Stability Analysis and Robust Stabilization of Uncertain Fuzzy Time-Delay Systems

: New sufﬁcient conditions for delay-independent and delay-dependent robust stability of uncertain fuzzy time-delay systems based on uncertain fuzzy Takagi-Sugeno (T-S) models are presented by using the properties of matrix and norm measurements. Further sufﬁcient conditions are formulated, in terms of the linear matrix inequalities (LMIs) of robust stabilization, and are developed via the technique of parallel distributed compensation (PDC), and then the simpliﬁcation of the conditions for the controller design of uncertain fuzzy time-delay systems. The proposed methods are simple and effective. Some examples below are presented to illustrate our results.


Introduction
The stability analysis and control design of a nonlinear system is a difficult task. In fact, the controlled object of the control system is always nonlinear. Sometimes it is difficult to use linearization or recognition techniques to describe complex nonlinear systems. Recently, the so-called fuzzy model [1,2], which uses fuzzy implication to express the local linear input/output relationship of nonlinear systems, is used to approximate nonlinear objects. The study conducted by [1] used fuzzy logic to model two effective factors that affect melatonin regarding Major Depressive Disorder (MDD). The Fuzzy Logic Direct Model Reference Adaptive Control (DMRAC) is the choice of insulin infusion control for Type I diabetes, as shown in [2]. Takagi-Sugeno (TS)-type fuzzy is a special case of fuzzy systems, where Takagi-Sugeno (TS)-type fuzzy controllers have been fully applied to stabilize controller design. In addition, based on the parallel distributed compensation (PDC) technology, the control design of this type of fuzzy control system is derived according to [3][4][5][6][7][8][9][10][11][12][13]. A new, observer-based fuzzy controller design method with a time-delay adaptability is proposed in the study conducted by [3]. The designed controller contains both current and past state information of the system and can be derived by solving a set of linear matrix inequalities (LMI). In the study put forth by [4], the so-called fuzzy Lyapunov function (i.e., multiple Lyapunov function) is used to solve the stability analysis and stability problems of the Takagi-Sugeno fuzzy system. In order to make full use of the fuzzy Lyapunov function, we propose a new parallel distributed compensation (PDC) scheme, which feeds back the time derivative of the premise membership function. The study conducted by [5] considers the observer design for a class of nonlinear systems with unknown inputs. The Parallel Distributed Observer (PDO), composed of a local linear observer and an appropriate level of membership function, is a conventional observer of the Takagi-Sugeno fuzzy bilinear system. The study conducted by [6] used the Takagi-Sugeno (TS) fuzzy model to design a controller and assuming that uncertainty affects system behavior through state author of [19] proposes an adaptive fuzzy hierarchical sliding mode control method for a class of multi-input, multi-output unknown nonlinear time-delay systems with input saturation. Based on the Takagi-Sugeno fuzzy model method, an output feedback control strategy is developed for a class of continuous-time nonlinear time-delay systems through an improved repetitive control scheme [20]. The author of [21] studies the sampling data stabilization problem of Takagi-Sugeno (T-S) fuzzy system with time delay. By considering the state information in the interval from t k to t and t to t k+1 , a new two-sided delay-related cyclic functional is introduced, which can not only relax the monotonic constraint (LKF) of the Lyapunov-Krasovskii functional, but also make better use of the actual sampling mode.
In this paper, based on the properties of matrix and norm measurements, we first give sufficient conditions for the robust stability test of uncertain fuzzy time-delay systems, the conditions of test results include delay-independent and delay-dependent. It needs only to solve r (the number of rules of the fuzzy model) inequality, not to find a common positive definite matrix P. It is a difficult task to find such a P, especially when the number of rules of the fuzzy model is large. Three numerical examples and one particular example are provided to illustrate the effectiveness of the proposed method. The following notations will be used throughout the paper: the identity matrix with dimension n is denoted by I n , λ(A) stands for the eigenvalues of matrix A, A denotes the norm of matrix A,

System Description and Problem Statement
Consider a time-delay system described by the following T-S fuzzy model. This T-S fuzzy time-delay system is composed of r rules that can be represented as follows: And f i (x(t)) is a bounded nonlinear perturbation vector satisfying where b i > 0. By using a center-average defuzzifer, product inference and singleton fuzzifier, the fuzzy time-delay system (1) can be inferred as . where For simplicity, we use the symbol h i to denote h i (θ(t)). First we will derive the delayindependent stability condition of an unforced system (1). For this objective, some helpful lemmas are given below.
Consider the time-delay system .
where x ∈ R n , A and A d are matrices in proper dimensions, τ is the delay duration.
Lemma 1 [22]. The stability of system (4) implies the stability for the following systems and vice versa. .
In light of Lemma 1, it is obvious that the stability of the fuzzy time-delay system (3) is equivalent to that of the system .
Lemma 2 [23]. For matrix A ∈ R n×n , B ∈ R n×n , the following relation holds: Furthermore, we assume following relation holds: This completes the proof.

A. Delay-Independent Robust Stability
Now, we present a delay-independent condition of robust stability for the fuzzy time-delay system (1). Theorem 1. The fuzzy time-delay system (1) will have robust stability with a delayindependence if η < 0 holds. Furthermore, the max perturbation bound b satisfies b < −η (8) Proof 2. The trajectory response of system (6) can be written as follows: Applying the norm on both sides of Equation (9) and in view of Lemma 1, we can obtain the following equation.
Mathematics 2021, 9, 2441 5 of 13 Then, Equation (10) is a solution of the following system (11): Since w(t) ≤ v(t) (from the comparison theorem), the exponential stability of v(t) implies that of w(t). Hence, the system (6) is globally exponentially stable, if (η + b) is negative. Furthermore, the system (1) is also stable due to the system (6) and (1) having the same stability properties. This completes the proof. Corollary 1. Consider the fuzzy system (1) without delay and perturbation is asymptotically stable, if the following condition is satisfied Remark 1. In view of Theorem 1, the stability criterion of fuzzy system (1)doesn not need to find a common positive definite matrix to satisfy any inequality.

B. Delay-Dependent Robust Stability
Next, we will derive a sufficient stability condition for fuzzy time-delay system (1) with delay-dependence. One can obtain the following equation: (13) Hence, the system (3) can be written as .
Prior to the examination of robust stability problem for system (1), one useful concept is given below.
In the following, a sufficient condition for ensuring the asymptotic stability of a fuzzy time-delay system is proposed. Theorem 2. Suppose δ + b < 0, then the fuzzy time-delay system (1) will have robust stability with delay-dependence, if is satisfied.
The trajectory response of system (14) can be written as follows Applying the norm on both sides of Equation (16), we obtain Then, the Equation (18) is a solution of the following system (19).
Since x(t) ≤ v(t) (from the comparison theorem), the exponentially stable of v(t) implies that of x(t). From Lemma 4, system (1) is asymptotically stable, if the following equation is satisfied: This completes the proof.

C. Controller Design
In the subsection, this paper investigates the problem of designing a fuzzy state feedback controller to stabilize the system (1). Based on the concept of PDC, we consider the following fuzzy control rules: Hence, the overall output of this fuzzy controller is By substituting (21) into (3), the global closed-loop fuzzy system becomes .
In light of Lemma 1, it is obvious that the stability of the fuzzy time-delay system (22) is equivalent to that of the system .
Furthermore, we assume η ij = µ A i − B i F j + A di and η = max 1≤i,j≤r η ij . Next, we shall discuss the robust stabilization problem of the system (1).

Theorem 3.
Suppose the state feedback gains F i satisfy the following conditions: For i, j = 1, 2, . . . , r, then the perturbed fuzzy time-delay system (1) can be stabilized asymptotically by the fuzzy controller (21). Furthermore, the tolerable perturbation bound b satisfies b < −η (25) Proof 4. The trajectory response of system (22) can be written as follows: Applying the norm on both sides of Equation (26) and in view of lemma 1, we can obtain the following equation.
Then, Equation (28) is a solution of the following system (29). .
Since w(t) ≤ v(t) (from the comparison theorem), the exponential stability of v(t) implies that of w(t). Hence, the system (26) is globally exponential stability, if (η + b) is negative, Furthermore, the system (1) is also stable due to the system (26) and (1) having the same as stability properties. This completes the proof.

Remark 2.
In fact, we can simplify state feedback gains F i to F, for i = 1, 2, . . . ,r, from the condition (24). Hence, we only need to consider the r fuzzy rules for the system (1) then we can find the controller gain F, (i.e., the controller (20) and condition (24) can be turned into the following controller and condition, respectively).
Corollary 3. If the tolerable perturbation bound b can be known, and the state feedback gains F, satisfy the following problem of LMIs.
For i = 1,2, . . . ,r, then the perturbed fuzzy time-delay system (1) can be stabilized asymptotically by the fuzzy controller (30). x Now we want to analyze the system's stability. Using Theorem 1, we obtain A i A dj = 4.5351 and σ 2 = max 1≤i,j≤2 A di A dj = 1.5326. In view of Theorem 2, the System (36) is robust stable, if the relationship between the delay and the tolerable perturbation bound b satisfy the following inequality The trajectories for states of the T-S fuzzy time-delay System (36) is shown in Figure 2 with The solid line is the state of x 1 and the dotted line is the state of x 2 .
Rule2 : IFx 2 (t)isM 2 (e.g.Big)THEN .  The membership function for 2 is as the same for Example 1. We get controller gain matrices satisfying (32) by using the LMI technique as follows:  The membership function for x 2 is as the same for Example 1. We get controller gain matrices satisfying (32) by using the LMI technique as follows: The trajectories for states of the T-S fuzzy time-delay System (37) is shown in Figure 3 with initial value x 1 (0) x 2 (0) T = 2 3 T and τ = 1, f 1 = f 2 = 0.9 sin x 2 1 + x 2 2 . The solid line is the state of x 1 and the dotted line is the state of x 2 .  Furthermore, if we choose the gain matrices F 1 = F 2 = −1.8598 0.5742 , then η ij = −1.1179 and the perturbation bound b < −η = 1.1179 can be obtained via Corollary 2.
The trajectories for states of the T-S fuzzy time-delay system (37) is shown in Figure 4 with initial value x 1 (0) x 2 (0) T = 2 3 T and τ = 1, f 1 = f 2 = 0.9 sin x 2 1 + x 2 2 . The solid line is the state of x 1 and the dotted line is the state of x 2 . The trajectories for states of the T-S fuzzy time-delay system (37) is shown in Figure  4 with initial value [x 1 (0) x 2 (0)] T = [2 3] T and = 1, 1 = 2 = 0.9 sin √ 1 2 + 2 2 . The solid line is the state of 1 and the dotted line is the state of 2 .   x(t) = −1.4274 0.0757 −1.4189 −0.9442 Rule 2 : If the temperature is moderate (i.e., x 2 (t) is about 2.7520) then .
x(t) = −4.5279 0.3167 −26.2228 0.9837 The membership function for x 2 are x 2 (t) is about 0.8862 : By using corollary 3 under the perturbation bound b < 0.5, we get controller gain matrices (33) as follows: The trajectories for states of the T-S fuzzy time-delay system (38) is shown in Figure 5 with initial value x 1 (0) x 2 (0) T = 2 3 T and τ = 1, f 1 = f 2 = 0.4 sin x 2 1 + x 2 2 . The solid line is the state of x 1 and the dotted line is the state of x 2 . By using corollary 3 under the perturbation bound b < 0.5, we get controller gain matrices (33) as follows: The trajectories for states of the T-S fuzzy time-delay system (38) is shown in Figure  5 with initial value [x 1 (0) x 2 (0)] T = [2 3] T and = 1, 1 = 2 = 0.4 sin √ 1 2 + 2 2 . The solid line is the state of 1 and the dotted line is the state of 2 .

Conclusions
Based on the uncertain fuzzy Takagi-Sugeno (T-S) model, using the properties and norm measurements of the matrix, new sufficient conditions for the delay-independent and delay-dependent robust stability of uncertain fuzzy time-delay systems are proposed. Further sufficient conditions for robust stabilization of linear matrix inequality (LMI) are developed through parallel distributed compensation (PDC) technology, and then the

Conclusions
Based on the uncertain fuzzy Takagi-Sugeno (T-S) model, using the properties and norm measurements of the matrix, new sufficient conditions for the delay-independent and delay-dependent robust stability of uncertain fuzzy time-delay systems are proposed. Further sufficient conditions for robust stabilization of linear matrix inequality (LMI) are developed through parallel distributed compensation (PDC) technology, and then the conditions for the design of controllers for uncertain fuzzy time-delay systems are simplified. It is only required to solve the inequality of r (the number of rules of the fuzzy model) instead of finding the public positive definite matrix P. Finding such P is a difficult task, especially when the number of rules in the fuzzy model is large.