Relaxed stability conditions based on Taylor series membership functions for polynomial fuzzy-model-based control systems

© 2014 IEEE. In this paper, we investigate the stability of polynomial fuzzy-model-based (PFMB) control systems, aiming to relax stability conditions by considering the information of membership functions. To facilitate the stability analysis, we propose a general form of approximated membership functions, which is implemented by Taylor series expansion. Taylor series membership functions (TSMF) can be brought into stability conditions such that the relation between membership grades and system states is expressed. To further reduce the con-servativeness, different types of information are taken into account: the boundary of membership functions, the property of membership functions, and the boundary of operating domain. Stability conditions are obtained from Lyapunov stability theory by sum of squares (SOS) approach. Simulation examples demonstrate the effect of each piece of information

Stability analysis of polynomial fuzzy models via polynomial fuzzy Lyapunov functions

In this paper, the stability of continuous-time polynomial fuzzy models by means of a polynomial generalization of fuzzy Lyapunov functions is studied. Fuzzy Lyapunov functions have been fruitfully used in the literature for local analysis of Takagi-Sugeno models, a particular class of the polynomial fuzzy ones. Based on a recent Taylor-series approach which allows a polynomial fuzzy model to exactly represent a nonlinear model in a compact set of the state space, it is shown that a refinement of the polynomial Lyapunov function so as to make it share the fuzzy structure of the model proves advantageous. Conditions thus obtained are tested via available SOS software. © 2011 Elsevier B.V. All rights reserved.Bernal Reza, MÁ.; Sala, A.; Jaadari, A.; Guerra, T. (2011). Stability analysis of polynomial fuzzy models via polynomial fuzzy Lyapunov functions. Fuzzy Sets and Systems. 185(1):5-14. doi:10.1016/j.fss.2011.07.008S514185

H∞ Control of Polynomial Fuzzy Systems: a Sum of Squares Approach

This paper proposes the control design ofa nonlinear polynomial fuzzy system with H∞ performance objective using a sum of squares (SOS) approach. Fuzzy model and controller are represented by a polynomial fuzzy model and controller. The design condition is obtained by using polynomial Lyapunov functions that not only guarantee stability but also satisfy the H∞ performance objective. The design condition is represented in terms of an SOS that can be numerically solved via the SOSTOOLS. A simulation study is presented to show the effectiveness of the SOS-based H∞ control designfor nonlinear polynomial fuzzy systems

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