BIBO stabilisation of continuous time takagi sugeno systems under persistent perturbations and input saturation

[EN] This paper presents a novel approach to the design of fuzzy state feedback controllers for continuous-time non-linear systems with input saturation under persistent perturbations. It is assumed that all the states of the Takagi¿Sugeno (TS) fuzzy model representing a non-linear system are measurable. Such controllers achieve bounded input bounded output (BIBO) stabilisation in closed loop based on the computation of inescapable ellipsoids. These ellipsoids are computed with linear matrix inequalities (LMIs) that guarantee stabilisation with input saturation and persistent perturbations. In particular, two kinds of inescapable ellipsoids are computed when solving a multiobjective optimization problem: the maximum volume inescapable ellipsoids contained inside the validity domain of the TS fuzzy model and the smallest inescapable ellipsoids which guarantee a minimum *-norm (upper bound of the 1-norm) of the perturbed system. For every initial point contained in the maximum volume ellipsoid, the closed loop will enter the minimum *-norm ellipsoid after a finite time, and it will remain inside afterwards. Consequently, the designed controllers have a large domain of validity and ensure a small value for the 1-norm of closed loop.The authors wish to thank the Editor-in-Chief and the anonymous reviewers for their valuable comments and suggestions. This work has been funded by Ministerio de Economia y Competitividad (Spain) through the research project DPI2015-71443-R and by Generalitat Valenciana (Valencia, Spain) through the research project GV/2017/029.Salcedo-Romero-De-Ávila, J.; Martínez Iranzo, MA.; Garcia-Nieto, S.; Hilario Caballero, A. (2018). BIBO stabilisation of continuous time takagi sugeno systems under persistent perturbations and input saturation. International Journal of Applied Mathematics and Computer Science (Online). 28(3):457-472. https://doi.org/10.2478/amcs-2018-0035S45747228

Contributions to analysis and control of Takagi-Sugeno systems via piecewise, parameter-dependent, and integral Lyapunov functions

Esta tesis considera un enfoque basado en Lyapunov para el análisis y control de sistemas no lineales cuyas ecuaciones dinámicas son reescritas como un modelo Takagi-Sugeno o uno polinomial convexo. Estas estructuras permiten resolver problemas de control mediante técnicas de optimización convexa, más concretamente desigualdades matriciales lineales y suma de cuadrados, que son eficientes herramientas desde un punto de vista computacional. Después de proporcionar una visión general básica del estado actual en el campo de los modelos Takagi-Sugeno, esta tesis aborda cuestiones sobre las funciones de Lyapunov por trozos, dependiente de parámetros e integral de línea, con las siguientes contribuciones: Un algoritmo mejorado para estimaciones del dominio de atracción de sistemas no lineales para sistemas de tiempo continuo. Los resultados se basan en funciones de Lyapunov por trozos, desigualdades matriciales lineales y argumentaciones geométricas; enfoques basados en conjuntos de nivel en la literatura previa se han mejorado significativamente. Una función Lyapunov generalizada dependiente de parámetros para la síntesis de controladores para sistemas Takagi-Sugeno. El enfoque propone una ley de control multi-índice que retroalimenta la derivada del tiempo de las funciones de membresía del modelo Takagi-Sugeno para anular los términos que causan localidad a priori en el análisis de Lyapunov. Una nueva función integral de Lyapunov para el análisis de estabilidad de sistemas no lineales. Estos resultados generalizan aquellos basados en funciones de Lyapunov integral de línea al marco polinomial; resulta que los requisitos de independencia del camino pueden ser anulados por una definición adecuada de una función Lyapunov con términos integrales.This thesis considers a Lyapunov-based approach for analysis and control of nonlinear systems whose dynamical equations are rewritten as a Takagi-Sugeno model or a convex polynomial one. These structures allow solving control problems via convex optimisation techniques, more specifically linear matrix inequalities and sum-of-squares, which are efficient tools from the computational point of view. After providing a basic overview of the state of the art in the field of Takagi-Sugeno models, this thesis address issues on piecewise, parameter-dependent and line-integral Lyapunov functions, with the following contributions: An improved algorithm to estimate the domain of attraction of nonlinear systems for continuous-time systems. The results are based on piecewise Lyapunov functions, linear matrix inequalities, and geometrical argumentations; level-set approaches in prior literature are significantly improved. A generalised parameter-dependent Lyapunov function for synthesis of controllers for Takagi-Sugeno systems. The approach proposed a multi-index control law that feeds back the time derivative of the membership function of the Takagi-Sugeno model to cancel out the terms that cause a priori locality in the Lyapunov analysis. A new integral Lyapunov function for stability analysis of nonlinear systems. These results generalise those based on line-integral Lyapunov functions to the polynomial framework; it turns out path-independency requirements can be overridden by an adequate definition of a Lyapunov function with integral terms.Aquesta tesi considera un enfocament basat en Lyapunov per a l'anàlisi i control de sistemes no lineals les equacions dinàmiques dels quals són reescrites com un model Takagi-Sugeno o un de polinomial convex. Aquestes estructures permeten resoldre problemes de control mitjançant tècniques d'optimització convexa, més concretament desigualtats matricials lineals i suma de quadrats, que són eines eficients des d'un punt de vista computacional. Després de proporcionar una visió general bàsica de l'estat actual en el camp dels models Takagi-Sugeno, aquesta tesi aborda qüestions sobre les funcions de Lyapunov per trossos, dependent de paràmetres i integral de línia, amb les següents contribucions: Un algoritme millorat per a estimar el domini d'atracció de sistemes no lineals per a sistemes de temps continu. Els resultats es basen en funcions de Lyapunov per trossos, desigualtats matricials lineals i argumentacions geomètriques; enfocaments basats en conjunts de nivell en la literatura prèvia s'han millorat significativament. Una funció Lyapunov generalitzada dependent de paràmetres per a la síntesi de controladors per a sistemes Takagi-Sugeno. L'enfocament proposa una llei de control multi-índex que retroalimenta la derivada del temps de les funcions de membres del model Takagi-Sugeno per anul·lar els termes que causen localitat a priori en l'anàlisi de Lyapunov. Una nova funció integral de Lyapunov per a l'anàlisi d'estabilitat de sistemes no lineals. Aquests resultats generalitzen aquells basats en funcions de Lyapunov integral de línia al marc polinomial; resulta que els requisits d'independència del camí poden ser anul·lats per una definició adequada d'una funció Lyapunov amb termes integrals.González Germán, IT. (2018). Contributions to analysis and control of Takagi-Sugeno systems via piecewise, parameter-dependent, and integral Lyapunov functions [Tesis doctoral no publicada]. Universitat Politècnica de València. https://doi.org/10.4995/Thesis/10251/101282TESI

Polytopic invariant and contractive sets for closed-loop discrete fuzzy systems

In this work a procedure for obtaining polytopic lambda-contractive sets for Takagi Sugeno fuzzy systems is presented, adapting well-known algorithms from literature on discrete-time linear difference inclusions (LDI) to multi-dimensional summations. As a complexity parameter increases, these sets tend to the maximal invariant set of the system when no information on the shape of the membership functions is available. lambda-contractive sets are naturally associated to level sets of polyhedral Lyapunov functions proving a decay-rate of lambda. The paper proves that the proposed algorithm obtains better results than a class of Lyapunov methods for the same complexity degree: if such a Lyapunov function exists, the proposed algorithm converges in a finite number of steps and proves a larger lambda-contractive set.This work has been supported by Projects DPI2011-27845-C02-01 and DPI2011-27845-C02-02, both from Spanish Government.Arino, C.; Perez, E.; Sala Piqueras, A.; Bedate, F. (2014). Polytopic invariant and contractive sets for closed-loop discrete fuzzy systems. Journal of The Franklin Institute. 351(7):3559-3576. https://doi.org/10.1016/j.jfranklin.2014.03.014S35593576351