Subspace-Based Takagi-Sugeno Modeling for Improved LMI Performance

"© 2017 IEEE. Personal use of this material is permitted. Permissíon from IEEE must be obtained for all other uses, in any current or future media, including reprinting/republishing this material for advertisíng or promotional purposes, creating new collective works, for resale or redistribution to servers or lists, or reuse of any copyrighted component of this work in other works."[EN] Given a nonlinear system, the sector-nonlinearity methodology provides a systematic way of transforming it in an equivalent Takagi-Sugeno (T-S) model. However, such transformation is not unique: conservatism of shape-independent performance conditions in the form of linear matrix inequalities results in some models yielding better results than others. This paper provides some guidelines on choosing a sector-nonlinearity T-S model, with provable optimality (in a particular sense) in the case of quadratic nonlinearities. The approach is based on Hessian and restrictions of a function onto a subspace.This work was supported by the following institutions: Project Ciencia Basica SEP-CONACYT CB-168406, Project DPI2016-81002, (Spanish government, MINECO), Grant PROMETEOII/2013/004 (Generalitat Valenciana) and, the Scholarship GRISOLIA/2014/006.Robles-Ruiz, R.; Sala, A.; Bernal Reza, MÁ.; Gonzalez-German, IT. (2017). Subspace-Based Takagi-Sugeno Modeling for Improved LMI Performance. IEEE Transactions on Fuzzy Systems. 25(4):754-767. https://doi.org/10.1109/TFUZZ.2016.2574927S75476725

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

Parallel Distributed Compensation for Voltage Controlled Active Magnetic Bearing System using Integral Fuzzy Model

Parallel Distributed Compensation (PDC) for current-controlled Active Magnetic Bearing System (AMBS) has been quite effective in recent years. However, this method does not take into account the dynamics associated with the electromagnet. This limits the method to smaller scale applications where the electromagnet dynamics can be neglected. Voltage-controlled AMBS is used to overcome this limitation but this comes with serious challenges such as complex mathematical modelling and higher order system control. In this work, a PDC with integral part is proposed for position and input tracking control of voltage-controlled AMBS. PDC method is based on nonlinear Takagi-Sugeno (T-S) fuzzy model. It is shown that the proposed method outperforms the conventional fuzzy PDC. It stabilizes the bearing shaft at any chosen operating point and tracks any chosen smooth trajectory within the air gap with a high external disturbance rejection capability

A CENTER MANIFOLD THEORY-BASED APPROACH TO THE STABILITY ANALYSIS OF STATE FEEDBACK TAKAGI-SUGENO-KANG FUZZY CONTROL SYSTEMS

The aim of this paper is to propose a stability analysis approach based on the application of the center manifold theory and applied to state feedback Takagi-Sugeno-Kang fuzzy control systems. The approach is built upon a similar approach developed for Mamdani fuzzy controllers. It starts with a linearized mathematical model of the process that is accepted to belong to the family of single input second-order nonlinear systems which are linear with respect to the control signal. In addition, smooth right-hand terms of the state-space equations that model the processes are assumed. The paper includes the validation of the approach by application to stable state feedback Takagi-Sugeno-Kang fuzzy control system for the position control of an electro-hydraulic servo-system

Fuzzy control turns 50: 10 years later

In 2015, we celebrate the 50th anniversary of Fuzzy Sets, ten years after the main milestones regarding its applications in fuzzy control in their 40th birthday were reviewed in FSS, see [1]. Ten years is at the same time a long period and short time thinking to the inner dynamics of research. This paper, presented for these 50 years of Fuzzy Sets is taking into account both thoughts. A first part presents a quick recap of the history of fuzzy control: from model-free design, based on human reasoning to quasi-LPV (Linear Parameter Varying) model-based control design via some milestones, and key applications. The second part shows where we arrived and what the improvements are since the milestone of the first 40 years. A last part is devoted to discussion and possible future research topics.Guerra, T.; Sala, A.; Tanaka, K. (2015). Fuzzy control turns 50: 10 years later. Fuzzy Sets and Systems. 281:162-182. doi:10.1016/j.fss.2015.05.005S16218228

A generalised integral polynomial Lyapunov function for nonlinear systems

[EN] This work generalises the line-integral Lyapunov function in (Rhee and Won, 2006) for stability analysis of continuous-time nonlinear models expressed as fuzzy systems. The referred result applied only to Takagi¿Sugeno representations, and required memberships to be a tensor-product of functions of a single state; these are generalised here so that membership arguments can be arbitrary polynomials of the state variables; in this way, systems for which earlier results cannot be applied are now covered. Both the modelling and the integral terms appearing in the Lyapunov functions are generalised to a fuzzy polynomial case. Illustrative examples show the advantage of the proposed method against previous literature, even in the TS case.The authors gratefully to the financial support of Spanish ministry of Economy and European Union, grant DPI2016-81002-R (AEI/FEDER, UE), the CONACyT/COECyT Sonora scholarship 383252, and Project ITSON-PROFAPI-CA 2017-0088.Gonzalez-German, IT.; Sala, A.; Bernal Reza, MÁ. (2019). A generalised integral polynomial Lyapunov function for nonlinear systems. Fuzzy Sets and Systems. 356:77-91. https://doi.org/10.1016/j.fss.2018.02.005S779135

Robust stabilization for discrete-time Takagi-Sugeno fuzzy system based on N4SID models

Nonlinear systems identification from experimental data without any prior knowledge of the system parameters is a challenge in control and process diagnostic. It determines mathematical model pa-rameters that are able to reproduce the dynamic behavior of a system. This paper combines two fun-damental research areas: MIMO state space system identification and nonlinear control system. This combination produces a technique that leads to robust stabilization of a nonlinear Takagi-Sugeno fuzzy system (T-S). Design/methodology/approach The first part of this paper describes the identification based on the Numerical algorithm for Subspace State Space System IDentification (N4SID). The second part, from the identified models of first part, explains how we use the interpolation of Linear Time Invariants (LTI) models to build a nonlinear multiple model system, T-S model. For demonstration purposes, conditions on stability and stabiliza-tion of discrete time, Takagi-Sugeno (T-S) model were discussed. Findings Stability analysis based on the quadratic Lyapunov function to simplify implementation was ex-plained in this paper. The LMIs (Linear Matrix Inequalities) technique obtained from the linearization of the BMIs (Bilinear Matrix Inequalities) was computed. The suggested N4SID2 algorithm had the smallest error value compared to other algorithms for all estimated system matrices. Originality The stabilization of the closed-loop discrete time T-S system, using the improved PDC control law (Parallel Distributed Compensation), was discussed to reconstruct the state from nonlinear Luen-berger observers

Asymptotically exact stabilisation for constrained discrete Takagi-Sugeno systems via set-invariance

[EN] Given a Takagi-Sugeno (TS) system, this paper proposes a novel methodology to obtain the state feedback controller guaranteeing, asymptotically as a Polya-related complexity parameter grows, the largest (membership-shape independent) possible domain-of-attraction with contraction-rate performance lambda, based on polyhedral lambda-contractive sets from constrained linear systems literature. The resulting controller is valid for any realisation of the memberships, as usual in most TS literature. For a finite complexity parameter, an inner estimate of such largest set is obtained; the frontier of such approximation can be understood as the level set of a polyhedral control-Lyapunov function. Convergence of a proposed iterative algorithm is asymptotically necessary and sufficient for TS system stabilisation: for a high-enough value of the complexity parameter, any conceivable shape-independent Lyapunov controller design procedure will yield a proven domain of attraction smaller or equal to the algorithm's output. (C) 2016 Elsevier B.V. All rights reserved.This work has been supported by grants DPI2015-70433- P and DPI2016-81002-R, from Spanish Government (MINECO) and grant PROMETEOII/2013/004 from Generalitat Valenciana.Ariño-Latorre, CV.; Sala, A.; Pérez Soler, E.; Bedate Boluda, F.; Querol-Ferrer, A. (2017). Asymptotically exact stabilisation for constrained discrete Takagi-Sugeno systems via set-invariance. Fuzzy Sets and Systems. 316:117-138. https://doi.org/10.1016/j.fss.2016.10.004S11713831

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