This paper proposes a new class of real-time optimization schemes to overcome
system-model mismatch of uncertain processes. This work's novelty lies in
integrating derivative-free optimization schemes and multi-fidelity Gaussian
processes within a Bayesian optimization framework. The proposed scheme uses
two Gaussian processes for the stochastic system, one emulates the (known)
process model, and another, the true system through measurements. In this way,
low fidelity samples can be obtained via a model, while high fidelity samples
are obtained through measurements of the system. This framework captures the
system's behavior in a non-parametric fashion while driving exploration through
acquisition functions. The benefit of using a Gaussian process to represent the
system is the ability to perform uncertainty quantification in real-time and
allow for chance constraints to be satisfied with high confidence. This results
in a practical approach that is illustrated in numerical case studies,
including a semi-batch photobioreactor optimization problem.