Synchronization of Partial Differential Systems via ... - Semantic Scholar
IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—I: REGULAR PAPERS, VOL. 59, NO. 11, NOVEMBER 2012
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Synchronization of Partial Differential Systems via Diffusion Coupling Kaining Wu and Bor-Sen Chen, Fellow, IEEE
Abstract—In this paper, we address the synchronization problem of coupled partial differential systems (PDSs). First, the asympsynchronization of N-coupled totical synchronization and the PDSs with space-independent coefficients are considered without or with spatio-temporal disturbance, respectively. The sufficient conditions to guarantee the asymptotical synchronization and the synchronization are derived. The effect of the spatial domain on the synchronization of the coupled PDSs is also presented. Then the problem of asymptotical synchronization of N-coupled PDSs with space-dependent coefficients is dealt with and the sufficient condition to guarantee the asymptotical synchronization is obtained by using the Lyapunov-Krasovskii method. The condition of the synchronization for N-coupled PDSs with space-dependent coefficients is also presented. Both conditions are given by integral inequalities, which are difficult to be verified. In order to avoid solving these integral inequalities, we adopt the semi-discrete difference method to turn the PDSs into an equivalent spatial space state system, then the sufficient condition of the synchronization for N-coupled PDSs is given by an LMI, which is easier to be verified. Further, the relationship between synchronization, obtained the sufficient conditions for the by the Lyapunov-Krasovskii method and semi-discrete difference method respectively, is investigated. Finally, two examples of coupled PDSs are given to illustrate the correctness of our results obtained in this paper. synchronization, LMI, N-coupled Index Terms—Diffusion, PDSs, Partial differential systems(PDSs), synchronization.
I. INTRODUCTION
S
YNCHRONIZATION appears within a widespread field, ranging from natural network to artificial network, such as flushing fireflies [1], brainweb [2], yeast cell [3], semiconductor lasers [4], sensor networks [5] and so on. Due to the ubiquitousness of synchronization, scientists attempt to understand the mechanism behind the phenomena and how it works to get its advantages. For instances, the semiconductor laser array can be synchronized to generate larger power laser, the clock of sensor network needs to be synchronized so that the sensor network Manuscript received August 22, 2011; revised December 10, 2011; accepted February 09, 2012. Date of publication April 05, 2012; date of current version October 24, 2012. This work was supported in part by the National Science Council of Taiwan under Contract NSC 99-2745-E-007-001-ASP, NSC 100-2745-E-007-001-ASP, in part by the National Natural Science Foundation of China under Grant 11026189 and in part by the Natural Scientific Research Innovation Foundation in Harbin Institute of Technology (HIT.NSRIF.201014). This paper was recommended by Associate Editor X. Yu. K. Wu is with the Department of Mathematics, Harbin Institute of Technology at Weihai, 264209 Weihai, China (e-mail: [email protected]). B.-S. Chen is with the Department of Electrical and Computer Engineering, National Tsinghua University, Hsinchu, Taiwan (e-mail: [email protected]. tw). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TCSI.2012.2190670
can process data more correctly or, to achieve secure communication [6], [7], the transmitter masks the signals with fuzzy chaotic oscillator and the receiver recovers the signals using fuzzy chaotic synchronization [8]. Recently, several studies have investigated the synchronization of coupled networks, using the master stability function(MSF) method [9] and the Lyapunov direct method [10]. Both methods provide the criteria of synchronization for nonlinear coupled networks. The criterion of MSF method can only guarantee the linearized stability of the synchronous state because the variational equation is just the linearized dynamic of coupled network along the synchronous state. The Lyapunov direct method guarantees the synchronization for coupled networks in a larger region and the range depends on the Lyapunov function that has been found. However, in general it is hard to find an appropriate Lyapunov function. Based on the above two methods, many research works have been done on the synchronization from different points of view. In [11], the network is considered as a fixed topology and the coupled network with maximal synchronization has non-diagonal coupling matrix. In [12], the interaction of oscillators is considered as the information of diffusion process. Some research works investigated the synchronization for networks with statistics feature, such as the small-wold network [13] or the scale-free network [14]. In [15]–[17], the authors have shown that randomly adding shortcuts can translate a regular network to a small-world network and improve the network’s synchronization [18]. In [19], Wang and Chen discussed the synchronization of scale-free network. In [20], [21], the synchronization of the scale-free network with degree-degree correlation is considered. The time-delayed coupling effects have also been considered for different classes of systems such as complex network with time-varying delayed coupling in [21]–[23]. The master-slave synchronization problems subject to communication channel noise and bandwidth limitations were studied in [24] and [25]. In [26] and [27], the authors dealt with the master-slave Lurie systems by using time-delayed feedback control and sample-feedback control, respectively. In the real world, external disturbances are ubiquitous and may lead a given system to an unanticipated state and even destroy the synchronization. Hence, it is deserving to investigate the ability of synchronization to resist the external disturbance, synchronization problem. The synchronizai.e., the tion problems have been studied by many authors [28]–[35]. synchronizaIn [28] and [29], the authors considered the tion between the master and slave of chaotic oscillators. Fursynchronization has been applied to time-dethermore the layed chaotic system [30], [31], [35], and neural systems [32]. method is also utilized to design an antisynchronizaThe tion controller between drive and response system [36].
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In recent years, a great deal of concern has been raised regarding the study of PDSs [37]–[48]. Many phenomena in science, engineering and biology have been modeled in the spatio-temporal domain by PDSs such as in chemical engineering, biology, population dynamics, neurophysiology and biodynamics, etc., therefore, the synchronization problem of PDSs is addressed in this paper. In this study, N-coupled linear PDSs with constant coefficients via spatial diffusion coupling are discussed at first, then a synchronization error dynamic of coupled PDSs is defined in the spatial domain. Based on the synchronization error dynamic of the coupled PDSs, the asymptotical synchronization problem becomes an asymptotical stabilization problem. Using the Lyapunov-Krasovskii techniques, the asymptotical synchronization criteria for these coupled linear PDSs are derived. Aside from synchronization, the disturbance-resisting ability of synchronous PDSs has also been investigated in this study. Based on the disturbance attenuation theory in temporal domain [49]–[54], the spatio-temporal disturbance attenuation level of a coupled PDSs network is considered as the upper bound of the gain from any finite energy disturbance to the synchronization error in the spatio-temporal domain. In this study, based on the synchronization error dynamic with disturbances, sufficient conditions are obtained for the synchronization of N-coupled PDSs with disturbances in the spatio-temporal domain. Except the coupled PDSs with constant coefficients, the coupled PDSs with space-dependent coefficients appear in electrostatics, stationary heat transfer and other diffusion problems for inhomogeneous media and the properties are studied by many researchers [55], [56]. This study also considers the synchronization for N-coupled PDSs with space-dependent coefficients. In term of Lyapunov-Krasovskii method, the sufficient condition to guarantee the synchronization is obtained. Similarly, the synchronization for N-coupled PDSs with space-dependent coefficients and disturbances is studied and the criterion of synchronization is also derived by virtue of Lyapunov-Krasovskii method. These criteria need to verify the corresponding integral inequalities, which can not be solved analytically at present, even if an appropriate Lyapunov-Krasovskii functional is given. In order to overcome this difficulty and make these criteria to be verified easier, the semi-discrete finite difference scheme is employed to represent the synchronization error dynamic. Under this situation, the synchronization criterion of coupled PDSs needed to be investigated becomes an LMI, which can be efficiently solved with the help of LMI toolbox in Matlab [57]. The relationship between the sufficient conditions for the synchronization, obtained by the Lyapunov-Krasovskii method and semi-discretedifferencemethodrespectively,isalsoinvestigated. Finally, two examples are given to illustrate the correctness of the results we got in this paper. The criteria of asymptotical synchronization with constant coefficients are verified by the numerical simulations. The effect of the spatial domain on the synchronization is also discussed. The synchronization of two-coupled PDSs with space-dependent coefficients is investigated when the spatio-temporal disturbances are considered. For convenience, we adopt the following standard notations: and denote, respectively, the dimensional Euclidean space and the set of all real matrices. . The superscript ‘ ’ denotes the
transpose and the notation , where and are symmetric matrices, means that is positive semidefinite(positive definite). is the identity matrix with compatible dimension. stands for the space of functions with continuous partial derivatives of order less than or equals to in . (or ) denotes the set of square integral functions on . II. MATHEMATICAL MODEL AND SYNCHRONIZATION ERROR DYNAMIC We consider the following N-coupled linear PDSs with constant coefficients:
(1) , for and , where . the state variables and are space and time variables respectively, are constant system matrices and we assume are symmetric matrices and . The Laplace operator for two-dimensional spatio-space is defined as follows:
.. .
.. .
(2)
. for The boundary conditions and initial values are given as follows: (3) The linear PDSs in (1) are coupled through diffusion in spatio-space with constant diffusion . coefficients be a function to which all ’s are expected Let satisfies the following equation: to synchronize and (4) subject to the boundary conditions and initial values as follows: (5) Let us define the synchronization error as follows: (6) Definition 1: The coupled linear PDSs in (1) are asymptotically synchronous if the synchronization error satisfies for all and . By transforming the coupled PDSs in (1) with state vector to synchronization errors in (6), the synchronization problem of PDSs in (1) turns to the stabilization
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problem of the corresponding synchronization error dynamical system in the following
(7) subject to the boundary conditions and initial values as follows: Take ..
.. . and
..
.
.. . . Obviously,
..
.
.
.. .
is symmetric. Denote
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III. SYNCHRONIZATION AND SYNCHRONIZATION COUPLED PDSS WITH CONSTANT COEFFICIENTS
OF
Based on the synchronization error dynamic of PDSs constructed in the last section, the asymptotical synchronization problem of coupled PDSs becomes the stabilization problem of synchronization error dynamic of PDSs. For the N-coupled linear PDSs in (1) and their synchronization error dynamic in (8), the asymptotical synchronization of coupled PDSs in (1) implies the asymptotical stability of synchronization error dynamic in (8). of the synDefinition 3: The equilibrium point chronization error dynamic in (8) is asymptotically stable for all , if as for all . Remark 2: The definition 3 of the asymptotical stability is essentially the strong stability in view of the distributed parameter system. For the distributed parameter systems, there are other definitions of stability, for the details, we refer to the reference [58] and the references therein. Now we present an important lemma which is stated in [59]. . Here, we assume ( , 2) and let Lemma 1: Let be a cube: be a real-valued function belonging to which vanishes of , i.e., . Then on the boundary (12)
then the synchronization error dynamic of N-coupled PDSs with constant coefficients in (7) can be represented by (8) If the coefficients depend on the space variable, i.e., are space-dependent, then the system (1) becomes the N-coupled PDSs with space-dependent coefficients
Using lemma 1, we can get the following lemma, which will be used in the sequel. ( , 2) and let Lemma 2: Let be a cube: be a function belonging to which vanishes on the boundary of . Then
Proof: From Green’s identity [60], using Lemma 1, we (9)
have
The synchronization error dynamic (7) becomes
(10) Similarly, the synchronization error dynamic of N-coupled PDSs with space-dependent coefficients in (10) can be represented by (11) and are defined similarly as and , respecwhere tively. , but the reRemark 1: Here we study the case: , sults in the sequel can be extended to the case: without any difficulty. To end this section, we present a definition, which will be used in the sequel. is positive-definite and raDefinition 2: The function dially unbounded if is positive except at where and as .
where indicates differentiation in the direction of the exterior normal to . Now we are in the position to state our first theorem on the asymptotical synchronization of N-coupled PDSs with constant coefficients:
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Theorem 1: Suppose there exists a positive symmetric matrix such that
consider the robustness to resist external disturbance of the coupled PDSs. We consider the following coupled PDSs with disturbance
(13) and
(18) (14)
then the synchronization error system (8) is asymptotically stable, that is the N-coupled PDSs (1) are of asymptotical synchronization. Proof: Define a Lyapunov-Krasovskii functional . Keeping in mind that is symmetric, we have
where stands for the external disturbance in is square intethe spatio-temporal domain and for any gral, i.e., and some positive constant . , then we get the following Let coupled synchronization error dynamic
(19) Let
Take , and as before, then we get the disturbed synchronization error dynamic as follows: (15) Since
, there exists a matrix , then
such that .
Let
, noting , using Lemma 2, we have
for
(20) Assuming that the coupled linear PDSs are in the synchro(i.e., ), we consider the nization state at disturbance attenuation of the synchronization error dynamic in (20) as follows:
or (16) (21)
Substituting (16) into (15) and using (14), we can get If the initial condition is considered, then
(17) , we get that, along the solution of system (8), as . Take is the smallest eigenvalue of , we have , then we get as , as . We get the asymptotical which implies stability of system (8), that is, the system (1) is asymptotically synchronous. Remark 3: From (14), the asymptotical synchronization can be verified by the systems’ coefficients, i.e., and and the range of space domain, i.e., and . Since the external disturbance is inevitable, and disturbance may destroy the synchronization of the coupled PDSs, we must Since
(22) . for some positive function If the synchronization error dynamic in (20) satisfies the disturbance attenuation (21) or (22), we say the coupled PDSs are synchronization with a prescribed attenuation level of the . Now, we state a theorem on the synchronization, which synchropresents the sufficient conditions to guarantee the nization of systems (18). , Theorem 2: Given a disturbance attenuation level suppose that there exists a positive matrix such that (23)
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and (24) synchronization in then the coupled PDSs (18) are of the (21) or (22). . Using Proof: Let Lemma 2 and the techniques of completing the squares, we have
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, and the inequality (21) holds, therefore, we conclude that the synchronization. coupled PDSs (18) are of the Remark 4: syn(i) The conditions (23) and (24) to guarantee the chronization are also sufficient for the asymptotical synchronization since means
which implies that the synchronization of (18) indicates the asymptotical synchronization of (1) when . synchronization in (23) and (24) can be verified (ii) The by the systems coefficients and , the range of space domain and and the prescribed disturbance attenuation level . In addition, the optimal disturbance attenuation level of synchronous PDSs can be measured by
(25) that is the inequality (22). If , i.e., the coupled PDSs , then (18) are in the synchronization state at
(iii) From the conditions (14) and (24), we see that for the becomes same system coefficients, if the constant is smaller, then insmaller, i.e., the space domain equalities of (14) and (24) are easier to hold, i.e., it is more easy to achieve the synchronization through spatial diffusion coupling in a small area. Therefore, we can get the following proposition: Proposition 1: The asymptotical synchronization and synchronization of coupled PDSs depend on the scale of the spatial domain: the smaller the spatial domain, the easier to achieve synchronization. the asymptotical synchronization or the Remark 5: In many paper, the structure of the network was considered for the synchronization problem, see, for example, [19], [32]. The synchronization of a coupled network system is not only determined by the structure of the coupled network, represented by the matrix in this paper, but also the dynamical properties of the subsystem without coupling, represented by the matrices and . In this paper, the sufficient conditions for the synchronization for the PDSs are dependent on the ma. It means trices and , which contain the matrices the structure and the properties of the subsystem are considered both for the synchronization. and the disturbance attenuation level are If the matrices given, then we can discuss what conditions should be imposed to make the inequalities (23) and (24) have on the matrix a positive solution. That is the design problem of the coupled network to achieve the synchronization. To see it clearly, we , where is a given constant, assume the matrix is a fixed diagonal matrix, if there is a connection between , , otherwise, node and node . This form of matrix was adopted in [19]. Take , then . Now, the topology of the coupled network can be completely presented by the matrix . For a strong connected graph, we know that the eigenvalues of are negative, then if (positive diffusion will harm the synchronization), from inequalities (23) and (24), we can see
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the larger to be, which represents the strength of coupling, the easer to achieve the synchronization. Moreover, if the network is globally coupled, then the eigenvalues of the matrix are 0 (the eigenvalue 0 is simple, see [19] for the details), and , the more nodes of the network have, the thus for a given easer to achieve the synchronization. To end this section, we figure out other results. In the literatures, many researchers studied the synchronization of coupled , where is systems using the error the state of the coupled systems, see [10]. Now, we also take the , and for the synchronization error simplicity, we just consider the two-coupled PDSs. We will give synthe criteria for the asymptotical synchronization and chronization for two-coupled PDSs and present the relationship between these results and the results we got before. We consider the following two-coupled PDSs
Theorem 4: Given a disturbance attenuation level , if the following inequalities hold (33) and (34) then, if
, we have
(35) synchronization. that is, the coupled PDSs (28) are of and , we can easily verify this Take theorem using the techniques in the proof of Theorem 2. Now we point out the relationship of Theorems 1, 2 and the Theorems 3, 4. First, we claim that
(26) subject to the boundary conditions and initial values Now we give a brief proof. Since (27) and the disturbed two-coupled PDSs
holds for any
(28)
where and are the external disturbances and , and we also impose the satisfy boundary conditions and initial values (27). From (26), (28) and keeping in mind , we have the following synchronization error dynamics (29) and
(30) where
. Obviously,
. Now we present the following theorem on synchronization of system (26). Theorem 3: For the synchronization error dynamic of two coupled PDSs in (29), if the following inequalities hold (31) (32) of (29) is asymptotically then the equilibrium point stable, that is, the two-coupled linear PDSs in (26) are asymptotically synchronous. The proof is similar to the proof of Theorem 1, just take and in the proof of Theorem 1, so we omit it. synchronization, we have the following theorem. On the
which implies
, take
, we get
. Using the same technique, we get
implies
From these results, we can get, without any difficulty, that if we in Theorem 1, then the conditions (13), (14) imply take (31), (32), i.e., the Theorem 1 implies the Theorem 3. The same relationship exists for Theorem 2 and Theorem 4. IV. ASYMPTOTICAL SYNCHRONIZATION AND SYNCHRONIZATION OF N-COUPLED PDSS WITH SPACE-DEPENDENT COEFFICIENTS In this section, we consider the N-coupled PDSs with spacedependent coefficients. We must point out that the technique used in (16) is not suitable under this situation since the coefficients are space-dependent. Therefore, we can not get analogous results as the constant coefficient case. Now, we employ the Lyapunov-Krasovskii method to present a theorem on the asymptotical synchronization of N-coupled PDSs with space-dependent coefficients. Theorem 5: For the synchronization error dynamic of N-coupled PDSs with space-dependent coefficients (11), if there ex, ists a positive-definite and radially unbounded function , such that the following integral inequality holds for : all
(36)
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then the equilibrium point of (11) is asymptotically stable, that is, the coupled linear PDSs in (9) is asymptotically synchronous. . Proof: Let
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Proof: Using the technique of completing the squares, we have
(37) , we can get as By virtue of , then as . We can get that the of (11) is asymptotically stable, i.e., the equilibrium N-coupled linear PDSs in (9) are asymptotically synchronous. Remark 6: If the following inequality holds for some positive ,
for all , then the asymptotical synchronization result in Theorem 5 still holds but with stricter condition than (37). synchronization of the Now we turn our attention to the following N-coupled PDSs with space-dependent coefficients
(38) subject to the boundary conditions and initial values (39) where
is the external disturbance and , for some positive constant . We modify the synchronization performance in (22) as
(40) We can get the synchronization error dynamic as follows:
Similarly, the above system can be presented by the following system (41) Theorem 6: Given a disturbance attenuation level , if there , such that the folexists a positive differentiable function lowing integral inequality holds
(42) then the coupled PDSs (38) are of
synchronization.
Then the inequality (40) holds. If and , synchronization in (21) holds, which completes then the the proof. Remark 7: If the following inequality holds for some positive , differentiable function
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for all and , then the synchronization result in Theorem 6 still holds but with stricter condition than (42). Remark 8: It is easy to see inequality (42) implies inequality (36), since that is, the synchronization guarantees the asymptotical . synchronization when Remark 9: Both the above results, Theorem 5 and Theorem 6, present sufficient conditions to guarantee the asymptotical synsynchronization of coupled PDSs, rechronization and the spectively. However, as we all know, they are not practical: It is not easy to find a suitable Lyapunov-Krasovskii functional such that the inequalities (36) and (42) hold, respectively. Therefore, we must proceed to find a practical method to deal with the synchronization of coupled PDSs with space-dependent coefficients. SYNCHRONIZATION CRITERIA BASED ON LMI V. synchronization criteria for coupled To simplify the PDSs with space-dependent coefficients, we need to develop a more suitable spatial state space model to represent the PDSs. For this purpose, the semi-discrete finite difference scheme is synchronization problem of coupled employed to treat the PDSs with space-dependent coefficients. The finite difference scheme is used to approximate the partial differential operator to simplify the synchronization criteria of coupled PDSs. The finite-difference scheme has been widely used to obtain numerical solutions of PDSs [45], [46]. In this paper, we use the finite-difference method to represent the PDSs (38) to get the simpler synchronization criteria. Consider a typical grid mesh, is presented by at the as shown in Fig. 1, the state , where grid node and , i.e., . Note , , or that the grid nodes are the grid nodes at the boundary. At the interior points of grid, the central-difference approximation for the differential operator can be written as follows:
(43) is the local truncation error. The remainder term A finite-difference model can be constructed to represent of synchronization error dynamic at in (41) as follows:
(44)
Fig. 1. Finite-difference grids on the spatio-domain.
Let us denote
(45) then (44) can be rewritten as follows:
(46) , . where For the simplification of the asymptotical synchronization to criterion for the coupled PDSs, we define a spatial vector at all gird nodes in Fig. 1. From the boundary collect the at the boundary are conditions, we know the values of for fixed. Therefore, the spatial state vector at all grid nodes is defined as
(47) where . Note that is the dimension of the for each grid node and is the number of vector grid nodes. in In order to simplify the index of the node , we denote the symbol the spatial state vector to replace . Note that the index from 1 to , i.e., , , , , , where in (47). The difference model of the two indexes in (46) could be represented with only one index as follows: (48) , with and in (49), shown at the bottom of the page, where zero matrix and denotes the matrix.
is defined as denote the identity
(49)
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We collect all synchronization errors of the grid nodes to the synchronization state vector in (47). Take
..
..
.
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Proof: Let of completing the squares, we get
. In term of the technique
. (50)
and .. .
.. .
then (48) can be represented by the following spatial space state system
(51) represents at all grid points on the spatial domain , . , (51) can be rewritten as follows: Take
where
(52) Suppose the external disturbance is absent and the trunof the finite difference scheme approaches cation error zero as the number of grid nodes is large enough, i.e., . Then the asymptotical stability of system (52) can be guaranteed by the following theorem. Theorem 7: For the augmented system in (52) with , suppose there exists a symmetric positive matrix such that the following inequality holds (53) of system (52) is asymptotithen the equilibrium cally stable. The proof is standard, so we omit it. From the practical point of view, the external disturbance and the truncation error will not be zero, then we turn our attention synchronization. For the spatial state space system as to the synchronization in (52), for some positive constant , the performance is described by (54) for some positive function . Theorem 8: For the augmented system in system (52) with a prescribed noise attenuation level , if there exists a positive matrix such that the following LMI holds: (55) then the inequality (54) holds, and the coupled PDSs (38) are of synchronization. Furthermore, the asymptotical synchro. nization could be achieved when
Obviously, if , we can get the desired result (54). By Schur Complement, the LMI (55) implies the above relationship. The last desired result is obvious, since
implies From Theorem 7, the asymptotical synchronization is achieved . when The optimal noise attenuation level can be obtained by solving the following constrained optimization
(56) This constrained optimization can be easily solved by decreasing until no existence of positive solution in the LMI (55), which can be easily solved by the LMI toolbox in Matlab. Now we present the relationship between the integral inequality (42) and the LMI (55).
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First, we take in (42), then with the semi-discrete difference approximation and approximation of integration by summation, the left of integral inequality (42) becomes the first equation shown at the bottom of the page. For the simplicity, we denote the symbols , , and to replace , , and , where . Noting , we have the second equation shown at the bottom of the page. Take , we have the last equation shown at the are defined in botton of the page, where , , and . (50) and We can read from the last equality, for small enough , results in
that indicates the integral inequality (42) with We also know that the inequality
.
is equivalent to the LMI (55). Then we get the following statement: Proposition 2: For small enough , if the LMI (55) holds, synchronizathen the integral inequality (42) holds and the tion of the coupled disturbed PDSs (36) is guaranteed. VI. NUMERICAL EXAMPLES In this section, we give two examples to illustrate the correctness of our results. Example 1: First, we consider the following three-coupled PDSs with constant coefficients, shown in (57) at the bottom , of the next page, where and
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Fig. 2. Simulation results of , and right) of coupled PDSs in Example 1 at different time: (a) . (c)
(from left to , (b) ,
We put the boundary conditions and initial conditions for system (57) as follows:
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Fig. 3. Simulation results of , and right) of coupled PDSs in Example 1 at different time: (a) . (c)
(from left to , (b) ,
Now we will illustrate the effect of the domain on the syn, and we provide the chronization. We take in Fig. 5. It can difference between each two states at be seen that the larger domain makes the coupled PDSs more time achieve asymptotical synchronization, and this can show the correctness of Proposition 1. Now, we present another example of disturbed two-coupled PDSs with space-dependent coefficients. Example 2: We consider the following disturbed two-coupled PDSs with space-dependent coefficients:
(58) where
Take in the Theorem 1, it is easy to verify the conditions (13) and (14) hold under this situation. The state , are shown in the Fig, 2, 3 at , , , respectively. We can read from these figures that the coupled PDSs are of asymptotical synchronization. Here we also provide figures to illustrate the difference between each two states at some fixed space point. We take , from Fig. 4, we can also see the asymptotical synchronization of the coupled PDSs (57).
(57)
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Fig. 5. Effect of the space domain on the synchronization in Example 1. It is seen that a larger space domain makes the coupled PDSs take more time to ; (b) achieve asymptotical synchronization. (a) .
Fig. 4. Difference between each two states at at ; (b) at ; (c) at .
in Example 1. (a) at ; (d)
We consider the synchronization of two-coupled PDSs , and (58). We take , then . According (49) and (50), we and . From the LMI (55), we can get a get the matrix , positive-definite matrix , that is, the coupled systems in (58) synchronization. Moreover, it can be obtained that are of by solving (56). the optimal noise attenuation level Take the boundary conditions and initial values as follows:
We show the synchronization of the coupled PDSs (58) by Fig. 6, where the difference between state and are illus. trated at
Fig. 6. Difference between two coupled PDSs with space-dependent coeffiand at ; (b) Difcients in Example 2. (a) Difference between and at . ference between
The synchronization performance can be computed by simulation as follows:, here we take , is obtained by the LMI (55).
VII. CONCLUSIONS This paper considered the asymptotical synchronization and synchronization for the coupled PDSs with constant coef-
WU AND CHEN: SYNCHRONIZATION OF PARTIAL DIFFERENTIAL SYSTEMS
ficients and with space-dependent coefficients respectively. We constructed the synchronization error dynamic and turned the synchronization problems of coupled PDSs to the stabilization problems of the synchronization error dynamic. Based on the synchronization error dynamic, we presented sufficient conditions to guarantee the asymptotical synchronization which can be verified using the system’s coefficients of the coupled PDSs and the range of the space domain. We also give the criterion synchronization for N-coupled PDSs with constant of the coefficients. The effect of spatial domain on the synchronization can be read from these criteria. We found that it needs to take more time for coupled PDSs to achieve synchronization in a lager rang of space domain. For the N-coupled PDSs with space-dependent coefficients, we first studied the sufficient consynditions for the asymptotical synchronization and the chronization with the Lyapunov-Krasovskii method. These sufficient conditions were given by integral inequalities. Generally, they are difficult to be solved. To overcome these difficulties, we adopted the semi-discrete difference method to turn the coupled PDSs into an equivalent spatial state space system, then a sufficient condition is presented based on an LMI, which guarsynchronization and is easy to be verified. Two antees the numerical simulations are given to illustrate the correctness of our results. REFERENCES [1] A. Pikovsky, M. Rosenblum, and J. Kurths, Synchronization: A Universal Concept in Nonlinear Sciences. Cambridge, U.K.: Cambridge Univ. Press, 2002. [2] F. Varela, J. Lachaux, E. Rodriguez, and J. Martinerie, “The brainweb: Phase synchronization and large-scale integration,” Nat. Rev. Neurosci., vol. 2, no. 4, pp. 229–239, Apr. 2001. [3] M. Bier, B. Bakker, and H. Westerhoff, “How yeast cells synchronize their glycolytic oscillations: A perturbation analytic treatment,” Biophys. J., vol. 78, no. 3, pp. 1087–1093, Mar. 2000. [4] G. Kozyreff, A. Vladimirov, and P. Mandel, “Global coupling with time delay in an array of semiconductor laser,” Phys. Rev. Lett., vol. 85, no. 18, pp. 3809–3812, Oct. 2000. [5] Q. Li and D. Rus, “Global clock synchronization in sensor networks,” IEEE Trans. Comput., vol. 55, no. 2, pp. 214–226, Feb. 2006. [6] K. Lian, C. Chiu, T. Chiang, and P. Liu, “LIM-based fuzzy chaotic synchronization and communications,” IEEE Trans. Fuzzy Syst., vol. 9, no. 4, pp. 539–553, Aug. 2001. [7] K. Lian, T. Chiang, C. Chiua, and P. Liu, “Synthesis of fuzzy model-based designs to synchronization and secure communications for chaotic systems,” IEEE Trans. System, Man Cybern., B. Cybern., vol. 31, no. 1, pp. 66–83, Feb. 2001. [8] M. Chen, “Chaos synchronization in complex networks,” IEEE Trans. Circuits Syst. I, Reg. Papers, vol. 55, no. 5, pp. 1335–346, Jun. 2008. [9] L. Pecora and T. Carroll, “Master stability functions for synchronized coupled systems,” Phys. Rev. Lett., vol. 80, no. 10, pp. 2109–2112, Mar. 1998. [10] C. W. Wu and L. Chua, “Synchronization in an array of linear coupled dynamical systems,” IEEE Trans. Circuits Syst. I, Fundam. Theory Appl., vol. 42, no. 8, pp. 430–447, Aug. 1995. [11] T. Nishikawa and A. Motter, “Synchronization is optimal in nondiagonalizable networks,” Phys. Rev. E, vol. 73, no. 6, p. 065106, Jun. 2006. [12] A. Motter, C. Zhou, and J. Kurthus, “Network synchronization, diffusion and the paradox of hetergeneity,” Phys. Rev. E, vol. 71, no. 1, p. 016116, Jan. 2005. [13] D. Watts and S. Strogatz, “Colletive dynamics of small-world networks,” Nature, vol. 393, no. 6684, pp. 440–442, Jun. 1998. [14] A. Barabasi and R. Albert, “Emergency of scaling in random networks,” Science, vol. 286, no. 5439, pp. 509–512, Oct. 1999. [15] M. Barahona and L. Pecora, “Synchronization in small-world systems,” Phys. Rev. Lett., vol. 89, no. 5, p. 054101, Jul. 2002. [16] X. Wang and G. Chen, “Synchronization in small-world dynamical networks,” Int. J. Bifurc. Chaos, vol. 12, no. 1, pp. 187–192, 2002.
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[17] J. Lu, X. Yu, G. Chen, and D. Cheng, “Characterizing the synchronizability of small-world dynamical networks,” IEEE Trans. Circuits Syst. I. Reg. Paper, vol. 51, no. 4, pp. 787–796, Apr. 2004. [18] C. Yin, B. Wang, W. Wang, and G. Chen, “Geographical effect on small-world network synchronization,” Phys. Rev. E, vol. 77, no. 2, p. 027102, Feb. 2008. [19] X. Wang and G. Chen, “Synchronization in scale-free dynamical networks: Robustness and fragility,” IEEE Trans. Circuits Syst. I, Fundam. Theory Appl., vol. 49, no. 1, pp. 54–62, Jan. 2002. [20] Z. Fei, H. Gao, and W. X. Zheng, “New synchronization stability of complex networks with an internal time-varying coupling delay,” IEEE Trans. Circuits Syst. II, Exp. Briefs, vol. 56, no. 6, pp. 499–503, Jun. 2009. [21] W. He and J. Cao, “Exponential synchronization of hybrid coupled networks with delay coupling,” IEEE Trans. Neural Netw., vol. 21, no. 4, pp. 571–583, Apr. 2010. [22] J. Liang, Z. Wang, Y. Liua, and X. Liu, “Robust synchronization of an array of coupled stochastic discrete-time delayed neural networks,” IEEE Trans. Neural Netw., vol. 19, no. 11, pp. 1910–1921, Nov. 2008. [23] A. Kruszewski, W. Jianga, E. Fridman, J. P. Richard, and A. Toguyeni, “A switched system approach to exponential stabilization through communication network,” IEEE Control Syst. Technol., 2011, 10.1109/TCST.2011.2159793. [24] B. Liu, D. J. Hill, and J. Yao, “Global uniform synchronization with estimated error under transmission channel noise,” IEEE Trans. Circuits Syst. I, Reg. Papers, vol. 56, no. 12, pp. 2689–2702, Dec. 2009. [25] A. Fradkov, B. Andrievsky, and B. J. Evans, “Synchronization of passifiable Lurie systems via limited-capacity communication channel,” IEEE Trans. Circuits Syst. I, Reg. Papers, vol. 56, no. 2, pp. 430–439, Feb. 2009. [26] J. G. Lu and D. J. Hill, “Global asymptotical synchronization of chaotic Lur’s systems using sampled data: A linear matrix inequality approach,” IEEE Trans. Circuits Syst. II, Exp. Briefs, vol. 55, no. 6, pp. 586–590, Jun. 2008. [27] M. Zhong and Q. L. Han, “Fault-tolerant master-slave synchronization for Lur’s systems using time-delay feedback control,” IEEE Trans. Circuits Syst. I, Reg. Papers, vol. 56, no. 7, pp. 1391–1404, Jul. 2009. [28] Y. Hou, T. Liao, and J. Yan, “ synchronization of chaotic systems using output feedback control design,” Phys. A: Stat. Mech. Appl., vol. 379, no. 1, pp. 81–89, Jun. 2007. [29] S. M. Lee, D. H. Ji, J. H. Park, and S. C. Won, “ synchronization of chaotic systems via dynamic feedback approach,” Phys. Lett. A, vol. 372, no. 29, pp. 4905–4912, Jul. 2008. [30] J. H. Park, D. H. Ji, S. C. Won, and S. M. Lee, “ synchronization of time-delayed chaotic systems,” Appl. Math. Compt., vol. 204, no. 1, pp. 170–177, Oct. 2008. [31] C. K. Ahn, “T-S fuzzy synchronization for chaotic systems via delayed output feedback control,” Nonlinear Dyn., vol. 59, no. 4, pp. 535–543, Mar. 2010. [32] B. S. Chen, C. H. Chiang, and S. K. Nguang, “Robust synchronization design of nonlinear coupled network via fuzzy interpolation methods,” IEEE Trans. Circuits Syst., I, Reg. Papers, vol. 58, no. 2, pp. 349–362, Feb. 2011. [33] H. Huang and G. Feng, “Robust synchronization of chaotic Lurie’s systems,” Chaos, vol. 18, no. 3, p. 033113, Sep. 2008. [34] J. H. Park, D. H. Ji, S. C. Won, and S. M. Lee, “Adaptive synchronization of unified chaotic systems,” Mod. Phys. Lett. B, vol. 23, no. 9, pp. 1157–1169, 2009. [35] H. R. Karimi and H. Gao, “New delay-dependent exponential synchronization for uncertain neural networks with mixed time delays,” IEEE Trans. Syst., Man Cybern., B. Cybern., vol. 40, no. 1, pp. 173–185, Feb. 2010. [36] C. K. Ahn, “An approach to antisynchronization for chaotic systems,” Phys. Lett. A, vol. 373, no. 20, pp. 1729–1733, Apr. 2009. [37] K. Liu, Stability of Infinite Dimensional Stochastic Differential Equations With Applications. New York: Chapman & Hall/CRC, 2006. [38] P. L. Chow, Stochastic Partial Differential Equations. New York: Chapman & Hall/CRC, 2007. [39] B. S. Chen and Y. T. Chang, “Fuzzy state space modeling and robust observer-based control design for nonlinear partial differential systems,” IEEE Trans. Fuzzy Syst., vol. 17, no. 5, pp. 1025–1043, Oct. 2009. [40] Y. T. Chang and B. S. Chen, “A fuzzy approach for robust reference tracking control design of nonlinear distributed parameter time-delay systems and its application,” IEEE Trans. Fuzzy Syst., vol. 18, no. 6, pp. 1041–1057, Dec. 2010.
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[41] C. I. Byrnes, I. G. Lauk’o, D. S. Gilliam, and V. I. Shabov, “Output regulation for linear distributed parameter systems,” IEEE Trans. Automat. Contr., vol. 45, no. 12, pp. 2236–2252, Dec. 2000. [42] C. L. Lin and B. S. Chen, “Robust observer-based control of large flexible structures,” J. Dyn. Syst. Maes. Control-Trans. ASME, vol. 116, pp. 713–722, Dec. 1994. [43] H. T. Banks, Modeling and Control in the Biomedical Sciences. New York: Springer-Verlag, 1975. [44] C. V. Pao, Nonlinear Parabolic and Elliptic Equations. New York: Plenum Press, 1992. [45] J. C. Strikwerda, Finite Difference Schemes and Partial Differential Equations, 2nd ed. Philadelphia, PA: SIAM, 2004. [46] G. Evans, J. Blackledge, and P. Yardley, Numerical Methods for Partial Differential Equations. : Springer-Verlag, 2000. [47] Y. Y. Chen, Y. T. Chang, and B. S. Chen, “Fuzzy solutions to partial differential equations: Adaptive approach,” IEEE Trans. Fuzzy Syst., vol. 17, no. 1, pp. 116–127, Feb. 2009. [48] E. Fridman and Y. Orlov, “An LMI approach to boundary control of semilinear parabolic and hyperbolic systems,” Automatica, vol. 45, no. 9, pp. 2060–2066, Sep. 2009. [49] A. Isidori and A. Astolfi, “Disturbance attenuation and control via measurment feedback in nonlinear systems,” IEEE Trans. Autom. Contr., vol. 37, no. 9, pp. 1283–1293, Sep. 1992. [50] H. Gao, J. Lam, L. Xie, and C. Wang, “New approach to mixed filtering for polytopic discrete-time systems,” IEEE Trans.Signal Process., vol. 53, no. 8, pp. 3183–3192, Aug. 2005. [51] B. S. Chen, W. H. Chen, and H. Wu, “Robust global linearization filter design for nonlinear stochastic systems,” IEEE Trans. Circuits Syst. I, Reg. Paper, vol. 56, no. 7, pp. 1441–1454, Jul. 2009. [52] C. L. Chen, G. Feng, D. Sun, and X. P. Guan, “ output feedback control of discrete-time fuzzy systems with application to chaos control,” IEEE Trans. Fuzzy Syst., vol. 13, no. 4, pp. 531–543, Aug. 2005. [53] D. Huang and S. K. Nguang, “Robust output feedback control of fuzzy systems: An LMI approach,” IEEE Trans. Syst., Man Cybern., B. Cybern., vol. 36, no. 1, pp. 216–222, Feb. 2006. [54] B. S. Chen, C. T. Tseng, and H. C. Wang, “Mixed fuzzy output feedback control for nonlinear uncertain systems: LMI approach,” IEEE Trans. Fuzzy Syst., vol. 8, no. 3, pp. 249–265, Apr. 2000. [55] S. Mikhailov, “Analysis of united boundary-domain integro-differential and integral equations for a mixed BVP with variable coeffcient,” Math. Meth. Appl. Sci., vol. 29, pp. 715–739, 2006. [56] J. Chang and B. Guo, “Identification of variable spacial coefficients for a beam equation from boundary measurements,” Automatica, vol. 43, pp. 732–737, 2007. [57] S. Boyd, L. EI Ghaoui, E. Feron, and V. balakrishnan, Linear Matrix Inequalities in Systems and Control Theory. Philadelphia, PA: SIAM, 1994. [58] J. Oostveen, Strongly Stabilizable Distributed Parameter Systems. Philadelphia, PA: SIAM, 2000.
[59] J. G. Lu, “Global exponential stability and periodicity of reaction-diffusion delayed recurrent neural networks with Dirichlet boundary conditions,” Chaos, Solitons and Fractals, vol. 35, pp. 116–125, 2008. [60] F. John, Partial Differential Equations. New York: Springer-Verlag, 1982.
Kaining Wu received the B.S., M.S., and Ph. D. degrees from the Harbin Institute of Technology, Harbin, Helongjiang, China, in 2003, 2005 and 2009 respectively, all in mathematics. He is currently with the Department of Mathematics, Harbin Institute of Technology (Weihai), Weihai, China, as a lecture. His research interests include robust control, stochastic control and delay differential systems.
Bor-Sen Chen (F’01) received the B.S. degree from the Tatung Institute of Technology, Taipei, Taiwan, in 1970, the M.S. degree from National central University, Chungli, Taiwan, in 1973, and the Ph.D. degree from the University of Southern California, Los Angeles, in 1982. From 1973 to 1987, he was a Lecturer, an Associate Professor and a Professor with the Tatung Institute of Technology. He is currently with the Laboratory of Control and Systems Biology, Department of Electrical Engineering, National Tsing Hua University, Hsinchu, Taiwan, as a Professor of electrical engineering and computer science. He is a Research Fellow with the National Science Council of Taiwan and is the holder of the excellent scholar Chair in engineering. According to the statistics of International Statistical Institute, he had published seven highly cited papers in the last ten years. One of them in the Society for Industrial and Applied Mathematics has been selected by Essential Science Indicator as the most cited paper in the research area of mathematics. His current research interests are control engineering, signal processing and system biology Dr. Chen is the editor of Asian Journal of Control. He is a member of the Editorial Advisory Board of the International Journal of Control, Automation and Systems. He is a member of the Editorial Board of Fuzzy Sets and Systems and BMC Systems Biology. He had been the Editor-in-Chief of the International Journal of Fuzzy Systems from 2006 to 2008 and is currently the Editor-in-Chief of the International Journal of Systems and Synthetic Biology.. Dr. Chen was four times the recipient of the Distinguished Research Award from the National Science Council of Taiwan. He was also the recipient of the Automatic Control medal from the Automatic Control Society of Taiwan in 2001. He was an Associate Editor of IEEE TRANSACTIONS ON FUZZY SYSTEMS from 2001 to 2006.
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Synchronization of Partial Differential Systems via Diffusion Coupling Kaining Wu and Bor-Sen Chen, Fellow, IEEE
Abstract—In this paper, we address the synchronization problem of coupled partial differential systems (PDSs). First, the asympsynchronization of N-coupled totical synchronization and the PDSs with space-independent coefficients are considered without or with spatio-temporal disturbance, respectively. The sufficient conditions to guarantee the asymptotical synchronization and the synchronization are derived. The effect of the spatial domain on the synchronization of the coupled PDSs is also presented. Then the problem of asymptotical synchronization of N-coupled PDSs with space-dependent coefficients is dealt with and the sufficient condition to guarantee the asymptotical synchronization is obtained by using the Lyapunov-Krasovskii method. The condition of the synchronization for N-coupled PDSs with space-dependent coefficients is also presented. Both conditions are given by integral inequalities, which are difficult to be verified. In order to avoid solving these integral inequalities, we adopt the semi-discrete difference method to turn the PDSs into an equivalent spatial space state system, then the sufficient condition of the synchronization for N-coupled PDSs is given by an LMI, which is easier to be verified. Further, the relationship between synchronization, obtained the sufficient conditions for the by the Lyapunov-Krasovskii method and semi-discrete difference method respectively, is investigated. Finally, two examples of coupled PDSs are given to illustrate the correctness of our results obtained in this paper. synchronization, LMI, N-coupled Index Terms—Diffusion, PDSs, Partial differential systems(PDSs), synchronization.
I. INTRODUCTION
S
YNCHRONIZATION appears within a widespread field, ranging from natural network to artificial network, such as flushing fireflies [1], brainweb [2], yeast cell [3], semiconductor lasers [4], sensor networks [5] and so on. Due to the ubiquitousness of synchronization, scientists attempt to understand the mechanism behind the phenomena and how it works to get its advantages. For instances, the semiconductor laser array can be synchronized to generate larger power laser, the clock of sensor network needs to be synchronized so that the sensor network Manuscript received August 22, 2011; revised December 10, 2011; accepted February 09, 2012. Date of publication April 05, 2012; date of current version October 24, 2012. This work was supported in part by the National Science Council of Taiwan under Contract NSC 99-2745-E-007-001-ASP, NSC 100-2745-E-007-001-ASP, in part by the National Natural Science Foundation of China under Grant 11026189 and in part by the Natural Scientific Research Innovation Foundation in Harbin Institute of Technology (HIT.NSRIF.201014). This paper was recommended by Associate Editor X. Yu. K. Wu is with the Department of Mathematics, Harbin Institute of Technology at Weihai, 264209 Weihai, China (e-mail: [email protected]). B.-S. Chen is with the Department of Electrical and Computer Engineering, National Tsinghua University, Hsinchu, Taiwan (e-mail: [email protected]. tw). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TCSI.2012.2190670
can process data more correctly or, to achieve secure communication [6], [7], the transmitter masks the signals with fuzzy chaotic oscillator and the receiver recovers the signals using fuzzy chaotic synchronization [8]. Recently, several studies have investigated the synchronization of coupled networks, using the master stability function(MSF) method [9] and the Lyapunov direct method [10]. Both methods provide the criteria of synchronization for nonlinear coupled networks. The criterion of MSF method can only guarantee the linearized stability of the synchronous state because the variational equation is just the linearized dynamic of coupled network along the synchronous state. The Lyapunov direct method guarantees the synchronization for coupled networks in a larger region and the range depends on the Lyapunov function that has been found. However, in general it is hard to find an appropriate Lyapunov function. Based on the above two methods, many research works have been done on the synchronization from different points of view. In [11], the network is considered as a fixed topology and the coupled network with maximal synchronization has non-diagonal coupling matrix. In [12], the interaction of oscillators is considered as the information of diffusion process. Some research works investigated the synchronization for networks with statistics feature, such as the small-wold network [13] or the scale-free network [14]. In [15]–[17], the authors have shown that randomly adding shortcuts can translate a regular network to a small-world network and improve the network’s synchronization [18]. In [19], Wang and Chen discussed the synchronization of scale-free network. In [20], [21], the synchronization of the scale-free network with degree-degree correlation is considered. The time-delayed coupling effects have also been considered for different classes of systems such as complex network with time-varying delayed coupling in [21]–[23]. The master-slave synchronization problems subject to communication channel noise and bandwidth limitations were studied in [24] and [25]. In [26] and [27], the authors dealt with the master-slave Lurie systems by using time-delayed feedback control and sample-feedback control, respectively. In the real world, external disturbances are ubiquitous and may lead a given system to an unanticipated state and even destroy the synchronization. Hence, it is deserving to investigate the ability of synchronization to resist the external disturbance, synchronization problem. The synchronizai.e., the tion problems have been studied by many authors [28]–[35]. synchronizaIn [28] and [29], the authors considered the tion between the master and slave of chaotic oscillators. Fursynchronization has been applied to time-dethermore the layed chaotic system [30], [31], [35], and neural systems [32]. method is also utilized to design an antisynchronizaThe tion controller between drive and response system [36].
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In recent years, a great deal of concern has been raised regarding the study of PDSs [37]–[48]. Many phenomena in science, engineering and biology have been modeled in the spatio-temporal domain by PDSs such as in chemical engineering, biology, population dynamics, neurophysiology and biodynamics, etc., therefore, the synchronization problem of PDSs is addressed in this paper. In this study, N-coupled linear PDSs with constant coefficients via spatial diffusion coupling are discussed at first, then a synchronization error dynamic of coupled PDSs is defined in the spatial domain. Based on the synchronization error dynamic of the coupled PDSs, the asymptotical synchronization problem becomes an asymptotical stabilization problem. Using the Lyapunov-Krasovskii techniques, the asymptotical synchronization criteria for these coupled linear PDSs are derived. Aside from synchronization, the disturbance-resisting ability of synchronous PDSs has also been investigated in this study. Based on the disturbance attenuation theory in temporal domain [49]–[54], the spatio-temporal disturbance attenuation level of a coupled PDSs network is considered as the upper bound of the gain from any finite energy disturbance to the synchronization error in the spatio-temporal domain. In this study, based on the synchronization error dynamic with disturbances, sufficient conditions are obtained for the synchronization of N-coupled PDSs with disturbances in the spatio-temporal domain. Except the coupled PDSs with constant coefficients, the coupled PDSs with space-dependent coefficients appear in electrostatics, stationary heat transfer and other diffusion problems for inhomogeneous media and the properties are studied by many researchers [55], [56]. This study also considers the synchronization for N-coupled PDSs with space-dependent coefficients. In term of Lyapunov-Krasovskii method, the sufficient condition to guarantee the synchronization is obtained. Similarly, the synchronization for N-coupled PDSs with space-dependent coefficients and disturbances is studied and the criterion of synchronization is also derived by virtue of Lyapunov-Krasovskii method. These criteria need to verify the corresponding integral inequalities, which can not be solved analytically at present, even if an appropriate Lyapunov-Krasovskii functional is given. In order to overcome this difficulty and make these criteria to be verified easier, the semi-discrete finite difference scheme is employed to represent the synchronization error dynamic. Under this situation, the synchronization criterion of coupled PDSs needed to be investigated becomes an LMI, which can be efficiently solved with the help of LMI toolbox in Matlab [57]. The relationship between the sufficient conditions for the synchronization, obtained by the Lyapunov-Krasovskii method and semi-discretedifferencemethodrespectively,isalsoinvestigated. Finally, two examples are given to illustrate the correctness of the results we got in this paper. The criteria of asymptotical synchronization with constant coefficients are verified by the numerical simulations. The effect of the spatial domain on the synchronization is also discussed. The synchronization of two-coupled PDSs with space-dependent coefficients is investigated when the spatio-temporal disturbances are considered. For convenience, we adopt the following standard notations: and denote, respectively, the dimensional Euclidean space and the set of all real matrices. . The superscript ‘ ’ denotes the
transpose and the notation , where and are symmetric matrices, means that is positive semidefinite(positive definite). is the identity matrix with compatible dimension. stands for the space of functions with continuous partial derivatives of order less than or equals to in . (or ) denotes the set of square integral functions on . II. MATHEMATICAL MODEL AND SYNCHRONIZATION ERROR DYNAMIC We consider the following N-coupled linear PDSs with constant coefficients:
(1) , for and , where . the state variables and are space and time variables respectively, are constant system matrices and we assume are symmetric matrices and . The Laplace operator for two-dimensional spatio-space is defined as follows:
.. .
.. .
(2)
. for The boundary conditions and initial values are given as follows: (3) The linear PDSs in (1) are coupled through diffusion in spatio-space with constant diffusion . coefficients be a function to which all ’s are expected Let satisfies the following equation: to synchronize and (4) subject to the boundary conditions and initial values as follows: (5) Let us define the synchronization error as follows: (6) Definition 1: The coupled linear PDSs in (1) are asymptotically synchronous if the synchronization error satisfies for all and . By transforming the coupled PDSs in (1) with state vector to synchronization errors in (6), the synchronization problem of PDSs in (1) turns to the stabilization
WU AND CHEN: SYNCHRONIZATION OF PARTIAL DIFFERENTIAL SYSTEMS
problem of the corresponding synchronization error dynamical system in the following
(7) subject to the boundary conditions and initial values as follows: Take ..
.. . and
..
.
.. . . Obviously,
..
.
.
.. .
is symmetric. Denote
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III. SYNCHRONIZATION AND SYNCHRONIZATION COUPLED PDSS WITH CONSTANT COEFFICIENTS
OF
Based on the synchronization error dynamic of PDSs constructed in the last section, the asymptotical synchronization problem of coupled PDSs becomes the stabilization problem of synchronization error dynamic of PDSs. For the N-coupled linear PDSs in (1) and their synchronization error dynamic in (8), the asymptotical synchronization of coupled PDSs in (1) implies the asymptotical stability of synchronization error dynamic in (8). of the synDefinition 3: The equilibrium point chronization error dynamic in (8) is asymptotically stable for all , if as for all . Remark 2: The definition 3 of the asymptotical stability is essentially the strong stability in view of the distributed parameter system. For the distributed parameter systems, there are other definitions of stability, for the details, we refer to the reference [58] and the references therein. Now we present an important lemma which is stated in [59]. . Here, we assume ( , 2) and let Lemma 1: Let be a cube: be a real-valued function belonging to which vanishes of , i.e., . Then on the boundary (12)
then the synchronization error dynamic of N-coupled PDSs with constant coefficients in (7) can be represented by (8) If the coefficients depend on the space variable, i.e., are space-dependent, then the system (1) becomes the N-coupled PDSs with space-dependent coefficients
Using lemma 1, we can get the following lemma, which will be used in the sequel. ( , 2) and let Lemma 2: Let be a cube: be a function belonging to which vanishes on the boundary of . Then
Proof: From Green’s identity [60], using Lemma 1, we (9)
have
The synchronization error dynamic (7) becomes
(10) Similarly, the synchronization error dynamic of N-coupled PDSs with space-dependent coefficients in (10) can be represented by (11) and are defined similarly as and , respecwhere tively. , but the reRemark 1: Here we study the case: , sults in the sequel can be extended to the case: without any difficulty. To end this section, we present a definition, which will be used in the sequel. is positive-definite and raDefinition 2: The function dially unbounded if is positive except at where and as .
where indicates differentiation in the direction of the exterior normal to . Now we are in the position to state our first theorem on the asymptotical synchronization of N-coupled PDSs with constant coefficients:
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Theorem 1: Suppose there exists a positive symmetric matrix such that
consider the robustness to resist external disturbance of the coupled PDSs. We consider the following coupled PDSs with disturbance
(13) and
(18) (14)
then the synchronization error system (8) is asymptotically stable, that is the N-coupled PDSs (1) are of asymptotical synchronization. Proof: Define a Lyapunov-Krasovskii functional . Keeping in mind that is symmetric, we have
where stands for the external disturbance in is square intethe spatio-temporal domain and for any gral, i.e., and some positive constant . , then we get the following Let coupled synchronization error dynamic
(19) Let
Take , and as before, then we get the disturbed synchronization error dynamic as follows: (15) Since
, there exists a matrix , then
such that .
Let
, noting , using Lemma 2, we have
for
(20) Assuming that the coupled linear PDSs are in the synchro(i.e., ), we consider the nization state at disturbance attenuation of the synchronization error dynamic in (20) as follows:
or (16) (21)
Substituting (16) into (15) and using (14), we can get If the initial condition is considered, then
(17) , we get that, along the solution of system (8), as . Take is the smallest eigenvalue of , we have , then we get as , as . We get the asymptotical which implies stability of system (8), that is, the system (1) is asymptotically synchronous. Remark 3: From (14), the asymptotical synchronization can be verified by the systems’ coefficients, i.e., and and the range of space domain, i.e., and . Since the external disturbance is inevitable, and disturbance may destroy the synchronization of the coupled PDSs, we must Since
(22) . for some positive function If the synchronization error dynamic in (20) satisfies the disturbance attenuation (21) or (22), we say the coupled PDSs are synchronization with a prescribed attenuation level of the . Now, we state a theorem on the synchronization, which synchropresents the sufficient conditions to guarantee the nization of systems (18). , Theorem 2: Given a disturbance attenuation level suppose that there exists a positive matrix such that (23)
WU AND CHEN: SYNCHRONIZATION OF PARTIAL DIFFERENTIAL SYSTEMS
and (24) synchronization in then the coupled PDSs (18) are of the (21) or (22). . Using Proof: Let Lemma 2 and the techniques of completing the squares, we have
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, and the inequality (21) holds, therefore, we conclude that the synchronization. coupled PDSs (18) are of the Remark 4: syn(i) The conditions (23) and (24) to guarantee the chronization are also sufficient for the asymptotical synchronization since means
which implies that the synchronization of (18) indicates the asymptotical synchronization of (1) when . synchronization in (23) and (24) can be verified (ii) The by the systems coefficients and , the range of space domain and and the prescribed disturbance attenuation level . In addition, the optimal disturbance attenuation level of synchronous PDSs can be measured by
(25) that is the inequality (22). If , i.e., the coupled PDSs , then (18) are in the synchronization state at
(iii) From the conditions (14) and (24), we see that for the becomes same system coefficients, if the constant is smaller, then insmaller, i.e., the space domain equalities of (14) and (24) are easier to hold, i.e., it is more easy to achieve the synchronization through spatial diffusion coupling in a small area. Therefore, we can get the following proposition: Proposition 1: The asymptotical synchronization and synchronization of coupled PDSs depend on the scale of the spatial domain: the smaller the spatial domain, the easier to achieve synchronization. the asymptotical synchronization or the Remark 5: In many paper, the structure of the network was considered for the synchronization problem, see, for example, [19], [32]. The synchronization of a coupled network system is not only determined by the structure of the coupled network, represented by the matrix in this paper, but also the dynamical properties of the subsystem without coupling, represented by the matrices and . In this paper, the sufficient conditions for the synchronization for the PDSs are dependent on the ma. It means trices and , which contain the matrices the structure and the properties of the subsystem are considered both for the synchronization. and the disturbance attenuation level are If the matrices given, then we can discuss what conditions should be imposed to make the inequalities (23) and (24) have on the matrix a positive solution. That is the design problem of the coupled network to achieve the synchronization. To see it clearly, we , where is a given constant, assume the matrix is a fixed diagonal matrix, if there is a connection between , , otherwise, node and node . This form of matrix was adopted in [19]. Take , then . Now, the topology of the coupled network can be completely presented by the matrix . For a strong connected graph, we know that the eigenvalues of are negative, then if (positive diffusion will harm the synchronization), from inequalities (23) and (24), we can see
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the larger to be, which represents the strength of coupling, the easer to achieve the synchronization. Moreover, if the network is globally coupled, then the eigenvalues of the matrix are 0 (the eigenvalue 0 is simple, see [19] for the details), and , the more nodes of the network have, the thus for a given easer to achieve the synchronization. To end this section, we figure out other results. In the literatures, many researchers studied the synchronization of coupled , where is systems using the error the state of the coupled systems, see [10]. Now, we also take the , and for the synchronization error simplicity, we just consider the two-coupled PDSs. We will give synthe criteria for the asymptotical synchronization and chronization for two-coupled PDSs and present the relationship between these results and the results we got before. We consider the following two-coupled PDSs
Theorem 4: Given a disturbance attenuation level , if the following inequalities hold (33) and (34) then, if
, we have
(35) synchronization. that is, the coupled PDSs (28) are of and , we can easily verify this Take theorem using the techniques in the proof of Theorem 2. Now we point out the relationship of Theorems 1, 2 and the Theorems 3, 4. First, we claim that
(26) subject to the boundary conditions and initial values Now we give a brief proof. Since (27) and the disturbed two-coupled PDSs
holds for any
(28)
where and are the external disturbances and , and we also impose the satisfy boundary conditions and initial values (27). From (26), (28) and keeping in mind , we have the following synchronization error dynamics (29) and
(30) where
. Obviously,
. Now we present the following theorem on synchronization of system (26). Theorem 3: For the synchronization error dynamic of two coupled PDSs in (29), if the following inequalities hold (31) (32) of (29) is asymptotically then the equilibrium point stable, that is, the two-coupled linear PDSs in (26) are asymptotically synchronous. The proof is similar to the proof of Theorem 1, just take and in the proof of Theorem 1, so we omit it. synchronization, we have the following theorem. On the
which implies
, take
, we get
. Using the same technique, we get
implies
From these results, we can get, without any difficulty, that if we in Theorem 1, then the conditions (13), (14) imply take (31), (32), i.e., the Theorem 1 implies the Theorem 3. The same relationship exists for Theorem 2 and Theorem 4. IV. ASYMPTOTICAL SYNCHRONIZATION AND SYNCHRONIZATION OF N-COUPLED PDSS WITH SPACE-DEPENDENT COEFFICIENTS In this section, we consider the N-coupled PDSs with spacedependent coefficients. We must point out that the technique used in (16) is not suitable under this situation since the coefficients are space-dependent. Therefore, we can not get analogous results as the constant coefficient case. Now, we employ the Lyapunov-Krasovskii method to present a theorem on the asymptotical synchronization of N-coupled PDSs with space-dependent coefficients. Theorem 5: For the synchronization error dynamic of N-coupled PDSs with space-dependent coefficients (11), if there ex, ists a positive-definite and radially unbounded function , such that the following integral inequality holds for : all
(36)
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then the equilibrium point of (11) is asymptotically stable, that is, the coupled linear PDSs in (9) is asymptotically synchronous. . Proof: Let
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Proof: Using the technique of completing the squares, we have
(37) , we can get as By virtue of , then as . We can get that the of (11) is asymptotically stable, i.e., the equilibrium N-coupled linear PDSs in (9) are asymptotically synchronous. Remark 6: If the following inequality holds for some positive ,
for all , then the asymptotical synchronization result in Theorem 5 still holds but with stricter condition than (37). synchronization of the Now we turn our attention to the following N-coupled PDSs with space-dependent coefficients
(38) subject to the boundary conditions and initial values (39) where
is the external disturbance and , for some positive constant . We modify the synchronization performance in (22) as
(40) We can get the synchronization error dynamic as follows:
Similarly, the above system can be presented by the following system (41) Theorem 6: Given a disturbance attenuation level , if there , such that the folexists a positive differentiable function lowing integral inequality holds
(42) then the coupled PDSs (38) are of
synchronization.
Then the inequality (40) holds. If and , synchronization in (21) holds, which completes then the the proof. Remark 7: If the following inequality holds for some positive , differentiable function
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for all and , then the synchronization result in Theorem 6 still holds but with stricter condition than (42). Remark 8: It is easy to see inequality (42) implies inequality (36), since that is, the synchronization guarantees the asymptotical . synchronization when Remark 9: Both the above results, Theorem 5 and Theorem 6, present sufficient conditions to guarantee the asymptotical synsynchronization of coupled PDSs, rechronization and the spectively. However, as we all know, they are not practical: It is not easy to find a suitable Lyapunov-Krasovskii functional such that the inequalities (36) and (42) hold, respectively. Therefore, we must proceed to find a practical method to deal with the synchronization of coupled PDSs with space-dependent coefficients. SYNCHRONIZATION CRITERIA BASED ON LMI V. synchronization criteria for coupled To simplify the PDSs with space-dependent coefficients, we need to develop a more suitable spatial state space model to represent the PDSs. For this purpose, the semi-discrete finite difference scheme is synchronization problem of coupled employed to treat the PDSs with space-dependent coefficients. The finite difference scheme is used to approximate the partial differential operator to simplify the synchronization criteria of coupled PDSs. The finite-difference scheme has been widely used to obtain numerical solutions of PDSs [45], [46]. In this paper, we use the finite-difference method to represent the PDSs (38) to get the simpler synchronization criteria. Consider a typical grid mesh, is presented by at the as shown in Fig. 1, the state , where grid node and , i.e., . Note , , or that the grid nodes are the grid nodes at the boundary. At the interior points of grid, the central-difference approximation for the differential operator can be written as follows:
(43) is the local truncation error. The remainder term A finite-difference model can be constructed to represent of synchronization error dynamic at in (41) as follows:
(44)
Fig. 1. Finite-difference grids on the spatio-domain.
Let us denote
(45) then (44) can be rewritten as follows:
(46) , . where For the simplification of the asymptotical synchronization to criterion for the coupled PDSs, we define a spatial vector at all gird nodes in Fig. 1. From the boundary collect the at the boundary are conditions, we know the values of for fixed. Therefore, the spatial state vector at all grid nodes is defined as
(47) where . Note that is the dimension of the for each grid node and is the number of vector grid nodes. in In order to simplify the index of the node , we denote the symbol the spatial state vector to replace . Note that the index from 1 to , i.e., , , , , , where in (47). The difference model of the two indexes in (46) could be represented with only one index as follows: (48) , with and in (49), shown at the bottom of the page, where zero matrix and denotes the matrix.
is defined as denote the identity
(49)
WU AND CHEN: SYNCHRONIZATION OF PARTIAL DIFFERENTIAL SYSTEMS
We collect all synchronization errors of the grid nodes to the synchronization state vector in (47). Take
..
..
.
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Proof: Let of completing the squares, we get
. In term of the technique
. (50)
and .. .
.. .
then (48) can be represented by the following spatial space state system
(51) represents at all grid points on the spatial domain , . , (51) can be rewritten as follows: Take
where
(52) Suppose the external disturbance is absent and the trunof the finite difference scheme approaches cation error zero as the number of grid nodes is large enough, i.e., . Then the asymptotical stability of system (52) can be guaranteed by the following theorem. Theorem 7: For the augmented system in (52) with , suppose there exists a symmetric positive matrix such that the following inequality holds (53) of system (52) is asymptotithen the equilibrium cally stable. The proof is standard, so we omit it. From the practical point of view, the external disturbance and the truncation error will not be zero, then we turn our attention synchronization. For the spatial state space system as to the synchronization in (52), for some positive constant , the performance is described by (54) for some positive function . Theorem 8: For the augmented system in system (52) with a prescribed noise attenuation level , if there exists a positive matrix such that the following LMI holds: (55) then the inequality (54) holds, and the coupled PDSs (38) are of synchronization. Furthermore, the asymptotical synchro. nization could be achieved when
Obviously, if , we can get the desired result (54). By Schur Complement, the LMI (55) implies the above relationship. The last desired result is obvious, since
implies From Theorem 7, the asymptotical synchronization is achieved . when The optimal noise attenuation level can be obtained by solving the following constrained optimization
(56) This constrained optimization can be easily solved by decreasing until no existence of positive solution in the LMI (55), which can be easily solved by the LMI toolbox in Matlab. Now we present the relationship between the integral inequality (42) and the LMI (55).
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First, we take in (42), then with the semi-discrete difference approximation and approximation of integration by summation, the left of integral inequality (42) becomes the first equation shown at the bottom of the page. For the simplicity, we denote the symbols , , and to replace , , and , where . Noting , we have the second equation shown at the bottom of the page. Take , we have the last equation shown at the are defined in botton of the page, where , , and . (50) and We can read from the last equality, for small enough , results in
that indicates the integral inequality (42) with We also know that the inequality
.
is equivalent to the LMI (55). Then we get the following statement: Proposition 2: For small enough , if the LMI (55) holds, synchronizathen the integral inequality (42) holds and the tion of the coupled disturbed PDSs (36) is guaranteed. VI. NUMERICAL EXAMPLES In this section, we give two examples to illustrate the correctness of our results. Example 1: First, we consider the following three-coupled PDSs with constant coefficients, shown in (57) at the bottom , of the next page, where and
WU AND CHEN: SYNCHRONIZATION OF PARTIAL DIFFERENTIAL SYSTEMS
Fig. 2. Simulation results of , and right) of coupled PDSs in Example 1 at different time: (a) . (c)
(from left to , (b) ,
We put the boundary conditions and initial conditions for system (57) as follows:
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Fig. 3. Simulation results of , and right) of coupled PDSs in Example 1 at different time: (a) . (c)
(from left to , (b) ,
Now we will illustrate the effect of the domain on the syn, and we provide the chronization. We take in Fig. 5. It can difference between each two states at be seen that the larger domain makes the coupled PDSs more time achieve asymptotical synchronization, and this can show the correctness of Proposition 1. Now, we present another example of disturbed two-coupled PDSs with space-dependent coefficients. Example 2: We consider the following disturbed two-coupled PDSs with space-dependent coefficients:
(58) where
Take in the Theorem 1, it is easy to verify the conditions (13) and (14) hold under this situation. The state , are shown in the Fig, 2, 3 at , , , respectively. We can read from these figures that the coupled PDSs are of asymptotical synchronization. Here we also provide figures to illustrate the difference between each two states at some fixed space point. We take , from Fig. 4, we can also see the asymptotical synchronization of the coupled PDSs (57).
(57)
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Fig. 5. Effect of the space domain on the synchronization in Example 1. It is seen that a larger space domain makes the coupled PDSs take more time to ; (b) achieve asymptotical synchronization. (a) .
Fig. 4. Difference between each two states at at ; (b) at ; (c) at .
in Example 1. (a) at ; (d)
We consider the synchronization of two-coupled PDSs , and (58). We take , then . According (49) and (50), we and . From the LMI (55), we can get a get the matrix , positive-definite matrix , that is, the coupled systems in (58) synchronization. Moreover, it can be obtained that are of by solving (56). the optimal noise attenuation level Take the boundary conditions and initial values as follows:
We show the synchronization of the coupled PDSs (58) by Fig. 6, where the difference between state and are illus. trated at
Fig. 6. Difference between two coupled PDSs with space-dependent coeffiand at ; (b) Difcients in Example 2. (a) Difference between and at . ference between
The synchronization performance can be computed by simulation as follows:, here we take , is obtained by the LMI (55).
VII. CONCLUSIONS This paper considered the asymptotical synchronization and synchronization for the coupled PDSs with constant coef-
WU AND CHEN: SYNCHRONIZATION OF PARTIAL DIFFERENTIAL SYSTEMS
ficients and with space-dependent coefficients respectively. We constructed the synchronization error dynamic and turned the synchronization problems of coupled PDSs to the stabilization problems of the synchronization error dynamic. Based on the synchronization error dynamic, we presented sufficient conditions to guarantee the asymptotical synchronization which can be verified using the system’s coefficients of the coupled PDSs and the range of the space domain. We also give the criterion synchronization for N-coupled PDSs with constant of the coefficients. The effect of spatial domain on the synchronization can be read from these criteria. We found that it needs to take more time for coupled PDSs to achieve synchronization in a lager rang of space domain. For the N-coupled PDSs with space-dependent coefficients, we first studied the sufficient consynditions for the asymptotical synchronization and the chronization with the Lyapunov-Krasovskii method. These sufficient conditions were given by integral inequalities. Generally, they are difficult to be solved. To overcome these difficulties, we adopted the semi-discrete difference method to turn the coupled PDSs into an equivalent spatial state space system, then a sufficient condition is presented based on an LMI, which guarsynchronization and is easy to be verified. Two antees the numerical simulations are given to illustrate the correctness of our results. REFERENCES [1] A. Pikovsky, M. Rosenblum, and J. Kurths, Synchronization: A Universal Concept in Nonlinear Sciences. Cambridge, U.K.: Cambridge Univ. Press, 2002. [2] F. Varela, J. Lachaux, E. Rodriguez, and J. Martinerie, “The brainweb: Phase synchronization and large-scale integration,” Nat. Rev. Neurosci., vol. 2, no. 4, pp. 229–239, Apr. 2001. [3] M. Bier, B. Bakker, and H. Westerhoff, “How yeast cells synchronize their glycolytic oscillations: A perturbation analytic treatment,” Biophys. J., vol. 78, no. 3, pp. 1087–1093, Mar. 2000. [4] G. Kozyreff, A. Vladimirov, and P. Mandel, “Global coupling with time delay in an array of semiconductor laser,” Phys. Rev. Lett., vol. 85, no. 18, pp. 3809–3812, Oct. 2000. [5] Q. Li and D. Rus, “Global clock synchronization in sensor networks,” IEEE Trans. Comput., vol. 55, no. 2, pp. 214–226, Feb. 2006. [6] K. Lian, C. Chiu, T. Chiang, and P. Liu, “LIM-based fuzzy chaotic synchronization and communications,” IEEE Trans. Fuzzy Syst., vol. 9, no. 4, pp. 539–553, Aug. 2001. [7] K. Lian, T. Chiang, C. Chiua, and P. Liu, “Synthesis of fuzzy model-based designs to synchronization and secure communications for chaotic systems,” IEEE Trans. System, Man Cybern., B. Cybern., vol. 31, no. 1, pp. 66–83, Feb. 2001. [8] M. Chen, “Chaos synchronization in complex networks,” IEEE Trans. Circuits Syst. I, Reg. Papers, vol. 55, no. 5, pp. 1335–346, Jun. 2008. [9] L. Pecora and T. Carroll, “Master stability functions for synchronized coupled systems,” Phys. Rev. Lett., vol. 80, no. 10, pp. 2109–2112, Mar. 1998. [10] C. W. Wu and L. Chua, “Synchronization in an array of linear coupled dynamical systems,” IEEE Trans. Circuits Syst. I, Fundam. Theory Appl., vol. 42, no. 8, pp. 430–447, Aug. 1995. [11] T. Nishikawa and A. Motter, “Synchronization is optimal in nondiagonalizable networks,” Phys. Rev. E, vol. 73, no. 6, p. 065106, Jun. 2006. [12] A. Motter, C. Zhou, and J. Kurthus, “Network synchronization, diffusion and the paradox of hetergeneity,” Phys. Rev. E, vol. 71, no. 1, p. 016116, Jan. 2005. [13] D. Watts and S. Strogatz, “Colletive dynamics of small-world networks,” Nature, vol. 393, no. 6684, pp. 440–442, Jun. 1998. [14] A. Barabasi and R. Albert, “Emergency of scaling in random networks,” Science, vol. 286, no. 5439, pp. 509–512, Oct. 1999. [15] M. Barahona and L. Pecora, “Synchronization in small-world systems,” Phys. Rev. Lett., vol. 89, no. 5, p. 054101, Jul. 2002. [16] X. Wang and G. Chen, “Synchronization in small-world dynamical networks,” Int. J. Bifurc. Chaos, vol. 12, no. 1, pp. 187–192, 2002.
2667
[17] J. Lu, X. Yu, G. Chen, and D. Cheng, “Characterizing the synchronizability of small-world dynamical networks,” IEEE Trans. Circuits Syst. I. Reg. Paper, vol. 51, no. 4, pp. 787–796, Apr. 2004. [18] C. Yin, B. Wang, W. Wang, and G. Chen, “Geographical effect on small-world network synchronization,” Phys. Rev. E, vol. 77, no. 2, p. 027102, Feb. 2008. [19] X. Wang and G. Chen, “Synchronization in scale-free dynamical networks: Robustness and fragility,” IEEE Trans. Circuits Syst. I, Fundam. Theory Appl., vol. 49, no. 1, pp. 54–62, Jan. 2002. [20] Z. Fei, H. Gao, and W. X. Zheng, “New synchronization stability of complex networks with an internal time-varying coupling delay,” IEEE Trans. Circuits Syst. II, Exp. Briefs, vol. 56, no. 6, pp. 499–503, Jun. 2009. [21] W. He and J. Cao, “Exponential synchronization of hybrid coupled networks with delay coupling,” IEEE Trans. Neural Netw., vol. 21, no. 4, pp. 571–583, Apr. 2010. [22] J. Liang, Z. Wang, Y. Liua, and X. Liu, “Robust synchronization of an array of coupled stochastic discrete-time delayed neural networks,” IEEE Trans. Neural Netw., vol. 19, no. 11, pp. 1910–1921, Nov. 2008. [23] A. Kruszewski, W. Jianga, E. Fridman, J. P. Richard, and A. Toguyeni, “A switched system approach to exponential stabilization through communication network,” IEEE Control Syst. Technol., 2011, 10.1109/TCST.2011.2159793. [24] B. Liu, D. J. Hill, and J. Yao, “Global uniform synchronization with estimated error under transmission channel noise,” IEEE Trans. Circuits Syst. I, Reg. Papers, vol. 56, no. 12, pp. 2689–2702, Dec. 2009. [25] A. Fradkov, B. Andrievsky, and B. J. Evans, “Synchronization of passifiable Lurie systems via limited-capacity communication channel,” IEEE Trans. Circuits Syst. I, Reg. Papers, vol. 56, no. 2, pp. 430–439, Feb. 2009. [26] J. G. Lu and D. J. Hill, “Global asymptotical synchronization of chaotic Lur’s systems using sampled data: A linear matrix inequality approach,” IEEE Trans. Circuits Syst. II, Exp. Briefs, vol. 55, no. 6, pp. 586–590, Jun. 2008. [27] M. Zhong and Q. L. Han, “Fault-tolerant master-slave synchronization for Lur’s systems using time-delay feedback control,” IEEE Trans. Circuits Syst. I, Reg. Papers, vol. 56, no. 7, pp. 1391–1404, Jul. 2009. [28] Y. Hou, T. Liao, and J. Yan, “ synchronization of chaotic systems using output feedback control design,” Phys. A: Stat. Mech. Appl., vol. 379, no. 1, pp. 81–89, Jun. 2007. [29] S. M. Lee, D. H. Ji, J. H. Park, and S. C. Won, “ synchronization of chaotic systems via dynamic feedback approach,” Phys. Lett. A, vol. 372, no. 29, pp. 4905–4912, Jul. 2008. [30] J. H. Park, D. H. Ji, S. C. Won, and S. M. Lee, “ synchronization of time-delayed chaotic systems,” Appl. Math. Compt., vol. 204, no. 1, pp. 170–177, Oct. 2008. [31] C. K. Ahn, “T-S fuzzy synchronization for chaotic systems via delayed output feedback control,” Nonlinear Dyn., vol. 59, no. 4, pp. 535–543, Mar. 2010. [32] B. S. Chen, C. H. Chiang, and S. K. Nguang, “Robust synchronization design of nonlinear coupled network via fuzzy interpolation methods,” IEEE Trans. Circuits Syst., I, Reg. Papers, vol. 58, no. 2, pp. 349–362, Feb. 2011. [33] H. Huang and G. Feng, “Robust synchronization of chaotic Lurie’s systems,” Chaos, vol. 18, no. 3, p. 033113, Sep. 2008. [34] J. H. Park, D. H. Ji, S. C. Won, and S. M. Lee, “Adaptive synchronization of unified chaotic systems,” Mod. Phys. Lett. B, vol. 23, no. 9, pp. 1157–1169, 2009. [35] H. R. Karimi and H. Gao, “New delay-dependent exponential synchronization for uncertain neural networks with mixed time delays,” IEEE Trans. Syst., Man Cybern., B. Cybern., vol. 40, no. 1, pp. 173–185, Feb. 2010. [36] C. K. Ahn, “An approach to antisynchronization for chaotic systems,” Phys. Lett. A, vol. 373, no. 20, pp. 1729–1733, Apr. 2009. [37] K. Liu, Stability of Infinite Dimensional Stochastic Differential Equations With Applications. New York: Chapman & Hall/CRC, 2006. [38] P. L. Chow, Stochastic Partial Differential Equations. New York: Chapman & Hall/CRC, 2007. [39] B. S. Chen and Y. T. Chang, “Fuzzy state space modeling and robust observer-based control design for nonlinear partial differential systems,” IEEE Trans. Fuzzy Syst., vol. 17, no. 5, pp. 1025–1043, Oct. 2009. [40] Y. T. Chang and B. S. Chen, “A fuzzy approach for robust reference tracking control design of nonlinear distributed parameter time-delay systems and its application,” IEEE Trans. Fuzzy Syst., vol. 18, no. 6, pp. 1041–1057, Dec. 2010.
2668
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[41] C. I. Byrnes, I. G. Lauk’o, D. S. Gilliam, and V. I. Shabov, “Output regulation for linear distributed parameter systems,” IEEE Trans. Automat. Contr., vol. 45, no. 12, pp. 2236–2252, Dec. 2000. [42] C. L. Lin and B. S. Chen, “Robust observer-based control of large flexible structures,” J. Dyn. Syst. Maes. Control-Trans. ASME, vol. 116, pp. 713–722, Dec. 1994. [43] H. T. Banks, Modeling and Control in the Biomedical Sciences. New York: Springer-Verlag, 1975. [44] C. V. Pao, Nonlinear Parabolic and Elliptic Equations. New York: Plenum Press, 1992. [45] J. C. Strikwerda, Finite Difference Schemes and Partial Differential Equations, 2nd ed. Philadelphia, PA: SIAM, 2004. [46] G. Evans, J. Blackledge, and P. Yardley, Numerical Methods for Partial Differential Equations. : Springer-Verlag, 2000. [47] Y. Y. Chen, Y. T. Chang, and B. S. Chen, “Fuzzy solutions to partial differential equations: Adaptive approach,” IEEE Trans. Fuzzy Syst., vol. 17, no. 1, pp. 116–127, Feb. 2009. [48] E. Fridman and Y. Orlov, “An LMI approach to boundary control of semilinear parabolic and hyperbolic systems,” Automatica, vol. 45, no. 9, pp. 2060–2066, Sep. 2009. [49] A. Isidori and A. Astolfi, “Disturbance attenuation and control via measurment feedback in nonlinear systems,” IEEE Trans. Autom. Contr., vol. 37, no. 9, pp. 1283–1293, Sep. 1992. [50] H. Gao, J. Lam, L. Xie, and C. Wang, “New approach to mixed filtering for polytopic discrete-time systems,” IEEE Trans.Signal Process., vol. 53, no. 8, pp. 3183–3192, Aug. 2005. [51] B. S. Chen, W. H. Chen, and H. Wu, “Robust global linearization filter design for nonlinear stochastic systems,” IEEE Trans. Circuits Syst. I, Reg. Paper, vol. 56, no. 7, pp. 1441–1454, Jul. 2009. [52] C. L. Chen, G. Feng, D. Sun, and X. P. Guan, “ output feedback control of discrete-time fuzzy systems with application to chaos control,” IEEE Trans. Fuzzy Syst., vol. 13, no. 4, pp. 531–543, Aug. 2005. [53] D. Huang and S. K. Nguang, “Robust output feedback control of fuzzy systems: An LMI approach,” IEEE Trans. Syst., Man Cybern., B. Cybern., vol. 36, no. 1, pp. 216–222, Feb. 2006. [54] B. S. Chen, C. T. Tseng, and H. C. Wang, “Mixed fuzzy output feedback control for nonlinear uncertain systems: LMI approach,” IEEE Trans. Fuzzy Syst., vol. 8, no. 3, pp. 249–265, Apr. 2000. [55] S. Mikhailov, “Analysis of united boundary-domain integro-differential and integral equations for a mixed BVP with variable coeffcient,” Math. Meth. Appl. Sci., vol. 29, pp. 715–739, 2006. [56] J. Chang and B. Guo, “Identification of variable spacial coefficients for a beam equation from boundary measurements,” Automatica, vol. 43, pp. 732–737, 2007. [57] S. Boyd, L. EI Ghaoui, E. Feron, and V. balakrishnan, Linear Matrix Inequalities in Systems and Control Theory. Philadelphia, PA: SIAM, 1994. [58] J. Oostveen, Strongly Stabilizable Distributed Parameter Systems. Philadelphia, PA: SIAM, 2000.
[59] J. G. Lu, “Global exponential stability and periodicity of reaction-diffusion delayed recurrent neural networks with Dirichlet boundary conditions,” Chaos, Solitons and Fractals, vol. 35, pp. 116–125, 2008. [60] F. John, Partial Differential Equations. New York: Springer-Verlag, 1982.
Kaining Wu received the B.S., M.S., and Ph. D. degrees from the Harbin Institute of Technology, Harbin, Helongjiang, China, in 2003, 2005 and 2009 respectively, all in mathematics. He is currently with the Department of Mathematics, Harbin Institute of Technology (Weihai), Weihai, China, as a lecture. His research interests include robust control, stochastic control and delay differential systems.
Bor-Sen Chen (F’01) received the B.S. degree from the Tatung Institute of Technology, Taipei, Taiwan, in 1970, the M.S. degree from National central University, Chungli, Taiwan, in 1973, and the Ph.D. degree from the University of Southern California, Los Angeles, in 1982. From 1973 to 1987, he was a Lecturer, an Associate Professor and a Professor with the Tatung Institute of Technology. He is currently with the Laboratory of Control and Systems Biology, Department of Electrical Engineering, National Tsing Hua University, Hsinchu, Taiwan, as a Professor of electrical engineering and computer science. He is a Research Fellow with the National Science Council of Taiwan and is the holder of the excellent scholar Chair in engineering. According to the statistics of International Statistical Institute, he had published seven highly cited papers in the last ten years. One of them in the Society for Industrial and Applied Mathematics has been selected by Essential Science Indicator as the most cited paper in the research area of mathematics. His current research interests are control engineering, signal processing and system biology Dr. Chen is the editor of Asian Journal of Control. He is a member of the Editorial Advisory Board of the International Journal of Control, Automation and Systems. He is a member of the Editorial Board of Fuzzy Sets and Systems and BMC Systems Biology. He had been the Editor-in-Chief of the International Journal of Fuzzy Systems from 2006 to 2008 and is currently the Editor-in-Chief of the International Journal of Systems and Synthetic Biology.. Dr. Chen was four times the recipient of the Distinguished Research Award from the National Science Council of Taiwan. He was also the recipient of the Automatic Control medal from the Automatic Control Society of Taiwan in 2001. He was an Associate Editor of IEEE TRANSACTIONS ON FUZZY SYSTEMS from 2001 to 2006.
