Although nonlinear model predictive control (NMPC) might be the best choice for a nonlinear plant, it is still not widely used. This is mainly due to the computational burden associated with solving online a set of nonlinear differential equations and a nonlinear dynamic optimization problem in real time. This thesis is concerned with strategies aimed at reducing the computational burden involved in different stages of the NMPC such as optimization problem, state estimation, and nonlinear model identification. A major part of the computational burden comes from function and derivative evaluations required in different parts of the NMPC algorithm. In this work, the problem is tackled using a recently introduced efficient tool, the automatic differentiation (AD). Using the AD tool, a function is evaluated together with all its partial derivative from the code defining the function with machine accuracy. A new NMPC algorithm based on nonlinear least square optimization is proposed. In a first–order method, the sensitivity equations are integrated using a linear formula while the AD tool is applied to get their values accurately. For higher order approximations, more terms of the Taylor expansion are used in the integration for which the AD is effectively used. As a result, the gradient of the cost function against control moves is accurately obtained so that the online nonlinear optimization can be efficiently solved. In many real control cases, the states are not measured and have to be estimated for each instance when a solution of the model equations is needed. A nonlinear extended version of the Kalman filter (EKF) is added to the NMPC algorithm for this purpose. The AD tool is used to calculate the required derivatives in the local linearization step of the filter automatically and accurately. Offset is another problem faced in NMPC. A new nonlinear integration is devised for this case to eliminate the offset from the output response. In this method, an integrated disturbance model is added to the process model input or output to correct the plant/model mismatch. The time response of the controller is also improved as a by–product. The proposed NMPC algorithm has been applied to an evaporation process and a two continuous stirred tank reactor (two–CSTR) process with satisfactory results to cope with large setpoint changes, unmeasured severe disturbances, and process/model mismatches. When the process equations are not known (black–box) or when these are too complicated to be used in the controller, modelling is needed to create an internal model for the controller. In this thesis, a continuous time recurrent neural network (CTRNN) in a state–space form is developed to be used in NMPC context. An efficient training algorithm for the proposed network is developed using AD tool. By automatically generating Taylor coefficients, the algorithm not only solves the differentiation equations of the network but also produces the sensitivity for the training problem. The same approach is also used to solve online the optimization problem of the NMPC. The proposed CTRNN and the predictive controller were tested on an evaporator and two–CSTR case studies. A comparison with other approaches shows that the new algorithm can considerably reduce network training time and improve solution accuracy. For a third case study, the ALSTOM gasifier, a NMPC via linearization algorithm is implemented to control the system. In this work a nonlinear state–space class Wiener model is used to identify the black–box model of the gasifier. A linear model of the plant at zero–load is adopted as a base model for prediction. Then, a feedforward neural network is created as the static gain for a particular output channel, fuel gas pressure, to compensate its strong nonlinear behavior observed in open–loop simulations. By linearizing the neural network at each sampling time, the static nonlinear gain provides certain adaptation to the linear base model. The AD tool is used here to linearize the neural network efficiently. Noticeable performance improvement is observed when compared with pure linear MPC. The controller was able to pass all tests specified in the benchmark problem at all load conditions.

<p>Direkt viktoptimering (Direct Weight Optimization, DWO) är en ickeparamterisk systemidentifieringsmetod. DWO bygger på att man skattar ett funktionsvärde i en viss punkt genom en viktad summa av mätvärden, där vikterna optimeras fram. Det faktum att DWO har en inparameter som man måste veta i förväg leder till att man på något sätt vill skatta denna inparameter. Det finns många sätt man kan göra denna skattning på men det centrala i denna uppsats är att skatta inparametern lokalt. Fördelen med detta är att metoden anpassar sig om till exempel systemet ändrar beteende från att variera långsamt till att variera snabbare. Denna typ av metoder brukar kallas adaptiva metoder.Det finns flera metoder för att skatta en inparameter lokalt och anpassningen till DWO är redan klar för ett fåtal som lämpar sig bra. Det är dock inte undersökt vilken av dessa metoder som ger det bästa resultatet för just DWO. Syftet med denna uppsats är alltså att ta reda på hur man lokalt kan skatta en inparameter till DWO på bästa sätt och om DWO är en bra grund att basera en adaptiv metod på.Det har visat sig att DWO kanske är för känslig för en lokalt vald inparameter för att vara en bra grund att basera en adaptiv metod på. Däremot utmärker sig en av metoderna för att skatta inparametern genom att vara mycket bättre än de andra metoderna när den kanske inte borde vara det. Varför den är så bra kan vara ett bra ämne för vidare forskning.</p> / <p>Direct Weight Optimization (DWO) is a nonparametric system identification meth\-od. In DWO the value of a function in a certain point is estimated by a weighted sum of measured values. The weights are obtained as a solution to a convex optimization problem. DWO has a design parameter which has to be chosen or estimated a priori. There are many ways to estimate this parameter. The main focus of this thesis is to estimate this parameter locally. The advantage of estimating the parameter locally is that the estimate will adapt if the system changes behavior from slowly varying to rapidly varying. Estimation methods of this type are usually called adaptive estimation methods.There are a number of adaptive estimation methods and the adaptation of some of these methods to DWO has already been done. There are however no evaluation studies done. The goal with this thesis is therefore to find out how to estimate the parameter in DWO in the best way and to find out whether DWO is a good base for an adaptive method.It turned out that DWO might be too sensitive to local changes in the design parameter to be a good base for an adaptive method. However, one of the adaptive estimation methods stands out from the rest because it is much better than the other methods when it, perhaps, should not. Why this method is good might be a good subject for further research.</p>

Direkt viktoptimering (Direct Weight Optimization, DWO) är en ickeparamterisk systemidentifieringsmetod. DWO bygger på att man skattar ett funktionsvärde i en viss punkt genom en viktad summa av mätvärden, där vikterna optimeras fram. Det faktum att DWO har en inparameter som man måste veta i förväg leder till att man på något sätt vill skatta denna inparameter. Det finns många sätt man kan göra denna skattning på men det centrala i denna uppsats är att skatta inparametern lokalt. Fördelen med detta är att metoden anpassar sig om till exempel systemet ändrar beteende från att variera långsamt till att variera snabbare. Denna typ av metoder brukar kallas adaptiva metoder.Det finns flera metoder för att skatta en inparameter lokalt och anpassningen till DWO är redan klar för ett fåtal som lämpar sig bra. Det är dock inte undersökt vilken av dessa metoder som ger det bästa resultatet för just DWO. Syftet med denna uppsats är alltså att ta reda på hur man lokalt kan skatta en inparameter till DWO på bästa sätt och om DWO är en bra grund att basera en adaptiv metod på.Det har visat sig att DWO kanske är för känslig för en lokalt vald inparameter för att vara en bra grund att basera en adaptiv metod på. Däremot utmärker sig en av metoderna för att skatta inparametern genom att vara mycket bättre än de andra metoderna när den kanske inte borde vara det. Varför den är så bra kan vara ett bra ämne för vidare forskning. / Direct Weight Optimization (DWO) is a nonparametric system identification meth\-od. In DWO the value of a function in a certain point is estimated by a weighted sum of measured values. The weights are obtained as a solution to a convex optimization problem. DWO has a design parameter which has to be chosen or estimated a priori. There are many ways to estimate this parameter. The main focus of this thesis is to estimate this parameter locally. The advantage of estimating the parameter locally is that the estimate will adapt if the system changes behavior from slowly varying to rapidly varying. Estimation methods of this type are usually called adaptive estimation methods.There are a number of adaptive estimation methods and the adaptation of some of these methods to DWO has already been done. There are however no evaluation studies done. The goal with this thesis is therefore to find out how to estimate the parameter in DWO in the best way and to find out whether DWO is a good base for an adaptive method.It turned out that DWO might be too sensitive to local changes in the design parameter to be a good base for an adaptive method. However, one of the adaptive estimation methods stands out from the rest because it is much better than the other methods when it, perhaps, should not. Why this method is good might be a good subject for further research.

A novel structural health assessment (SHA) technique is proposed. It is a finite element-based time domain nonlinear system identification technique. The procedure is developed in two stages to incorporate several desirable features and increase its implementation potential. First, a weighted global iteration with an objective function is introduced in the unscented Kalman filter (UKF) procedure in order to obtain stable, convergent, and optimal solution. Furthermore, it also improves the capability of the UKF procedure to identify a large structural system using only a short duration of responses measured at a limited number of dynamic degrees of freedom (DDOFs). The combined procedure is denoted as unscented Kalman filter with weighted global iteration (UKF-WGI). Then, UKF-WGI is integrated with iterative least-squares with unknown input (ILS-UI) in order to increase its implementation potential. The substructure concept is also incorporated in the procedure. The integrated procedure is denoted as unscented Kalman filter with unknown input and weighted global iteration (UKF-UI-WGI). The two most important features of the method are that it does not need information on input excitation and uses only limited number of noise-contaminated response information to identify structural systems. Also, the method is able to identify the defects at the local element level by tracking the changes in the stiffness of the structural elements in the finite element representation. The UKF-UI-WGI procedure is implemented in two stages. In Stage 1, based on the location of input excitation, the substructure is selected. Using only responses at all DDOFs in the substructure, ILS-UI can identify the input excitation time-histories, stiffness parameters of all the elements in the substructure, and two Rayleigh damping coefficients. The outcomes of the first stage are necessary to initiate UKF-WGI. Using the information from Stage 1, the stiffness parameters of all the elements in the structure are identified using UKF-WGI in Stage 2. To demonstrate the effectiveness of the procedure, health assessment of relatively large structural systems is presented. Small and relatively large defects are introduced at different locations in the structure and the capability of the method to detect the health of the structure is examined. The optimum number and location of measured responses are also investigated. It is demonstrated that the method is capable of identifying defect-free and defective states of the structures using minimum information. Furthermore, it can locate defect spot within a defective element accurately. The comparative studies are also conducted between the proposed methods and available methods in the literature. First, it is between the UKF-WGI and extended Kalman filter with weighted global iteration (EKF-WGI) procedure. Then, it is between UKF-UI-WGI and generalized iterative least-squares extended Kalman filter with unknown input (GILS-EKF-UI) procedure, developed earlier by the research team. It is demonstrated that the proposed UKF-based procedures are superior to the EKF-based procedures for SHA.

MATTOS, C. L. C. Recurrent gaussian processes and robust dynamical modeling. 2017. 189 f. Tese (Doutorado em Engenharia de Teleinformática)–Centro de Tecnologia, Universidade Federal do Ceará, Fortaleza, 2017. / Submitted by Renato Vasconcelos (ppgeti@ufc.br) on 2017-09-09T02:26:38Z No. of bitstreams: 1 2017_tes_clcmattos.pdf: 5961013 bytes, checksum: fc44d8b852e28fa0e1ebe0c87389c0da (MD5) / Rejected by Marlene Sousa (mmarlene@ufc.br), reason: Prezado César; Prezado Pedro: Existe uma orientação para que normalizemos as dissertações e teses da UFC, em suas paginas pré-textuais e lista de referencias, pelas regras da ABNT. Por esse motivo, sugerimos consultar o modelo de template, para ajudá-lo nesta tarefa, disponível em: http://www.biblioteca.ufc.br/educacao-de-usuarios/templates/ Vamos agora as correções sempre de acordo com o template: 1. A partir da folha de aprovação as informações devem ser em língua inglesa. 2. A dedicatória deve ter a distancia até o final da folha observado. 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No. of bitstreams: 1 2017_tes_clcmattos.pdf: 6102699 bytes, checksum: 0a85b8841d77f0685b1153ee8ede0d85 (MD5) Previous issue date: 2017-08-25 / The study of dynamical systems is widespread across several areas of knowledge. Sequential data is generated constantly by different phenomena, most of them we cannot explain by equations derived from known physical laws and structures. In such context, this thesis aims to tackle the task of nonlinear system identification, which builds models directly from sequential measurements. More specifically, we approach challenging scenarios, such as learning temporal relations from noisy data, data containing discrepant values (outliers) and large datasets. In the interface between statistics, computer science, data analysis and engineering lies the machine learning community, which brings powerful tools to find patterns from data and make predictions. In that sense, we follow methods based on Gaussian Processes (GP), a principled, practical, probabilistic approach to learning in kernel machines. We aim to exploit recent advances in general GP modeling to bring new contributions to the dynamical modeling exercise. Thus, we propose the novel family of Recurrent Gaussian Processes (RGPs) models and extend their concept to handle outlier-robust requirements and scalable stochastic learning. The hierarchical latent (non-observed) structure of those models impose intractabilities in the form of non-analytical expressions, which are handled with the derivation of new variational algorithms to perform approximate deterministic inference as an optimization problem. The presented solutions enable uncertainty propagation on both training and testing, with focus on free simulation. We comprehensively evaluate the proposed methods with both artificial and real system identification benchmarks, as well as other related dynamical settings. The obtained results indicate that the proposed approaches are competitive when compared to the state of the art in the aforementioned complicated setups and that GP-based dynamical modeling is a promising area of research. / O estudo dos sistemas dinâmicos encontra-se disseminado em várias áreas do conhecimento. Dados sequenciais são gerados constantemente por diversos fenômenos, a maioria deles não passíveis de serem explicados por equações derivadas de leis físicas e estruturas conhecidas. Nesse contexto, esta tese tem como objetivo abordar a tarefa de identificação de sistemas não lineares, por meio da qual são obtidos modelos diretamente a partir de observações sequenciais. Mais especificamente, nós abordamos cenários desafiadores, tais como o aprendizado de relações temporais a partir de dados ruidosos, dados contendo valores discrepantes (outliers) e grandes conjuntos de dados. Na interface entre estatísticas, ciência da computação, análise de dados e engenharia encontra-se a comunidade de aprendizagem de máquina, que fornece ferramentas poderosas para encontrar padrões a partir de dados e fazer previsões. Nesse sentido, seguimos métodos baseados em Processos Gaussianos (PGs), uma abordagem probabilística prática para a aprendizagem de máquinas de kernel. A partir de avanços recentes em modelagem geral baseada em PGs, introduzimos novas contribuições para o exercício de modelagem dinâmica. Desse modo, propomos a nova família de modelos de Processos Gaussianos Recorrentes (RGPs, da sigla em inglês) e estendemos seu conceito para lidar com requisitos de robustez a outliers e aprendizagem estocástica escalável. A estrutura hierárquica e latente (não-observada) desses modelos impõe expressões não- analíticas, que são resolvidas com a derivação de novos algoritmos variacionais para realizar inferência determinista aproximada como um problema de otimização. As soluções apresentadas permitem a propagação da incerteza tanto no treinamento quanto no teste, com foco em realizar simulação livre. Nós avaliamos em detalhe os métodos propostos com benchmarks artificiais e reais da área de identificação de sistemas, assim como outras tarefas envolvendo dados dinâmicos. Os resultados obtidos indicam que nossas propostas são competitivas quando comparadas ao estado da arte, mesmo nos cenários que apresentam as complicações supracitadas, e que a modelagem dinâmica baseada em PGs é uma área de pesquisa promissora.

Ce rapport porte sur le sujet de recherche de l'identification boîte noire du système non linéaire. En effet, parmi toutes les techniques nombreuses et variées développées dans ce domaine de la recherche ces dernières décennies, il semble toujours intéressant d'étudier l'approche réseau de neurones dans l'estimation de modèle de système complexe. Même si des modèles précis ont été obtenus, les principaux inconvénients de ces techniques restent le grand nombre de paramètres nécessaires et, en conséquence, le coût important de calcul nécessaire pour obtenir le niveau de pratique de la précision du modèle désiré. Par conséquent, motivés pour remédier à ces inconvénients, nous avons atteint une méthodologie complète et efficace du système d'identification offrant une précision équilibrée, la complexité et les modèles de coûts en proposant, d'une part, de nouvelles structures de réseaux de neurones particulièrement adapté à une utilisation très large en matière de modélisation système pratique non linéaire, d'autre part, un simple et efficace technique de réduction de modèle, et, troisièmement, une procédure de réduction de coût de calcul. Il est important de noter que ces deux dernières techniques de réduction peut être appliquée à une très large gamme d'architectures de réseaux de neurones sous deux simples hypothèses spécifiques qui ne sont pas du tout contraignant. Enfin, la dernière contribution importante de ce travail est d'avoir montré que cette phase d'estimation peut être obtenue dans un cadre robuste si la qualité des données d'identification qu'il oblige. Afin de valider la procédure d'identification système proposé, des exemples d'applications entraînées en simulation et sur un procédé réel, de manière satisfaisante validé toutes les contributions de cette thèse, confirmant tout l'intérêt de ce travail. / This report concerns the research topic of black box nonlinear system identification. In effect, among all the various and numerous techniques developed in this field of research these last decades, it seems still interesting to investigate the neural network approach in complex system model estimation. Even if accurate models have been derived, the main drawbacks of these techniques remain the large number of parameters required and, as a consequence, the important computational cost necessary to obtain the convenient level of the model accuracy desired. Hence, motivated to address these drawbacks, we achieved a complete and efficient system identification methodology providing balanced accuracy, complexity and cost models by proposing, firstly, new neural network structures particularly adapted to a very wide use in practical nonlinear system modeling, secondly, a simple and efficient model reduction technique, and, thirdly, a computational cost reduction procedure. It is important to notice that these last two reduction techniques can be applied to a very large range of neural network architectures under two simple specific assumptions which are not at all restricting. Finally, the last important contribution of this work is to have shown that this estimation phase can be achieved in a robust framework if the quality of identification data compels it. In order to validate the proposed system identification procedure, application examples driven in simulation and on a real process, satisfactorily validated all the contributions of this thesis, confirming all the interest of this work.

No / Extrusion is one of the major methods for processing polymeric materials and the thermal homogeneity of the process output is a major concern for manufacture of high quality extruded products. Therefore, accurate process thermal monitoring and control are important for product quality control. However, most industrial extruders use single point thermocouples for the temperature monitoring/control although their measurements are highly affected by the barrel metal wall temperature. Currently, no industrially established thermal profile measurement technique is available. Furthermore, it has been shown that the melt temperature changes considerably with the die radial position and hence point/bulk measurements are not sufficient for monitoring and control of the temperature across the melt flow. The majority of process thermal control methods are based on linear models which are not capable of dealing with process nonlinearities. In this work, the die melt temperature profile of a single screw extruder was monitored by a thermocouple mesh technique. The data obtained was used to develop a novel approach of modelling the extruder die melt temperature profile under dynamic conditions (i.e. for predicting the die melt temperature profile in real-time). These newly proposed models were in good agreement with the measured unseen data. They were then used to explore the effects of process settings, material and screw geometry on the die melt temperature profile. The results showed that the process thermal homogeneity was affected in a complex manner by changing the process settings, screw geometry and material. (C) 2013 Elsevier Inc. All rights reserved.

In recent years, response analysis of complex structures under impact loads has attracted a great deal of attention. For example, a collision or an accident that produces impact loads that exceed the design load can cause severe damage on the structural components. Although the AASHTO specification is used for impact-resistant bridge design, it has many limitations. The AASHTO specification does not incorporate complex and uncertain factors. Thus, a well-designed structure that can survive a collision under specific conditions in one region may be severely damaged if it were impacted by a different vessel, or if it were located elsewhere with different in-situ conditions. With these limitations in mind, we propose different solutions that use smart control technology to mitigate impact hazard on structures. However, it is challenging to develop an accurate mathematical model of the integrated structure-smart control systems. The reason is due to the complicated nonlinear behavior of the integrated nonlinear systems and uncertainties of high impact forces. In this context, novel algorithms are developed for identification, control and monitoring of nonlinear responses of smart structures under high impact forces. To evaluate the proposed approaches, a smart aluminum and two smart reinforced concrete beam structures were designed, manufactured, and tested in the High Impact Engineering Laboratory of Civil and Environmental Engineering at WPI. High-speed impact force and structural responses such as strain, deflection and acceleration were measured in the experimental tests. It has been demonstrated from the analytical and experimental study that: 1) the proposed system identification model predicts nonlinear behavior of smart structures under a variety of high impact forces, 2) the developed structural health monitoring algorithm is effective in identifying damage in time-varying nonlinear dynamic systems under ambient excitations, and 3) the proposed controller is effective in mitigating high impact responses of the smart structures.

The linear source-filter model (LSFM) has been used as a primary model for speech processing since 1960 when G. Fant presented acoustic speech production theory. It assumes that the source of voiced speech sounds, glottal flow, is independent of the filter, vocal tract. However, acoustic simulations based on the physical speech production models show that, especially when the fundamental frequency (F0) of source harmonics approaches to the first formant frequency (F1) of vocal tract filter, the filter has significant effects on the source due to the nonlinear coupling between them. In this thesis, as an alternative to linear source-filter model, nonlinear interactive source-filter models are proposed for voiced speech. This thesis has two parts, in the first part, a framework for the coupling of the source and the filter is presented. Then, two interactive system models are proposed assuming that glottal flow is a quasi-steady Bernoulli flow and acoustics in vocal tract is linear. In these models, instead of glottal flow, glottal area is used as a source for voiced speech. In the proposed interactive models, the relation between the glottal flow, glottal area and vocal tract is determined by the quasi-steady Bernoulli flow equation. It is theoretically shown that linear source-filter model is an approximation of the nonlinear models. Estimation of ISFM&rsquo / s parameters from only speech signal is a nonlinear blind deconvolution problem. The problem is solved by a robust method developed based on the acoustical interpretation of the systems. Experimental results show that ISFMs produce source-filter coupling effects seen in the physical simulations and the parameter estimation method produce always stable and better performing models than LSFM model. In addition, a framework for the incorporation of the source-filter interaction into classical source-filter model is presented. The Rosenberg source model is extended to an interactive source for voiced speech and its performance is evaluated on a large speech database. The results of the experiments conducted on vowels in the database show that the interactive Rosenberg model is always better than its noninteractive version. In the second part of the thesis, LSFM and ISFMs are compared by using not only the speech signal but also HSV (High Speed Endocopic Video) of vocal folds in a system identification approach. In this case, HSV and speech are used as a reference input-output data for the analysis and comparison of the models. First, a new robust HSV processing algorithm is developed and applied on HSV images to extract the glottal area. Then, system parameters are estimated by using a modified version of the method proposed in the first part. The experimental results show that speech signal can contain some harmonics of the fundamental frequency of the glottal area other than those contained in the glottal area signal. Proposed nonlinear interactive source-filter models can generate harmonics components in speech and produce more realistic speech sounds than LSFM.

Cette thèse se concentrera exclusivement sur l’application de méthodes non paramétriques basées sur le noyau à des problèmes d’identification non-linéaires. Comme pour les autres méthodes non-linéaires, deux questions clés dans l’identification basée sur le noyau sont les questions de comment définir un modèle non-linéaire (sélection du noyau) et comment ajuster la complexité du modèle (régularisation). La contribution principale de cette thèse est la présentation et l’étude de deux critères d’optimisation (un existant dans la littérature et une nouvelle proposition) pour l’approximation structurale et l’accord de complexité dans l’identification de systèmes non-linéaires basés sur le noyau. Les deux méthodes sont basées sur l’idée d’intégrer des contraintes de complexité basées sur des caractéristiques dans le critère d’optimisation, en pénalisant les dérivées de fonctions. Essentiellement, de telles méthodes offrent à l’utilisateur une certaine souplesse dans la définition d’une fonction noyau et dans le choix du terme de régularisation, ce qui ouvre de nouvelles possibilités quant à la facon dont les modèles non-linéaires peuvent être estimés dans la pratique. Les deux méthodes ont des liens étroits avec d’autres méthodes de la littérature, qui seront examinées en détail dans les chapitres 2 et 3 et formeront la base des développements ultérieurs de la thèse. Alors que l’analogie sera faite avec des cadres parallèles, la discussion sera ancrée dans le cadre de Reproducing Kernel Hilbert Spaces (RKHS). L’utilisation des méthodes RKHS permettra d’analyser les méthodes présentées d’un point de vue à la fois théorique et pratique. De plus, les méthodes développées seront appliquées à plusieurs «études de cas» d’identification, comprenant à la fois des exemples de simulation et de données réelles, notamment : • Détection structurelle dans les systèmes statiques non-linéaires. • Contrôle de la fluidité dans les modèles LPV. • Ajustement de la complexité à l’aide de pénalités structurelles dans les systèmes NARX. • Modelisation de trafic internet par l’utilisation des méthodes à noyau / This thesis will focus exclusively on the application of kernel-based nonparametric methods to nonlinear identification problems. As for other nonlinear methods, two key questions in kernel-based identification are the questions of how to define a nonlinear model (kernel selection) and how to tune the complexity of the model (regularisation). The following chapter will discuss how these questions are usually dealt with in the literature. The principal contribution of this thesis is the presentation and investigation of two optimisation criteria (one existing in the literature and one novel proposition) for structural approximation and complexity tuning in kernel-based nonlinear system identification. Both methods are based on the idea of incorporating feature-based complexity constraints into the optimisation criterion, by penalising derivatives of functions. Essentially, such methods offer the user flexibility in the definition of a kernel function and the choice of regularisation term, which opens new possibilities with respect to how nonlinear models can be estimated in practice. Both methods bear strong links with other methods from the literature, which will be examined in detail in Chapters 2 and 3 and will form the basis of the subsequent developments of the thesis. Whilst analogy will be made with parallel frameworks, the discussion will be rooted in the framework of Reproducing Kernel Hilbert Spaces (RKHS). Using RKHS methods will allow analysis of the methods presented from both a theoretical and a practical point-of-view. Furthermore, the methods developed will be applied to several identification ‘case studies’, comprising of both simulation and real-data examples, notably: • Structural detection in static nonlinear systems. • Controlling smoothness in LPV models. • Complexity tuning using structural penalties in NARX systems. • Internet traffic modelling using kernel methods