Robust stabilization of T-S fuzzy systems via improved integral inequality

This paper addressed the robust stabilization performance of Takagi–Sugeno (T-S) fuzzy systems under a state feedback controller. To attain this, an integral inequality is proposed by rearranging the quadratic matrix-vector form combined with Jensen’s inequality to handle the single integral terms obtained by taking the derivative of the concerned Lyapunov–Krasovskii functional. By employing this integral inequality and by using integral inequality techniques, some improved delay-dependent stability and stabilization results are established in terms of linear matrix inequalities for the proposed T-S fuzzy model. Finally, some numerical examples are provided to facilitate the feasibility and less conservativeness of the proposed theoretical results.


Introduction
In the field of control engineering (Takagi and Sugeno 1985), the fuzzy logic control has been regarded as a powerful tool to analyze the performance of the nonlinear systems.Among various fuzzy models, Takagi-Sugeno (T-S) fuzzy model is widely accepted as a simple but effective model to describe the behaviors of uncertain nonlinear systems, where a set of fuzzy IF-THEN rules is introduced to describe the local linear input-output relations of a nonlinear system (Li et al. 2015;Subramaniam et al. 2020;Moodi and Farrokhi 2014;Hui and Xie 2016;Yang and Wang 2014;Gao et al. 2013;Subramaniam and Joo 2019;Zhang et al. 2014;Nagamani and Ramasamy 2016;Araujo et al. 2019;Nithya et al. 2020).On the other hand, over the past decades, the stability and stabilization performance of uncertain T-S fuzzy systems turned out to be a more sensitive research theme (Xia et al. 2017;Wang and Lam 2017;Gu 2000;Zhang et al. 2016;Li et al.B G. Nagamani nagamanigru@gmail.com 1 Department of Mathematics, The Gandhigram Rural Institute (Deemed to be University), Dindigul, Tamil Nadu, India 2015).Currently, a plenty of approaches have been developed to examine the themes of such systems, and these proposed techniques are of highly relevant to both theory and application.For example, the problem of T-S fuzzy systems with observer-based non-parallel distributed compensation control throughout an irregular sampling interval is investigated in Peng et al. (2017).Recently, the stabilization of T-S fuzzy model has been studied in Wang and Lam (2017) via fuzzy control approach.
In view of the practical engineering systems, there arise a couple of challenging problems such as the existence of the time-delay and the control directions probably unknown.Both always yield the control design more complicated and complex.In such cases, the relevant LKF mechanism is always a powerful tool to deal with these shortcomings.Based on these useful techniques, many important results have been proposed through numerous inequality approaches such as Wirtinger-based inequality, weighted integral inequality, and relaxed integral inequality (Gu 2000;Zhang et al. 2016;Li et al. 2015).With the help of these inequalities, the derivatives of the concerned LKF are carried out to establish the desired stability conditions in terms of LMIs.To be more specific, an integral inequality proposed in Seuret and Gouaisbaut (2013) results in the reduction in the conservatism based on the treatment in Fourier theory, particularly in the view of well-known Wirtinger inequalities.In Gong et al. (2016), the authors developed new integral inequalities by means of the proposed integral inequality in Park et al. (2015) in order to investigate the problem of exponential stability performance of the delayed systems.In Zhang et al. (2016), the stability criterion for delayed system is studied with the aid of bounding inequalities and augmented LKF.
On consideration of the aforementioned issues into account, in this paper, an attempt has been made to propose an improved integral inequality to investigate the stable performance of the considered nonlinear model under fuzzy-based state feedback controller.To obtain the desired stabilization criteria, an appropriate LKF has been constructed involving delay-dependent single and double integral quadratic terms.Hence, the derivative of such LKF will result in single integral terms.By introducing an appropriate dimensional matrix, the obtained single integral term can be divided into two single integral terms.Then, by combining the rearrangement of Jensen's inequality and the desired quadratic matrix vector form based on the appropriate dimensioned matrix introduced earlier, a new integral inequality is proposed which yields a broad foundation for computing the single integral terms.As far as we know, the problems of stabilization for uncertain nonlinear systems via novel integral inequality have not been fully examined, which motivates us to derive the new integral inequality.
The main novelty and contribution of this manuscript can be summarized as follows: • An improved integral inequality is acquired with help of Jensen's inequality to handle the considered LKF.• Based on the fuzzy control approach, the stability and stabilization criteria are obtained for the considered uncertain nonlinear models through LKF whose derivative can be solved by means of the improved integral inequality.• Appropriate control gain matrices K j are calculated from the LMIs to guarantee the stabilization of the proposed model via state feedback control.• Finally, a continuous stirred-tank reactor (CSTR) system is adapted to verify the applicability and virtue of our theoretical findings.Furthermore, numerical simulations are appropriately designated to illustrate the applicability of the designed control to reduce the conservatism.
This manuscript is organized as follows.In Sect.2, we outlined the preliminaries and system descriptions.In Sect.3, an improved integral inequality is introduced.Based on this integral inequality, in Sect.4, the exponential stabilization of the uncertain T-S fuzzy model has been analyzed.Numerical simulation on the considered nonlinear model is provided to evaluate and illustrate the feasibility of the designed controller in Sect. 5. Finally, Sect.6 gives the conclusion.

Notations
The notations in this manuscript are quiet standard.A T stands for the transpose of the matrix A. M ≥ 0 (M ≤ 0) denotes that M is a positive (negative) semi-definite symmetric matrix.R n represents an n-dimensional Euclidean space and R n×m denotes the set of all n × m real matrices.C ([−τ, 0], R n ) denotes the space of all n-valued continuous functions defined on [−τ, 0].A block-diagonal matrix is denoted by diag{. ..}.For each positive integer q, let I q = {1, 2, 3, . . .q}. Use sym{M} to represent M + M T , for any matrix M. The symbol 0 m×n represents null matrices of order m × n.The notation * is commonly used to denote the symmetric block of a symmetric matrix.I denotes the identity matrix.Matrices, if not specifically mentioned, are supposed to be appropriate dimensions.

System formulation
We consider the following fuzzy system characterized by the fuzzy IF-THEN rules as Rule i: and . . . , and where the premise variables are represented by ϑ 1 (x(t)), ϑ 2 (x(t)), . . ., ϑ p (x(t)); the fuzzy sets are denoted by H i ς , (i = 1, 2, . . ., q; ς = 1, 2, . . ., p), q represents the number of IF-THEN rules.The state vector is denoted by x(t) ∈ R n , the control input is represented by u(t) ∈ R m , the output vector is denoted as y(t) ∈ R v .The known constant system matrices are represented by and A τ i (t) denote the unknown real-valued matrices involving uncertainties of time-varying parameter defined as follows: here D i , E i and E τ i are known constant matrices and the unknown matrix F i (t) is follows F T i (t)F i (t) ≤ I , ∀t.Then, by applying fuzzy approach, one can deduce the system dynamics (1) as follows: (2) here m i (x(t)), i = 1, 2, . . ., q, denote the normalized membership grades satisfying is the membership grade of ϑ υ (x(t)) subject to the fuzzy set H i υ .Based on system (2), a state feedback fuzzy model controller is designed as Fuzzy Controller Rule l: IF θ 1 (x(t)) is J l 1 , and . . ., and where (4) Let us make the transformation as follows: where α is decay rate of system (5).From ( 5) and ( 4), one can get the following T-S fuzzy model: where

Improved integral inequality
The following lemma will be derived to investigate the stability and stabilization condition of the considered nonlinear model: where Proof By applying Jensen's inequality to the integral term Also, we have the quadratic matrix vector form as After a small computation, one can rewrite the inequality as follows: Using ( 8) in ( 7), we obtain 123 Lemma 3.2 For any real matrices D, M, and F(t) with appropriate dimensions satisfying F T (t)F(t) ≤ I , the subsequent inequality holds for any constant > 0: On view of the above system configuration, the major objective of the manuscript lies in Problem 1 as follows.
Problem 1 Given the T-S fuzzy system (1), under the state feedback control, one can estimate the controller gain matrices K l , l = 1, 2, . . .q considered in (3) such that system (6) is exponentially stable.

Main results
In this section, for obtaining the solution of the above Problem 1, the following time-dependent LKF is considered. where Theorem 4.1 For given scalars τ d > 0 and τ > 0, the nominal system ( 6) is exponentially stable with decay rate α, if there exist matrices P, Q, R, M 33 , J il , H il , i, l ∈ I q which are symmetric positive definite and appropriately dimensioned matrices M 11 , M 12 , M 13 , M 22 , M 23 , and K l ∈ R m×n under which the subsequent LMIs hold for all i, l ∈ I q : where Proof Differentiating V (t), one can obtain the following: ṀT (s)M 33 Ṁ(s)ds. (10) where Adding inequalities ( 10) and ( 11), it follows that where il is defined in Theorem 4.1.Denoting q l=1 h il , by making use of some algebraic manipulations, we have where hil is the upper bound of h il , h il is the lower bound of h il , H il = H T il ≥ 0, and J il = J T il ≥ 0. This completes the proof.
Remark 4.2 By taking Āi = A i − B i K l + α I , the matrix il can be rewritten as Now, Theorem 4.1 can be extended to obtain the exponential stability criterion for system (6) under the case of timevarying uncertainties as in the following theorem.
Theorem 4.3 For given scalars τ d > 0 and τ > 0, the uncertain system ( 6) is exponentially stable with decay rate α, if there exist matrices P, Q, R, M 33 , J il , H il , i, l ∈ I q which are symmetric positive definite and appropriately dimensioned matrices M11 , M12 , M13 , M22 , M23 , and K l ∈ R m×n under which the subsequent LMIs hold for all i, l ∈ I q : where h il ≤ h il ≤ hil Proof By continuing the same procedure as in Theorem 4.1 and by considering Āi = A i − B i K l + α I , from Remark 4.2, it can be obtained that By rewriting ¯ il , we obtain the following inequality: 123 By using the Schur complement lemma, inequality ( 12) can be equivalently expressed as Hence, by Lemma 3.2 it follows that Now by applying Schur complement to ( 16) one can obtain that 11 il This inequality together with the inequality R − M33 ≥ 0 implies that V (t) < 0. Hence, the existence of LMI ( 12) ensures that the exponential stability of the system (6).This completes the proof.
Remark 4.4 Based on Theorem 4.3, the gain matrices K l ∈ R m×n (l ∈ I q ) can be obtained by making the transformation K l = Y l G −1 , (l ∈ I q ) as in the subsequent theorem.
Theorem 4.5 Under given scholars a > 0, τ d > 0, τ > 0 and 0 < τ(t) ≤ τ , the system formulated in ( 6) is exponentially stable with decay rate α under the controller gain matrices K l = Y l G −1 , (l ∈ I q ), if there exist symmetric positive definite matrices P, Q, R, M 33 , and J il , H il , i, l ∈ I q , and appropriately dimensioned matrices M 11 , M 12 , M 13 , M 22 , M 23 under which the subsequent inequalities hold simultaneously for all i, l ∈ I q : where h il ≤ h il ≤ hil , Applying the congruence transformation to ( 12)-( 14) with diag G, G, G, G , diag G, G, G , and G, respectively, we can obtain LMIs ( 17)-( 19).This completes the proof.

Remark 4.6
For facilitating the importance of the stability conditions established in this paper, we attempt to propose the less conservative stability conditions for the following system obtained from the system (20) obtained from system (4) by removing the uncertain parameter matrices A i (t) and A τ i (t).From Theorem 4.5 and by taking α = 0, the required asymptotic stability conditions for system (20) can be achieved by adapting the following LKF: x T (s) Qx(s)ds where By making use of the same procedure as in Theorem 4.5, the following corollary can be given.

Numerical examples
Five examples are provided to demonstrate the feasibility and effectiveness of the developed control design technique in this section.In the first example, borrowed from Li et al. (2015); Hua et al. (2008), a CSTR model is given to show the applicability of the considered model and the other examples are considered to show the less conservativeness of the developed technique against the conventional approaches (Kwon et al. 2012;An et al. 2012;Zeng et al. 2014;Zhang et al. 2015;Kwon et al. 2016;An and Wen 2011).
Example 5.1 Consider the following dynamic equations governing CSTR model previewed in Hua et al. (2008), which is illustrated in Fig. 1: where x = [x 1 (t) x 2 (t)] T in which x 1 represents the conversion rate of the reaction, whereas x 2 refers to the dimensionless temperature.The functions g 1 (x) and g 2 (x) are described by
Similar to Hua et al. (2008), the fuzzy model of the above system is expressed under the fuzzy rules defined as follows: where Based on Hua et al. (2008) and Zhou et al. (2019), the concerned membership functions are given as With respect to mismatched premise design procedure, a tworule fuzzy controller is adapted as follows: The membership functions of fuzzy controllers are considered from Zhou et al. (2019) to be w 1 (x 1 ) = 1 1 + e −x 1 /2 , w 2 (x 1 ) = 1 − w 1 (x 1 ).
By fixing α = 0.1, τ = 0.625, τ d = 0.02 and = 0.01, it is easy to show that LMIs (17)-( 19) estimated in Theorem 4.5 can be made feasible by involving MATLAB LMI toolbox.This yields that the CSTR model ( 21) is exponentially stable under the fuzzy controller (3) with the decay rate α = 0.1.Correspondingly, the fuzzy-based control scheme of the control gains K l (l = 1, 2, 3) are calculated, as follows: The obtained theoretical results are further examined through numerical simulation.Under the initial condition Let us consider the following membership functions: As a result of parameter uncertainty η(t), the membership functions designed above become the uncertain grades of membership.Therefore, the minimum and maximum membership functions are calculated as Let us consider the membership functions of (3) as follows: with the parameters α = 0.1, τ = 0.625, τ d = 0.02 and = 0.01, Figs. 4 and 5 show that control input and the state response of the corresponding control system with the following controller gains Moreover, on solving LMIs ( 17)-( 19) obtained in Theorem 4.5, the corresponding larger delay upper bounds τ are calculated for different values of α as given in Table 1.In view of Table 1, one can conclude that when the value of τ decreases, simultaneously α increases.The obtained theoretical results are further examined through numerical simulations.Under the initial condition by applying the fuzzybased control developed in (3), the trajectories of the states of the considered nonlinear system together with the corresponding control parameters are plotted in Figs. 4 and 5, respectively.
Example 5.3 Consider T-S fuzzy system (20) having the subsequent parameters with two plant rule: In this example, when τ d is chosen as 0 and 0.1, the appropriate maximum values of time-delay bound τ for guaranteeing the asymptotic stability can be estimated as a result of the proposed LMIs in Corollary 4.7 and it is compared with recent studies (Kwon et al. 2012;Zeng et al. 2014;Yang et al. 2014;Zhang et al. 2015) as summarized in Table 2.In addition, Corollary 4.7 provides maximum delay bounds than the existing works (Zeng et al. 2014;An and Wen 2011) in which the delay-partitioning approach is involved.
Example 5.4 Consider the T-S fuzzy model (20) having the following system parameters with u(t) = 0 under two plant rule: In this example, the stability of open loop fuzzy system defined in (20) has been analyzed extremely in Yang and Wang (2014) and the goal is to calculate the larger upper bound τ with which the fuzzy system is exponentially stable.Table 3 summarizes the values of the larger upper bound τ estimated by various approaches.Through this table, it is witnessed that the performance of the developed results provides maximum delay bounds than the existing ones (An et al. 2012;Zeng et al. 2014;Zhang et al. 2015;Kwon et al. 2016).
In Kwon et al. (2016), Mozelli et al. (2011), Souza et al. (2009), and Zhang et al. (2015), the two-rule fuzzy system has been widely reviewed and the objective is to produce the larger upper bound τ for which the fuzzy system could be stabilized by designing the appropriate controller.To obtain the less conservatism, we compare the derived results with the traditional techniques (Zhang et al. 2015;Kwon et al. 2016;Mozelli et al. 2011;Souza et al. 2009).Table 4 gives the upper bound τ obtained by means of various approaches.Hence, it can be viewed that Corollary 4.7 establishes the higher delay bounds than the existing ones as shown in Table 4.

Conclusion
The problem of exponential stabilization for the nonlinear system via fuzzy-based controller under the framework of Lyapunov approach has been investigated in this paper.First, novel integral inequality is introduced with the help of Jensen's inequality to limit the considered LKF.Based on the fuzzy control approach, the stability and stabilization criteria have been obtained for the considered uncertain nonlinear models through suitable LKF whose derivative can be solved via improved integral inequality.Appropriate control gain matrices have been calculated from the LMIs to guarantee the stabilization of the proposed model via state feedback control.Finally, the CSTR system has been adapted to verify the applicability and merits of our theoretical findings.Furthermore, numerical simulations have been designated appropriately to illustrate the applicability of the designed control by reducing the conservatism.

Fig. 2
Fig. 2 State responses of x(t) for CSTR model