Fuzzy membership function-dependent switched control for nonlinear systems with memory sampled-data information

In this paper, a fuzzy memory-based coupling sampled-data control (SDC) is designed for nonlinear systems through the switched approach. Compared with the usual SDC scheme, by employing the Bernoulli sequence, a more general coupling switched SDC that involving the signal transmission delay is designed. The Lyapunov–Krasovskii functional (LKF) is presented with the available characteristics of the membership function, and a coupling sampling pattern, for the T-S fuzzy systems. Based on LKF, together with time derivative information of membership function, and the generalized N-order free-matrix-based inequality, the suitable conditions are obtained in terms of linear matrix inequalities (LMIs) for guaranteeing the asymptotic stability and stabilization of the concerned system. Then, the desired fuzzy coupling SDC gain is attained from the solvable LMIs. In the end, two examples are given to validate the derived theoretical results.


Introduction
The analysis and synthesis of nonlinear dynamical systems have gradually become the focus of attention due to extensive applications in physics, engineering communities, and so on (Haddad and Chellaboina 2011;Precup et al. 2014). In real life, stability analysis and control design for nonlinear dynamical systems are quite difficult. Recently, the T-S fuzzy model has been dealt with the intrinsic nonlinear systems because the T-S fuzzy model can depict the considered nonlinear systems as a weighted sum of some linear subsystems. From this idea, the stabilization for T-S fuzzy systems has been established in Aslam and Li (2019), Shen et al. (2020) Luo et al. (2020), Kalat (2019), Zhao et al. (2018). Moreover, the various nonlinear dynamical systems, such as Rossler's system (Precup et al. 2014), Chua's circuit (Mkaouar and Boubaker 2012), mass-spring system (Sakthivel et al. 2016), have been formulated as T-S fuzzy systems to study the stabilization problem. To tackle the stabilization issue of T-S fuzzy systems, several control techniques have been designed in the literature, for example, adaptive control (Chen et al. 2021), impulsive control (Zhang 2016), mixed H ∞ passive control (Shen et al. 2017), SDC (Wang and Wu 2015), and so on.
With the rapid development of communication technology and digital networks, SDC becomes an attractive field of research in the control system. The main scope of the SDC is updating the control signal information only at the sampling instants, not for the whole time interval, which reduces the communication bandwidth (Wang et al. 2017). Also, the SDC has many advantages when compared with continuous-time controllers, such as efficiency, maintenance with low cost, and simple installation. According to these applications, the SDC has been utilized to investigate the various problems in T-S fuzzy systems (Wang et al. 2017;Qu et al. 2020;Yoneyama 2012;Ali et al. 2020). For instance, the dissipativity and extended dissipativity for T-S fuzzy systems have been investigated via SDC in Zeng et al. (2019a) and Tian and Wang (2020), respectively. Moreover, the asymp-totic stabilization of T-S fuzzy-based chaotic systems has been studied via the SDC scheme in Zeng et al. (2019b).
Further, the updating signal successfully transmitted from sampler to the controller and to the zero-order-hold (ZOH), the control signal may experience the constant transmission delay at any time instant t k , which leads to the necessity of the memory-based SDC ). According to this viewpoint, the memory-based SDC has been designed for the stabilization of T-S fuzzy systems in Zhang et al. (2019), Ge et al. (2019), . Also, the dissipativity of T-S fuzzy systems has been analyzed via memory SDC in Ge et al. (2021). Very recently, the stabilization problem has been studied via memory SDC for T-S fuzzy time-delay system. From the above literature, memory SDC has improved the stability performance of the proposed systems with less conservative results. Inspired from the above, the memory-based coupling SDC which combines both the traditional SDC and memory-based SDC with the help of the Bernoulli sequence is designed to improve the asymptotic stability of T-S fuzzy systems in the present study.
Meanwhile, when investigating the stabilization of T-S fuzzy systems under SDC scheme, the less conservative results through the choosing of proper sampling interval are significant. Since, the maximum sampling interval gives some superiorities, like lower communication channel occupation, less signal transmission, and less actuation of the controller. Hence, several methods have been established in the literature (Coutinho et al. 2020;Lee and Park 2018;Xia et al. 2020;Wu et al. 2014; to get the less conservatism via largest upper bounds (LUB) of sampling interval which ensures the stabilization of T-S fuzzy systems. In particular, a looped-functional has been introduced in Seuret (2012), which relax the positivity condition of LKF and also obtained the less conservative results. Nowadays, a new two-sided looped-functional has been employed to improve the stability condition from the information of the whole sampling interval and greatly enlarge the maximum sampling intervals (Hua et al. 2020;Tang and Ma 2019;. Moreover, in fuzzy systems, membership functions play a crucial role to analyze the stability of systems and acquire the less conservatism (Kim et al. 2018;Kim and Lee 2020). For example, the output-feedback exponential stabilization condition for fuzzy control system has been obtained via membership function-dependent LKFs in Kim et al. (2018). In the meantime, the time derivative of fuzzy membership function-dependent LKFs has been employed for the stability analysis of T-S fuzzy systems in Zhao et al. (2020). More recently, the stability problem has been studied for T-S fuzzy systems via membership function-dependent LKF in Wang et al. (2020). Furthermore, when compared with usual control method (Wang and Wu 2015), switched control approach for T-S fuzzy systems diminishes the number of LMIs and attains maximum sampling interval, which has been studied in Kuppusamy and Joo (2020), Wang et al. (2020). Although a lot of attempts had made on the investigation of T-S fuzzy systems stability, when the memory-based coupling switched SDC information is taken into account, the stability of fuzzy systems has been drawn very little research, which motivates us to carry on the current research. Hence, this paper aims to propose SDC strategy for T-S fuzzy systems together with the coupling sampling pattern, membership function-dependent LKF, and applying the switching topology. Through this, we attempt to ensure better performance for T-S fuzzy systems when compared to the existing works with the complete information about the time derivative of chosen membership function.
Inspired from the above, the stabilization of T-S fuzzy systems is investigated in this paper via memory-based coupling SDC. The main contributions of the paper lie in the following aspects: 1. Different from the conventional SDC and memory-based SDC scheme (Wang and Wu 2015;Wang et al. 2017;Ge et al. 2019Ge et al. , 2021) a more general memory-based coupling SDC is designed by Bernoulli sequence for the T-S fuzzy systems. 2. Unlike from the LKF (Wang et al. 2017;Zeng et al. 2019a, b;Lee and Park 2018), the membership functiondependent LKF, which includes the available information of the coupling sampling pattern, signal transmission delay, and full state information from t to t k and t to t k+1 , is constructed in the present study. 3. By utilizing the generalized N -order free-matrix-based inequality, the stabilization criteria are derived with the aid of the time derivative of the membership function via switched approach for the considered systems in the form of LMIs. 4. To show the effectiveness of the designed control method, the derived stabilization conditions are compared with existing works (Wang and Wu 2015;Wang et al. 2017;Wu et al. 2014;Zheng et al. 2021;Wu et al. 2019), and the designed controller achieves the LUB of sampling interval with less conservative results.
The paper has the following structure: Sect. 2 devotes the formulation of T-S fuzzy system. In Sect. 3, the memorybased coupling SDC is designed for T-S fuzzy system. Two examples are considered in Sect. 4. Conclusions are drawn in Sect. 5. Notations R n×m and R n denote n × m real matrix and the ndimensional Euclidean space, respectively. The matrix U > 0 (< 0) denotes a positive (negative) definite. E{·} indicates the mathematical expectation operator. Sym{A} = A + A T . I and 0 represent the identity and zero matrix with appropri-ate dimensions, respectively. diag{· · · } is a block diagonal matrix.

T-S fuzzy system formulation
Let us consider the nonlinear systems as follows: where x(t) ∈ R n and u(t) ∈ R m represent the state and control input vector, respectively. h(x(t), u(t)) denotes a known nonlinear function which satisfies h(0, 0) = 0. Then, based on the T-S fuzzy modeling approach, nonlinear system (1) can be represented by a series of IF-THEN rules: where A i ∈ R n×n and B i ∈ R n×m are constant matrices; w 1 (t), w 2 (t), . . . , w p (t) are the premise variables; θ i 1 , θ i 2 , . . . , θ i p denote the fuzzy sets. By utilizing product inference, singleton fuzzifier and center average defuzzifier, the whole T-S fuzzy system (2) can be inferred aṡ where is the grade of membership of w g (t) in θ i g .

Design of memory-based Coupling SDC strategy
In this subsection, the memory-based coupling SDC is designed for the system (3). For this, the control input signal is assumed to be generated by utilizing a ZOH function with where K j and L j are the control gain matrices and τ is constant signal transmission delay. Now, define d(t) = t − t k withḋ(t) = 1 for t = t k and which satisfies 0 < d(t) ≤ d k = t k+1 − t k ≤d,d is the maximum sampling interval. Here, β(t) is the Bernoulli stochastic variable coupling the conventional SDC and memory-based SDC with Then, the overall memory-based coupling SDC is represented as Based on the control input (4) and system (3), we get the fuzzy system as follows: Remark 1 In the existing literature, the conventional SDC and memory-based SDC have been separately designed for T-S fuzzy systems. Distinct from this, the above-said controllers are coupled by the Bernoulli sequence in this present study. From the memory-based coupling SDC (4), when the stochastic variable β(t) = 0, the designed control technique (4) reduces to the conventional SDC input , which has been widely established for T-S fuzzy systems in Wang and Wu (2015), Wang et al. (2017). Meantime, if β(t) = 1, then the designed controller becomes a memory-based SDC u(t) = r j=1 δ j (w(t k ))K j x(t k − τ ), which also has significantly designed for T-S fuzzy systems, recently (see, Ge et al. 2019Ge et al. , 2021. From the above, the employed control technique for T-S fuzzy systems is more general than the works of Wang and Wu (2015)

Switching approach for designing control
The following switching approach is constructive for obtaining the proposed fuzzy system's stability and stabilization conditions. Let us consideṙ are matrix variables to be obtained; andδ i (w(t)) will be positive or negative which dependence upon of time. In order to ensureṖ δ ≤ 0,Q δ ≤ 0,Ṡ δ ≤ 0,U δ ≤ 0,Ṙ δ ≤ 0 andẆ δ ≤ 0, the following switching idea is utilized: From (6), there are 2 r −1 possible cases exist. Define μ ∈ χ = {1, 2, . . . , 2 r −1 }, then, equation (6) can be expressed as where A μ and B μ represent the sets that contains the possible permutations ofδ(w(t)) for μ ∈ χ and all the possible con- Based on the above, we get the following lemma as in Wang et al. (2020).

Lemma 1 Consider the system (3) with fuzzy membership function δ i (w(t)). For the symmetric matrices
If the switching rules (7) holds, then we havė Now, from the Lemma 1 for different A μ and B μ , we will design the corresponding coupling memory-based sampleddata switched control as follows: The final memory-based coupling sampled-data switched control (4) is expressed as Meanwhile, the block diagram of the T-S fuzzy systems with memory-based coupling sampled-data switched control is presented in Fig. 1. Hence, the closed-loop system with the control input (8) is given as follows: Fig. 1 Block diagram of memory-based coupling sampled-data switched control system 1, 2, . . . , N ).

Main results
The stability and stabilization problem of system (9) are analyzed in this section by the memory-based coupling sampled-data switched control. To maintain the representation simplify, the following notations are utilized:

Stability analysis
The following theorem is proposed to achieve the asymptotic stability of T-S fuzzy system (9).
Theorem 1 For given gain matrices K μ, j , L μ, j , the scalars d > 0, τ , β, and integer N ≥ 0, system (9) is globally hold, where Proof Let us choose the following LKF: where The time derivative of (15) is written aṡ Applying the Lemma 2 for the integral terms of (21), we get Now, for any appropriate dimensional matrices Z 1 , Z 2 , Z 3 , we have the following equation: with

Remark 2
Note that the fuzzy membership functions are the main characteristic of T-S fuzzy-based control systems. Compared to traditional LKF, the fuzzy membership function-dependent LKF gives less conservative results in the existing literature (Kim et al. 2018;Kim and Lee 2020;Zhao et al. 2020;Wang et al. 2020). Unlike the previous studies, the constructed LKF in Theorem 1 contains the information of the sampling pattern and includes more details about constant signal transmission delay and fuzzy membership functions. Moreover, inspired by Hua et al. (2020), Tang and Ma (2019), , we have considered the looped LKF V l (t) in this current study as in (15) satisfies V l (t k ) = V l (t k+1 ) = 0, (l = 2, 3, 4). Therefore, V (t) is continuous in time and at the sampling instants . Thus, it should be mentioned the great strength of the looped functional is not required to be positive definite between the sampling times and involves the full information of x(t) to x(t k ) and x(t) to x(t k+1 ). Hence, the considered novel membership function-dependent looped LKF will relax the stability condition and give less conservatism.

Remark 3
In order to avoid the time derivative of the membership function in LKF, we use switching rules similar to that in Zhao et al. (2018), Zhao et al. (2020), Wang et al. (2020), Kuppusamy and Joo (2020). In other words, the information of the time derivative of the membership function in LKF is obtained with the help of the inequalities (11) and (12) in Theorem 1. These constraints ( (11) and (12)) are obtained directly from the switching rules (6) and (7), stated as Lemma 1. Therefore, this study fully utilizes the information on the membership function and its time derivative via the switching approach.

Remark 4
In the existing studies, the novel integral inequalities have been utilized to obtain the less conservatism for stabilization of T-S fuzzy systems, such as Jensen's inequality (Yoneyama 2012), Wirtinger's inequality (Wang et al. 2017;Ge et al. 2019), second-order Bessel inequality (Hua et al. 2020). Different from those studies, the generalized N -order free-matrix-based inequality handles the stabilization problem of T-S fuzzy systems in the present study, which has some inequalities as particular cases. For example, substituting N = 0 in (10), it can be reduced to Jensen's inequality which has been utilized for T-S fuzzy systems in Yoneyama (2012). Similarly, when N = 1, the inequality becomes Wirtinger's inequality as in Wang et al. (2017), Ge et al. (2019). Moreover, the second-order Bessel inequality (N = 2) has been examined in Hua et al. (2020). Thus, the proposed method utilizes the generalized N -order free-matrix-based inequality which is more general than the aforesaid previous studies.

Controller design
In this subsection, we will analyze the stabilization of system (9) via memory-based coupling SDC by the results in Theorem 1.
Remark 5 It should be pointed out that the purpose of the following corollary is to affirm the advantage of the proposed method under conventional SDC input, that is, the stochastic variable β(t) vanishes in (4) Hence, the corresponding closed-loop system (9) is presented as follows: Also, the LKF can be constructed from (15) without the constant signal transmission delay and the stochastic variable as follows: where

Stabilization of T-S fuzzy systems under conventional SDC
In this subsection, we derive the adequate conditions to assure the stabilization of T-S fuzzy systems under conventional SDC scheme which is a special case of previous subsection.
Corollary 1 For given scalars λ,d > 0, and nonnegative integer N , system (31) is globally asymptotically stable, if there exist P i > 0, X > 0, Y > 0, Q i = Q T i , R i = R T i , nonsingular matrix Z , and appropriate dimensional matrices W i , J z N (z = 1, 2), H μ, j such thaṫ hold for d k ∈ (0,d]. Here, The control gain matrices are calculated by L μ, j = H μ, j Z −1 .
Proof With a similar procedure as Theorems 1 and 2, the proof can be derived.

Numerical examples
This section presents the superiority of derived theoretical results by using two numerical examples.
Choosing τ = 0.02, β = 0.5, λ 1 = λ 2 = 1 and from the switching rule (6), solving the LMIs (26)-(29) for the constraints B 1 : { P 1 > P 2 , S 1 > S 2 , Q 1 > Q 2 , R 1 > R 2 , U 1 > U 2 , W 1 > W 2 }, we obtain the control gain matrices with the LUB of sampling intervald = 0.0141 as follows: Similarly, for the same parameter values, we obtain the LUB of sampling intervald = 0.0090 under the constraints In addition, the control gains are calculated as follows: Hence, the final LUB of sampling interval is obtained as d = 0.0090. Moreover, the state trajectories of the closedloop system are displayed in Fig. 3, which is asymptotically stable by the designed control scheme (8). Figure 4 represents the time evolution of β(t). Figures 5 and 6, respectively, present the evolution ofδ 1 (x 1 (t)) and the stabilizing controller. From Fig. 5, we observe that the obtained control gain matrices are switched between the constraints B 1 and B 2 based onδ 1 (x 1 (t)) < 0 andδ 1 (x 1 (t)) ≥ 0, respectively. In addition, the LUB of the sampling interval for various values of N and β is tabulated in Table 1. From this table, we can conclude that the parameter N plays a vital role in obtaining the less conservative results.
According to the structure of conventional SDC in (30), by using the same aforesaid parameter values in Corollary 1, the LUB of sampling interval is calculated asd = 0.1094 andd = 0.1091 for system (36) based on the constraints It is clear thatd = 0.1090 is the final LUB of sampling interval for (36) by the conventional SDC (30). To show the superiority of the obtained results, the comparison of LUB of sampling interval by different methods with Corollary 1 is tabulated in Table 2.
As seen in Table 2, the obtained LUB of sampling interval in Corollary 1 is larger than the various methods in Wang et al. (2017), Wang and Wu (2015), Wu et al. (2014), Zheng et al. (2021), Wu et al. (2019), which indicates the advantages of proposed theoretical results.

Example 2
The nonlinear mass-spring system ) is considered as follows: where M represents the mass, u(t) denotes the control force, and y(t) is the position vector of the spring. g(y(t)) and (ẏ(t)) are the nonlinear terms of spring and input, respectively.

Conclusion
In this work, the stability and stabilization analysis for nonlinear systems through the fuzzy memory-based coupling SDC has been investigated. Under the consideration of signal transmission delay, Bernoulli sequence and switching approach, a fuzzy memory-based coupling SDC has been designed. To do this, the LKF has been constructed with a fuzzy membership function and utilized the generalized N -order free-matrix inequality, and the adequate conditions have been established in terms of LMIs for guaranteeing the stability of T-S fuzzy systems with less conservativeness. In comparison with the existing studies, the LUB of the sampling interval has been calculated by the proposed method.
At last, the simulation results for the nonlinear system have demonstrated the merits of the presented theoretical results.