Qiang Gan

Fuzzy local linearization and local basis function expansion in nonlinear system modeling

Fuzzy local linearization is compared with local basis function expansion for modeling unknown nonlinear processes. First-order Takagi-Sugeno fuzzy model and the analysis of variance (ANOVA) decomposition are combined for the fuzzy local linearization of nonlinear systems, in which B-splines are used as membership functions of the fuzzy sets for input space partition. A modified algorithm for adaptive spline modeling of observation data (MASMOD) is developed for determining the number of necessary B-splines and their knot positions to achieve parsimonious models. This paper illustrates that fuzzy local linearization models have several advantages over local basis function expansion based models in nonlinear system modeling.

Linearization and state estimation of unknown discrete-time nonlinear dynamic systems using recurrent neurofuzzy networks

Model-based methods for the state estimation and control of linear systems have been well developed and widely applied. In practice, the underlying systems are often unknown and nonlinear. Therefore, data based model identification and associated linearization techniques are very important. Local linearization and feedback linearization have drawn considerable attention in recent years. In this paper, linearization techniques using neural networks are reviewed, together with theoretical difficulties associated with the application of feedback linearization. A recurrent neurofuzzy network with an analysis of variance (ANOVA) decomposition structure and its learning algorithm are proposed for linearizing unknown discrete-time nonlinear dynamic systems. It can be viewed as a method for approximate feedback linearization, as such it enlarges the class of nonlinear systems that can be feedback linearized using neural networks. Applications of this new method to state estimation are investigated with realistic simulation examples, which shows that the new method has useful practical properties such as model parametric parsimony and learning convergence, and is effective in dealing with complex unknown nonlinear systems.

State estimation and multi-sensor data fusion using data-based neurofuzzy local linearisation process models

Abstract Modelling unknown non-linear dynamic processes is an essential prerequisite for model-based state estimation and fusion. Fuzzy local linearisation (FLL) is a useful divide-and-conquer method for coping with complex problems such as data-based non-linear process modelling. In this paper, a hybrid learning scheme which combines a modified adaptive spline modelling (MASMOD) algorithm and the expectation-maximisation (EM) algorithm is developed for FLL modelling, based on which Kalman filter type algorithms for state estimation and multi-sensor data fusion are investigated. Two commonly used measurement fusion methods are analytically compared. A hierarchical multi-sensor data fusion architecture is proposed, with an example of non-linear trajectory estimation to validate the proposed method, which integrates the techniques for FLL modelling, neurofuzzy state estimation and multi-sensor data fusion. Whilst this paper mainly focuses on state estimation and data fusion for unknown non-linear dynamic processes, maneuvering targets are also briefly considered.

A hybrid learning scheme combining EM and MASMOD algorithms for fuzzy local linearization modeling

Fuzzy local linearization (FLL) is a useful divide-and-conquer method for coping with complex problems such as modeling unknown nonlinear systems from data for state estimation and control. Based on a probabilistic interpretation of FLL, the paper proposes a hybrid learning scheme for FLL modeling, which uses a modified adaptive spline modeling (MASMOD) algorithm to construct the antecedent parts (membership functions) in the FLL model, and an expectation-maximization (EM) algorithm to parameterize the consequent parts (local linear models). The hybrid method not only has an approximation ability as good as most neuro-fuzzy network models, but also produces a parsimonious network structure (gain from MASMOD) and provides covariance information about the model error (gain from EM) which is valuable in applications such as state estimation and control. Numerical examples on nonlinear time-series analysis and nonlinear trajectory estimation using FLL models are presented to validate the derived algorithm.

Fuzzy and neurofuzzy modelling

Fuzzy logic and fuzzy systems have received considerable attention both in scientific and popular media, yet the basic techniques of vagueness go back to at least the 1920s. Zadeh’s seminal paper in 1965 on fuzzy logic introduced much of the terminology that is used conventionally in fuzzy logic today. The considerable success of fuzzy logic products, as in automobiles, cameras, washing machines, rice cookers, etc. has done much to temper much of the scorn poured out by the academic community on the ideas first postulated by Zadeh. The existing fuzzy logic literature, number of international conferences and academic journals, together with a rapidly increasing number and diversity of applications is a testament to the vitality and importance of this subject, despite the continuing debate over its intellectual viability.

An introduction to modelling and learning algorithms

Conventional science and engineering is based on a set of abstract physical or phenomenological principles based on an evolving scientific theory such as Newtonian mechanics, wave theory or thermodynamics, which are then validated by experimental or observational data. However for analytic tractability and ease of comprehension much of this theory is linear and time invariant, yet as the performance requirement and the range of operation of controlled processes increases, nonlinear, nonstationary and stochastic system behaviours need to be adequately represented. The power of current parallel computers allows simulation methods such as computational fluid dynamics, computational electromagnetics, or finite element methods to be integrated with observational data to resolve some of these nonlinear modelling problems. However such approaches are inappropriate to processes that are too complex to have ‘local’ phenomenological representations (such as Navier Stokes, or Maxwell equations), or the underlying causal processes or physical relationships are a priori unknown. If measurement data or information is available in sufficient quantities and richness, then models or relationships between a systems independent and dependent (input-output) variables can be estimated by empirical or data driven paradigms. This is not a new approach, since it is the basis of classical statistical estimation theory whereby the properties of some a priori unknown system output y(t) = f(x(t)) is estimated from available sets of input-output data \( {D_N} = \{ x(t), y(t)\} _{t = 1}^N \), drawn from an unknown probability distribution, which is then utilised to predict future outputs to unseen input data (x(t)) (the principle of generalisation), where the observed data set is the collection of all observed input-output regressors of the unknown process f(·).

Delaunay input space partitioning modelling

We have already demonstrated in previous chapters that data-b ased modelling is a complex and demanding process especially if there is little prior knowledge about the underlying process. In practice there is usually some process structural knowledge that can be exploited in model construction, additionally specific model structures lend themselves to subsequent uses such as a control design and estimation (see Chapter 8). For many processes the derived models are only valid across a limited domain (for empirical data modelling this is usually determined by the data gathering process) , and are often implicitly represented in the model. In this regard the concept of local models is a promising te chnique since they combine conventional system theory approaches (especially for locally linear models — see Chapter 6) with adaptive learning algorithms.

Local neurofuzzy modelling

The previously described model construction algorithms of Chapter 5 resolved model complexity by utilising the divide and conquer strategy, by decomposing high dimensional problems into a number of lower dimensional submodels each with variable dependencies whose composite solution yields the original complex problem. Central to the conventional neurofuzzy approach is an orthogonal axis partitioning of the input space, which is the main cause of the curse of dimensionality. Clearly, other decompositions or partitioning are possible including irregular simplexes (see Figure 6.1(a)) and data clustering with centred Gaussians (see Figure 6.1(b)).

Support vector neurofuzzy models

The class of models considered so far are generalised linear models that construct nonlinear models by linear combinations of nonlinear basis functions such as B-splines, or Gaussian radial basis in the input or observed variables x. The power of these models is their ability to incorporate prior knowledge by structurally designing the network through choice of the type, number and position of the basis functions. This form of structural regularisation is the basis of many of the construction algorithms introduced in this book. All of these generalised linear model are examples of parametric models in which weights or parameters are identified by using linear optimisation techniques (see Chapter 3). A further class of models can be defined which do not explicitly depend on a set of parameters, the so-called nonparametric models, are applicable to sparse data sets in relatively high dimensional spaces. In nonparametric models, the parameters are not predetermined, but are determined by the training data so that the model capacity reflects complexity contained in the data. Of particular importance in this context is the class of nonparametric models whose output is a linear combination of functions of the observations x, where the linear weighting functions are determined by the characteristics of the kernel functions. The approximation of the approximant is taken over a functional including some measure of the data fit and a functional penalising certain characteristics of the approximant, this ensures a well posed solution.

Basic concepts of data-based modelling

In this chapter we provide the basic model mathematical representations that are used throughout the book. Both parametric and nonparametric models require structural representations which reflect the modeller’s prior knowledge of the underlying process, or is selected so as to provide a form that ensures process identification easily from observed data, or is selected with another end process in mind such as control or condition monitoring. Generally in control and fault diagnosis problems, the model representation is in state space form (see Section 2.2) as knowledge of the unknown systems states are required for detection of process faults, or for state feedback control, or for state vector data fusion (see Section 9.3). Fundamental to this book is the representation of nonlinear observable processes by additive basis function expansions for which the basis functions are locally defined (i.e. have compact local support) rather than global basis functions such as general polynomials. In the sequel this representation coupled with linearly adjustable parameters is shown to have many knowledge and computational based advantages such as parameterisation via linear optimisation, easy incorporation of prior knowledge, and model transparency with direct links to fuzzy rule base representations.