Christian Harkort

I4Copter: an adaptable and modular quadrotor platform

Quadrotor helicopters are micro air vehicles with vertical take-off and landing capabilities controlled by varying the rotation speed of four fixed pitch propellers. Due to their rather simple mechanical design they have grown to popularity as platform for various research projects. Despite most of them being individually highly successful, they are typically tailored to a specific purpose making it hard to utilise them for further research and education. In this paper, we present the novel design of the I4Copter quadrotor. It has been developed to provide a stable demonstration quadrotor platform for various kinds of research and education projects targeting cross-field challenges in real-time and embedded systems, distributed systems, robotics and cybernetics. The modular and open architecture of our platform allows an application-specific, fine-grained extension, adaption and replacement of software and hardware components. The safe extensibility is supported by strict temporal and spatial isolation between the software modules. We validated our approach by two distinct cross-field use cases: an evaluation platform for modularised control algorithms enabling trajectory tracking and an implementation that is resilient to transient hardware errors.

Krylov Subspace Methods for Linear Infinite-Dimensional Systems

The well-known Krylov subspace methods for model order reduction of large-scale lumped parameter systems are generalized such that they can be applied directly to a large class of linear infinite-dimensional systems including distributed parameter systems as well as delay systems. The proposed approach allows to derive finite-dimensional approximations of these infinite-dimensional systems without recourse to a large-scale lumped parameter approximation. The resulting finite-dimensional model has the usual property that prescribed moments of its transfer function coincide with the moments of the infinite-dimensional system. As in the finite-dimensional case the approach allows for a numerical efficient implementation. The results of the article are demonstrated by means of a simple example.

Parametric state feedback design of linear distributed-parameter systems

The parametric approach for the design of state feedback controllers has been formulated so far only for linear lumped-parameter systems. It yields an explicit parametric expression for the state feedback gain given the closed-loop eigenvalues and the set of corresponding parameter vectors. This contribution presents a parameterisation of state feedback controllers for linear distributed-parameter systems with scalar state and distributed control. By introducing the closed-loop eigenvalues and the parameter vectors as design parameters, an explicit expression for the state feedback is obtained. In contrast to the pure eigenvalue assignment, the parameterisation allows the assignment not only of the closed-loop eigenvalues but also of the closed-loop eigenfunctions. The usefulness of the proposed parametric approach is demonstrated by decoupling the transfer behaviour of a MIMO diffusion system with respect to its dominant modes.

Finite-dimensional observer-based control of linear distributed parameter systems using cascaded output observers

The synthesis of compensators for linear distributed-parameter systems on the basis of a finite-dimensional approximation is a classical technique. This so-called early-lumping approach suffers from the occurrence of spillover which means that the closed-loop dynamics is deteriorated by the neglected dynamics. In this contribution, an early-lumping compensator design is presented that overcomes the spillover problem by using certain fictitious outputs which can be reconstructed without spillover and that suppress the contributions of the unmodelled dynamics. When the new outputs are used for the compensator instead of the available measurements, the perturbation of the closed-loop spectrum, caused by spillover, can be reduced to an arbitrary extent. It is shown that the worst-case eigenvalue perturbation decreases exponentially with respect to the compensator order so that the spillover can be reduced systematically. In addition, an a priori estimate for the compensator order, that guarantees a prescribed maximal eigenvalue perturbation, is presented. The proposed design procedure is demonstrated for the control of an Euler–Bernoulli beam with Kelvin–Voigt damping.

Entwurf endlich-dimensionaler Regler für lineare verteilt-parametrische Systeme durch Ausgangsbeobachtun

Zusammenfassung In diesem Beitrag wird der endlich-dimensionale Reglerentwurf für lineare verteilt-parametrische Systeme basierend auf Ausgangsbeobachtern betrachtet. Da für diese Beobachter beim Entwurf das Separationsprinzip gilt, lassen sich statische Rückführungen der rekonstruierten Ausgänge und der verfügbaren Messgrößen systematisch entwerfen. Die besondere Eigenschaft der zusätzlichen Ausgänge, dass sie nur wenig “spillover”-behaftet sind, wird auch für den Entwurf endlich-dimensionaler Beobachter mittels der “early-lumping”-Methode genutzt. Durch Kaskadierung der Ausgangsbeobachter lässt sich dann eine systematische “spillover”-Vermeidung erreichen. Die vorgestellten Verfahren werden anhand eines Wärmeleiters erprobt. Abstract In this contribution the finite-dimensional control of linear distributed-parameter systems using output observers is considered. Since the separation principle holds for these observers, it is possible to systematically design a static output feedback of the reconstructed outputs and the available measurements. The special property of the additional outputs, that the spillover problem related to these outputs is mitigated, is used for the design of finite-dimensional observers using the early lumping approach. A systematic prevention of spillover is achieved by cascading output observers. The proposed design procedures are demonstrated for the control of a heat conductor.

A parametric approach to finite-dimensional control of linear distributed-parameter systems

This article considers the design of finite-dimensional compensators for distributed-parameter systems using eigenvalue assignment. The proposed compensator consists of an observer estimating additional outputs and a static feedback of the measurable and the estimated outputs. Since the additional outputs can be asymptotically reconstructed, the compensator can be designed using the separation principle, i.e. the closed-loop eigenvalues are given by the observer eigenvalues and the eigenvalues resulting from the static output feedback control. In order to solve the corresponding eigenvalue assignment problem, the parametric approach for the design of static output feedback controllers in finite-dimensions is extended to distributed-parameter systems. By using a parameter optimisation it is possible to assign all closed-loop eigenvalues within specified regions of the complex plane in order to stabilise the system and to assure a desired control performance. A heat conductor is used to demonstrate the proposed design procedure.

Taking control: Modular and adaptive robotics process control systems

Robotics systems usually comprise sophisticated sensor and actuator systems with no less complex control applications. These systems are subject to frequent modifications and extensions and have to adapt to their environment. While automation systems are tailored to particular production processes, autonomous vehicles must adaptively switch their sensors and controllers depending on environmental conditions. However, when designing and implementing the process control system, traditional control theory focuses on the control problem at hand without having this variability in mind. Thus, the resulting models and implementation artefacts are monolithic, additionally complicating the real-time system design. In this paper, we present a modularisation approach for the design of robotics process control systems, which not only aims for variability at design-time but also for adaptivity at run-time. Our approach is based on a layered control architecture, which includes an explicit interface between the two domains involved: control engineering and computer science. Our architecture provides separation of concerns in terms of independent building blocks and data flows. For example, the replacement of a sensor no longer involves the tedious modification of downstream filters and controllers. Likewise, the error-prone mapping of high-level application behaviour to the process control system can be omitted. We validated our approach by the example of an autonomous vehicle use case. Our experimental results demonstrate ease of use and the capability to maintain quality of control on par with the original monolithic design.

Stability and passivity preserving Petrov-Galerkin approximation of linear infinite-dimensional systems

This contribution presents two approximation methods for linear infinite-dimensional systems that ensure the preservation of stability and passivity. The first approach allows one to approximate internal source free infinite-dimensional systems such that the resulting approximation is a port-controlled Hamiltonian system with dissipation. The second method deals with the class of systems that are not required to have conjugated outputs but only a dissipative system operator. It yields approximations with a dissipative system matrix for which bounds of their stability margin are provided. Both approaches are based on a state space formulation of the infinite-dimensional system. This makes it possible to use the Petrov-Galerkin approximation whose free parameters are partly used for achieving the structure preservation. Since still free parameters remain, further application specific objectives, such as, e.g., moment matching, can be achieved. Both approaches are applied to the approximation of an Euler-Bernoulli beam.

Discrete-Time Modal State Reconstruction for Infinite-Dimensional Systems Using Generalized Sampling

Abstract In this contribution a novel approach for discrete-time state reconstruction for infinite-dimensional systems is presented. It is shown that an exact reconstruction of a finite number of modal states of the infinite-dimensional system can be obtained at the sampling time instances. This is achieved by aid of generalized sampling which generates the output samples by weighted averaging over one sampling interval. The reconstruction is done in finite time and is thus non-asymptotic. Since the method makes it possible to implement state feedback controls without involving a Luenberger observer, the spillover problem arising from the early-lumping approach can be avoided.

Vollständige Modale Synthese eines Wärmeleiters Parametric State Feedback Design for a Heat Conductor

Zusammenfassung Die Vollständige Modale Synthese wurde bisher nur für lineare zeitinvariante Systeme mit konzentrierten Parametern formuliert. Sie liefert einen expliziten Formelausdruck für die Zustandsrückführung in Abhängigkeit von den Regelungseigenwerten und Parametervektoren. In diesem Beitrag wird ein Wärmeleiter als Beispiel verwendet, um die Vollständige Modale Synthese auf lineare zeitinvariante verteilt-parametrische Systeme mit skalarem Zustand und verteiltem Eingriff zu erweitern. Durch Einführung der Regelungseigenwerte und Parametervektoren als Entwurfsparameter lässt sich ein expliziter Formelausdruck für die Zustandsrückführung des verteilt-parametrischen Systems herleiten. Neben der Eigenwertvorgabe erlaubt die neue Parametrierung eine gezielte Vorgabe der Regelungseigenfunktionen. Die Vorteile der neuen Entwurfsmethode werden anhand der Führungsentkopplung des Wärmeleiters diskutiert.