A Sum of Squares Approach to Stability Analysis of Polynomial Fuzzy Systems

Tanaka, Kazuo; Yoshida, Hibiki; Ohtake, Hiroshi; Wang, H.O.

doi:10.1109/acc.2007.4282579

Cited by 80 publications

(61 citation statements)

References 6 publications

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“…The extension to fuzzy polynomial models in the formẋ = r i=1 h i (z(t)) (p i (x) + q i (x)u), with p i , q i polynomial matrices, was first proposed in 2007 [88,89,90]. Such polynomial fuzzy models include the well-known Takagi-Sugeno fuzzy systems and controllers as special cases.…”

Section: Fuzzy-polynomial Techniquesmentioning

confidence: 99%

Fuzzy control turns 50: 10 years later

Guerra

Sala

Tanaka

2015

Fuzzy Sets and Systems

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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.

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Section: Fuzzy-polynomial Techniquesmentioning

confidence: 99%

Fuzzy control turns 50: 10 years later

Guerra

Sala

Tanaka

2015

Fuzzy Sets and Systems

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“…A polynomial continuous fuzzy model has been proposed in [4]. In this section, we propose a polynomial discrete fuzzy model.…”

Section: A Polynomial Discrete Fuzzy Modelmentioning

confidence: 99%

“…To obtain more relaxed stability results, we employ a polynomial Lyapunov function [4] represented bŷ…”

Section: B Polynomial Lyapunov Functionmentioning

confidence: 99%

An SOS-based stable control of polynomial discrete fuzzy systems

Tanaka

Ohtake

Wang

2008

2008 American Control Conference

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Abstract-This paper presents stability and stabilization conditions of polynomial discrete fuzzy systems. First, based on polynomial Lyapunov functions, we derive a stability condition of polynomial discrete fuzzy systems without inputs. Secondly, we derive a stabilization condition to design a stable polynomial fuzzy controller. Both of them are represented in terms of SOS and are numerically (partially symbolically) solved via the recent developed SOSTOOLS. In addition, polynomial discrete fuzzy systems and controllers contain discrete Takagi-Sugeno (T-S) fuzzy systems and controllers as a special case. Hence, the approach discussed in this paper is more general than that based on the existing LMI approaches to discrete T-S fuzzy control system designs. To illustrate the validity of the design approach, a design example is provided. The example shows the utility of our approach.

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“…However, because of the existence of monomials, the LMI-based approach cannot be used to get the solution for the stability conditions. Then, a SOSbased approach [20] is employed [16,21], where a feasible solution (if any) can be found by, for example, SOSTOOLS [22]. In the literature such as the work mentioned above, most of the stability analysis are MFI that the information of membership functions is not taken into account and thus it potentially leads to conservative analysis results.…”

Section: Introductionmentioning

confidence: 99%

Membership‐Function‐Dependent Stability Analysis of Interval Type‐2 Polynomial Fuzzy‐Model‐Base Control Systems

Song

Lam

Yang

2017

IET Control Theory & Applications

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Abstract:In this paper, the stability analysis for interval type-2 (IT2) polynomial fuzzy-model-based (PFMB) control system using the information of membership functions is investigated. In order to tackle uncertainties, IT2 membership functions are used in the IT2 polynomial fuzzy model and IT2 polynomial fuzzy controller. To improve the design flexibility and reduce the implementation costs, the IT2 polynomial fuzzy controller does not need to share the same premise membership functions nor the same number of rules with the IT2 polynomial fuzzy model. The stability of IT2 PFMB control system is investigated based on the Lyapunov stability theory and both sets of membership function independent (MFI) and membership function dependent (MFD) stability conditions are derived on the basis of the sum-of-squares (SOS) approach. To make the stability conditions membership function dependent, the boundary information of IT2 membership functions is used in the stability analysis. To extract richer information of IT2 membership functions, the operating domain is partitioned into sub-domains. In each sub-domain, the boundary information of IT2 membership functions and those of the upper and lower membership function are obtained. Furthermore, to further relax the conservativeness, a switching polynomial fuzzy controller, together with the informations obtained in each sub-domain, is employed in investigating the stability analysis. Numerical Examples and simulation results are given to demonstrate the validity of MFD and MFD switching methods.

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A Sum of Squares Approach to Stability Analysis of Polynomial Fuzzy Systems

Cited by 80 publications

References 6 publications

Fuzzy control turns 50: 10 years later

Fuzzy control turns 50: 10 years later

An SOS-based stable control of polynomial discrete fuzzy systems

Membership‐Function‐Dependent Stability Analysis of Interval Type‐2 Polynomial Fuzzy‐Model‐Base Control Systems

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