Mathematical Problems for Complex NetworksView this Special Issue
Enhancement of the Quality and Robustness in Synchronization of Nonlinear Lur'e Dynamical Networks
In order to improve the synchronous reliability and dependability of complex dynamical networks, methods need to be proposed to enhance the quality and robustness of the synchronization scheme. The present study focuses on the robust fault detection issue within the synchronization for a class of nonlinear dynamical networks composed by identical Lur'e systems. Sufficient conditions in terms of linear matrix inequalities (LMIs) are established to guarantee global robust synchronization of the network. Under such a synchronization scheme, the error dynamical system is globally asymptotically stable, the effect of external disturbances is suppressed, and at the same time, the network is sensitive to possible faults based on a mixed performance. The fault sensitivity index, moreover, can be optimized via a convex optimization algorithm. The effectiveness and applicability of the analytical results are demonstrated through a network example composed by the Chua's circuit, and it shows that the quality and robustness of synchronization has been greatly enhanced.
In daily life, many physical systems can be characterized by various complex network models whose nodes are the elements of the network and the edges represent the interactions among them . Treated as typical versions of large-scale systems, the notion of complex dynamical networks has drawn more and more attentions in recent years [2, 3]. One of the interesting and significant phenomena in complex dynamical networks is the synchronization of all dynamical nodes, which is a kind of typical collective behaviors and basic motions in nature [4–9]. Aiming at deriving global synchronization conditions, attempts have been made to consider the synchronization for a special class of networks composed of nonlinear Lur’e systems [10–12]. The main reason is that, in various fields of theory and engineering applications, vast amounts of nonlinear systems can be represented as the Lur’e type, including the Chua’s circuit , the Goodwin model , and the swarm model . Primary methods of dealing with such problems, among others, are developed under the framework of absolute stability theory .
In order to improve the synchronous reliability and dependability, methods have been proposed to enhance the quality and robustness of the synchronization scheme. Due to the instability and poor performance that caused by noise or disturbances, it is reasonable to take the noise phenomenon into account during the synchronization process of complex dynamical networks [17, 18]. On the other hand, research in fault diagnosis has been gaining increasing consideration worldwide in the past decades [19–23]. One of the key issues related to fault detection is concerned with its robustness. Large amounts of the relevant jobs have been done for the linear systems in order to examine the robust fault detection (RFD) problem (see [22, 23] and the references therein). In a recent work, we have investigated the robust fault sensitive synchronization of nonlinear Lur’e systems coupled in a master-slave fashion . Similarly, in complex dynamical networks, since it is inevitable for faults to happen within each of the single node, a fault-free synchronization process cannot always be guaranteed. Even though, there is a few work concentrating upon the RFD problem of large-scale nonlinear systems, and hardly there is any previous work that brought the notion “fault” into physical aspects such as synchronization of nonlinear dynamical networks.
Based on these considerations, this present study considers the fault detection and disturbance rejection problem within robust synchronization for a class of dynamical networks. The network model is composed by identical nodes with each node being a perturbed nonlinear Lur’e system, while at the same time, subject to possible faults. The main challenge in evaluating the synchronization scheme is to distinguish failures from other disturbances, and accordingly, the paradigm is introduced . For the purpose of description, the robustness objectives during synchronization are considered in virtue of the norm, while the fault sensitivity specifications are expressed by utilizing the formulation of index. In this manner, the closed-loop error system is asymptotically stable with the -norm from the disturbance input to controlled output reduced to a prescribed level, and at the same time, with the performance index maximized. By transforming the synchronization problem of dynamical networks into absolute stability problem of corresponding error systems as well as applying Lur’e system method in control theory , sufficient conditions to the global robust synchronization within nonlinear Lur’e networks are developed in terms of sets of linear matrix inequalities (LMI) . Furthermore, the derived high-dimensional LMI condition is simplified into three groups of lower-dimensional LMIs, which are easier to handle. It should be pointed out that no linearization technique is involved through derivation of all the synchronization criteria.
The rest of the paper is organized as follows. Section 2 proposes the model to be examined in this study, and gives the mathematical formulations of the global robust synchronization problem to be solved. In Section 3, the global robust synchronization scheme of the networks is firstly studied, based on which the criteria on synchronization are then proposed in virtue of the LMI technique. Moreover, performance analysis of the network is also discussed in this part. The dynamical network composed by ten identical Chua’s circuits is adopted as a numerical example in Section 4, and Section 5 closes the paper.
2. Notations and Preliminaries
The notations used in this study are fairly standard. is the set of real matrices. For a matrix , denotes its transpose. He is the Hermit operator with . If is a real symmetric negative definite matrix, it is shown by . implies a diagonal or block-diagonal matrix. indicates the Kronecker product of an matrix and a matrix , that is, If not explicitly stated, matrices are assumed to have compatible dimensions, and the terms replaced by of a matrix refer to the terms in a symmetric position that do not need to be written out.
2.1. Basic Knowledge on Lur’e Systems
The basic model of nonlinear Lur’e systems subject to input noise and possible faults considered in this paper is described by where is the state vector and represents the measurement output vector. is an unknown input vector (including disturbance, uninterested fault as well as some norm-bounded unstructured model uncertainty) belonging to , while denotes the process, sensor, or actuator fault vector to be detected and isolated. Depending on specific situations under consideration, and can be modeled as different types of signals. The matrices and , are known constant matrices with appropriate dimensions. Nonlinearity is continuous and locally Lipschitz in the first argument with and , where the functions are assumed to satisfy the following inequalities: where . Denoting , it is obvious to get and the nonlinearity is said to be in the sector if it satisfies (2.4).
In order to characterize the influence of the disturbance and fault input, several definitions are introduced.
Definition 2.2. Consider the following transfer function of system (2.2): Then its norm is defined as , where represents the maximal singular value.
Definition 2.3. For system (2.2), the transfer function from the input to output is given as: whose index is defined by =, where stands for the minimum singular value and denotes the frequency band .
Remark 2.4. The index defined has been widely adopted to measure the sensitivity of residual to fault in the frequency domain. A system is said to possess a better level of RFD, if the norm of its transfer function from the disturbances to the performance variable is small; meanwhile, the index of the transfer function from fault to the output variable is large . Various kinds of performance criteria have been proposed to determine the RFD issue , and the performance is mostly adopted as a trade-off between robustness and sensitivity. In this study, for the sake of simplicity, we will consider the case of maximizing the fault sensitivity with disturbance attenuation being a prescribed constant.
2.2. Dynamical Networks Composed of Lur’e Nodes
Consider a class of complex dynamical network model with each node being a general Lur’e system (2.2) shown as follows: where are the state and measurement output of the th node, respectively. and are defined as in system (2.2), which are supposed to be the same with respect to each node. The inner coupling matrix denotes the coupling pattern between two nodes. is the outer coupling matrix, standing for the coupling configuration of the network. If there is a connection between node and node , then ; otherwise, . The row sums of are zero, that is, . Let and with the following properties: Denote then the nonlinear function belongs to the sector .
Lemma 2.5 (Wu ). The eigenvalues of an irreducible matrix with satisfy the following.(i)0 is an eigenvalue of associated with the eigenvector .(ii)If for all , then the real parts of all eigenvalues of are less than or equal to 0 and all possible eigenvalues with zero part are 0. In fact, 0 is its eigenvalue of multiplicity 1.
Assume that the network (2.7) has no isolate clusters; namely, the network is connected. Under this circumstance, the coupling matrix is symmetric and irreducible; hence it satisfies all the properties given in Lemma 2.5. Besides, suppose that the coupling matrix has distinct different eigenvalues ; then there exists a nonsingular matrix with such that , where is in the following form: Here, is the maximum eigenvalue of multiply 1 and is the eigenvalue of multiply satisfying and .
Definition 2.6. When , the dynamical network (2.7) is said to achieve global (asymptotical) synchronization if where means the Euclidean norm. is a solution of an isolate node given by which can be an equilibrium point, a periodic orbit, or even a nonperiodic orbit.
From the properties of the internal coupling matrix , the following condition holds: Define error signals and residual signals for . By subtracting (2.12) from (2.7), one arrives at the dynamics of synchronization residual error: where . Let ; it is not difficult to derive from (2.8) that for , the nonlinear functions , , satisfy the following sector restrictions: which leads to and thus also belongs to the sector .
Remark 2.7. Based on the basic knowledge of synchronization, the residual error dynamics must be asymptotically stable in order for the whole process to work. Note that the dynamics of the residual error signal depends not only on , and but also on the states of each isolated node . In consequence, this study aims at ensuring the residual error dynamical system to be sensitive to possible faults in the regard of index, but the error dynamics also remain robustly asymptotically stable to external disturbance in the sense. Under such circumstances, the dynamical network composed of Lur’e nodes is said to achieve global synchronization with a guaranteed performance.
Reformulating system (2.13) in virtue of the Kronecker product  as where with and belonging to the sector . accordingly, the residual error dynamical system (2.16) can be treated as an -dimensional nonlinear Lur’e system, and the synchronization of the nonlinear dynamical networks (2.7) can be transformed into the performance analysis and stabilization problem of the corresponding residual error dynamics (2.16).
For system (2.16), the transfer function is given by , while denotes the transfer function . Specifically speaking, the main objective of this present study is to determine under what condition the residual error dynamics (2.16) could be asymptotically stable and, at the same time, satisfy the following conditions: where is a prescribed positive constant, and is a constant to be optimized. By applying the well-known Parseval theorem to the frequency-domain expressions (2.18), where the ratios norm and index are presented in Definitions 2.2 and 2.3, respectively, we arrive at the equivalent statements as follows:
Accordingly, the definition of robust synchronization is derived as follows.
Definition 2.8. The dynamical networks composed of nonlinear Lur’e nodes in (2.7) are said to achieve global robust synchronization with disturbance attenuation and fault sensitivity over the frequency range (where could be both finite and infinite), if with zero disturbance and zero fault, the synchronization residual error signal (2.16) is asymptotically stable, while with zero initial condition and given constants , conditions (2.19)-(2.20) hold.
3. Main Results
The intention of this part is to investigate the fault sensitivity as well as disturbance rejection ability of the complex dynamical network (2.7). In order to quantify these two performance indices, one borrows the concept of index and -norm defined in the previous section.
3.1. Global Synchronization of Nonlinear Lur’e Networks
In this subsection, we first consider the case that there is no fault existed in the network by extending previous results on synchronization between two identical Lur’e systems to that of nonlinear Lur’e dynamical networks. Accordingly, the network model is described by and the corresponding error dynamics in form of Kronecker product is expressed as The disturbance rejection problem within the synchronization of nonlinear dynamical network (3.1) is summarized in the following definition.
Definition 3.1. Given constant scalar , the dynamical network (3.1) is said to achieve global robust synchronization, if system (3.2) is globally asymptotically stable with zero disturbance, and meanwhile, the performance index (2.19) is satisfied with zero initial conditions.
The robust synchronization can be determined in virtue of the following criterion.
Theorem 3.2. Suppose that is a prescribed constant. For a given scalar , if there exist positive-definite matrices , diagonal matrices , , and , and matrices and such that the following LMI holds, where , then the dynamical network (3.1) achieves global robust synchronization with disturbance attenuation .
Proof. See Appendix A.
Remark 3.3. Theorem 3.2 has provided a sufficient condition for the global robust synchronization of nonlinear Lur’e networks by introducing slack matrices and into LMI (3.3). It is thus expected that Theorem 3.2 will be less conservative than some existing results due to the increasing freedom of these slack variables . With the derived synchronization conditions on Lur’e networks, the fault detection issue will then be examined in the next subsection. However, if the number of nodes is large, condition (3.3) would become a high-dimensional LMI, which is rather tedious to verify. To this end, both of these criteria will be further simplified to the test of three groups of lower-dimensional LMIs.
3.2. Fault Detection within Global Synchronization
The RFD within a synchronization configuration can be treated as a multiple objective design task; that is, the design objective is not only being as sensitive as possible to faults such that early detection of faults is possible, but on the other hand, the sensitivity of possible faults is maximized, also suppressing the effect of disturbances and modeling errors on the synchronization error and subsequently on the residual, in order to prevent the synchronization process from being destroyed. Next theorem gives an LMI formulation for global robust synchronization.
Theorem 3.4. Suppose that are prescribed constant scalars. For a given constant , if there exist a positive-definite matrix , diagonal matrices ,and , and matrices and such that LMIs (3.3) as well as hold, then the dynamical network in (2.7) achieves global robust synchronization with disturbance attenuation and fault sensitivity .
Proof. See Appendix B.
Theorem 3.5. Suppose that and are given scalars. If there exist matrices , and diagonal matrices ,and such that the following conditions for and hold: where and , then the conditions given in Theorem 3.4 are ensured.
Proof. See Appendix C.
Corollary 3.6. For a constant , let and be prescribed constant scalars. The dynamical network (2.7) is said to achieve global robust synchronization with disturbance attenuation and fault sensitivity , if there exist matrices ,, and diagonal matrices such that the LMI conditions (3.5)-(3.6) hold for and (corresponding to the largest, second largest, and smallest eigenvalues, resp.).
Remark 3.7. If the number of nodes is large, the synchronization criterion of the dynamical network would become a group of LMIs with rather high dimensions. In order to tackle this problem, the synchronization of the -dimensional network has been disposed in a lower -dimensional space through verifying three groups of -dimensional LMIs in Corollary 3.6, and the derived conditions are quite convenient to use.
As an immediate consequence, we arrive at the simplified criterion for global robust synchronization of nonlinear dynamical network (3.1) summarized as in the following corollary.
Corollary 3.8. For a constant , let and be prescribed constant scalars. If there exist matrices , , and diagonal matrices such that the LMIs (3.5) for and are feasible, then the dynamical network (2.7) is said to achieve global robust synchronization.
3.3. Performance Analysis
It comes from Corollary 3.6 that the synchronization within a dynamical network can be cast into that of three sets of independent systems whose dimensions are the same as that of each isolate node. Namely, if the following systems satisfy (3.5)-(3.6) for and, then the conditions given in Definition 2.8 will be guaranteed. Suppose the transfer function of system (3.7) from and for as and , respectively. Then denote where are in the following form:
On the other hand, consider the following system: where . It can be found that the transfer functions from and of system (3.10) are just those defined in (3.8). Moreover, by carrying out unitary transformation, is similar to , and so do and . Recall the definition of the norm and index previously stated in Definitions 2.2 and 2.3; then we arrive at the following relationships between the norms of and as well as the indexes of and for :
Conditions (3.11) show that the norm of the transfer function from in (2.16) equals to the maximum of those of the systems (3.7), whilst the corresponding index is the minimum value within those of (3.7). Accordingly, the RFD of the network (2.7) can be cast into those of (3.7); thus we have the following corollary.
The following corollary presents a method of deriving the maximum value of fault sensitivity and, at the same time, suppresses the external disturbance to a prescribed level for the global robust synchronization of network (2.7).
Corollary 3.10. The nonlinear dynamical networks (2.7) are said to achieve global synchronization with guaranteed performance and the maximum fault detection sensitivity , where is the global minimum of the following generalized eigenvalue minimization problem with respect to matrices for as well as diagonal matrices , and : as well as the LMI condition (3.5) holds. Here, and are described in Theorem 3.5 with constant scalars and prescribed.
4. Numerical Examples
A lower-dimensional dynamical network model is concerned in this part so as to demonstrate the applicability and effectiveness of the approaches proposed in the previous sections. Throughout our numerical simulations, each node of the network is supposed to be a concrete Chua’s circuit, which is frequently observed in various fields of theory and engineering applications .
In the first stage, it will be shown that how the results derived in Section 3.1 can be used to guarantee the global robust synchronization of the dynamical network (2.7). Let us take a group of ten dimensionless state equations of Chua’s oscillators, for example, where one of the node system is shown as system in Figure 1, : Here, and are linear resistors. The voltages across the capacitors and are denoted by and is the current through the inductances , and is an external disturbance current that system subjects to. The nonlinear characteristic represents the current through the nonlinear resistor , which is a piecewise-linear function expressed as and it satisfies .
Suppose that each node of the dynamical network developed by (2.7) is a circuit in the form of (4.1). The possible coupling between two arbitrary Chua’s circuits, as shown in Figure 1, indicates that there is a connection from to but none from to , where the element plays the role of unicommunication. Depending on different values of the controller gain, the resistor can be adjusted. It is straightforward to reformulate system (4.1) into the Lur’e form as where and the nonlinear function satisfies the sector condition on . Furthermore, suppose the output equation to be as with parameter matrix . Choose system parameters as , . In the following, Ω is taken. The network topology is assumed as star-like with ten nodes; thus has the eigenvalues as follows: Picking , and prescribing disturbance attenuation , we arrive at the feasible solutions given in Appendix B by solving the LMIs (3.5), which, according to Corollary 3.8, implies that the dynamical network composed of Chua’s circuits has achieved the global robust synchronization.
Simulation results also confirm the effectiveness of the design. Figure 2 depicts the time response of synchronization error of the nominal dynamical network without disturbance signal , and it shows that the synchronization error converges to zero exponentially. Herein, initial values are taken arbitrarily.
To observe the performance with disturbance attenuation, assume the unknown input noise disturbance to be as Accordingly, the time response of the output residual error of Lur’e dynamical network with the above disturbance signals and zero initial conditions are shown in Figure 3.
In what follows, let us consider the global robust synchronization of the dynamical network (4.4) in the presence of fault signal . For the purpose of illustration, the process fault is supposed to be a faulty current flowing in the same direction as along with the leftmost branch of each of the circuits, which will be simulated as two different types. Accordingly, it leads to with and .
Remaining and picking the fault sensitivity , we arrive at solution of the LMI (3.5)-(3.6) with presented in Appendix C, which on its turn ensures that the network (4.4) has achieved global robust synchronization in the presence of possible faults and external disturbances.
As for the corresponding simulation results, first let the process fault be a pulse of unit amplitude occurred from to (and is zero otherwise). The generated residual signals , are depicted in Figure 4(a), from which one observes that the effect of the disturbance input on the residual error signal , has been greatly reduced, and the residuals have rather large amplitudes so that the synchronization process remains sensitive to fault. With the same disturbance , then redo the simulation with a different: and the results are plotted in Figure 4(b).
By solving the generalized eigenvalue problem corresponding to the minimization problem given in Corollary 3.6, we get estimates of the maximum values of fault sensitivity as , , and ., which also guarantees in terms of Corollary 3.6 that Lur’e dynamical networks achieve synchronization with , where .
5. Conclusion and Future Work
Aiming at enhancing the reliability and robustness of synchronization, the global robust synchronization scheme has been introduced into a class of nonlinear dynamical networks in the existence of possible faults and external disturbances. The criterion on synchronization was developed in virtue of the LMI technique such that each of the node systems of the network is robustly synchronized as well as sensitive to faults according to a mixed performance. Since both of the external disturbance and system fault are, respectively, considered, such synchronization scheme proposed here may be more practical than the synchronization in the previous literature. Moreover, the fault sensitivity index could be optimized via a convex optimization algorithm. In order to demonstrate the effectiveness and applicability of the derived results, a low-dimensional dynamical network with each node being a Chua’s circuit has been adopted as an example.
As for future work, it will be interesting to study the synchronization of complex networks with different disturbance from various sources. Also, it is possible to extend the present results to stochastic complex networks [31–34].
A. Proof of Theorem 3.2
Proof. First, we will show the global asymptotical stability of the residual error dynamics (3.2) with (no fault is taken into account here), and accordingly (3.2) is represented as
Under the given conditions, the performance indexes in condition (2.19) are then proven to be satisfied.
Choose a Lyapunov functional candidate in the form of where and with need to be determined. By calculating the time derivative of along with the trajectory of the residual error dynamics (3.2), it yields where . Then consider the sector restrictions that nonlinearities and satisfy, namely, for any diagonal matrices diag and diag: with . The results are obtained with the assumption that each subsystem has the same diagonal matrices , , and which does not affect the feasibility of inequality (2.18). For the sake of simplicity, denote in the following contexts. Moreover, it is known from (A.1) that there exist free-weighting matrices with appropriate dimensions such that Incorporating formulations (A.4)-(A.5) into equality (A.3) derives where and it follows that is guaranteed by the upper left block of LMI (3.3); hence the synchronization residual error dynamics (3.2) is globally asymptotically stable.
In the following, we will show that the restriction on performance index given in (2.19) is satisfied under zero initial conditions for all nonzero . In this case, the error dynamics (3.2) is expressed by Based on (A.6) and (A.8), it is not difficult to derive where , and is described in condition (3.3) with . It further implies that for any , . Under zero initial condition, the Lyapunov function defined in (A.2) satisfies and for , hence and (2.19) is satisfied, which completes the proof.
B. Proof of Theorem 3.4
Proof. On the basis of Theorem 3.2, it is known that if there exist solutions to LMI (3.3), the network achieves global synchronization and robust to input disturbances. As for the condition of fault detection within the synchronization, namely, the synchronization process should be sensitive to possible input faults, it then comes to an extra verification of condition (2.20) under zero initial conditions for all nonzero . In this situation, the error dynamics is given by Following the same line of the proof of in Theorem 3.2, we know that if holds, then the constraint (2.20) will be satisfied where is defined in (A.2), and further, the inequality condition (B.2) is guaranteed by where with given in (3.4). Thus the performance index is satisfied, and the proof is completed.
C. Proof of Theorem 3.5
Proof. To facilitate the design of the coupling matrix , we designate and , respectively, where is a constant scalar; also it can be seen from (3.3) that , and thus is nonsingular.
Recall that there exists a unitary matrix such that with defined in (2.9). Pre- and postmultiplying to both sides of LMIs (3.3) by and yields where , and . It implies from (C.1) that all the matrices appearing in this LMI are diagonal except for matrices . To this end, suppose that there exist matrices and such that for , the LMIs (3.5) hold; then there must exist diagonal matrices and as solutions to condition (C.1), and accordingly (3.3) holds. In a similar pattern, the feasibility of LMI (3.6) means that condition (3.4) holds.
Moreover, since the coupling matrix has distinct different eigenvalues as (2.9), it is evident to find that the number of LMI groups to be examined in (3.5)-(3.6) can be reduced from to . On the other hand, it is noted that due to the convex property of LMI , each of the rest groups of LMIs for can be written as a linear combination of the two groups of LMIs corresponding to the second-maximum and the minimum eigenvalue . In this situation, the synchronization condition only requires the feasibility of three groups LMIs (3.5)-(3.6) with and ; thus it completes the proof.
D. Solution of LMIs (3.5) for and
This work is supported by the National Science and Technology Infrastructure Program (Grant no. 2008BAA13B07), the China Postdoctoral Science Foundation-funded project (Grant no. 20100480242), the Science Technology Project of State Grid Corporation of China (Research on Safety and Stability of large-scale power system based on complex system theory), and the National Science Foundation of China (Grant no. 60874011).
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