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

Multicriteria fuzzy-polynomial observer design for a 3DoF nonlinear electromechanical platform

This paper proposes local fuzzy-polynomial observer discrete-time designs for state estimation of a nonlinear 3DoF electromechanical platform (fixed quadrotor). A trade-off between H∞ norm bounds and speed of convergence performance is taken into account in the design process. Actual experimental data are used to compare performance of the fuzzy polynomial design with classical ones based on the Takagi–Sugeno and linearized models, both using the same optimization criteria and design parameters.The authors are grateful to the financial support of the Spanish government under research project DPI2011-27845-C02-01 and FPI Grant BES-2009-013882, as well as to Generalitat Valenciana grant PROMETEOII/2013/004. The authors are also grateful to Ph.D. students A. Berna, J. Guzman and associate professor P.J. Garcia for their laboratory data acquisition work.Pitarch Pérez, JL.; Sala Piqueras, A. (2014). Multicriteria fuzzy-polynomial observer design for a 3DoF nonlinear electromechanical platform. Engineering Applications of Artificial Intelligence. 30:96-106. https://doi.org/10.1016/j.engappai.2013.11.006S961063

Polynomial Fuzzy Observer Designs: A Sum-of-Squares Approach

This paper presents a simple passive attitude stabilizer (PAS) for vision-based stabilization of palm-size aerial vehicles. First, a mathematical dynamic model of a palm-size aerial vehicle with the proposed PAS is constructed. Stability analysis for the dynamics is carried out in terms of Lyapunov stability theory. The analysis results show that the proposed stabilizer guarantees passive stabilizing behavior, i.e., passive attitude recovering, of the aerial vehicle for small perturbations from a stability theory point of view. Experimental results demonstrate the utility of the proposed PAS for the aerial vehicle

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

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

Filter Design for Positive T-S Fuzzy Continuous-Time Systems with Time Delay Using Piecewise-Linear Membership Functions

The file attached to this record is the author's final peer reviewed version. The Publisher's final version can be found by following the DOI link.This work focuses on the filtering problem and stability analysis for positive Takagi-Sugeno (T-S) fuzzy systems with time delay under L1-induced performance. Due to the importance of estimation of system states but the few filter design results on positive nonlinear systems, it is an attractive and meaningful topic well worth studying. In order to fully exploit and take advantage of the positivity of positive T-S fuzzy systems, many commonly used methods, for instance free-weighting matrix approach and similarity transformation are probably not suitable for positive systems. To address the hard-nut-to-crack problem, an auxiliary variable is introduced so that the augmentation approach can be employed to carry out the positivity and stability analysis of filtering error systems. In addition, another obstacle that cannot be ignored is the existence of non-convex terms in the stability and positivity conditions. For getting around this barrier, some iterative linear matrix inequality (ILMI) algorithms have been proposed in the literature. However, considering the weakness that these methods cannot guarantee the convergence to a numerical solution and the iterative process is exhaustive, we present an effective matrix decoupling method to convert the nonconvex conditions into convex ones in this paper. Furthermore, a linear co-positive Lyapunov function which incorporates the positivity of system states and time delay at the same time is chosen so that the positivity characteristic of filtering error systems can be captured further. However, because of plenty of valuable information of membership functions (MFs) being ignored, hence, the obtained results are conservative. For the sake of relaxing the conservativeness, the advanced piecewise-linear membership functions (PLMFs) approximate method is utilized to facilitate the stability and positivity analysis. Therefore, the relaxed stability and positivity conditions which are cast as sum of squares (SOS) are obtained and can be solved numerically. Finally, the effectiveness of the designed fuzzy filtering strategy with satisfying L1-induced performance are demonstrated by a simulation example