# Control of an inverted pendulum with an application of differential flatness

The goal of this bachelor thesis is to design and to implement a flatness control for an inverted pendulum built on a sledge driven by a linear motor. The thesis begins with three shorter chapters. In the first chapter are the detailed description of this goal and the analysis of the task to be found. The second chapter introduces briefly beside the different versions of inverted pendulums some technical results related to the topic, emphasizing the benefits and drawbacks. The idea, the concept and the emerging challenges of the flatness control are outlined in the third chapter.

The theoretical basics needed to understand the operation of the controller and to begin the design are described in the fourth chapter, with special regard to the condition of differential flatness, the properties of the differentially flat systems, the definition flat output, the connection between flatness and linearization. The chapter ends with a theoretical supplement required to the pendulum swing-up control (the Control-Ljapunov-Function).

The fifth chapter concerns the design of the controllers. The mechanical model of the system gets firstly introduced, the flatness of which can be recognized after linearization, and so the state space of the system can be transformed into the so-called space of the flat coordinates. In so doing, the design of the controller takes place by means of pole placement, similar to linear systems. The performance of the proposed flatness controller has been examined considering its properties regarding the trajectory tracking, which are illustrated by means of simulation results. The chosen procedure for the pendulum swing-up is an energy-based controller. After describing the details of the technique, an optimization process gets introduced, the results of which lead to two different natures of the swing-up control. The design of this controller is also subsequently followed by simulation results, then the combination of the two controllers takes place.

The proposed controllers could successfully be implemented at the experimental set-up at the Institute for Control Systems of the KIT. The discussion about the problems occurring during the realization and the measurement results are in the sixth chapter to be found. The seventh chapter comprises the consequences and a suggestion is made regarding the future development of the method.

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