Nonparametric adaptive control using Gaussian Processes with online hyperparameter estimation

Many current model reference adaptive control methods employ parametric adaptive elements in which the number of parameters are fixed a-priori and the hyperparameters, such as the bandwidth, are pre-defined, often through expert judgment. Typical examples include the commonly used Radial Basis Function (RBF) Neural Networks (NNs) with pre-allocated centers. As an alternative to these methods, a nonparametric model using Gaussian Processes (GPs) was recently proposed. Using GPs, it was shown that it is possible to maintain constant coverage over the operating domain by adaptively selecting new kernel locations without any previous domain knowledge. However, even if kernel locations provide good coverage of the input domain, incorrect bandwidth selection can result in poor characterization of the model uncertainty, leading to poor performance. In this paper, we propose methods for learning hyperparameters online in GP-MRAC by optimizing a modified likelihood function. We prove the stability and convergence of our algorithm in closed loop. Finally, we evaluate our methods in simulation on an example of wing rock dynamics. Results show learning hyperparameters online robustly reduces the steady state modeling error and improves control smoothness over other MRAC schemes.