procedure showing that an exact linearization can be achieved by prelongations of suitably chosen functions of the system variables. In this way, we extend previously established results for two-input systems to the general m-input case. Finally, we illustrate the proposed linearization procedures by an academic example and a discrete-time model of a quadrotor.

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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.

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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.

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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.

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arXiv2026

Testing Backward-Flatness of Nonlinear Discrete-Time Systems

Despite ongoing research, testing the flatness of discrete-time systems remains a challenging problem. To date, only the property of forward-flatness - a special case of difference-flatness - can be checked in a computationally efficient manner. In this paper, we propose a systematic approach for testing backward-flatness, which is another special case of difference-flatness, and for deriving a corresponding backward-flat output. Additionally, we discuss the relationship between the Jacobian matrices associated with the flat parameterization of backward- and forward-flat systems and illustrate our results by an academic example.

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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.

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arXiv2025

Flatness of Two-Input Discrete-Time Systems and their Linearization

In this contribution we discuss flat discrete-time nonlinear systems in a general setting including two special subclasses, namely, forward- and backward-flat systems. We relate rank conditions for certain submatrices of the Jacobian of the flat parameterization to the mentioned subclasses. Motivated by these rank conditions, for the case of two-input systems that possess an (x,u)-flat output, we derive a simple type of dynamic extension for the purpose of an exact linearization.

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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.

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arXiv2025

A Dual Geometric Test for Forward-Flatness

Forward-flatness is a generalization of static feedback linearizability and a special case of a more general flatness concept for discrete-time systems. Recently, it has been shown that this practically quite relevant property can be checked by computing a unique sequence of involutive distributions which generalizes the well-known static feedback linearization test. In this paper, a dual test for forward-flatness based on a unique sequence of integrable codistributions is derived. Since the main mathematical operations for determining this sequence are the intersection of codistributions and the calculation of Lie derivatives of 1-forms, it is computationally quite efficient. Furthermore, the formulation with codistributions also facilitates a comparison with the existing discrete-time literature regarding the closely related topic of dynamic feedback linearization, which is mostly formulated in terms of 1-forms rather than vector fields. The presented results are illustrated by two examples.

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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 Brunovský states.

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arXiv2024

Duality of Geometric Tests for Forward-Flatness

Recently it has been shown that the property of forward-flatness for discrete-time systems, which is a generalization of static feedback linearizability and a special case of a more general concept of flatness, can be checked by two different geometric tests. One is based on unique sequences of involutive distributions, while the other is based on a unique sequence of integrable codistributions. In this paper, the relation between these sequences is discussed and it is shown that the tests are in fact dual. The presented results are illustrated by an academic example.

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arXiv2024

Tracking Control for $(x,u)$-Flat Systems by Quasi-Static Feedback of Classical States

It is well known that for flat systems the tracking control problem can be solved by utilizing a linearizing quasi-static feedback of generalized states. If measurements (or estimates) of a so-called generalized Brunovský state are available, a linear, decoupled and asymptotically stable tracking error dynamics can be achieved. However, from a practical point of view, it is often desirable to achieve the same tracking error dynamics by feedback of a classical state instead of a generalized one. This is due to the fact that the components of a classical state typically correspond to measurable physical quantities, whereas a generalized Brunovský state often contains higher order time derivatives of the (fictitious) flat output which are not directly accessible by measurements. In this paper, a systematic solution for the tracking control problem based on quasi-static feedback and measurements of classical states only is derived for the subclass of $(x,u)$-flat systems.

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arXiv2024

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.

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arXiv2024

A Triangular Normal Form for x-Flat Control-Affine Two-Input Systems

This paper is devoted to normal forms for x-flat control-affine systems with two inputs. We propose a general triangular normal form which contains several other normal forms discussed in the literature as special cases. We derive conditions under which a system with given x-flat output can be transformed into the proposed triangular form. Based on the triangular form we motivate a simple algorithm for identifying candidates for flat outputs.

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arXiv2023

Data-driven control and transfer learning using neural canonical control structures*

An indirect data-driven control and transfer learning approach based on a data-driven feedback linearization with neural canonical control structures is proposed. An artificial neural network auto-encoder structure trained on recorded sensor data is used to approximate state and input transformations for the identification of the sampled-data system in Brunovsky canonical form. The identified transformations, together with a designed trajectory controller, can be transferred to a system with varied parameters, where the neural network weights are adapted using newly collected recordings. The proposed approach is demonstrated using an academic and an industrially motivated example.

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arXiv2022

On the Exact Linearization and Control of Flat Discrete-time Systems

The paper addresses the exact linearization of flat nonlinear discrete-time systems by generalized static or dynamic feedbacks which may also depend on forward-shifts of the new input. We first investigate the question which forward-shifts of a given flat output can be chosen in principle as a new input, and subsequently how to actually introduce the new input by a suitable feedback. With respect to the choice of a feasible input, easily verifiable conditions are derived. Introducing such a new input requires a feedback which may in general depend not only on this new input itself but also on its forward-shifts. This is similar to the continuous-time case, where feedbacks which depend on time derivatives of the closed-loop input - and in particular quasi-static ones - have already been used successfully for the exact linearization of flat systems since the nineties of the last century. For systems with a flat output that does not depend on forward-shifts of the input, it is shown how to systematically construct a new input such that the total number of the corresponding forward-shifts of the flat output is minimal. Furthermore, it is shown that in this case the calculation of a linearizing feedback is particularly simple, and the subsequent design of a discrete-time flatness-based tracking control is discussed. The presented theory is illustrated by the discretized models of a wheeled mobile robot and a 3DOF helicopter.

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arXiv2022

Necessary and Sufficient Conditions for Difference Flatness

We show that the flatness of a nonlinear discrete-time system can be checked by computing a unique sequence of involutive distributions. The well-known test for static feedback linearizability is included as a special case. Since the computation of the sequence of distributions requires only the solution of algebraic equations, it allows an efficient implementation in a computer algebra program. In case of a positive result, a flat output can be obtained by straightening out the involutive distributions with the Frobenius theorem.

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