State-dependent parameter modelling and identification of stochastic non-linear sampled-data systems

State-dependent parameter representations of stochastic non-linear sampled-data systems are studied. Velocity-based linearization is used to construct state-dependent parameter models which have a nominally linear structure but whose parameters can be characterized as functions of past outputs and inputs. For stochastic systems state-dependent parameter ARMAX (quasi-ARMAX) representations are obtained. The models are identified from input–output data using feedforward neural networks to represent the model parameters as functions of past inputs and outputs. Simulated examples are presented to illustrate the usefulness of the proposed approach for the modelling and identification of non-linear stochastic sampled-data systems.     

Constructing NARMAX models using ARMAX models

This paper outlines how it is possible to decompose a complex non-linear modelling problem into a set of simpler linear modelling problems. Local ARMAX models valid within certain operating regimes are interpolated to construct a global NARMAX (non-linear NARMAX) model. Knowledge of the system behaviour in terms of operating regimes is the primary basis for building such models, hence it should not be considered as a pure black-box approach, but as an approach that utilizes a limited amount of a priori system knowledge. It is shown that a large class of non-linear systems can be modelled in this way, and indicated how to decompose the systems range of operation into operating regimes. Standard system identification algorithms can be used to identify the NARMAX model, and several aspects of the system identification problem are discussed and illustrated by a simulation example.     

Prediction-error estimation algorithm for non-linear output-affine systems

A prediction-error estimation algorithm is developed for non-linear discrete-time systems which can be represented by the output-affine difference equation model. The theory of hypothesis testing is employed to select a model with the correct structure. Problems relating to the use of output-affine models in the identification of non-linear systems are discussed, and a comparison with the NARMAX (nonlinear ARMAX) model is given.     

An investigation into the characteristics of non-linear frequency response functions. Part 1: Understanding the higher dimensional frequency spaces

The characteristics of generalized frequency response functions (GFRFs) of non-linear systems in higher dimensional space are investigated using a combination of graphical and symbolic decomposition techniques. It is shown how a systematic analysis can be achieved for a wide class of non-linear systems in the frequency domain using the proposed methods. The paper is divided into two parts. In Part 1, the concepts of input and output frequency subdomains are introduced to give insight into the relationship between one dimensional and multi-dimensional frequency spaces. The visualization of both magnitude and phase responses of third order generalized frequency response functions is presented for the first time. In Part 2 symbolic expansion techniques are introduced and new methods are developed to analyse the properties of generalized frequency response functions of non-linear systems described by the NARMAX class of models. Case studies are included in Part 2 to illustrate the application of the new methods.     

Input-output modeling of nonlinear systems with time-varying linearmodels

Time varying ARMA (autoregressive moving average) and ARMAX (autoregressive moving average with exogenous inputs) models are proposed for input-output modeling of nonlinear deterministic and stochastic systems. The coefficients of these models are estimated by a random walk Kalman filter (RWKF). This method requires no prior assumption on the nature of the model coefficients, and is suitable for real-time implementation since no off-line training is needed. A simulation example illustrates the method. Goodness of performance is judged by the quality of the residuals, histograms, autocorrelation functions and the Kolmogorov-Smirnoff test     

Identification of multi-input systems: variance analysis and input design issues

This paper examines the identification of multi-input systems. Motivated by an experiment design problem (should one excite the various inputs simultaneously or separately), we examine the effect of an additional input on the variance of the estimated coefficients of parametrized rational transfer function models, with special emphasis on the commonly used FIR, ARX, ARMAX, OE and BJ model structures. We first show that, for model structures that have common parameters in the input-output and noise models (e.g. ARMAX), any additional input contributes to a reduction of the covariance of all parameter estimates. We then show that the accuracy improvement extends beyond the case of common parameters in all transfer functions, and we show exactly which parameter estimates are improved when a new input is added. We also conclude that it is always better to excite all inputs simultaneously.     

Optimal tracking design for stochastic fuzzy systems

In general, fuzzy control design for stochastic nonlinear systems is still difficult since the fuzzy bases are stochastic so as to increase the difficulty and complexity of the fuzzy tracking control design. In this study, a fuzzy stochastic moving-average model with control input (fuzzy ARMAX model) is introduced to describe nonlinear stochastic systems. From the fuzzy ARMAX model, a fuzzy one-step ahead prediction model is developed. Based on a fuzzy one-step ahead prediction stochastic model, optimal design algorithms are proposed to achieve the optimal tracking of nonlinear stochastic systems. In this study, the minimum variance tracking control, generalized minimum variance tracking control, and the optimal model reference tracking control are developed for stochastic fuzzy systems. We construct some basic stability conditions for general stochastic fuzzy systems and use these conditions to verify the stability of the fuzzy tracking control systems. Finally, two simulation examples are given to indicate the performance of the proposed methods.     

Discrete-time fractional-order PID controller: Definition,tuning, digital realization and some applications

In some of the complicated control problems we have to use the controllers that apply nonlocal operators to the error signal to generate the control. Currently, the most famous controller with nonlocal operators is the fractional-order PID (FOPID). Commonly, after tuning the parameters of FOPID controller, its transfer function is discretized (for realization purposes) using the so-called generating function. This discretization is the origin of some errors and unexpected results in feedback systems. It may even happen that the controller obtained by discretizing a FOPID controller works worse than a directly-tuned discrete-time classical PID controller. Moreover, FOPID controllers cannot directly be applied to the processes modeled by, e.g., the ARMA or ARMAX model. The aim of this paper is to propose a discrete-time version of the FOPID controller and discuss on its properties and applications. Similar to the FOPID controller, the proposed structure applies nonlocal operators (with adjustable memory length) to the error signal. Two methods for tuning the parameters of the proposed controller are developed and it is shown that the proposed controller has the capacity of solving complicated control problems.     

Detection and classification of non-linearities based on Volterra kernels processing

Modelling and identification of the real systems, which are more or less non-linear, are important activities not only for scientific research but also for real-time applications like fault detection and compensation. In this paper the non-linear system is defined by a serial combination of a linear part followed by a variable static parameterised non-linearity. A structure to detect and classify some non-linearities based on Volterra kernels is proposed. Such kernels can be estimated from the I–O measurements of the system under study. Processing Volterra kernels means estimation followed by classification. Mainly artificial neural networks to obtain a fast answer process the Volterra kernels. The neural methods and the detection structure can be developed to study more complicated non-linear models for fault detection purposes and structural identification of complex non-linear systems. The feasibility of the method is demonstrated using a simulated example.     

Diagnosability of fuzzy discrete event systems

In this paper, discrete event systems (DESs) are reformulated as fuzzy discrete event systems (FDESs) and fuzzy discrete event dynamical systems (FDEDSs). These frameworks include fuzzy states, events and IF-THEN rules. In these frameworks, all events occur at the same time with different membership degrees. Fuzzy states and events have been introduced to describe uncertainties that occur often in practical problems, such as fault diagnosis applications. To measure a diagnoser’s fault discrimination ability, a fuzzy diagnosability degree is proposed. If the diagnosability of the degree of the system yields one a diagnoser can be implemented to identify all possible fault types related to a system. For any degree less than one, researchers should not devote their time to distinguish all possible fault types correctly. Thus, two different diagnosability definitions FDEDS and FDES are introduced. Due to the specialized fuzzy rule-base embedded in the FDEDS, it is capable of representing a class of non-linear dynamic system. Computationally speaking, the framework of diagnosability of the FDEDS is structurally similar to the framework of diagnosability of a non-linear system. The crisp DES diagnosability has been turned into the term fuzzy diagnosability for the FDES. The newly proposed diagnosability definition allows us to define a degree of diagnosability in a class of non-linear systems. In addition, a simple fuzzy diagnosability checking method is introduced and some numerical examples are provided to illustrate this theoretical development. Finally, the potential applications of the proposed method are discussed.     

Identification of non-linear characteristics of a class of block-oriented non-linear systems via Daubechies wavelet-based models

The paper deals with the identification of non-linear characteristics of a class of block-oriented dynamical systems. The systems are driven by random stationary white processes (i.i.d. random input sequences) and disturbed by a zero-mean stationary, white or coloured, random noise. The prior knowledge about non-linear characteristics is non-parametric excluding implementation of standard parametric identification methods. To recover non-linearities, a class of Daubechies wavelet-based models using only input-output measurement data is introduced and their accuracy is investigated in the global MISE error sense. It is shown that the proposed models converge with a growing collection of data to the true non-linear characteristics (or their versions), provided that the complexity of the models is appropriately fitted to the number of measurements. Suitable rules for optimum model size selection, maximizing the convergence speed, are given and the asymptotic rate of convergence of the MISE error for optimum models is established. It is shown that in some circumstances the rate is the best possible that can be achieved in non-parametric inference. We also show that the convergence conditions and the asymptotic rate of convergence are insensitive to the correlation of the noise and are the same for known and unknown input probability density function (assumed to exist). The theory is illustrated by simulation examples.     

Discrete time subharmonic modelling and analysis

Traditionally the Volterra time and frequency domain analysis tools cannot be applied to severely non-linear systems. In this paper, a new method of building a time-domain NARX MISO model for a class of severely SISO non-linear systems that exhibit subharmonics is introduced and it is shown how this allows the Volterra time and frequency domain analysis to be extended to this class of non-linear systems. The new approach is based on decomposing the original single input based on a Fourier analysis to provide a set of modified input signals which have the same period as the output signal. A MISO NARX model can then be constructed from the decomposed multiple inputs and the single output signal. The resulting MISO model is shown to meet the basic requirement for the existence of a Volterra series representation from which important frequency domain properties can be derived, explained and discussed. This is done by first introducing the derivation of generalized frequency response functions (GFRFs) from time domain MISO NARX models. The steady state response synthesis problem using the input spectrum and the MISO GFRFs is then investigated in order to verify the effectiveness and accuracy of the MISO modelling approach for severely non-linear systems. Finally a new frequency domain analysis method is introduced for systems that exhibit subharmonic oscillations.     

基于模糊ARMAX模型的模糊建模

提出了一种利用MGS(modified Gram-Schmidt)算法建立模糊ARMAX模型的方法, 给出了基于MGS算法的模型结构和参数辨识的一体化方法. 利用MGS正交变换对通过GK模糊聚类的聚类结果进行变换, 确定对模型贡献大的规则, 删除对模型贡献小的规则, 同时对模型中的参数进行估计. 本文提出的方法能够实现模糊模型的结构和参数的优化. 仿真结果表明, 本文提出的方法能够建立非线性系统的模糊ARMAX模型.     

Inference in HIV dynamics models via hierarchical likelihood

HIV dynamical models are often based on non-linear systems of ordinary differential equations (ODE), which do not have an analytical solution. Introducing random effects in such models leads to very challenging non-linear mixed-effects models. To avoid the numerical computation of multiple integrals involved in the likelihood, a hierarchical likelihood (h-likelihood) approach, treated in the spirit of a penalized likelihood is proposed. The asymptotic distribution of the maximum h-likelihood estimators (MHLE) for fixed effects is given. The MHLE are slightly biased but the bias can be made negligible by using a parametric bootstrap procedure. An efficient algorithm for maximizing the h-likelihood is proposed. A simulation study, based on a classical HIV dynamical model, confirms the good properties of the MHLE. The method is applied to the analysis of a clinical trial.     

Adaptive stabilisation for a large class of high-order uncertain non-linear systems

The problem of global adaptive stabilisation by state-feedback is investigated for a class of high-order non-linear systems with uncertain control coefficients and zero dynamics. First, some appropriate unknown parameters are introduced to obtain the updating laws when adapting control design. Then, by the flexible way of combining adding a power integrator with adaptive technique and the idea of changing supply functions, the requirement on the uncertain control coefficients is relaxed, and a recursive design procedure is successfully developed to achieve a continuous adaptive stabilising controller. Finally, an example is provided to illustrate the correctness of the theoretical results.     

Application of a modified semismooth Newton method to some elasto-plastic problems

Some elasto-plasticity models with hardening are discussed and some incremental finite element methods with different time discretisation schemes are considered. The corresponding one-time-step problems lead to variational equations with various non-linear operators. Common properties of the non-linear operators are derived and consequently a general problem is formulated. The problem can be solved by Newton-like methods. First, the semismooth Newton method is analysed. The local superlinear convergence is proved in dependence on the finite element discretisation parameter. Then it is introduced a modified semismooth Newton method which contain suitable “damping” in each Newton iteration in addition. The determination of the damping coefficients uses the fact that the investigated problem can be formulated as a minimisation one. The method is globally convergent, independently on the discretisation parameter. Moreover the local superlinear convergence also holds. The influence of inexact inner solvers is also discussed. The method is illustrated on a numerical example.     

Identification and prediction of stochastic dynamical systems in a polynomial chaos basis

Non-parametric system identification techniques have been proposed for constructing predictive models of dynamical systems without detailed knowledge of the mechanisms of energy transfer and dissipation. In a class of such models, multi-dimensional Chebychev polynomials in the state variables are fitted to the observed dynamical state of the system. Due to the approximative nature of this non-parametric model as well as to various other sources of uncertainty such as measurement errors and non-anticipative excitations, the parameters of the model exhibit a scatter that is treated here in a probabilistic context. The statistics of these coefficients are related to the physical properties of the model being analyzed, and are used to endow the model predictions with a probabilistic structure. They are also used to obtain a parsimonious characterization of the predictive model while maintaining a desirable level of accuracy. The proposed methodology is quite simple and robust.     

On total stability of competitive equilibrium

Despite the fact that the demand and supply characteristics of n competitive market are inherently non-stationary, stability properties of time-varying models of multiple markets have remained largely unexplored. The purpose of this work is to introduce a time-dependent non-linear formulation of the excess demand function, and consider time-varying effects on stability of the competitive equilibrium caused by structural changes in the interactions among individual markets. It will be shown that the classical Hicks conditions are sufficient (and sometimes necessary) for stability of a large class of non-linear non-stationary models of multiple market systems.     

Model selection approaches for non-linear system identification: a review

The identification of non-linear systems using only observed finite datasets has become a mature research area over the last two decades. A class of linear-in-the-parameter models with universal approximation capabilities have been intensively studied and widely used due to the availability of many linear-learning algorithms and their inherent convergence conditions. This article presents a systematic overview of basic research on model selection approaches for linear-in-the-parameter models. One of the fundamental problems in non-linear system identification is to find the minimal model with the best model generalisation performance from observational data only. The important concepts in achieving good model generalisation used in various non-linear system-identification algorithms are first reviewed, including Bayesian parameter regularisation and models selective criteria based on the cross validation and experimental design. A significant advance in machine learning has been the development of the support vector machine as a means for identifying kernel models based on the structural risk minimisation principle. The developments on the convex optimisation-based model construction algorithms including the support vector regression algorithms are outlined. Input selection algorithms and on-line system identification algorithms are also included in this review. Finally, some industrial applications of non-linear models are discussed.     

On hybrid control of a class of stochastic non-linear Markovian switching systems

In this paper, we address the problem of hybrid control for a class of stochastic non-linear Markovian switching systems. First, a hybrid controller is introduced for the systems. Then under some appropriate assumptions, the stabilization condition for the systems under pure impulsive control is given. Further under impulsive control, the output feedback stabilization problem of the systems is discussed and linear output feedback controllers are designed. Finally a numerical example is provided to illustrate the effectiveness of the proposed methods.