Control design for interval type‐2 polynomial fuzzy‐model‐based systems with time‐varying delay

Abstract

Citing this paper Please note that where the full-text provided on King's Research Portal is the Author Accepted Manuscript or Post-Print version this may differ from the final Published version. If citing, it is advised that you check and use the publisher's definitive version for pagination, volume/issue, and date of publication details. And where the final published version is provided on the Research Portal, if citing you are again advised to check the publisher's website for any subsequent corrections. General rightsCopyright and moral rights for the publications made accessible in the Research Portal are retained by the authors and/or other copyright owners and it is a condition of accessing publications that users recognize and abide by the legal requirements associated with these rights.•Users may download and print one copy of any publication from the Research Portal for the purpose of private study or research.•You may not further distribute the material or use it for any profit-making activity or commercial gain •You may freely distribute the URL identifying the publication in the Research Portal Take down policy If you believe that this document breaches copyright please contact librarypure@kcl.ac.uk providing details, and we will remove access to the work immediately and investigate your claim. Abstract: In this paper, the problems of stabilization for interval type-2 polynomial fuzzy systems with time-varying delay and parameter uncertainties are investigated. The objective is to design a state-feedback interval type-2 polynomial fuzzy controller such that the closed-loop control system is asymptotically stable. The conditions for the existence of such a controller are delay dependent and membership function dependent in terms of sum-of-squares (SOS). Based on a basic lemma to deal with the delay terms, we formulate and solve the problem with more flexibility due to imperfect premise matching that the number of rules and premise membership functions are not necessary the same between the interval type-2 polynomial fuzzy model and interval type-2 polynomial fuzzy controller. Piecewise linear membership functions approximations enclosing the original lower and upper membership functions are employed to facilitate the stability analysis. A numerical example indicates the effectiveness of the derived results.

Admissibilisation of singular interval type‐2 Takagi–Sugeno fuzzy systems with time delay

“…Remark 6: The research ideas of (38) and (39) of this study are firstly proposed in [23], which can alleviate the conservativeness of the system by introducing some slack matrices under the imperfect premise matching.…”

“…The common methods include Park's inequality [30], Jensen inequality [31], Wirtinger-based inequality [32], auxiliary function-based integral inequality [33], Bessel-Legendre inequality [34], generalised integral inequality [35], delay partitioning technique [36] etc. According to [35], we know that Jensen inequality, Wirtinger-based inequality, auxiliary function-based integral inequality, Bessel-Legendre inequality can be regarded as special cases of generalised integral inequality with k = 0 and different values of l. The study of time-delay systems has gone deep into various types of systems, such as normal time-delay system [30][31][32][33][34][35]37], T-S fuzzy systems [38], T-S fuzzy singular systems [13,14,16,19,20,36], IT2 fuzzy systems [39,40]. The delay partitioning technique was adopted to handle the delay term in [36].…”

A review on stability analysis of continuous-time fuzzy-model-based control systems: From membership-function-independent to membership-function-dependent analysis

“…Remark 6. From the research point of view, the partially and imperfectly matched premises address the fundamental issues of some emerging research topics of FMB/PFMB control systems, such as time-delayed [104], sampleddata [140,68,71], observer-based [141,65], IT2 [105,106], networked [142,143,144] fuzzy control systems. For all these control systems, the grade of membership functions are not the same as those of the fuzzy model due to the system states obtained by the fuzzy controller are altered, say, by the time delay, sampling process, or observer.…”

“…Since then, MFD stability analysis results using various membership function information and techniques can be found for the FMB control systems [91,92,93,75,94,95,96,97,98] and the PFMB control systems [76,77,79,82,85]. The MFI stability analysis has also extended to interval type-2 (IT2) FMB/PFMB control systems [99,100,101,102,103,104,105,106]. Further details regarding MFD stability analysis will be provided in the later sections.…”

Interval Type-2 Fuzzy Systems as Deep Neural Network Activation Functions

“…To define nonlinear input-output mappings, it has been shown that Fuzzy Logic Systems (FLSs), especially Interval T2 (IT2) FLSs, are very powerful tools [13][14][15][16][17][18][19][20][21]. This lies due to the fact that IT2-FLSs use and employ IT2 Fuzzy Sets (FSs) that have more extra degree of freedom provided by the Footprint of Uncertainty (FOU) [13][14][15][16][17][18][19][20][21]. Hence, many theoretical studies have been presented on how the size/shape of the FOU affects the IT2 Fuzzy Mappings (FMs) [20][21][22][23].…”