Closed-form estimates of the domain of attraction for nonlinear systems via fuzzy-polynomial models

In this work, the domain of attraction of the origin of a nonlinear system is estimated in closed-form via level sets with polynomial boundary, iteratively computed. In particular, the domain of attraction is expanded from a previous estimate, such as, for instance, a classical Lyapunov level set. With the use of fuzzy-polynomial models, the domain-of-attraction analysis can be carried out via sum of squares optimization and an iterative algorithm. The result is a function wich bounds the domain of attraction, free from the usual restriction of being positive and decrescent in all the interior of its level sets

Contributions to fuzzy polynomial techniques for stability analysis and control

The present thesis employs fuzzy-polynomial control techniques in order to improve the stability analysis and control of nonlinear systems. Initially, it reviews the more extended techniques in the field of Takagi-Sugeno fuzzy systems, such as the more relevant results about polynomial and fuzzy polynomial systems. The basic framework uses fuzzy polynomial models by Taylor series and sum-of-squares techniques (semidefinite programming) in order to obtain stability guarantees. The contributions of the thesis are: ¿ Improved domain of attraction estimation of nonlinear systems for both continuous-time and discrete-time cases. An iterative methodology based on invariant-set results is presented for obtaining polynomial boundaries of such domain of attraction. ¿ Extension of the above problem to the case with bounded persistent disturbances acting. Different characterizations of inescapable sets with polynomial boundaries are determined. ¿ State estimation: extension of the previous results in literature to the case of fuzzy observers with polynomial gains, guaranteeing stability of the estimation error and inescapability in a subset of the zone where the model is valid. ¿ Proposal of a polynomial Lyapunov function with discrete delay in order to improve some polynomial control designs from literature. Preliminary extension to the fuzzy polynomial case. Last chapters present a preliminary experimental work in order to check and validate the theoretical results on real platforms in the future.Pitarch Pérez, JL. (2013). Contributions to fuzzy polynomial techniques for stability analysis and control [Tesis doctoral no publicada]. Universitat Politècnica de València. https://doi.org/10.4995/Thesis/10251/34773TESI

A review of convex approaches for control, observation and safety of linear parameter varying and Takagi-Sugeno systems

This paper provides a review about the concept of convex systems based on Takagi-Sugeno, linear parameter varying (LPV) and quasi-LPV modeling. These paradigms are capable of hiding the nonlinearities by means of an equivalent description which uses a set of linear models interpolated by appropriately defined weighing functions. Convex systems have become very popular since they allow applying extended linear techniques based on linear matrix inequalities (LMIs) to complex nonlinear systems. This survey aims at providing the reader with a significant overview of the existing LMI-based techniques for convex systems in the fields of control, observation and safety. Firstly, a detailed review of stability, feedback, tracking and model predictive control (MPC) convex controllers is considered. Secondly, the problem of state estimation is addressed through the design of proportional, proportional-integral, unknown input and descriptor observers. Finally, safety of convex systems is discussed by describing popular techniques for fault diagnosis and fault tolerant control (FTC).Peer ReviewedPostprint (published version

Piecewise-Takagi-Sugeno asymptotically exact estimation of the domain of attraction of nonlinear systems

[EN] This report generalises recent results on stability analysis and estimation of the domain of attraction of nonlinear systems via exact piecewise affine Takagi Sugeno models. Algorithms in the form of linear matrix inequalities are proposed that produce progressively better estimates which are proved to asymptotically render the actual domain of attraction; regions already proven to belong to such domain of attraction can be removed and the estimate can contain significant portions of the modelling region boundary; in this way, level-set approaches in prior literature can be significantly improved. Illustrative examples and comparisons are provided. (C) 2016 The Franklin Institute. Published by Elsevier Ltd. All rights reserved.The authors gratefully acknowledge the support of the following institutions: Project Ciencia Basica SEP-CONACYT CB-168406, the CONACyT/COECYT Sonora scholarship 383252, project DPI2016-81002-R (Spanish government, MINECO), and the scholarship GRISOLIA/2014/006 from Generalitat Valenciana (regional government).Gonzalez-German, IT.; Sala, A.; Bernal Reza, MÁ.; Robles-Ruiz, R. (2017). Piecewise-Takagi-Sugeno asymptotically exact estimation of the domain of attraction of nonlinear systems. Journal of the Franklin Institute. 354(3):1514-1541. https://doi.org/10.1016/j.jfranklin.2016.11.033S15141541354

Distributed Saturated Control for a Class of Semilinear PDE Systems: A SOS Approach

Producción CientíficaThis paper presents a systematic approach to deal with the saturated control of a class of distributed parameter systems which can be modeled by first-order hyperbolic partial differential equations (PDE). The approach extends (also improves over) the existing fuzzy Takagi-Sugeno (TS) state feedback designs for such systems by applying the concepts of the polynomial sum-of-squares (SOS) techniques. Firstly, a fuzzy-polynomial model via Taylor series is used to model the semilinear hyperbolic PDE system. Secondly, the closed-loop exponential stability of the fuzzy-PDE system is studied through the Lyapunov theory. This allows to derive a design methodology in which a more complex fuzzy state-feedback control is designed in terms of a set of SOS constraints, able to be numerically computed via semidefinite programming. Finally, the proposed approach is tested in simulation with the standard example of a nonisothermal plug-flow reactor (PFR).The research leading to these results has received funding from the European Union and from the Spanish Government (MINECO/FEDER DPI2015-70975-P)

Control synthesis for polynomial discrete-time systems under input constraints via delayed-state Lyapunov functions

This paper presents a discrete-time control design methodology for input-saturating systems using a Lyapunov function with dependence on present and past states. The approach is used to bypass the usual difficulty with full polynomial Lyapunov functions of expressing the problem in a convex way. Also polynomial controllers are allowed to depend on both present and past states. Furthermore, by considering saturation limits on the control action, the information about the relationship between the present and past states is introduced via Positivstellensatz multipliers. Sum-of-squares techniques and available semi-definite programming (SDP) software are used in order to find the controller.The research work by J.L. Pitarch and A. Sala has been partially supported by the Spanish government under research project [grant number DPI2011-27845-C02-01 (MINECO)]; Generalitat Valenciana [grant number PROMETEOII/2013/004]. The work by T.M. Guerra and J. Lauber has been supported by the International Campus on Safety and Intermodality in Transportation, the European Community, Delegation Regionale a la Recherche et a la Technologie, Ministere de l'Enseignement superieur et de la Recherche, Region Nord Pas de Calais and the Centre National de la Recherche Scientifique.Pitarch Pérez, JL.; Sala Piqueras, A.; Lauber, J.; Guerra, TM. (2016). Control synthesis for polynomial discrete-time systems under input constraints via delayed-state Lyapunov functions. International Journal of Systems Science. 47(5):1176-1184. https://doi.org/10.1080/00207721.2014.915357S1176118447

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

Optimisation of transient and ultimate inescapable sets with polynomial boundaries for nonlinear systems

[EN] This paper addresses the problem of bounding the trajectories of nonlinear systems (transient and ultimate bounds) from initial conditions in given sets, when subject to possibly nonvanishing disturbances constrained by some finite-interval integral bounds, with a suitable controller. The so-called robustly inescapable sets are determined from such initial conditions and disturbance bounds. In order to get numerical results, the approach considers embedding the nonlinear dynamics in a convex combination of polynomials, and solving sum-of-squares (SOS) problems on them, optimising some inescapable-set size parameters. Determination of approximate (locally) optimal solutions usually requires an iterative evaluation of SOS problems, because of products of decision variables. (C) 2016 Elsevier Ltd. All rights reserved.The research leading to these results has received funding from the European Union (FP7/2007-2013 no 604068) and from the Spanish Government (MINECO/FEDER DPI2015-70975-P, DPI2016-81002). The material in this paper was not presented at any conference. This paper was recommended for publication in revised form by Associate Editor Graziano Chesi under the direction of Editor Richard Middleton.Sala, A.; Pitarch, JL. (2016). Optimisation of transient and ultimate inescapable sets with polynomial boundaries for nonlinear systems. Automatica. 73:82-87. https://doi.org/10.1016/j.automatica.2016.06.017S82877