Introduction

Nonlinear control is difficult even in the case that the system dynamics are known. If the dynamics are not known, the traditional approach is to make a model of the dynamics (system identification) and then try to control the simulated model (nonlinear model-predictive control). The model learned from data is of course not perfect, but these imperfections are often ignored. The modern view of control sees feedback as a tool for uncertainty management [11], but managing it already in the modelling might have advantages. For instance, the controller can avoid regions where the confidence in model is not high enough [9]. The idea of studying uncertainty in control is not new. It is known that the magnitude of motor noise in human hand motion is proportional to muscle activation [10]. In control theory, the theoretical foundations are already well covered in [3]. In [14], a nonlinear state-space model is used for control. The nonlinearities are modelled using piecewise affine mappings. Parameters are estimated using the prediction error method, which is equivalent to the maximum likelihood estimate in the Bayesian framework. Nonlinear dynamical factor analysis (NDFA) [17] is a state-of-the-art tool for finding nonlinear state-space models with variational Bayesian learning. This paper is about using NDFA for control. In NDFA, the parameters, the states, and the observations are real-valued vectors that are modelled with parametrised probability distributions. Uncertainties from noisy observations and model imperfections are thus taken explicitly into account. Learning is extremely important for control of complex systems [2]. The proposed method involves learning in more than one way. The original NDFA is based on unsupervised learning. That is, it creates a model of the underlying dynamics by passively making observations. When used for control, though, control signals need to be selected either by following an example or by maximising a reward. The model should thus not only learn the dynamics, but also learn to help control. The rest of the paper is structured as follows: In Section II, a nonlinear state-space model is reviewed and in Section III its use as a controller is presented. After experiments in Section IV matters are discussed and concluded.