Bayesian optimization has become a powerful tool for safe online optimization of systems, due to its high sample efficiency and noise robustness. For further speed-up reduced physical models of the system can be incorporated into the optimization to accelerate the process, since the models are able to offer an approximation of the actual system, and sampling from them is significantly cheaper. The similarity between model and reality is represented by additional hyperparameters and learned within the optimization process. Safety is an important criteria for online optimization methods like Bayesian optimization, which has been addressed by recent literature, which provide safety guarantees under the assumption of known hyperparameters. However, in practice this is not applicable. Therefore, we extend the robust Gaussian process uniform error bounds to meet the multi-task setting, which involves the calculation of a confidence region from the hyperparameter posterior distribution utilizing Markov chain Monte Carlo methods. Then, using the robust safety bounds, Bayesian optimization is applied to safely optimize the system while incorporating measurements of the models. Simulations show that the optimization can be significantly accelerated compared to other state-of-the-art safe Bayesian optimization methods depending on the fidelity of the models.

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