# Research

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. Our research focuses on

Bridging the gap between machine learning and control theory to develop new algorithms towards safe, robust, and intelligent control of physical systems.

For this purpose, we focus on Bayesian models as learning method due to many beneficial properties such as the bias-variance trade-off and the inherent probabilistic nature. The group's research allows to answer the following questions:

How to model nonlinear dynamical systems with Bayesian approaches in an effective and accurate way

How to include physical prior knowledge into the learning process for more data-efficient and trustworthy models

How to exploit Bayesian models for control to establish safety and performance guarantees for systems with unknown dynamics

In the following, you find a selection of our work. Some code is publicly available at the Github repo.

Switching physical systems are ubiquitous in modern control applications, for instance, locomotion behavior of robots and animals, power converters with switches and diodes. The dynamics and switching conditions are often hard to obtain or even inaccessible in case of a-priori unknown environments and nonlinear components. Black-box neural networks can learn to approximately represent switching dynamics, but typically require a large amount of data, neglect the underlying axioms of physics, and lack of uncertainty quantification. We propose a Gaussian process based learning approach enhanced by switching Port-Hamiltonian systems (GP-SPHS) to learn physical plausible system dynamics and identify the switching condition. The Bayesian nature of Gaussian processes uses collected data to form a distribution over all possible switching policies and dynamics that allows for uncertainty quantification. A simulation with a hopping robot validates the effectiveness of the proposed approach.

T. Beckers, T. Z. Jiahao, and G. J. Pappas. “Learning Switching Port-Hamiltonian Systems with Uncertainty Quantification”. In: Proceedings of the IFAC World Congress. 2023. (accepted)

Data-driven approaches achieve remarkable results for the modeling of complex dynamics based on collected data. However, these models often neglect basic physical principles which determine the behavior of any real-world system. This omission is unfavorable in two ways: The models are not as data-efficient as they could be by incorporating physical prior knowledge, and the model itself might not be physically correct. We propose Gaussian Process Port-Hamiltonian systems (GPPHS) as a physics-informed Bayesian learning approach with uncertainty quantification. The Bayesian nature of GP-PHS uses collected data to form a distribution over all possible Hamiltonians instead of a single point estimate. Due to the underlying physics model, a GP-PHS generates passive systems with respect to designated inputs and outputs. Further, the proposed approach preserves the compositional nature of Port-Hamiltonian systems.

T. Beckers, J. H. Seidman, P. Perdikaris, G. J. Pappas. “Gaussian Process Port-Hamiltonian Systems: Bayesian Learning with Physics Prior”. In: Proceedings of the Conference on Decision and Control. 2022. doi: 10.1109/CDC51059.2022.9992733

Underactuated vehicles have gained much attention in the recent years due to the increasing amount of aerial and underwater vehicles as well as nanosatellites. Trajectory tracking control of these vehicles is a substantial aspect for an increasing range of application domains. However, external disturbances and parts of the internal dynamics are often unknown or very time-consuming to model. To overcome this issue, we present a tracking control law for underactuated rigid-body dynamics using an online learning-based oracle for the prediction of the unknown dynamics. We show that Gaussian process models are of particular interest for the role of the oracle. The presented approach guarantees a bounded tracking error with high probability where the bound is explicitly given.

T. Beckers, L. Colombo, S. Hirche, and G. Pappas. “Online learning-based trajectory tracking for underactuated vehicles with uncertain dynamics”.

In: IEEE Control Systems Letters (L-CSS). 2022.

doi: 10.1109/LCSYS.2021.3138546. [Preprint]

The modeling and simulation of dynamical systems is a necessary step for many control approaches. Using classical, parameter-based techniques for modeling of modern systems, e.g., soft robotics or human-robot interaction, is often challenging or even infeasible due to the complexity of the system dynamics. In contrast, data-driven approaches need only a minimum of prior knowledge and scale with the complexity of the system. In particular, Gaussian process dynamical models (GPDMs) provide very promising results for the modeling of complex dynamics. However, the control properties of these GP models are just sparsely researched, which leads to a "blackbox" treatment in modeling and control scenarios. In addition, the sampling of GPDMs for prediction purpose respecting their non-parametric nature results in non-Markovian dynamics making the theoretical analysis challenging. In this article, we present approximated GPDMs which are Markov and analyze their control theoretical properties. Among others, the approximated error is analyzed and conditions for boundedness of the trajectories are provided. The outcomes are illustrated with numerical examples that show the power of the approximated models while the the computational time is significantly reduced.

T. Beckers and S. Hirche. “Prediction with Approximated Gaussian Process Dynamical Models”. In: Transaction on Automatic Control. 2022. (to appear). doi: 10.1109/TAC.2021.3131988. [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.

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]