New Stabilization Conditions for Fuzzy-Based Sampled-Data Control Systems Using a Fuzzy Lyapunov Functional

This paper focuses on the stabilization problem of Takagi-Sugeno fuzzy systems via a sampled-data controller using a fuzzy dependent functional. By employing a property of convex combination, a new approach to deal with the time derivatives of the fuzzy membership functions (FMFs) in the stabilization conditions is proposed, and less conservative conditions are derived in the form of linear matrix inequalities (LMIs). Moreover, the proposed approach introduces a switching function which opens up possibilities to use a switched controller and take advantage of its benefits well known in the literature. Therefore, two sampled-data control strategies are proposed, where the first one is a fuzzy controller and the second is a robust switched controller, that does not require the expressions of the FMFs to implement the control law, which guarantees the robustness of the controlled system in cases where the FMFs depend on uncertain parameters. Finally, the effectiveness of the proposed strategies is verified by two examples.

On the other hand, with the increasing development of digital technologies and communication networks, digital controllers play an important role in the design of new control strategies. The sampled-data control is a type of digital control which feedback the state variables obtained from the system at its sampling instants to the controller during a certain sampling interval, generating a control signal by a zero-orderhold (ZOH) function. In fact, sampled-data systems are a type of hybrid system, where the plant is a continuous-time system while the controller is discrete-time system. The main goal of the sampled-data controller design criterion is to obtain the maximum permissible sampling interval to keep the sampled-data system stable. This fact follows due to at higher sampled intervals, the efficiency in the system is increased, as for example, bandwidth resources are economised and relaxed the communication capacity between the controller and the system. Three main approaches have been used to obtain the design criteria for sampled-data controllers: a discrete-time, a input delay, and an impulsive system approach, among which input delay approach [17], is usually employed due to it can be used in systems with nonuniform sampling intervals. Based on the input-delay approach, many strategies have been proposed in the literature in order to reduce the conservatism of the design conditions, as in [16] where was proposed a time-dependent Lyapunov functional, later based on Wirtinger's inequality a discontinuous term in the Lyapunov functional was introduced in [18] considerably improving the results achieved until then. Another strategy is the use of looped-functional-based method [19], [20], which has the benefit of not needing the Lyapunov functional to be positive within the sampling intervals and can be improved by taking into account entire information of the intervals x(t k ) to x(t) and x(t) to x(t k+1 ), [21], [22], [23], [24], [25]. To further obtain improved stability conditions of sampled-data control systems, in [26] was introduced a free-matrix-based timedependent discontinuous Lyapunov functional which is based on free-matrix-approach in time delay systems [27], but modifies the term β α x(s) ds instead of 1 β−α β α x(s) ds, applying to to sampled data systems by the convex combination technique. Among other recent approaches to address the design of sampled-data systems include, by transforming them into discrete-time switched polytopic system [28], by use the triple integral terms in the Lyapunov functional [29], or by include a delay term in the signal successfully transmitted from the sampler to the controller and to the ZOH, obtaining a memory sampled-data controller [30].
From the above techniques, the Lyapunov functional was derived from constant symmetric matrices, which can be improved using fuzzy dependent matrices instead of constant symmetric matrices. However, the main difficulty of using a fuzzy dependent Lyapunov functional is to deal with the combination of the time derivative of FMFs with the symmetric matrices, which can not be introduced directly in the conditions given in the LMIs. Focused on that issue, in [39], was proposed a new method, by using the information of the time derivative of FMFs at each instant of time, a switching idea was applied to ensure the time derivative of the Lyapunov functional is negative. Therefore, based on [39], a fuzzy dependent Lyapunov functional was used in other recent works [21], [23], [31], [32], [33], [34], obtaining less conservative conditions. However, three principal disadvantages of this method can be mentioned; it is necessary to assume that the derivative of FMFs can be measured, which in some systems can be complicated; the number of possible switching cases increase exponentially with respect to the rules obtained, which can be large for systems with many rules; and the method requires that all the terms of the combination of the time derivatives of FMFs with symmetric matrices be negative, which leads to conservatism in the LMIs.
On the other hand, switched control based on T-S fuzzy model is another research field, which involves a state dependent switching function leading the switching of controllers. As for example, in [6] was introduce a type of switched control which focused on T-S fuzzy systems with parametric uncertainties in FMFs, which achieves the stabilisation of the system without the need to measure the FMFs in the controller. Furthermore, in [8] and [10], was included the minimization of the H ∞ norm in the stability conditions, and in [7] was addressed the problem of chattering and sliding mode in the switching of the controllers.
Motivated by the above discussions, in this paper, we consider the stabilization problem with sampleddata control for fuzzy systems, using a fuzzy dependent Lyapunov-Krasovskii functional and a switching function proposed in [6]. Briefly, the main contributions of this work are summarized as follows: • In order to use a fuzzy Lyapunov functional in the controller design conditions, a new method to deal with time derivatives of FMFs is proposed.
• The proposed method does not need to obtain the derivative of FMFs in each instant of time and only requires to know its maximum value.
• Based on a switching function, a switched sampled-data controller is proposed, which does not require to obtain the membership functions, being suitable for systems with parametric uncertainties. Finally, the mentioned contributions are verified in two simulation examples.
Notations: The notation used throughout is standard. R n is the n-dimensional Euclidean space and R n×m denotes the sets of real matrices with n × m dimension. K r represents the set {1, 2, . . . , r}, ∀r ∈ N. For a matrix X ∈ R m×n , X T means its transpose. The identity and zero matrices are represented by I and 0 with appropriate dimensions, respectively. For any symmetric matrix, X > 0(X ≥ 0) denotes a positive (semi) definite matrix, ( * ) represents the elements below the main diagonal, and Sym{X } is defined as X + X T . diag{B7 B7 B7} denotes block diagonal matrix. Finally, r ∈ R r is considered as a general convex set with dimension r, i.e.

II. PRELIMINARIES AND PROBLEM STATEMENT
Consider the following Takagi-Sugeno fuzzy model: where M i 1 , M i 2 , . . . , M i p are the fuzzy sets of the rule i, i ∈ K r ; x(t) ∈ R n is the state vector and u(t) ∈ R m is the control input vector; A i ∈ R n×n and B i ∈ R n×m are known constant matrices; z 1 (t), z 2 (t) , . . . , z p (t) are called premise variables, that are smooth nonlinear functions available in VOLUME 11, 2023 real time, whose values are used to determine which Rule i is active at time t. By construction, z p (t) can depend on state variables, external disturbances, and/or time [2]. Hence, applying the center average defuzzifier, product interferences and singleton fuzzifier, the form of T-S fuzzy model is given below, more details can be found in [1].
where α i (z(t)) are the weights of each local model i defined as , and satisfies Similar to the fuzzy model (1), the fuzzy sampled-data controller is designed as follows: where K j ∈ R m×n is the control gain matrix to be determined and the state vector x(t) is assumed to be generated by a ZOH function at the sampling instant t k , satisfying: In order to work with the time delay approach [17], we consider the function τ (t) = t − t k , such that from (5) satisfies: where τ denotes the upper bound of the time interval length between any two sampling instants. Thus, the fuzzy sampled data controller is inferred by (8) and the closed-loop system T-S fuzzy sampled-data system is obtaineḋ A. SWITCHED SAMPLED-DATA CONTROLLER In the sampled-data control, the states x(t) are feedback only at sampling instants t k , hence, we have only the information of membership functions α j (z(t k )) for the controller design. A important disadvantage occurs when the premise variables z(t k ) include complex nonlinear terms or parametric uncertainties that are difficult to know. Since α j (z(t k )) depends on the z(t k ), it leads to complexity in obtaining the membership functions and then increases the difficulty in the implementation of the controller. Focused on that problem, in [6] was proposed a switched controller that is designed for T-S fuzzy systems, and does not require the computation of membership functions, by using the switched controller defined as follows: whereQ j ∈ R n×n is a symmetric matrix and is designed to obtain the index σ (t) ∈ K r and arg min j∈K r {x T (t)Q j x(t)} returns the index j at which the function x T (t)Q j x(t) assumes its minimum value at instant t. The control gain K σ ∈ R m×n belongs to a set of linear controllers {K 1 , . . . , K r }, which are switching leaded by the function σ (t), and are obtained by solving optimisation problems formulated by LMIs. Substituting (10) in (2), gives the following closed-loop system: Therefore, in order to obtain the LMIs that guarantee the stability of the system using the switching function (11), the following property of convex combination is used: where α i (t) ∈ r is the membership function and i, j ∈ K r . In the literature of switched control, the function σ (t) is designed to lead the switching of a set of gains. However, the approach proposed in this paper uses the definition (11) to deal with the time derivative of FMFs in the Lyapunov functional candidate, independently of the controller chosen. Thus, in the case where a switched controller is not used (Example 1), inequality (13) will be satisfied ∀t ≥ 0, but when the sampled-data controller (10) is implemented (Example 2) we will only have the information of the states at sampling instants t k , and the state dependent switching function will be: as shown in Figure 1. Therefore, (13) will be satisfied only at the time instants t k .
Remark 1: Different from the Lyapunov functions for systems without sampled-data control discussed in [6], the Lyapunov-Krasovskii functionals include several state-dependent vectors such as: x(t),ẋ(t), x(t k ), etc. For each of them, a different switching function can be obtained, so it is important to guarantee a unique function σ (t) that satisfies (13) for all vectors.

B. DISCUSSION OF THE TIME DERIVATIVE OF MEMBERSHIP FUNCTIONS
The use of FMFs in the Lyapunov function can considerably reduces the conservatism of the LMIs, due to FMFs includes membership informations on the Lyapunov function candidate which reduces the conservatism of the stability conditions in comparison to quadratic Lyapunov functions. However, it generates the difficulty of introducing the time derivative of the FMFs in the stability conditions. For instance, let the Lyapunov function as follows: where, for i ∈ K r , P i is a positive definite matrix and α i (t) are the FMFs. Considering the following notation P α = r i=1 α i (t)P i for the remainder of the paper, the time derivative of (15) will be: whereṖ α = r i=1α i (t)P i . In the recent literature on sampled-data control [21], [23], [31], [32], [33], [34], switched rules have been introduced to guarantee thatṖ α ≤ 0 hold. Due the property of the membership functions, r i=1α i (t) = 0, it yields that: Then by [39], it is proposed an IF-THEN approach to guar-anteeṖ α ≤ 0: From ( Thus, the IF-THEN rules in (18) can be rewritten in a more compact form as follows: Based on the above discussions, some remarks are important to mention. If is required to satisfy additional conditions asQ α ≤ 0, the number of rules remains 2 r−1 but is necessary to include the conditions Q l − Q r > 0 or Q l − Q r ≤ 0, in each IF-THEN rule. Moreover, in order to satisfiedṖ α ≤ 0, all the termsα 1 (t)(P 1 − P r ),α 2 (t)(P 2 − P r ),α 3 (t)(P 3 − P r ), . . . , are required to be negative which leads to a conservatism in conditions.
In addition, the above procedure can requires assuming that the continuous time derivative of α i (t) can be measured for t ∈ [t k , t k+1 ). However, when the system present complex nonlinearities, the time derivative can be difficult to obtain or could present some parametric uncertainties in α i (t), it leads to increased difficulty in implementation.

III. PRELIMINARY RESULTS
In this section, we present a method, which assists in including the termṖ α in the LMIs. Before proceeding further, is necessary to introduce the following assumption to derive the preliminaries results.
Assumption 1: [37] It is assumed that the continuous first-order derivative of α i (t) exist for t ∈ [t k , t k+1 ), and there are known scalars For the system (2), based on (4) follows: Let the set which contains r elements, and ρ = max i∈K r {ρ i }. It is easy to verify that due to Assumption 1 and (21), ϒ r satisfy (4). Therefore, we can apply the property (13) on the set ϒ r , obtaining where X j ∈ R n×n is a symmetric matrix, ς (t) ∈ R n is any time dependent continuous vector, and We define the following switching functions: Thus, since σ and σ are independent of i, multiplying (23) by α i (t) and adding for all i ∈ K r , gives VOLUME 11, 2023 substituting (24) in (27), we obtain Therefore, in order to guaranteeV (t) < 0, for x(t) ̸ = 0, it will be possible to replace the expression by the right or left hand side of (28), according to required in the inequalities. The above discussion was developed for a single matrix X i but can be extended for more matrices. Therefore, the following Lemma is proposed: Lemma 1: Let any s ∈ N and the scalar ρ that satisfies Thus, for any symmetric matrices X m j ∈ R n×n , j ∈ K r , m ∈ K s , and functions α i (t) ∈ R that satisfies (4), the following inequality is satisfied: where σ ∈ K r represents the switching function: (30) and Therefore, by the right hand side of the inequality (28) and the switching function (30), it follows: Substituting (31), as shown in the equation at the bottom of the page.
In Lemma 1, is generalised the previous result, for the sum of several fuzzy dependent matrices, which is usually found in the design conditions of the sampled-data controller using the input delay approach. In addition, the switching function (30) is unique for all matrices X s i . Remark 2: Since the matrices X m j only require to be sym- (29), from the left side to (28) follows: As is known that just replace with (ρX m i − rρX m σ ) a m-term on the right-hand side of (29) and change the sign of X m j in the diagonal block matrix X j .
Remark 3: In Lemma 1, the conditions does not focus on ensuring s m=1 ς T m (t)Ṗ m α ς m (t) ≤ 0 orṖ m α ≤ 0, which can be a restrictive condition. In fact, the expression of the right side of (29) is always positive, for example due to the property (13), yields: Therefore, Lemma 1 introduces a positive term in the conditions, which is increased proportionally to the value of ρ as a consequence of using a fuzzy depended Lyapunov functional. However, this restriction does not require to know the time derivative of FMFs in order to lead with fuzzy depended matrices, and is less conservative than used in [21], [22], [23], [32], [33], and [34], as will be seen later.

VOLUME 11, 2023
Proof: Consider the following Lyapunov-Krasovskii functional: where and lim t→ t k V (t) = V 1 (t k ). Thus, the Lyapunov-Krasovskii functional V (t) is continuous.
Taking the first time derivative of V (t), along the trajectory of the system (9) yields: Since Q 11 > 0, by Schur complement is obtained which mean that for any matrix J ∈ R n×6n , is obtained where F 1 = [F 11 , F 12 , F 13 ]. Therefore, for any matrix M > 0, based on Schur complement, we have which means that for a matrix F 1 , is satisfied Thus, is obtained Now, we discuss about fuzzy dependent matrix P α , R α and S α . By summing the fuzzy dependent terms in (38) and (41) yields: Therefore, applying the Lemma 1 is obtained where the functions σ 1 (t) and σ 2 (t) are defined by where the switching functions σ 1 (t), σ 2 (t) ∈ {1, 2, . . . , r}. Then, from the system (9), we have Then, from (38)-(47), and replacing σ 1 (t) and σ 2 (t) by l 1 and l 2 respectively, it obtainṡ Finally, according to the convex combination technique, is equivalent to (τ k ,0) < 0 and (τ k ,τ k ) < 0. Thus, by Schur complement, one has from (35) and (36) thatV In order to prove that 3,4) and P i > 0, follows: which means that V ℓ (t), ℓ = (2, 3, 4) are not be positive definite on the sampling intervals, and V (t) is only required to be positive definite at sampling times.
Remark 4: In Theorem 1, a Lyapunov functional dependent on the FMFs was constructed. In contrast to most of the existing work on the sampled-data control [21], [23], [31], [32], [33], [34], [40], in this case only the Assumption 1 is required, and it is not necessary to obtain the time derivatives of the FMFs at each instant time.
Remark 5: Moreover, the functions σ 1 (t) and σ 2 (t) do not need to be considered as switching functions. In fact, since σ 1 (t), σ 1 (t) ∈ K r , just by replacing by index terms l 1 and l 2 respectively is obtain stability conditions of the system.
Based on Theorem 1, the sampled-data controllers design method for system (9) is provided by Theorem 2.
Theorem 2: Let scalars τ > 0, υ 1 > 0, υ 2 > 0, ρ > 0 that satisfies |α i (t)| ≤ ρ, ∀i ∈ K r . The T-S fuzzy system (9) is asymptotically stabilized by the controller (8), ∀τ k ∈ (0, τ ], if there exist positive definite matrices P i ∈ R n×n , Q 11 ∈ R n×n , M ∈ R n×n , symmetric matrices and any matrices J , G with appropriate dimensions, such that the following inequalities hold, ∀i, j, l 1 , where the controller gain matrix K j = K j G −1 , and Proof: Define Therefore, multiplying on the left and right side of (49) by diag{G T , G T , G T , G T , G T , G T , G T } and diag{G, G, G, G, G, G, G} respectivly, and similar to the prove of Theorem 1, applying the Schur complement we obtain (51) and (52).
Remark 6: In Theorem 2 the scalars υ 1 and υ 2 are introduced to conditions to provide more degrees of freedom in the LMIs. For every value of υ 1 and υ 2 we can obtain a different VOLUME 11, 2023 maximum value of τ . Focus on this discussion, a grid search Algorithm [21], [32], [33], [40], is used to obtain the highest value of τ in a bounded interval. Therefore, for each value of υ 1 , υ 2 , and τ , (51) and (52) become LMIs, that are easy to solve with any LMIs toolbox. Now, in order to obtain a switched controller which does not depend on FMFs in the controller implementation, we will obtain the stability conditions for the system (2) using the controller (10) as shown in Figure 1.
In Theorem 1 and Theorem 2 the fuzzy dependent matrices P α , R α , and S α were considered. However, this leads to two switching functions, which increases the number of controllers to be switched. Therefore, in order to reduce the number of controllers, matrices R α and S α will be considered as constant matrices.
Moreover, as mentioned previously, the switching function will be (14) and the inequality (13) will only be satisfied at sampling instants. Therefore, the expression x T (t)Ṗ α x(t) can not be reduced by using the Lemma 1, and matrix P α will be considered as constant. However, will be added to the Lyapunov functional the following expression: where X i are positive definite matrices and the Lyapunov functional given in (37) reduces to the following: Corollary 1: Let scalars τ > 0, υ 1 > 0, υ 2 > 0, ρ > 0 that satisfies |α i (t)| ≤ ρ, ∀i ∈ K r , the T-S fuzzy system (2) is asymptotically stabilized by the switching controller (10) and the switching function σ (t) = arg max j∈K r {x T (t)X j x(t)}, ∀τ k ∈ (0, τ ], if there exist positive definite matrices X j ∈ R n×n , P ∈ R n×n , Q 11 ∈ R n×n , M ∈ R n×n , symmetric matrices and any matrices J , G with appropriate dimensions, such that the following inequalities hold, ∀i, where the controller gain matrix K j = K j G −1 , and Proof: Since (54), the values ofṖ α ,Ṙ α andṠ α are null. Then in order to lead with the fuzzy dependent matrix X α , applying the Lemma 1, yields: where Following the same procedure as in the proof of Theorem 1, and replacing σ by j we obtaiṅ where (τ k ,τ (t)) = (τ k ,τ (t)) + τ (t)J T Q −1 Therefore, multiplying on the left and right side of (58) by diag{G T , G T , G T , G T , G T , G T , G T } and diag{G, G, G, G, G, G, G} respectively, applying the Schur complement, and taking the same definitions of the proof of Theorem 2, we obtain (55) and (56).
Based on Lemma 1 a design conditions of switched sampled-data controller was obtained, which does not require to feedback the FMFs to implement the controller.
Remark 7: Procedures to lead with controllers (8) and (10) are similar, due to from (4) the switched controller can also be expressed as , where the function σ (t) is replaced by j because not depend of i.

V. SIMULATION EXAMPLES
In this section, two simulation examples are given to show the effectiveness and less conservatism of our results.

A. EXAMPLE 1
In this subsection is presented the chaotic Rossler's system with an input signal [41]: where

t)] T is the state vector and u(t)
is the input control. In order to obtain the T-S fuzzy model, let following the fuzzy rules: with FMFs as α 1 (t) = c+d−x 1 (t) 2d and α 2 (t) = 1 − α 1 (t). For this example the parameters values will be a = 0.3, b = 0.5, c = 5, d = 10, and the initial conditions For u(t) ≡ 0, the time derivative of x 1 (t) and the state variables x(t) are shown in the Figure 2, where the lower and upper bound of |ẋ(t)| ≤ 6. Thus, in order to satisfy the Assumption 1, we can choose ρ = 0.3. Moreover, solving the LMIs conditions of Theorem 2 for υ 1 = 3.2 and υ 2 = 21.7, we obtain τ = 0.1412 s as the upper bound of the time interval length, and the following gains:   The state variables of the system and the input control using the obtained gains are shown in the Fig. 3, where it is verified the asymptotic stability of the system, and we can see that the system achieves the stability in a time of approximately 50 seconds. In addition, in the Table 1 it is shown a comparison of the maximum sampled time obtained with respect to [22] and [33], which use the rule (20) to deal with the fuzzy dependent matrices.

B. EXAMPLE 2
In this case, we consider the inverted pendulum [37], [38], which the model is shown in Fig. 4 and the dynamic model is given by (61), as shown at the bottom of the page, where x 1 (t) represent the angle of the pendulum from vertical and x 2 (t) is the angular velocity. Therefore the space state model x 2 (t) = g sin(x 1 (t)) − amlx 2 2 (t)sin(2x 1 (t))/2 − a cos(x 1 (t))u(t) 4l/3 − aml cos 2 (x 1 (t)) (61) VOLUME 11, 2023 with f 1 (t) = g sin(x 1 (t)) − amlx 2 2 (t) sin(2x 1 (t))/2 4l/3x 1 (t) − aml cos 2 (x 1 (t))x 1 (t) , and f 2 (t) = a cos(x 1 (t))u(t) aml cos 2 (x 1 (t)) − 4l/3 , as two non linear terms. Therefore, by achieving the T-S fuzzy model using the Taniguchi's method [2], it could generate FMFs with complex expressions, which generate a difficulty in the implementation of the fuzzy controller. For this reason, in most of the literature, the system is reduced to only two fuzzy rules considering that x 1 (t) is close to ± π 2 or 0, obtaining two membership functions of triangular shape. Focusing on this issue, a switched controller that does not require to obtain the FMFs for the implementation will be designed using Corollary 1.
Choose M = 8 kg, m = 2 kg, l = 0.5 m, a = 1 (m+M ) , g = 9.8 m/s 2 , and β =cos(88 • ). We consider x 1 (t) ∈ [− π 2 , π 2 ] and applying the T-S fuzzy model as [37] and [38], system (61) can be rewritten as where and α 2 (t) = 1 − α 1 (t). As in [37] and [38], choose |x 2 (t)| ≤ 10 rad/sec. Then, we obtain |α i (t)| ≤ ρ = 20/π. Solving the LMI's of the Corollary 1 for υ 1 = 2.3 and υ 2 = 23.7, we obtain τ = 0.1445 s as the upper bound of the time interval length, and the following gains The maximum sampled interval obtained is greater than those obtained in [37] and [38], which confirms the less conservatism of the conditions. By use the control strategy shown in the Fig. 1, the state variables of the system and the input control are shown in the Fig. 5, where the system achieves the stability in approximately 5 seconds. Moreover, in the Fig. 6 is shown the switched index σ (t), which change the value between 1 and 2, achieving the stability of system.

VI. CONCLUSION
Relaxed design conditions of sampled-data controller using fuzzy dependent Lyapunov functional was proposed in this work. By using the convex combination inequality, we derive new stability conditions to deal with the derivatives of the FMFs, which does not require to obtain the derivative of the membership functions to guarantee the stability of the system. Moreover, the convex combination property allows us to design a switched controller, that not depend of the FMFs for implemented the controller. The design conditions obtained are applied to two classic examples in the literature, verifying the improvement of the proposed strategy. Finally, this paper introduces a switched control approach based on state-dependent switching functions of type, argmin and argmax functions; expanding the possibilities of taking the benefits of switched systems that have been widely studied in the literature. Moreover, the Lemma 1 is not limited to being applied only in sampled-data control, but in several applications which involve Lyapunov functions dependent on FMFs. He is currently an Associate Professor in the Computer Engineering Course Coordination with Federal University of Technology-Paraná (COENC-UTFPR-AP). His research interests include control theory and its applications, fuzzy control using TS models, and LMI-based design for optimal and robust control. VOLUME 11, 2023