Synchronization Control Design Based On Observers For Time-delay Lur’e Systems

This paper researches synchronization control design for chaotic Lur’e systems (CLSs) with time-varying delay. Considering of the unmeasurable state variables, Luenberger-type observers are designed for the master-slave systems. The synchronization has been achieved by using the proposed controller with the estimated system states. Based on Bessel-Legendre inequality, improved reciprocally convex combination approach and a novel double integral inequality, new Lyapunov Krasovskii functionals(LKFs) are tailored and a synchronization control criterion with less conservatism is obtained. Some simulations are given to verify that this control strategy we proposed is effective.


I. INTRODUCTION
As is well known, chaos synchronization is a hot topic in recent years and receives much attention for the reason of its wide applicability in many scientific areas such as neural network [1]- [5], biomedical [6] and spacecraft [7]. Chaos demonstrates the oscillatory behavior which is unavoidable in dynamic nonlinear systems and shows a great sensitivity to initial conditions. A great deal of control methods have been developed to restrain chaos [8], [9]. Chaos synchronization could be seen as a special case of chaos control. The driving-response synchronization is by employing control strategies to make the response system gradually synchronize to the driving system. In the past few decades, a variety of approaches for chaos synchronization have been proposed. For example, adaptive control [1], [10], sliding mode control [11], adaptive fuzzy control [12].
It is noteworthy that large amounts of chaotic nonlinear systems can be clearly modeled into Lur'e system which consists of a linear system and a nonlinear term that satisfies a certain sector condition, for example, n-scroll attractor, networked control system and Chua's circuit [13]- [15]. Therefore, synchronization problems of CLSs have attracted increasing attention in recent years and The associate editor coordinating the review of this manuscript and approving it for publication was Zhongyang Fei . amounts of master-slave synchronization strategies have been explored. Reference [16] studies continuous time feedback control with time-varying delay in chaotic Lur'e system. References [17], [18] researches sampled-data synchronization, compared with the continuous time control, sampleddata control only requires the samples of the system states at discrete time instants. The synchronization of Lur'e networks with proportional delay which is a sort of unbounded time-varying delay has been concerned in [19]. The butterfly hysteresis nonlinearities are considered in [20] to make Lur'e system with less conservatism.
It is of significance to declare that in the above works which refer to Lur'e system, they are all depended on the condition that all the system states are measurable. As is well known, as a complex chaotic system, it's difficult to completely capture the state information of Lur'e system. Observers are well used to obtain the state estimates in the systems with unmeasurable states [21], [22]. From this point of view, applying observers to CLSs would be helpful to reduce the conservatism. From the existing results, the observer design for a single Lur'e system with unmeasurable system states has been focused. For example, in [23], a state observer is designed for the Lur'e singularly perturbed system. Based on the observer, a feedback control law is obtain to make the system stable. In [24], an H ∞ observer is developed for the singular Lur'e system in which the nonlinear functions satisfy one-sided Lipschitz condition.
However, the synchronization control strategy for masterslave Lur'e systems with unmeasurable system states has not been well concerned. The coupling between the state estimation errors and the synchronization error leads to the difficulty of calculating the observer gain matrices and the controller gain matrix. How to decouple is to be solved. Besides, how to make the LKFs contain more system information to reduce the conservatism deserves consideration. Moreover, the improved reciprocally convex inequality is mainly to extend the upper bound of the time-varying delay as well as reduce the computational complexity, it has been verified in linear systems, but its application to nonlinear systems like CLSs has not been well concerned so far.
Motivated by the discussions above, synchronization control design for CLSs with time-varying delay is explored in this paper. The main contributions are: 1. Compared with the existing literature that directly applies the precise state information of the Lur'e system for control design, we mainly research the observer-based synchronization control design for the master-slave systems with unmeasurable state variables to achieve state estimation. Apply estimated state instead of precise state to realize corresponding control design, which is more applicable.
2. A feedback control law is proposed for synchronization of the master-slave CLSs.
3. Based on Bessel-Legendre inequality and a double integral inequality, the LKFs which involve more system information are constructed. In order to further reduce the conservatism, the improved reciprocally convex approach is employed. The lager upper bound for time-delay and better performance of controller are obtained. 4. A decoupling approach which can be used to deal with the cross terms while calculating the observer and controller gain matrices is introduced.
In this strategy, a new control scheme is proposed for CLSs. The synchronization error and state estimation errors asymptotically converge to the origin by utilizing the observers and controller designed in this paper. Some simulations are given to show that this control approach can make the system with time-varying delay robust.

A. MAIN STRUCTURE
Section 2 is the system formulation and preliminaries. The main results are in Section 3. Section 4 and 5 express the simulation and conclusion respectively.

B. NOTATIONS
Standard notations through this paper are shown as: I is the identity matrix which has appropriate dimensions; R n×m and R n respectively represent the n × m dimensional matrices and n-dimensional Euclidean space; The matrix transposition of R is given by R T ; If matrix U is real symmetric positive definite, we define that U > 0(U ≥ 0), for U ∈ R n×n ; diag{l 1 , . . . , l n } means block diagonal matrix where l r (r = 1, . . . , n) are the diagonal elements, Sym{A} = A+A T

II. PROBLEM FORMULATION AND PRELIMINARIES
The CLSs with feedback control is given as: where x m (t) represents the state vector for ζ which is the master system, x s (t) represents the state vector for ς which is the slave system and u(t) indicates the control input. The output vectors are denoted as y m (t), y s (t). h(t) expresses the timevarying delay. R, L, V , T with appropriate dimensions are known real matrices. (.) represents the nonlinear function. Nevertheless, taking the actual condition into account, it's difficult to capture the full state variables of the master system and slave system. In order to implement synchronization, the observers are designed as: The estimates of x m (t), x s (t) are written asx m (t) andx s (t) respectively, the observer gain matrixes are given as C m , C s . Our aim is to propose a feedback controller to synchronize the drive-response systems. Taking time delay into consideration, u(t) can be designed as: K , as the controller gain matrix, is supposed to be calculated. The error systems are given as e m (t), e s (t), e o (t) respectively, according to (1)-(5), the following equations are established: The nonlinear terms are defined as: where m (t), can be given as zero, positive or negative. VOLUME 8, 2020 V = [ε 1 , . . . , ε n ] T with ε d ∈ R n , the following can be obtained: ≤ 0 (10) Remark 1: Similar to Assumption C in [16], we can regard k + d and k − d as positive, negative, or zero. If k + d > 0 and k − d = 0, the function described by Assumption C can be the class of monotone nondecreasing nonlinear and globally Lipschitz continuous. While both k + d and k − d are positive, the function can be the class of monotone increasing nonlinear and globally Lipschitz continuous. Hence, this kind of nonlinear function is more favourable to obtain less conservatism.

III. MAIN RESULTS
For the sake of synchronizing the master-slave CLSs and reduce the conservatism, novel observers(3)(4) and a feedback controller (5) will be proposed via using novel LKFs in this section.
However, the terms F T 3 N 3 C m Te m (t) and F T 3 N 3 C s Te s (t) in (38) can produce the cross terms between e m and e o , e s and e o respectively. That would result in the difficulty of seeking solutions by LMI directly. For this reason, a decoupling approach [28] which can be used to handle the coupling terms is introduced and the cross terms are included in m and s .
Remark 2: Applying the decoupling approach proposed in [28], the gain matrixes of the observers and the controller C m , C s and K can be obtained by 1 , 2 and 3 respectively, guaranteeing that the synchronization of masterslave systems is achieved while completing state estimation. Compared with solving < 0, the condition in theorem 1 is  easier to deal with because the terms m , s are eliminated. The specific proof process is shown in [28].

IV. SIMULATION
The correctness of the observers and the controller designed above is tested by the simulations.
Example 1: The system equation of Chua's circuit is: and the nonlinear term of Chua's diode is r(x 1 (t)) = 0.5 (I 0 −I 1 )[−|−B+x 1 (t)|+|B+x 1 (t)|]+I 1 x 1 (t) with parameters I 0 = − 1 7 , I 1 = 2 7 , λ = 9, γ = 14.28 and B = 1. It's obvious that the system discussed above could be shown as Lur'e system under the conditions:      Fig.5 and Fig.6, it's clear to see that the states reconstruction has been achieved within 4s. Fig.7 shows that the synchronization has   been completed within 6s. From the above analysis we can conclude that the observers and the control law we proposed above can synchronize the uncertain master-slave systems.
Example 2: The system parameters of (1)(2) are given as follows: The Lur'e system has been reduced to a neural network which has three neurons. The initial condition values of the master-slave systems are given by     and controller gain matrixes can be obtained as:  Fig.11 and Fig.12, the states reconstruction has been achieved within 4s. Fig.13 shows that the synchronization has been completed within 11s. The simulations verify the effectiveness of Theorem 1.

V. CONCLUSION
In this paper, considering of the unmeasurable state variables in CLSs, we research the observer-based synchronization control design. Based on the Bessel-Legendre inequality, improved reciprocally convex approach and double integral inequality, novel LKFs have been constructed. A new synchronization strategy with less conservatism is obtained and a decoupling approach is employed to obtain the observer and controller gain matrices. By utilizing the control method we proposed, the synchronization and state estimate of the master-slave systems have been accomplished in finite time. Some simulations are given to show our strategy can make the CLSs have good robustness.