Guaranteed Cost Control of Polynomial Fuzzy Systems via a Sum of Squares Approach

This paper presents the guaranteed cost control of polynomial fuzzy systems via a sum of squares (SOS) approach. First, we present a polynomial fuzzy model and controller that are more general representations of the well-known Takagi-Sugeno (T-S) fuzzy model and controller, respectively. Second, we derive a guaranteed cost control design condition based on polynomial Lyapunov functions. Hence, the design approach discussed in this paper is more general than the existing LMI approaches (to T-S fuzzy control system designs) based on quadratic Lyapunov functions. The design condition realizes a guaranteed cost control by minimizing the upper bound of a given performance function. In addition, the design condition in the proposed approach can be represented in terms of SOS and is numerically (partially symbolically) solved via the recent developed SOSTOOLS. To illustrate the validity of the design approach, two design examples are provided. The first example deals with a complicated nonlinear system. The second example presents micro helicopter control. Both the examples show that our approach provides more extensive design results for the existing LMI approach

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

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

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

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

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

An SOS-Based Control Lyapunov Function Design for Polynomial Fuzzy Control of Nonlinear Systems

This paper deals with a sum-of-squares (SOS)-based control Lyapunov function (CLF) design for polynomial fuzzy control of nonlinear systems. The design starts with exactly replacing (smooth) nonlinear systems dynamics with polynomial fuzzy models, which are known as universal approximators. Next, global stabilization conditions represented in terms of SOS are provided in the framework of the CLF design, i.e., a stabilizing controller with nonparallel distributed compensation form is explicitly designed by applying Sontag\u27s control law, once a CLF for a given nonlinear system is constructed. Furthermore, semiglobal stabilization conditions on operation domains are derived in the same fashion as in the global stabilization conditions. Both global and semiglobal stabilization problems are formulated as SOS optimization problems, which reduce to numerical feasibility problems. Five design examples are given to show the effectiveness of our proposed approach over the existing linear matrix inequality and SOS approaches