Stabilization of Polynomial Fuzzy Systems via a Sum of Squares Approach

Abstract-This paper presents a sum of squares (SOS, for brevity) based observer design for a more general class of polynomial fuzzy systems with the polynomial matrices Ai(x(t)) and Bi(x(t)) that are permitted to be dependent of the states x(t). First, we briefly summarize previous works on SOS-based observer designs for two limited classes of polynomial fuzzy systems. To overcome the difficulty of the fact that does not realize the so-called separation principle design for the more general class, this paper provides a practical design procedure of a polynomial fuzzy controller and a polynomial fuzzy observer without lack of guaranteeing the stability of the overall control system in addition to converging state estimation error (via the observer) to zero. The design approach discussed in this paper is more general than the existing LMI approaches (to T-S fuzzy controller and observer designs) and also than the previous SOS-based observer designs. To illustrate the validity of the design approach, a design example is provided. The example shows the utility of our SOS approach to the polynomial fuzzy observer-based control for the more general class of polynomial fuzzy systems.

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“…The SOS conditions (see Theorem 1 below) derived in [6], [8] are applied to determine the polynomial feedback gains F i (x).…”

Section: Main Results:class III Designmentioning

confidence: 99%

An SOS-based observer design for polynomial fuzzy systems

Tanaka

Ohtake

Seo

et al. 2011

Proceedings of the 2011 American Control Conference

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“…We drop the notation with respect to time . For example, we will use ,̂( ) to replace ( ),̂( ( )), respectively [10,11]. In addition, ( ) refers to the th row of ( ).…”

Section: Finite-time Stability Of Continuous-time Polynomialmentioning

confidence: 99%

Finite-Time Stability and Controller Design of Continuous-Time Polynomial Fuzzy Systems

Chen

2017

Abstract and Applied Analysis

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Finite-time stability and stabilization problem is first investigated for continuous-time polynomial fuzzy systems. The concept of finite-time stability and stabilization is given for polynomial fuzzy systems based on the idea of classical references. A sumof-squares-(SOS-) based approach is used to obtain the finite-time stability and stabilization conditions, which include some classical results as special cases. The proposed conditions can be solved with the help of powerful Matlab toolbox SOSTOOLS and a semidefinite-program (SDP) solver. Finally, two numerical examples and one practical example are employed to illustrate the validity and effectiveness of the provided conditions.

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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%

“…Reaching any possible smooth Lyapunov function using the above multi-dimensional fuzzy summations in fuzzy Lyapunov functions [59], or increasing the degree of a polynomial Lyapunov function at will [108,101] progressively increases the expressive power of the chosen Lyapunov function family. In the polynomial case, if there exists a smooth Lyapunov function (so that its Taylor series converges to it), the result is asymptotically exact as degree increases (well, actually, up to the gap between positive and SOS polynomials [87]).…”

Section: Asymptotical Exactnessmentioning

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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Stabilization of Polynomial Fuzzy Systems via a Sum of Squares Approach

Cited by 57 publications

References 7 publications

An SOS-based observer design for polynomial fuzzy systems

An SOS-based observer design for polynomial fuzzy systems

Finite-Time Stability and Controller Design of Continuous-Time Polynomial Fuzzy Systems

Fuzzy control turns 50: 10 years later

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