New Results of Fuzzy Sampled-Data Control for Nonlinear Time-Delay Systems

In the manuscript, synthesis and stability analysis for fuzzy <inline-formula> <tex-math notation="LaTeX">$H_{2}/H_{\infty }$ </tex-math></inline-formula> sampled-data control of dynamical systems with delay behavior are studied by employing the input delay method. By using Lyapunov theory, a new <inline-formula> <tex-math notation="LaTeX">$H_{2}/H_{\infty }$ </tex-math></inline-formula> standard is derived and the fuzzy sampled-data controller is proposed to ensure <inline-formula> <tex-math notation="LaTeX">$H_{\infty }$ </tex-math></inline-formula> performance and <inline-formula> <tex-math notation="LaTeX">$H_{2}$ </tex-math></inline-formula> performance simultaneously. Meanwhile, the control design is verified by two practical examples. Furthermore, experimental results also indicate that the <inline-formula> <tex-math notation="LaTeX">$H_{2}/H_{\infty }$ </tex-math></inline-formula> sampled-data control has a better performance.


I. INTRODUCTION
Recently, fuzzy control method has always been a hot topic in the field of control and can be widely employed. Because of its logical reasoning abilities and superior approximation performance, T-S fuzzy model was adopted to capture the nonlinear dynamical system in [1]. Subsequently, fuzzy controllers were also designed to guarantee the system stability in [2]- [6]. Generally, in the field of control, the role of computer is to control the controlled plants as a digital controller. When the continuous-time measurement signal is processed by the digital computer, the measurement signal is sampled and quantized firstly. Next, the discrete-time signal is generated by a zero-order holder, and then transmitted back to the continuous-time control input signal.
By now, there have been a lot of reports on the synthesis and analysis of sampled-data control with fuzzy form [7]- [37]. Among these existing literatures, [7], [9], [12], [21], [23], [24], [28]- [31] had analyzed the stability of the system, [18], [19], [23], [32], [33] had carried on the stabilization, [8], [10], [13], [14], [16], [25], [27] had discussed the H ∞ control, [15], [21] had studied the H 2 GC control, [11], [17], [26] had discussed the tracking control, and [20], [22] had considered the filtering. Meanwhile, delay phenomena often occur in some engineering systems, such The associate editor coordinating the review of this manuscript and approving it for publication was M. Venkateshkumar . as manual control, aircraft stability, ship stability and nuclear reactor. Moreover, in practical systems, the phenomenon of time delay leads to oscillations and instability frequently. In [9], [10], [14], [16], [34], [35], some fuzzy sampleddata control schemes were designed for nonlinear dynamical systems with delay behavior. The control performance design was described clearly in [9]. In [10], the issue of reliable non-uniform H ∞ sampling fuzzy control was analyzed, in which the generalized model transformation and input delay method were employed. Meanwhile, H ∞ control was also investigated by utilizing Leibniz-Newton formula and Lyapunov-Krasovskii functional [14]. Furthermore, a distinctive delay-dependent stabilization standard was proposed in [16]. For T-S fuzzy systems with time delay and parametric uncertainties, a robust guaranteed cost sampled-data fuzzy control design method was proposed in [34]. In [35], the stabilization of a T-S fuzzy system with time delay was explored, in which the designed fuzzy controller of sampleddata contained both the current and delayed state information. In addition, if the H 2 performance and H ∞ performance can be optimized simultaneously, the control system will show a better performance. Nevertheless, fuzzy H 2 /H ∞ sampleddata control issue of dynamical systems with delay behavior has not been explored yet.
H 2 performance focuses on the system state and the system input. H ∞ performance is mainly reflected in the gain between the system state and the external disturbance.
However, H 2 control or H ∞ control can only achieve a single control performance. In the existing literatures, fuzzy sampled-data H 2 control and fuzzy sampled-data H ∞ control have been explored respectively. But, the dimension of LMIs is very large, which can bring about the conservatism of the results. Therefore, for some engineering systems, how to design a more convenient and easier to implement sampled-data controller with larger sampling interval is still an unsolved problem.
Based on the above researches, fuzzy H 2 /H ∞ sampleddata control issue is proposed for dynamical systems with delay behavior. A new H 2 /H ∞ criterion is constructed and formulated as the linear matrix inequalities (LMIs) form. The feasibility of sampled-data control scheme is tested by simulation results. Moreover, by considering both H ∞ performance and H 2 performance, the global optimization algorithm is achieved and a fuzzy H 2 /H ∞ sampled-data control scheme is firstly investigated for dynamical systems with delay behavior. The presented control algorithm is less conservative, which reduces the dimension of the LMIs and shortens the implementation time.
The main innovation points of this paper are embodied in several aspects.
(I) The proposed control algorithm is less conservative, where the dimension of the LMIs is simplified, which adds the existence of the feedback gains, reduces the computational complexity and lowers the implementation time. (II) The proposed method achieves a better performance, where fuzzy sampled-data controller has a larger sampling interval and shows a better attenuated level and faster state responses. Notations: Throughout the manuscript, the notation W > 0(< 0) means the positivity (negativity) W T represent the transpose of a matrix W ; and * describes the symmetric term of a symmetric matrix. It is assumed that the matrices are compatible.

II. PROBLEM FORMULATION
Analyze the T-S fuzzy model capturing the dynamical system with delay behavior: Rule i : IF σ 1 (t) is N i1 and · · · and σ p (t) is N ip , THEṄ where the state, the input, and the disturbance are expressed by x(t), u(t), and ω(t), respectively; the matricesÂ i ,B i , and A id are time-invariant; and time delay d is a constant.
The global system is deduced aṡ where ξ i (σ (t)) ≥ 0, i = 1, 2, . . . , M , and The j th rule of fuzzy controller is as follows: Rule j : IF σ 1 (t k ) is N j1 and · · · and σ p (t k ) is N jp ,THEN where G j represents the gain, and 0 < t k+1 − t k = h k ≤ h. Therefore, the global controller is shown as Let Obviously, the derivativė τ (t) = 1 for t = t k . By employing the input delay method, (3) is rewritten as By considering the system (2) and the controller (4), the closed-loop system is obtained aṡ Let us take into account the following H ∞ control performance where ε denotes a attenuation level and the matrix S 1 is positive-definite.
The H 2 control performance is as follows: where S 2 and Q are positive-definite matrices. By analyzing a desired H ∞ disturbance rejection constraint in (6), the suboptimal H 2 control performance (7) is obtained. Our control objective aims to explore a fuzzy sampled-data controller to ensure the H 2 /H ∞ performance for the closedloop system (5).

III. FUZZY H2/H ∞ SAMPLED-DATA CONTROL
Based on the LMIs, a fuzzy sampled-data H 2 /H ∞ control criterion is proposed as follows.
Theorem 1: Considering the closed-loop system (5), for given matrices S 1 > 0, S 2 > 0, and Q > 0, scalars ε > 0, h > 0, and λ > 0, the H 2 /H ∞ control performance in (6) and (7) is satisfied, simultaneously, if there exist matrices Q 1 > 0 and Q 2 > 0 satisfying the LMIs (8) and (9) where And, the control gains in sampled-data controller are Proof: Selecting the Lyapunov-Krasovskii functional as the candidate where The derivative of V is as follows: By applying Lemma in [38], we can obtain By employing (13) and Leibniz-Newton formula, we geṫ It should be clarified that for a given µ > 0, we have By utilizing (11)-(13) and (15)-(16), we can acquire thaṫ wherẽ with Left-and right-multiplying where By adopting the Schur complement in (8), we haveĈ ij < 0. Therefore, C ij < 0. Substituting C ij < −S into (17), one haṡ Owing Integrating both sides of (22), there is Remark 1: Eq. (8) provides a new relaxed stability condition for the system (2). Unlike the existing works, in the proof, eq.(16) is introduced to consider the fuzzy relationship betweenẋ(t), x(t), x(t − τ (t)) and x(t − d). By using of the LMIs, it is easy to determine the feedback gains. Now, we continue to consider the H 2 control performance for the closed-loop system (5) in the absence of ω(t).
The derivative of V in the absence of ω(t) is as follows: By using (15), It is noted that for a given µ > 0, one has From (24)- (27), we can obtain thaṫ

Left-and right-multiplying
Applying the Schur complement to (9), there isˆ ij < 0. And, we have ij < 0 in (29). Thus, as for (28), we havė which implies thatV (x t ) < 0. Integrating (31) from t = 0 to t = ∞, there is Due to V (x t (∞)) = 0, one has The design of this paper is formulated as the following optimization problem.
Theorem 2: Considering the fuzzy closed-loop system (5), if the following problem (8), (9), and U I * W > 0 has a solution G j , j = 1, · · ·, M and W , then a whole optimal H 2 /H ∞ control performance is achieved. The control gains are G j =Ḡ jW −1 , j = 1, 2, · · ·, M . Remark 2: Fuzzy H 2 /H ∞ sampled-data control is firstly discussed for time-delay systems. In practical engineering systems, the H 2 /H ∞ sampled-data control is more appealing in achieving the desired control performance. And, the focus of H ∞ performance is mainly on the gain between the system state and the external disturbance. Meanwhile, both the system state and the system input are considered in H 2 performance. Furthermore, H 2 /H ∞ control synthesizes the merits of H ∞ control and H 2 control.
Remark 3: The proposed control algorithm simplifies the dimension of the LMI, reduces the computational complexity, and is less conservative. At the same time, it increases the existence of the feedback gain and shortens the implementation time. The designed method has better performance, in which the fuzzy sampled-data controller has a larger sampling interval, a faster state response and a better attenuation level.

IV. SIMULATION EXAMPLES
The proposed design is demonstrated by two practical dynamical systems with delay. In addition, the superiority is obvious in the obtained simulation results.
Example 1: The truck-trailer system in [39] iṡ The truck-trailer system is represented aṡ By employing the methods of [22], [10], and Theorem 1, a lower dimension in the LMIs is obtained in this paper, see, Table 1. That is to say, the computation burden of this paper is smaller than that of the methods in [22] and [10].
Using the methods of [10] and Theorem 1, a larger sampling interval is obtained, see, Table 2.   Experimental results are depicted in FIGUREs. 1-4. The system stability is confirmed and the piecewise continuous behavior of the control input is also plotted.
When h = 0.432, we find that this paper shows faster state responses than those of the H ∞ method in [22].
That is to say, the computation burden in this paper is smaller than that of the methods in [22] and [10].
When h = 0.1, there is ε min = 0.0497 under Theorem 1. The minimum disturbance attenuation is 0.0497, which is ineffective in the LMI conditions of H ∞ control [22]. This implies that the proposed fuzzy H 2 /H ∞ sampled-data control method is superior to the existing H ∞ control method.
When ε = 1.0, Theorem 1 gives h max = 0.222. The maximum sampling interval is 0.222, which is infeasible in the LMI conditions of H 2 control [10]. This implies that the proposed fuzzy H 2 /H ∞ sampled-data control method is superior to the existing H 2 control method.

V. CONCLUSION
Fuzzy H 2 /H ∞ sampled-data control for dynamical systems with delay behavior is addressed by using the input delay method. Based on Lyapunov theory, a fuzzy sampled-data controller is proposed, which can guarantee the H ∞ performance and H 2 performance concurrently. Moreover, a new H 2 /H ∞ standard is derived. Both stability analysis and simulation results demonstrate the proposed design and the superiority have also been verified through comparative analysis. In the future, this method will be applied to uncertain dynamical systems and provide theoretical support for further research. Furthermore, we will pay attention to more developments on the sampled-data control, the adaptive neural control, the adaptive fuzzy tracking control, the event-triggered scheme and the stochastic control in the literatures.