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

and Applied Analysis 3 Remark 5. Definitions 3 and 4 are a first attempt to give the concept of finite-time stability and stabilization for polynomial fuzzy systems based on the basic idea in classical paper [24, 25]. When x(x(t)) here reduces to x(t), the above two definitions reduce to the forms in [24]. Lemma 6 (see [30]). Let F(x(t)) be an N × N symmetric polynomial matrix of degree 2d in x(t) ∈ Rn. Furthermore, let x(x(t)) be a column vector whose entries are all monomials in x(t) with degree not greater than d. If VT(t)F(x(t))V(t) is an SOS, where V(t) ∈ RN, then F(x(t)) ≥ 0 for all x(t) ∈ Rn. 3. Main Results In this section, the finite-time stability and stabilization conditions for the continuous-time polynomial fuzzy system (3) are given, respectively. 3.1. Finite-Time Stability of Continuous-Time Polynomial Fuzzy System. Firstly, we will analyze the finite-time stability of the continuous-time polynomial fuzzy system (3) with u(t) = 0. We drop the notation with respect to time t. For example, we will use x, x(x) to replace x(t), x(x(t)), respectively [10, 11]. In addition, Aki (x) refers to the kth row of A i(x). Then system (3) with u = 0 has the following form: ?̇? = r ∑ i=1 hi (z (t)) A i (x) x (x) . (7) Theorem 7. Suppose there exist a symmetric polynomial matrixP(x) ∈ RN×N, a constantα ≥ 0, and two positive scalars Φ1 and Φ2 such that the following are satisfied: xT (x) (P (x) − ε1 (x) I) x (x) is SOS, (8) − xT (x)(P (x) T (x)A i (x) + ATi (x) TT (x) P (x) + n ∑ k=1 ∂P (x) ∂xk Aki (x) x (x) − αP (x) + ε2i (x) I)x (x) is SOS, ∀i, (9) xT (x) (P (x) − Φ1R − ε3 (x) I) x (x) is SOS, (10) xT (x) (Φ2R − P (x) − ε4 (x) I) x (x) is SOS, (11) xT (x) (Φ1c2 − Φ2c1eαTf − ε5 (x)) x (x) is SOS, (12) where polynomials ε1(x) > 0, ε2i(x) > 0, ε3(x) > 0, ε4(x) > 0, and ε5(x) > 0 for x ̸ = 0 and the (i, j)th entry of polynomial matrix T(x) ∈ RN×n is given by Tij (x) = ∂xi ∂xj (x) ; (13) then the continuous-time polynomial fuzzy system (3) with u = 0 is finite-time stable with respect to (c1, c2, Tf, R). Proof. Choose the following candidate polynomial Lyapunov functional: V (x) = xT (x) P (x) x (x) . (14) It can be seen from condition (8) that P(x) is positive definite for all x, and thus, V(x) is a positive definite function of x. The time derivation of V(x) along system (7) is given by ?̇? (x) = xT (x) P (x) ̇̂ x (x) + ̇̂ xT (x) P (x) x (x) + xT (x) ?̇? (x) x (x) = xT (x) P (x) T (x) ?̇? (x) + ?̇?T (x) TT (x) P (x) x (x) + xT (x)( n ∑ k=1 ∂P (x) ∂xk ?̇?k)x (x) . (15) On the other hand, ?̇?k can be represented as ?̇?k = r ∑ i=1 hi (z) Aki (x) x (x) . (16) From (7), (15), and (16), ?̇?(x) is rewritten as ?̇? (x) = r ∑ i=1 hi (z) xT (x)(P (x) T (x)A i (x) + ATi (x) TT (x) P (x) + n ∑ k=1 ∂P (x) ∂xk Aki (x) x (x)) ⋅ x (x) . (17) In addition, condition (9) implies that xT (x)(P (x) T (x)A i (x) + ATi (x) TT (x) P (x) + n ∑ k=1 ∂P (x) ∂xk Aki (x) x (x))x (x) < xT (x) (αP (x)) ⋅ x (x) . (18)


Introduction
Over the past decades, the fuzzy logic control has developed into a successful and fruitful branch of automation and control theory due to the fact that the fuzzy models are the appealing and efficient tools in approximating the complex nonlinear dynamical systems.Among different fuzzy models, the well-known Takagi-Sugeno (T-S) fuzzy model has attracted considerable research attention [1][2][3][4][5] and a large amount of literature has appeared on the fundamental issues of stability and stabilization for T-S fuzzy systems (see, e.g., [6,7] and the cited therein).Very recently, Choi et al. [8] put forward a new method to formulate a framework that can describe T-S fuzzy systems with time-varying input delay and output constraints based on (, , ) − -dissipativity, which not only synthesized  ∞ control and passivity control, but also applied the obtained results to different road conditions.In the field of the fuzzy controller design, the authors in [9] considered input saturation of the nonlinear systems and employed a variables separation approach and the smallgain approach to design an adaptive back-stepping fuzzy controller, which provided more extensive applications in practice.
Recently, the T-S fuzzy model has been extended to polynomial fuzzy model [10][11][12][13][14][15].SOS approach to polynomial fuzzy system has been presented simultaneously and provides more extensive and/or relaxed results for the existing LMI approaches to T-S fuzzy system.A feasible solution to the SOS-based stability conditions can be found numerically with the help of a powerful Matlab toolbox SOSTOOLS [16].
Stability and stabilization conditions in terms of SOS are central to the problem of stability analysis and control design for polynomial fuzzy model, which have been investigated in the past ten years.For example, Tanaka et al. [10][11][12][13][14] firstly proposed SOS-based framework to obtain stability and stabilization conditions of the polynomial fuzzy systems based on polynomial Lyapunov functions.Recently, a new SOS design framework for robust control of polynomial fuzzy systems with uncertainties is presented by the research group of professor Tanaka [17].During this period, professor Lam and his colleagues make great contributions to relax the stability and stabilization conditions.For instance, shapedependent SOS-based stability analysis of the polynomial fuzzy-control systems is investigated in [18].A piecewiselinear membership function [19] and a switching polynomial Lyapunov function [20] are separately proposed to facilitate the stability analysis.Some other related issues of the polynomial fuzzy model have also been investigated for closed-form estimates of the domain of attraction [21], for tracking control of polynomial fuzzy networked systems [22], and for stability analysis and controller design of discrete-time polynomial fuzzy time-varying delay systems [23].
As is well known to the control community, the behavior of a system over a fixed time interval is practicable and then widely concerned.In view of this, a concept of finite-time stability (FTS) is introduced in [24,25] and great attention has been paid to (T-S) fuzzy model [26][27][28][29].However, the finitetime stability analysis and controller design of polynomial fuzzy model are seldom discussed.
Motivated by above discussions, in this paper, the problem of finite-time stability and stabilization for a class of continuous-time polynomial fuzzy system is considered.The main contributions of this paper are twofold: (1) the concept of finite-time stability for polynomial fuzzy model is firstly proposed, and (2) finite-time stabilizing controller is designed to stabilize the nonlinear system represented in the form of SOS, which is superior to the form of LMIs.
The rest of this paper is organized as follows: some foundational descriptions, necessary definitions, and relevant lemmas are recalled in Section 2. The main results are obtained in Section 3. Firstly, finite-time stability conditions for the polynomial fuzzy system without any control input are derived by using a new polynomial Lyapunov-Krasovskii functional.Secondly, the stability conditions are extended to finite-time stabilization conditions for the continuous-time polynomial fuzzy system with a controller in terms of SOS.All conditions can be solved via the SOSTOOLS and an SDP solver.Section 4 provides three illustrative examples.Finally, conclusions are drawn in Section 5.
Notations.Throughout this paper, the notations employed are fairly standard.The superscripts −1 and  stand for the inverse and transpose of a matrix, respectively. > 0 ( ≥ 0,  < 0, and  ≤ 0) means that the matrix  is positive definite (positive semidefinite, negative definite, and negative semidefinite). × is the set of  ×  real matrices. denotes the identity matrix with compatible dimensions.

Preliminaries and Problem Formulation
Consider the following nonlinear system: where  is a nonlinear function, is the state vector, and is the input vector.Based on the sector nonlinearity concept [1], system (1) can be expressed in the following polynomial if-then rules.
The defuzzification process of model ( 2) can be calculated as below (for more details, see [10]): where Before proceeding, the following definitions and lemma are necessary.
Definition 1 (see [16]).A polynomial (),  ∈   , is an SOS, if there exist polynomials  1 (), . . .,   () such that Definition 2 (see [16]).The SOS condition ( 5) is equivalent to the existence of a positive semidefinite matrix , such that where () is some properly chosen vector of monomials.Remark 5. Definitions 3 and 4 are a first attempt to give the concept of finite-time stability and stabilization for polynomial fuzzy systems based on the basic idea in classical paper [24,25].When x(()) here reduces to (), the above two definitions reduce to the forms in [24].

Main Results
In this section, the finite-time stability and stabilization conditions for the continuous-time polynomial fuzzy system (3) are given, respectively.

Finite-Time Stability of Continuous-Time Polynomial
Fuzzy System.Firstly, we will analyze the finite-time stability of the continuous-time polynomial fuzzy system (3) with () = 0. We drop the notation with respect to time .
Proof.Choose the following candidate polynomial Lyapunov functional: It can be seen from condition (8) that () is positive definite for all , and thus, () is a positive definite function of .
Finite-time stability conditions of the continuous-time polynomial fuzzy system (3) with  = 0, which can be checked by the SOSTOOLS [16], have been derived in Theorem 7. When   () and () in Theorem 7 reduce to the constant matrices  and , we can get the following corollary.
Remark 9.It is well known that [25] is a classical paper in the concept of finite-time stability.As shown in Corollary 8, the SOS conditions ( 8)-( 12) for finite-time stability in Theorem 7 reduce to the well-known LMI conditions (24).In addition, two parameters Φ 1 and Φ 2 used in Theorem 7 instead of the role of eigenvalues of  in [25] and the condition like (25) in Corollary 8 are not needed, which help find more relaxed solutions for the range of () in our proposed method.Moreover, small-gain theorem in [9] is proposed to ensure the resulting closed-loop system to be bounded, but our method employs SOS approach to ensure the range of ().

Finite-Time Stabilization of Continuous-Time Polynomial
Fuzzy System.In this subsection, a finite-time fuzzy controller with polynomial rule consequence is designed for the system (3) and the control rule shares the same IF parts with system (3) as follows.
From condition (29), which implies that both ( x) and  −1 ( x) are positive definite for all , we can obtain that () is a positive definite function of .
Remark 11.When   (),   (), and ( x) in Theorem 10 reduce to constant matrices , , and , the finite-time stabilization conditions of the continuous-time polynomial fuzzy system (3) reduce to the one in Theorem 5 of [24], which means that the classical result is a special case of ours.

Numerical Simulations
To illustrate the effectiveness and validity of our main results, this section provides three illustrative examples.
Example 1.Consider the following nonlinear system: Using the sector nonlinearity technique [1], the nonlinear system (47) is exactly converted into the following two-rule polynomial fuzzy system: where ,  =  1 , and The corresponding parameters for the finite-time stability are given in this example as follows: By using the SOSTOOLS of Matlab and solving Theorem 7, we can get the following feasible (), and corresponding Φ 1 , Φ 2 for system (47): where the degree of the polynomial () is set to be 0. It is obvious that when the degree of () is set different from 0, different Φ  ,  = 1, 2, will be obtained accordingly.Notation.
Using the same technique as before, the nonlinear system (52) can be exactly converted into the following two-rule polynomial fuzzy system: where ,  =  1 , and The corresponding parameters for the finite-time stabilization are given in this example as follows: By using the SOSTOOLS of Matlab and solving Theorem 10, the solutions are found as follows: and the feedback gains  1 () and  2 () are obtained as follows: respectively, where the degree of the polynomial   () is set to be 0 to 1 (code for SOSTOOLS).When the degree is set to be different from what is set here, different   () will be obtained accordingly.Figures 3(b) and 4 show the validity and effectiveness of our proposed method.
Example 3. As shown in [31], a coaxial counter rotating helicopter dynamics can be written as where  = 1.5,   = 0.2857, and the other corresponding variables such as , V, , Ψ,   ,   , and   are the same as [31].For the lack of space, the details are omitted here.As above, nonlinear system (59) can be exactly converted into the following two-rule polynomial fuzzy system for Ψ ∈ [− ]: where  = x() = [ V        ]  ,  = Ψ, and .  ,   ,   , and   () matrices are defined the same as in [31].Besides,   () and the membership functions are given as follows: The corresponding parameters for the finite-time stabilization are given in this example as follows:   is set to be 0 to 1 (code for SOSTOOLS).When the degree is set to be different from what is set here, different   () will be obtained accordingly.
Figures 5 and 6 depict the trajectory of the coaxial counter rotating helicopter dynamical system (59) under finite-time stabilizing control, which shows the validity and effectiveness of our proposed method.

Conclusions
Although numerous works have been published in recent years on the finite-time stability and stabilization of T-S fuzzy model, little attention has been paid to the investigation on the problem of finite-time stability analysis and synthesis method for continuous-time polynomial fuzzy system.The proposed results in this paper fill in this gap by using the polynomial Lyapunov functional approach and the SOS theory.Three illustrative examples have been provided to demonstrate the validity and effectiveness of the proposed results.In the future, we will consider the similar problem for the discrete-time polynomial fuzzy system and the polynomial fuzzy model with time delays or (and) uncertainties [17].