A new direct approach of computing multi-step ahead predictions for non-linear models

A new direct approach of computing multi-step ahead predictions for non-linear time series is introduced. The covariance of the parameter estimates associated with, and the mean squared k -step ahead prediction errors of the new direct approach are smaller than those obtained using the conventional direct approach. Numerical examples are included to illustrate the application of the new direct approach.     

Non-linear output frequency response functions for multi-input non-linear Volterra systems

The concept of non-linear output frequency response functions (NOFRFs) is extended to the non-linear systems that can be described by a multi-input Volterra series model. A new algorithm is also developed to determine the output frequency range of non-linear systems from the frequency range of the inputs. These results allow the concept of NOFRFs to be applied to a wide range of engineering systems. The phenomenon of the energy transfer in a two degree of freedom non-linear system is studied using the new concepts to demonstrate the significance of the new results.     

Identification of MIMO non-linear systems using a forward-regression orthogonal estimator

An orthogonal least squares estimator, which was originally derived for single-input single-output systems, is extended to multi-input multi-output non-linear systems. The estimator can provide information about the structure, or which terms to include in the model, and final parameter estimates in a very simple and efficient manner. Multivariable non-linear model validity tests are also discussed and the results of applying the orthogonal estimator to both simulated and real data are included.     

New bound characteristics of NARX model in the frequency domain

New results about the bound characteristics of both the generalized frequency response functions (GFRFs) and the output frequency response for the NARX (Non-linear AutoRegressive model with eXogenous input) model are established. It is shown that the magnitudes of the GFRFs and the system output spectrum can all be bounded by a polynomial function of the magnitude bound of the first order GFRF, and the coefficients of the polynomial are functions of the NARX model parameters. These new bound characteristics of the NARX model provide an important insight into the relationship between the model parameters and the magnitudes of the system frequency response functions, reveal the effect of the model parameters on the stability of the NARX model to a certain extent, and provide a useful technique for the magnitude based analysis of nonlinear systems in the frequency domain, for example, evaluation of the truncation error in a volterra series expression of non-linear systems and the highest order needed in the volterra series approximation. A numerical example is given to demonstrate the effectiveness of the theoretical results.     

Neighborhood detection for the identification of spatiotemporal systems.

Neighborhood detection and local state vector construction for the identification of spatiotemporal systems is considered in this paper. Determining the neighborhood size both in the space and time domain can considerably reduce the complexity of the set of candidate model terms for the identification of coupled map lattice models. The computation requirements of the model identification algorithm can also be greatly reduced instead of the more direct identification approach of searching over the entire spatiotemporal neighborhood in the original space. In this paper, a new neighborhood detection method is introduced based on embedding theory for nonlinear dynamical systems to produce an initial spatiotemporal neighborhood for the identification of spatiotemporal systems. Numerical examples are provided to demonstrate the feasibility and applicability of the new neighborhood detection method.     

An investigation into the characteristics of non-linear frequency response functions. Part 2: New analysis methods based on symbolic expansions and graphical techniques

In Part 1 of this paper the concepts of input and output frequency subdomains were introduced to give insight into the relation between one dimensional and multi-dimensional frequency spaces. The visualisation of both magnitude and phase responses of third order generalized frequency response functions was also presented. In this, the second part, symbolic expansion techniques are introduced and together with the results achieved in Part 1, yield new methods for analysing the properties of generalized frequency response functions. Case studies are included to illustrate the application of the new methods.     

An application of Lyapunov stability analysis to improve the performance of NARMAX models

Previously we presented a novel approach to program a robot controller based on system identification and robot training techniques. The proposed method works in two stages: first, the programmer demonstrates the desired behaviour to the robot by driving it manually in the target environment. During this run, the sensory perception and the desired velocity commands of the robot are logged. Having thus obtained training data we model the relationship between sensory readings and the motor commands of the robot using ARMAX/NARMAX models and system identification techniques. These produce linear or non-linear polynomials which can be formally analysed, as well as used in place of “traditional robot” control code.In this paper we focus our attention on how the mathematical analysis of NARMAX models can be used to understand the robot’s control actions, to formulate hypotheses and to improve the robot’s behaviour. One main objective behind this approach is to avoid trial-and-error refinement of robot code. Instead, we seek to obtain a reliable design process, where program design decisions are based on the mathematical analysis of the model describing how the robot interacts with its environment to achieve the desired behaviour. We demonstrate this procedure through the analysis of a particular task in mobile robotics: door traversal.     

Multi-directional model validity tests for non-linear system identification

New multi-directional model validation tests are derived to provide improved statistical validation test procedures for a wide class of non-linear modelling methods.