Contributions to nonlinear system modelling and controller synthesis via convex structures

Esta tesis discute diferentes metodologías de modelado para extraer mejores prestaciones o resultados de estabilidad que aquéllas que el modelado convencional basado en sector no-lineal de sistemas Takagi-Sugeno (también denominados cuasi-LPV) es capaz de producir. En efecto, incluso si las LMIs pueden probar distintas cotas de prestaciones o márgenes de estabilidad (tasa de decaimiento, $\mathcal H_\infty$, etc.) para sistemas politópicos, es bien conocido que las prestaciones probadas dependen del modelo elegido y, dado un sistema no-lineal, dicho modelo politópico no es único. Por tanto, se presentan exploraciones hacia cómo obtener el modelo que es menos perjudicial para la medida de prestaciones elegida. Como una última contribución, mejores resultados son obtenidos mediante la extensión del modelado politópico Takagi-Sugeno a un marco de inclusiones en diferencias cuasi-convexas con planificación de ganancia. En efecto, una versión sin planificación de ganancia fue propuesta por un equipo de investigadores de la Universidad de Sevilla (Fiaccini, Álamo, Camacho) para generalizar el modelado politópico, y esta tesis propone una version aún más general de algunos de dichos resultados que incorpora planificación de ganancia.This thesis discusses different modelling methodologies to eke out best performance/stability results than conventional sector-nonlinearity Takagi-Sugeno (also known as quasi-LPV) systems modelling techniques are able to yield. Indeed, even if LMIs can prove various performance and stability bounds (decay rate, $\mathcal H_\infty$, etc.) for polytopic systems, it is well known that the proven performance depends on the chosen model and, given a nonlinear dynamic systems, the polytopic embeddings available for it are not unique. Thus, explorations on how to obtain the model which is less deletereous for performance are presented. As a last contribution, extending the polytopic Takagi-Sugeno setup to a gain-scheduled quasi-convex difference inclusion framework allows to improve the results over the polytopic models. Indeed, the non-scheduled convex difference inclusion framework was proposed by a research team in University of Seville (Fiacchini, Alamo, Camacho) as a generalised modelling methodology which included the polytopic one; this thesis poses a further generalised gain-scheduled version of some of these results.Aquesta tesi discuteix diferents metodologies de modelatge per extreure millors prestacions o resultats d'estabilitat que aquelles que el modelatge convencional basat en sector no-lineal de sistemes Takagi-Sugeno (també anomenats quasi-LPV) és capaç de produir. En efecte, fins i tot si les LMIs poden provar diferents cotes de prestacions o marges d'estabilitat (taxa de decaïment, $\mathcal H_\infty$, etc.) per a sistemes politòpics, és ben conegut que les prestacions provades depenen del model triat i, donat un sistema no-lineal, el dit model politòpic no és únic. Per tant, es presenten exploracions cap a com obtenir el model que és menys perjudicial per a la mesura de prestacions triada. Com una darrera contribució, millors resultats són obtinguts mitjançant l'extensió del modelatge politòpic Takagi-Sugeno a un marc d'inclusions en diferències quasi-convexes amb planificació de guany. En efecte, una versió sense planificació de guany va ser proposada per un equip d'investigadors de la Universitat de Sevilla (Fiaccini, Álamo, Camacho) per a generalitzar el modelatge politòpic, i aquesta tesi proposa una versió més general d'alguns d'aquests resultats que incorpora planificació de guany.Robles Ruiz, R. (2018). Contributions to nonlinear system modelling and controller synthesis via convex structures [Tesis doctoral no publicada]. Universitat Politècnica de València. https://doi.org/10.4995/Thesis/10251/100848TESI

A polytopic strategy for improved non-asymptotic robust control via implicit Lyapunov functions

International audienceThis paper is concerned with finite-and fixed-time robust stabilization of uncertain multi-input nonlinear systems via the implicit Lyapunov function method. Instead of splitting the system into a linear nominal model and an additive perturbation which gathers nonlinearities, parametric uncertainties , and exogenous disturbances, the methodology hereby proposed preserves some nonlinear terms in the nominal system via an exact polytopic representation which leads to design conditions in the form of linear matrix inequalities. As a result, feasible solutions are found where former approaches fail; these solutions have more accurate settling-time estimates with reduced control effort. The corresponding control law includes well-known high-order sliding modes as a particular case. Numerical simulations are provided to illustrate the advantages of the proposal

Relaxed LMI conditions for control of nonlinear Takagi-Sugeno models

Los problemas de optimización de desigualdades matriciales lineales en control borroso se han convertido en la herramienta más utilizada en dicha área desde los años 90. Muchos sistemas no lineales pueden ser modelados como sistemas borrosos de modo que el control borroso puede considerarse como una técnica de control no lineal. Aunque se han obtenido muchos y buenos resultados, quedan algunas fuentes de conservadurismo cuando se comparan con otros enfoques de control no lineal. Esta tesis discute dichas cuestiones de conservadurismo y plantea nuevos enfoques para resolverlas. La principal ventaja de la formulación mediante desigualdades matriciales lineales es la posibilidad de asegurar estabilidad y prestaciones de un sistema no lineal modelado como un sistema borroso Takagi-Sugeno. Estos modelos están formados por un conjunto de modelos lineales eligiendo el sistema a aplicar mediante el uso de unas reglas borrosas. Estas reglas se traducen en funciones de interpolación o de pertenecía que nos indican el grado de validez de un modelo lineal respecto del resto. El mayor problema que presentan estas técnicas basadas en desigualdades matriciales lineales es que las funciones de pertenencia no están incluidas en las condiciones de estabilidad del sistema, lo que significa que se prueba la estabilidad y prestaciones para cualquier forma de interpolación entre los diferentes modelos lineales. Esto genera una fuente de conservadurismo que sería conveniente limitar. En la tesis doctoral se presentan varias metodologías capaces de trasladar la información de las funciones de pertenencia del sistema al problema basado en desigualdades matriciales lineales de estabilidad y prestaciones. Las dos principales aportaciones propuestas se basan, respectivamente, en introducir una serie de matrices de relajación que permitan incorporar esta información y en aprovechar la descripción de una amplia clase de sistemas borrosos en productos tensoriales de...Ariño Latorre, CV. (2008). Relaxed LMI conditions for control of nonlinear Takagi-Sugeno models [Tesis doctoral no publicada]. Universitat Politècnica de València. https://doi.org/10.4995/Thesis/10251/8301Palanci

Estimation et commande des systèmes descripteurs

This thesis addresses the estimation and control for nonlinear descriptor systems. The developments are focused on a family of nonlinear descriptor models with a full-rank descriptor matrix. The proposed approaches are based on a Takagi-Sugeno (TS) descriptor representation of a given nonlinear descriptor model. This type of TS models is a generalization of the standard TS ones. One of the mains goals is to obtain conditions in terms of linear matrix inequalities (LMIs). In the existing literature, the observer design for TS descriptor models has led to bilinear matrix inequality (BMI) conditions. In addition, to the best of our knowledge, there are no results in the literature on controller/observer design for discrete-time TS descriptor models (with a non-constant and invertible descriptor matrix).Three problems have been addressed: state feedback controller design, observer design, and static output feedback controller design. LMI conditions have been obtained for both continuous and discrete-time TS descriptor models. In the continuous-time case, relaxed LMI conditions for the state feedback controller design have been achieved via parameterdependent LMI conditions. For the observer design, pure LMI conditions have been developed by using a different extended estimation error. For the static output feedback controller, LMI constraints can be obtained once an auxiliary matrix is fixed. In the discretetime case, results in the LMI form are provided for state/output feedback controller design and observer design; thus filling the gap in the literature. Several examples have been included to illustrate the applicability of the obtained results and the importance of keeping the original descriptor structure instead of computing a standard state-space.Cette thèse est consacrée au développement des techniques d’estimation et de commande pour systèmes descripteurs non linéaires. Les développements sont centrés sur une famille particulière de systèmes descripteurs non linéaires avec une matrice descripteur de rang plein. Toutes les approches présentées utilisent un formalisme de modélisation du type Takagi-Sugeno (TS) pour représenter les modèles descripteurs non linéaires. Un objectif très important est de développer des conditions sous la forme d’inégalités matricielles linéaires (LMI, en anglais). Dans la littérature, les conditions pour l’estimation des modèles TS descripteurs s’écrivent sous forme d’inégalités matricielles bilinéaires (BMI, en anglais). En plus, à notre connaissance, il n’y pas de résultats dans la littérature concernant la commande/estimation pour les modèles TS descripteurs en temps discret (avec une matrice descripteur régulière non linéaire).Trois problèmes ont été examinés : commande par retour d’état, estimation de l’état et commande statique par retour de la sortie. Dans le cas continu, des conditions moins conservatives ont été développées pour la commande par retour d’état. Pour l’estimation d’état, des conditions LMI ont été obtenues (au lieu des usuelles BMI) en utilisant un différent vecteur d’erreur augmenté. Pour la commande statique par retour de la sortie, des conditions LMI sont proposées si une matrice auxiliaire est fixée. Pour le temps discret, des nouveaux résultats sous la forme LMI ont été développées pour la commande/estimation, comblant ainsi certains manques de la littérature. Des exemples ont été inclus pour montrer l’applicabilité de tous les résultats que nous avons obtenus et ainsi l’importance de garder la structure originale des descripteurs

Recent Advances in Robust Control

Robust control has been a topic of active research in the last three decades culminating in H_2/H_\infty and \mu design methods followed by research on parametric robustness, initially motivated by Kharitonov's theorem, the extension to non-linear time delay systems, and other more recent methods. The two volumes of Recent Advances in Robust Control give a selective overview of recent theoretical developments and present selected application examples. The volumes comprise 39 contributions covering various theoretical aspects as well as different application areas. The first volume covers selected problems in the theory of robust control and its application to robotic and electromechanical systems. The second volume is dedicated to special topics in robust control and problem specific solutions. Recent Advances in Robust Control will be a valuable reference for those interested in the recent theoretical advances and for researchers working in the broad field of robotics and mechatronics

Stability analysis and controller synthesis for a class of piecewise smooth systems

This thesis deals with the analysis and synthesis of piecewise smooth (PWS) systems. In general, PWS systems are nonsmooth systems, which means their vector fields are discontinuous functions of the state vector. Dynamic behavior of nonsmooth systems is richer than smooth systems. For example, there are phenomena such as sliding modes that occur only in nonsmooth systems. In this thesis, a Lyapunov stability theorem is proved to provide the theoretical framework for the stability analysis of PWS systems. Piecewise affine (PWA) and piecewise polynomial (PWP) systems are then introduced as important subclasses of PWS systems. The objective of this thesis is to propose efficient computational controller synthesis methods for PWA and PWP systems. Three synthesis methods are presented in this thesis. The first method extends linear controllers for uncertain nonlinear systems to PWA controllers. The result is a PWA controller that maintains the performance of the linear controller while extending its region of convergence. However, the synthesis problem for the first method is formulated as a set of bilinear matrix inequalities (BMIs), which are not easy to solve. Two controller synthesis methods are then presented to formulate PWA and PWP controller synthesis as convex problems, which are numerically tractable. Finally, to address practical implementation issues, a time-delay approach to stability analysis of sampled-data PWA systems is presented. The proposed method calculates the maximum sampling time for a sampled-data PWA system consisting of a continuous-time plant and a discrete-time emulation of a continuous-time PWA state feedback controller

The optimal control of power electronic embedded networks in More Electric Aircraft

With the advancement of power electronic technologies over recent decades, there has been an overall increase in the utilisation of distributed generation and power electronic embedded networks in a large sphere of applications. Probably one of the most prominent areas of utilisation of new power electronics embedded systems is the use in power networks onboard military and civilian aircraft. With environmental concerns and increased competition in the civil aviation sector, more aircraft manufactures are replacing and interfacing electrical alternatives over heavier, less efficient and costly pneumatic, hydraulic and mechanical systems. In these modern power systems, the increased proliferation of power electronic converters and distributed generation raises important issues in regards to the performance, stability and robustness between interfaced switching units. These phenomena, such as power electronic sub-system interactions, become even more prominent in micro-grid applications or other low voltage distribution systems where interfaced converters are in close proximity to one another. In More Electric Aircraft (MEA), these interfaced power electronic converters are connected to the same non-stiff low power AC grid, which further increases the interactive effects between converter sub-systems. If these effects are not properly taken into account, then external disturbances to the system at given operating conditions can result in degradation of the system performance, failure in meeting the operating requirements of the grid, or in the worst case, instability of the whole grid. With much research in the area of decreasing the size and weight of systems, there is much literature proposing optimisation methods which decrease the size of filters between interfacing converters. Whilst effectively decreasing the size of these systems, interactions between interfaced converters gets worse, and is often improperly accounted for. The work presented in this thesis proposes a novel approach to the decentralisation and optimisation of converter controls on a power electronics embedded power network. In order to account for the interactive dynamics between sub-systems in the environment of reduced passive filter networks, all the system dynamics including the interactive terms are modelled globally. An optimal controller design approach based on the H2 optimisation is proposed to synthesise and generate automatically the controller gains for each power electronic sub-system. H2 optimisation is a powerful tool, which not only allows the submission, optimisation and development of closed loop controls for large dynamic systems, but offers the ability to the user to construct the controller for given structures. This enables the development of decentralised controllers for every sub-system with intrinsic knowledge of the closed loop dynamics of every other interconnect sub-system. It is shown through simulation and by experimental validation that this novel approach to grid control optimisation not only can improve overall dynamic performance of all sub-systems over 15traditional methods of design, but can also intrinsically reduce or better yet mitigate against the interactive effects between all converters. In addition, this method of controller design will be shown to not only be scalable to expanding sizes of grids, but the Phase-locked loops (PLLs) integrated to grid connected devices can also be considered in the optimisation procedure. PLLs are widely known to further cause interactive behaviours between grid interfaced devices. Including this into the optimisation also has been validated experimentally to prevent interactions on the grid, and improve performance over traditional design methods. Adaptations to the controller are performed to ensure operation in variable frequency environments (as is common in MEA), as well as methods of single converter optimisation when interfacing to an unknown grid. Additionally some initial research towards an adaption of the H2 controller to incorporate robustness as well as performance into the optimisation procedure is presented with mathematical concepts shown through simulation

The optimal control of power electronic embedded networks in More Electric Aircraft

With the advancement of power electronic technologies over recent decades, there has been an overall increase in the utilisation of distributed generation and power electronic embedded networks in a large sphere of applications. Probably one of the most prominent areas of utilisation of new power electronics embedded systems is the use in power networks onboard military and civilian aircraft. With environmental concerns and increased competition in the civil aviation sector, more aircraft manufactures are replacing and interfacing electrical alternatives over heavier, less efficient and costly pneumatic, hydraulic and mechanical systems. In these modern power systems, the increased proliferation of power electronic converters and distributed generation raises important issues in regards to the performance, stability and robustness between interfaced switching units. These phenomena, such as power electronic sub-system interactions, become even more prominent in micro-grid applications or other low voltage distribution systems where interfaced converters are in close proximity to one another. In More Electric Aircraft (MEA), these interfaced power electronic converters are connected to the same non-stiff low power AC grid, which further increases the interactive effects between converter sub-systems. If these effects are not properly taken into account, then external disturbances to the system at given operating conditions can result in degradation of the system performance, failure in meeting the operating requirements of the grid, or in the worst case, instability of the whole grid. With much research in the area of decreasing the size and weight of systems, there is much literature proposing optimisation methods which decrease the size of filters between interfacing converters. Whilst effectively decreasing the size of these systems, interactions between interfaced converters gets worse, and is often improperly accounted for. The work presented in this thesis proposes a novel approach to the decentralisation and optimisation of converter controls on a power electronics embedded power network. In order to account for the interactive dynamics between sub-systems in the environment of reduced passive filter networks, all the system dynamics including the interactive terms are modelled globally. An optimal controller design approach based on the H2 optimisation is proposed to synthesise and generate automatically the controller gains for each power electronic sub-system. H2 optimisation is a powerful tool, which not only allows the submission, optimisation and development of closed loop controls for large dynamic systems, but offers the ability to the user to construct the controller for given structures. This enables the development of decentralised controllers for every sub-system with intrinsic knowledge of the closed loop dynamics of every other interconnect sub-system. It is shown through simulation and by experimental validation that this novel approach to grid control optimisation not only can improve overall dynamic performance of all sub-systems over 15traditional methods of design, but can also intrinsically reduce or better yet mitigate against the interactive effects between all converters. In addition, this method of controller design will be shown to not only be scalable to expanding sizes of grids, but the Phase-locked loops (PLLs) integrated to grid connected devices can also be considered in the optimisation procedure. PLLs are widely known to further cause interactive behaviours between grid interfaced devices. Including this into the optimisation also has been validated experimentally to prevent interactions on the grid, and improve performance over traditional design methods. Adaptations to the controller are performed to ensure operation in variable frequency environments (as is common in MEA), as well as methods of single converter optimisation when interfacing to an unknown grid. Additionally some initial research towards an adaption of the H2 controller to incorporate robustness as well as performance into the optimisation procedure is presented with mathematical concepts shown through simulation