A kernel method for non-linear systems identification – infinite degree volterra series estimation

Volterra series expansions are widely used in analyzing and solving the problems of non-linear dynamical systems. However, the problem that the number of terms to be determined increases exponentially with the order of the expansion restricts its practical application. In practice, Volterra series expansions are truncated severely so that they may not give accurate representations of the original system. To address this problem, kernel methods are shown to be deserving of exploration. In this report, we make use of an existing result from the theory of approximation in reproducing kernel Hilbert space (RKHS) that has not yet been exploited in the systems identification field. An exponential kernel method, based on an RKHS called a generalized Fock space, is introduced, to model non-linear dynamical systems and to specify the corresponding Volterra series expansion. In this way a non-linear dynamical system can be modelled using a finite memory length, infinite degree Volterra series expansion, thus reducing the source of approximation error solely to truncation in time. We can also, in principle, recover any coefficient in the Volterra series

A kernel method for non-linear systems indentification - infinite degree volerra series estimation

Volterra series expansions are widely used in analyzing and solving the problems of non-linear dynamical systems. However, the problem that the number of terms to be determined increases exponentially with the order of the expansion restricts its practical application. In practice, Volterra series expansions are truncated severely so that they may not give accurate representations of the original system. To address this problem, kernel methods are shown to be deserving of exploration. In this report, we make use of an existing result from the theory of approximation in reproducing kernel Hilbert space (RKHS) that has not yet been exploited in the systems identification field. An exponential kernel method, based on an RKHS called a generalized Fock space, is introduced, to model non-linear dynamical systems and to specify the corresponding Volterra series expansion. In this way a non-linear dynamical system can be modelled using a finite memory length, infinite degree Volterra series expansion, thus reducing the source of approximation error solely to truncation in time. We can also, in principle, recover any coefficient in the Volterra series

Some lemmas on reproducing kernel Hilbert spaces

Reproducing kernal Hilbert spaces (RKHS) provide a framework for approximation from finite data using the idea of bounded linear functionals. The approximation problem in this case can be viewed as the inverse problem of finding the optimum operator from the Euclidean space of observations to some subspace of the RKHS. In constructing the appropriate invers operator, use is made of both adjoint operators in RKHS and various norms. In this report a number of lemmas are given with respect to such adjoint operators and norms

Iterative sparse interpolation in reproducing kernel Hilbert spaces

The problem of interpolating data in reproducing kernel Hilbert spaces is well known to be ill-conditioned. In the presence of noise, regularisation can be applied to find a good solution. In the noise-free case, regularisation has the effect of over-smoothing the function and few data points are interpolated. In this paper an alternative framework, based on sparsity, is proposed for interpolation of noise-free data. Iterative construction of a sparse sequence of interpolants is shown to be well defined and produces good results

Steepest descent for a linear operator equation of the second kind with application to Tikhonov regularisation

Let H1 H2 be Hilbert spaces, T a bounded linear operator on H1 into H2 such that the range of T, R (T), is closed. Lrt T* denote the adjoint of T. In this paper, we review the generalised solution and method of steepest descent, for the linear operator equation, Tx=b,b E H2. Further, we establish the convergence of the method of steepest descent to the unique solution (T*T=.......

Angle and position perception for exploration with active touch

Over the past few decades the design of robots has gradually improved, allowing them to perform complex tasks in interaction with the world. To behave appropriately, robots need to make perceptual decisions about their environment using their various sensory modalities. Even though robots are being equipped with progressively more accurate and advanced sensors, dealing with uncertainties from the world and their sensory processes remains an unavoidable necessity for autonomous robotics. The challenge is to develop robust methods that allow robots to perceive their environment while managing uncertainty and optimizing their decision making. These methods can be inspired by the way humans and animals actively direct their senses towards locations for reducing uncertainties from perception [1]. For instance, humans not only use their hands and fingers for exploration and feature extraction but also their movements are guided according to what it is being perceived [2]. This behaviour is also present in the animal kingdom, such as rats that actively explore the environment by appropriately moving their whiskers [3]. © 2013 Springer-Verlag Berlin Heidelberg

Reduction of kernel models

Kernel models can be expensive to compute and in a non-stationary environment can become unmanageably large. Here we present several previously reported techniques for reducing the complexity of these models in a common framework. This reformulation leads to the development of further related reduction techniques and clarifies the relationships between these and the existing techniques

Sampling-based stochastic optimal control with metric interval temporal logic specifications

This paper describes a method to find optimal policies for stochastic dynamic systems that maximise the probability of satisfying real-time properties. The method consists of two phases. In the first phase, a coarse abstraction of the original system is created. In each region of the abstraction, a sampling-based algorithm is utilised to compute local policies that allow the system to move between regions. Then, in the second phase, the selection of a policy in each region is obtained by solving a reachability problem on the Cartesian product between the abstraction and a timed automaton representing a real-time specification given as a metric interval temporal logic formula. In contrast to current methods that require a fine abstraction, the proposed method achieves computational tractability by modelling the coarse abstraction of the system as a bounded-parameter Markov decision process (BMDP). Moreover, once the BMDP is created, this can be reused for new specifications assuming the same stochastic system and workspace. The method is demonstrated with an autonomous driving example