Nonparametric Adaptive Control and Prediction: Theory and Randomized Algorithms

A key assumption in the theory of nonlinear adaptive control is that the uncertainty of the system can be expressed in the linear span of a set of known basis functions. While this assumption leads to efficient algorithms, it limits applications to very specific classes of systems. We introduce a novel nonparametric adaptive algorithm that learns an infinite-dimensional parameter density to cancel an unknown disturbance in a reproducing kernel Hilbert space. Surprisingly, the resulting control input admits an analytical expression that enables its implementation despite its underlying infinite-dimensional structure. While this adaptive input is rich and expressive – subsuming, for example, traditional linear parameterizations – its computational complexity grows linearly with time, making it comparatively more expensive than its parametric counterparts. Leveraging the theory of random Fourier features, we provide an efficient randomized implementation which recovers the computational complexity of classical parametric methods while provably retaining the expressiveness of the nonparametric input. In particular, our explicit bounds only depend polynomially on the underlying parameters of the system, allowing our proposed algorithms to efficiently scale to high-dimensional systems. As an illustration of the method, we demonstrate the ability of the algorithm to learn a predictive model for a 60-dimensional system consisting of ten point masses interacting through Newtonian gravitation.