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Bernd Kolar

Publications and source records attributed to Bernd Kolar.

At least 19 recordsLinked to original sources
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

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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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

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

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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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.

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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.

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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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arXiv2022

Discrete-time Flatness and Linearization along Trajectories

The paper studies the relation between a nonlinear time-varying flat discrete-time system and the corresponding linear time-varying systems which are obtained by a linearization along trajectories. It is motivated by the continuous-time case, where it is well-known that the linearization of flat systems along trajectories results in linear time-varying systems which are again flat. Since flatness implies controllability, this constitutes an important verifiable necessary condition for flatness. In the present contribution, it is shown that this is also true in the discrete-time case: We prove that the linearized system is again flat, and that a possible flat output is given by the linearization of a flat output of the nonlinear system. Analogously, the map that describes the parameterization of the system variables of the linear system by this flat output coincides with the linearization of the corresponding map of the nonlinear system. The results are illustrated by two examples.

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arXiv2022

A Flat System Possessing no (x,u)-Flat Output

In general, flat outputs of a nonlinear system may depend on the system's state and input as well as on an arbitrary number of time derivatives of the latter. If a flat output which also depends on time derivatives of the input is known, one may pose the question whether there also exists a flat output which is independent of these time derivatives, i.e., an (x,u)-flat output. Until now, the question whether every flat system also possesses an (x,u)-flat output has been open. In this contribution, this conjecture is disproved by means of a counterexample. We present a two-input system which is differentially flat with a flat output depending on the state, the input and first-order time derivatives of the input, but which does not possess any (x,u)-flat output. The proof relies on the fact that every (x,u)-flat two-input system can be exactly linearized after an at most dim(x)-fold prolongation of one of its (new) inputs after a suitable input transformation has been applied.

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arXiv2022

Flatness Analysis for the Sampled-data Model of a Single Mast Stacker Crane

We show that the Euler-discretization of the nonlinear continuous-time model of a single mast stacker crane is flat. The construction of the flat output is based on a transformation of a subsystem into the linear time-variant discrete-time controller canonical form. Based on the derived flat output, which is also a function of backward-shifts of the system variables, we discuss the planning of trajectories to achieve a transition between two rest positions and compute the corresponding discrete-time feedforward control.

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arXiv2022

Discrete-time Flatness-based Controller Design using an Implicit Euler-discretization

In this contribution, we present a constructive method to derive flat sampled-data models for continuous-time flat systems through an implicit Euler-discretization. We show how the sampled-data model can be used subsequently for a flatness-based controller design, and illustrate our results with the well-known planar VTOL aircraft example.

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arXiv2021

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 Brunovsk\'y 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 Brunovsk\'y 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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arXiv2021

Discrete-time Flatness-based Control of a Gantry Crane

This article addresses the design of a discrete-time flatness-based tracking control for a gantry crane and demonstrates the practical applicability of the approach by measurement results. The required sampled-data model is derived by an Euler-discretization with a prior state transformation in such a way that the flatness of the continuous-time system is preserved. Like in the continuous-time case, the flatness-based controller design is performed in two steps. First, the sampled-data system is exactly linearized by a discrete-time quasi-static state feedback. Subsequently, a further feedback enforces a stable linear tracking error dynamics. To underline its practical relevance, the performance of the novel discrete-time tracking control is compared to the classical continuous-time approach by measurement results from a laboratory setup. In particular, it turns out that the discrete-time controller is significantly more robust with respect to large sampling times. Moreover, it is shown how the discrete-time approach facilitates the design of optimal reference trajectories, and further measurement results are presented.

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arXiv2021

Differential-Geometric Decomposition of Flat Nonlinear Discrete-Time Systems

We prove that every flat nonlinear discrete-time system can be decomposed by coordinate transformations into a smaller-dimensional subsystem and an endogenous dynamic feedback. For flat continuous-time systems, no comparable result is available. The advantage of such a decomposition is that the complete system is flat if and only if the subsystem is flat. Thus, by repeating the decomposition at most $n-1$ times, where $n$ is the dimension of the state space, the flatness of a discrete-time system can be checked in an algorithmic way. If the system is flat, then the algorithm yields a flat output which only depends on the state variables. Hence, every flat discrete-time system has a flat output which does not depend on the inputs and their forward-shifts. Again, no comparable result for flat continuous-time systems is available. The algorithm requires in each decomposition step the construction of state- and input transformations, which are obtained by straightening out certain vector fields or distributions with the flow-box theorem or the Frobenius theorem. Thus, from a computational point of view, only the calculation of flows and the solution of algebraic equations is needed. We illustrate our results by two examples.

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arXiv2021

A Trajectory-Based Approach to Discrete-Time Flatness

For discrete-time systems, flatness is usually defined by replacing the time-derivatives of the well-known continuous-time definition by forward-shifts. With this definition, the class of flat systems corresponds exactly to the class of systems which can be linearized by a discrete-time endogenous dynamic feedback as it is proposed in the literature. Recently, verifiable necessary and sufficient differential-geometric conditions for this property have been derived. In the present contribution, we make an attempt to take into account also backward-shifts. This extended approach is motivated by the one-to-one correspondence of solutions of flat systems to solutions of a trivial system as it is known from the continuous-time case. If we transfer this idea to the discrete-time case, this leads to an approach which also allows backward-shifts. To distinguish the classical definition with forward-shifts and the approach of the present paper, we refer to the former as forward-flatness. We show that flat systems (in the extended sense with backward-shifts) still share many beneficial properties of forward-flat systems. In particular, they still are reachable/controllable, allow a straightforward planning of trajectories and can be linearized by a certain subclass of dynamic feedbacks.

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