In modern technologies such as autonomous vehicles and service robots, control engineering plays a crucial role for the overall performance and safety of the system. However, the control design becomes often very time-consuming or unfeasible due to the increasing complexity of recent technological advancements. The classical control approaches, which are based on models of the systems using first-order principles, are not satisfactory in the presence of complex dynamics, e.g., for highly nonlinear systems or interaction with prior unknown environment.

Recent findings in the area of computational intelligence and machine learning have shown that data-driven approaches lead to very promising results in a wide application range as they require only a minimal prior knowledge for the modeling of complex dynamics. However, the major drawback in data-driven approaches frequently manifests as unpredictable outcomes. Thus, guarantees about the stability and performance of the control loop are absent which is translated as compromised safety in control systems. As a consequence, the current application of data-driven models in control scenarios is limited to non-critical and low performance systems due to their unpredictable blackbox-behavior.

My research focuses on the safe, efficient and rational use of data-driven models in control.

For this purpose, I use Gaussian process models as a data-driven technique due to many beneficial properties such as the bias-variance trade-off and the strong connection to Bayesian mathematics. My research allows to answer the following questions:

  • How to use GP models as dynamical systems and what are the control-theoretic properties of dynamical GP models

  • How to select the underlying kernel function to give the maximum control performance and how to deal with a misselection

  • How to apply GP models in control approaches with safety and performance guarantees for the closed-loop

In the following, you can see a selection of my work. Some code is publicly available on my Github repo.

Gaussian Process State Space Models (GP-SSM) are a data-driven stochastic model class suitable to represent nonlinear dynamics. They have become increasingly popular in non-parametric modeling approaches since they provide not only a prediction of the system behavior but also an accuracy of the prediction. For the application of these models, the analysis of fundamental system properties is required. In this paper, we analyze equilibrium distributions and stability properties of the GP-SSM. The computation of equilibrium distributions is based on the numerical solution of a Fredholm integral equation of the second kind and is suitable for any covariance function. Besides, we show that the GP-SSM with squared exponential covariance function is always mean square bounded and there exists a set which is positive recurrent.

T. Beckers, and S. Hirche. “Equilibrium distributions and stability analysis of Gaussian Process State Space Models”. In: Proceedings of the Conference on Decision and Control. 2016. doi: 10.1109/CDC.2016.7799247. [Preprint] [bibtex]

Kernel-based nonparametric models have become very attractive for model-based control approaches for nonlinear systems. However, the selection of the kernel and its hyperparameters strongly influences the quality of the learned model. Classically, these hyperparameters are optimized to minimize the prediction error of the model but this process totally neglects its later usage in the control loop. In this work, we present a framework to optimize the kernel and hyperparameters of a kernel-based model directly with respect to the closed-loop performance of the model. Our framework uses Bayesian optimization to iteratively refine the kernel-based model using the observed performance on the actual system until a desired performance is achieved.

T. Beckers, S. Bansal, C. J. Tomlin, and S. Hirche. “Closed-loop model selection for kernel based models via Bayesian Optimization”. In: Proceedings of the Conference on Decision and Control. 2019. doi: 10.1109/CDC40024.2019.9029690. [Preprint] [bibtex]

Perfect tracking control for real-world Euler-Lagrange systems is challenging due to uncertainties in the system model and external disturbances. The magnitude of the tracking error can be reduced either by increasing the feedback gains or improving the model of the system. The latter is clearly preferable as it allows to maintain good tracking performance at low feedback gains. However, accurate models are often difficult to obtain. In this article, we address the problem of stable high-performance tracking control for unknown Euler-Lagrange systems. In particular, we employ Gaussian Process regression to obtain a data-driven model that is used for the feed-forward compensation of unknown dynamics of the system. The model fidelity is used to adapt the feedback gains allowing low feedback gains in state space regions of high model confidence. The proposed control law guarantees a globally bounded tracking error with a specific probability. Simulation studies demonstrate the superiority over state of the art tracking control approaches.

T. Beckers, D. Kulić, and S. Hirche. “Stable Gaussian Process based Tracking Control of Euler-Lagrange Systems”. In: Automatica 103 (2019), pp. 390–397.
doi: 10.1016/j.automatica.2019.01.023. [Preprint] [bibtex]