Abstract
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.
Highlights
The concept of flatness has been introduced by Fliess, Levine, Martin and Rouchon for nonlinear continuous-time systems in the 1990s
We have shown by means of the laboratory model of a gantry crane that the concept of discrete-time flatness is well-suited for the design of tracking controllers in practical applications
The difficulty lies in determining a flat sampled-data model of the plant
Summary
The concept of flatness has been introduced by Fliess, Levine, Martin and Rouchon for nonlinear continuous-time systems in the 1990s (see e.g. [1], [2] and [3]). Once a flat sampled-data model has been derived, the design of tracking controllers can be performed in a systematic way since discrete-time flat systems can be exactly linearized by a dynamic state feedback, see e.g. For the considered sampled-data model of the gantry crane, we show that even an exact linearization by a quasi-static state feedback – as it is proposed e.g. in [10] for continuous-time systems – is possible. With the exactly linearized system, the design of a tracking control for the stabilization of reference trajectories becomes a straightforward task An implementation of both the dynamic and the quasi-static controller on a laboratory setup has shown that especially the novel discrete-time quasi-static controller is quite robust with respect to low sampling rates. We present measurement results for the optimized trajectory as well as for the tracking of a closed path in the shape of a lying eight
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