Mathematical models are fundamental building blocks in the design of dynamical control systems. As control systems are becoming increasingly complex and networked, approaches for obtaining such models based on first principles reach their limits. Data-driven methods provide an alternative. However, without structural knowledge, these methods are prone to finding spurious correlations in the training data, which can hamper generalization capabilities of the obtained models. This can significantly lower control and prediction performance when the system is exposed to unknown situations. A preceding causal identification can prevent this pitfall. In this paper, we propose a method that identifies the causal structure of control systems. We design experiments based on the concept of controllability, which provides a systematic way to compute input trajectories that steer the system to specific regions in its state space. We then analyze the resulting data leveraging powerful techniques from causal inference and extend them to control systems. Further, we derive conditions that guarantee the discovery of the true causal structure of the system. Experiments on a robot arm demonstrate reliable causal identification from real-world data and enhanced generalization capabilities.