Online Nonlinear System Identification in High Dimensional Environments Duncan Potts and Claude Sammut The University of New South Wales, Australia {duncanp,claude}@cse.unsw.edu.au Abstract This paper compares a range of techniques from both the system identification and machine learning literature that can construct nonlinear models online. Strong parallels exist between the various representations giving insights into which methods can scale up to a high num- ber of dimensions. It is found that tree-based algorithms scale well, are easy to implement, and require minimal prior knowledge. The al- gorithms are empirically tested on a number of tasks ranging from the modelling of a simple il- lustrative test function up to the identification of complex high-dimensional aircraft dynamics. 1 Introduction A standard method for identifying systems is to compile a set of example data containing samples of the input and output signals. A batch process can then use this data set to optimise various model parameters. Some- times it is not possible to obtain this data set in advance, or there is too much data for the batch process, in which case some form of online algorithm is required that can update the system model as more data arrives. For ex- ample an autonomous system may have an approximate initial model of its environment, but would benefit from updating this model in an online fashion as new infor- mation is received from its sensors. In this paper it is assumed that a discrete time dy- namic environment can be fully described by the time- invariant state-space model z k+1 = f (z k , u k ) (1) where z k is the n-dimensional state vector and u k is the m-dimensional action vector at time k (this is a sim- ple extension of continuous time representations such as in [Slotine and Li, 1991] where the input is con- stant between sampling times). Furthermore it is as- sumed that the entire state and action vectors can be sensed (with observation noise) at each sampling instant k. The system identification task is to find a sufficiently accurate approximation to the multiple-input multiple- output (MIMO) function f given any available prior knowledge and a sequence of observations of the system’s behaviour. Common classical methods of system identification in- clude analysing either the frequency response of the sys- tem to sine waves of different frequencies or the time- domain response to impulse or step inputs. These tech- niques only apply to linear systems and are not suited to the MIMO domains in which we are interested. If the underlying system is linear and the loss function be- ing minimised is chosen to be a sum of the squared er- rors, the problem is a linear optimisation and many well known batch and online solutions exist. However most interesting real world domains exhibit some degree of non-linearity and system identification becomes signifi- cantly harder. The identification of nonlinear MIMO models usually involves a transformation of the inputs such that the model f is linear in the new feature space, and stan- dard online learning techniques can be applied [Ljung, 1987]. This transformation, however, requires detailed prior knowledge of the types of non-linearity present. When there is minimal prior knowledge a number of standard techniques from the literature can still be ap- plied in an online manner, although each suffers from particular drawbacks. This paper compares several methods from the system identification and machine learning literature that can construct nonlinear models online. With the exception of neural networks, strong parallels between the different representations are observed, whereas the learning algo- rithms themselves are very different. These insights in- dicate which techniques can scale to higher dimensional problems, and the algorithms are empirically tested on a simple 2D test function, a pendulum on a cart, and on the identification of complex aircraft dynamics. It is often advantageous to decompose the large MIMO 1