1. NanJing University of Posts & Telecommunications Synchronization and Fault Diagnosis of Complex Dynamical Networks Guo-Ping Jiang College of Automation, Nanjing University of Posts & Telecommunications Email:jianggp@njupt.edu.cn

2. NanJing University of Posts & Telecommunications outline 1. Motivation and Background 2. Synchronization of Network 3. Our Research Results 4. Conclusions

3. NanJing University of Posts & Telecommunications 1. Motivation and Background Network Synchronization Inner syn. and outer Syn. Identification Network topology is uncertain in real engineering Network topological identification Monitoring Fault diagnosis of networks

4. NanJing University of Posts & Telecommunications 2. Synchronization of Network -Inner Synchronization: a collective behaviour within a network Coupling with all state variablesCoupling with output variable

5. NanJing University of Posts & Telecommunications Coupling with all state variables The model of a dynamical complex network: State variables of the node: Inner coupling matrix: Coupling matrix: (connected) (network topology) (otherwise) [1]. X. Wang and G. Chen, “Complex network: Small-world, scale-free, and beyond,” IEEE Circuits Syst. Mag., vol. 3, no. 2, pp. 6-20, 2003. [2]. J. Lü and G. Chen, “A time-varying complex dynamical network model and its controlled synchronization criteria,” IEEE Trans. Autom. Control, vol. 50, no. 6, pp. 841-846, 2005. [1]. X. Wang and G. Chen, “Complex network: Small-world, scale-free, and beyond,” IEEE Circuits Syst. Mag., vol. 3, no. 2, pp. 6-20, 2003. [2]. J. Lü and G. Chen, “A time-varying complex dynamical network model and its controlled synchronization criteria,” IEEE Trans. Autom. Control, vol. 50, no. 6, pp. 841-846, 2005.

6. NanJing University of Posts & Telecommunications The model of a dynamical complex network: Outer coupling variable: Outer coupling matrix: (connected) (network topology) (otherwise) Observer gain matrix: Coupling with output variable [1] G. –P. Jiang, W. K. -S. Tang, G. Chen, “A state-observer-based approach for synchronization in complex dynamical networks,” IEEE Trans. on Circuits &Systems-I, vol. 53, pp. 2739-2745, 2006

7. NanJing University of Posts & Telecommunications [1] C. Li, W. Sun, J. Kurths, “Synchronization between two coupled complex networks,” Phys Rev. E vol. 76, 046204, 2007 [2] H. Tang, L. Chen, J.-A. Lu, C. K. Tse, “Adaptive synchronization between two complex networks with nonlidentical topological structure,” Physica A, vol. 387, pp. 5623-5630, 2008. [3] C.-X. Fan, G.-P. Jiang, F.-H. Jiang, “Synchronization between two complex networks using scalar signals under pinning control,” IEEE Transaction on Circuits and Systems-I, vol. 57, 2010 [1] C. Li, W. Sun, J. Kurths, “Synchronization between two coupled complex networks,” Phys Rev. E vol. 76, 046204, 2007 [2] H. Tang, L. Chen, J.-A. Lu, C. K. Tse, “Adaptive synchronization between two complex networks with nonlidentical topological structure,” Physica A, vol. 387, pp. 5623-5630, 2008. [3] C.-X. Fan, G.-P. Jiang, F.-H. Jiang, “Synchronization between two complex networks using scalar signals under pinning control,” IEEE Transaction on Circuits and Systems-I, vol. 57, 2010 Outer Synchronization: between two or more networks 状态耦合复杂动态网络输出耦合复杂动态网络

8. NanJing University of Posts & Telecommunications The drive network model: The response network model: Control law:

9. NanJing University of Posts & Telecommunications Identification of network topology using outer synchronization of network L. Zhu et al. assume that the dynamics of the network can be described by a linear stochastic model, But if a more complex network is considered, it may not be true. [1] L. Zhu, Y. C. Lai, F. C. Hoppensteadt, J. He, “Characterization of neural interaction during learning and adaptation from spike-train data,” Mathematical Biosciences and Engineering, vol. 2, pp. 1-23, Jan.2005.

10. NanJing University of Posts & Telecommunications W. K. -S. Tang et al. develop an adaptive observer approach that using a state variable to identify and monitor the topology of neural network with each ode being a HR model Effective for special dynamics of nodes, but difficult to be extended to a general case, where the node dynamics is a general nonlinear system [1] W. K. -S. Tang, Y. Mao, L. Kocarev. “Identification and monitoring of biological neural network,” IEEE International Synposium on Circuits and Systems, pp. 2646-2649, May. 2007.

11. NanJing University of Posts & Telecommunications The network model: Where:

12. NanJing University of Posts & Telecommunications The observer model: Where: is the condition constant we want to get.

13. NanJing University of Posts & Telecommunications J. Zhou et al. construct a state observer and use all the state variables to get network synchronization for topological identification. X. Q. Wu extends to time-delay networks. But if some state variables are not measurable, it may not be practical. [1] J. Zhou, J. A. Lu, “Topology identification of weighted complex dynamical networks,” Physica A, vol. 386, pp. 481-491, 2007. [2] X. Q. Wu, “Synchronization-based topology identification of weighted general complex dynamical networks with time-varying coupling delay,” Physica A, vol. 387, pp. 997-1008, 2008. [1] J. Zhou, J. A. Lu, “Topology identification of weighted complex dynamical networks,” Physica A, vol. 386, pp. 481-491, 2007. [2] X. Q. Wu, “Synchronization-based topology identification of weighted general complex dynamical networks with time-varying coupling delay,” Physica A, vol. 387, pp. 997-1008, 2008.

14. NanJing University of Posts & Telecommunications The drive network model: The response network model: where is any positive constant

15. NanJing University of Posts & Telecommunications Networks with time-varying coupling delay

16. NanJing University of Posts & Telecommunications 3.Our research results: Inner synchronization Outer synchronization Topological identification Fault diagnosis.

17. NanJing University of Posts & Telecommunications Inner synchronization: The model of a dynamical complex network: Outer coupling variable: Outer coupling matrix: (connected) (network topology) (otherwise) Observer gain matrix: [1] G. –P. Jiang, W. K. -S. Tang, G. Chen, “A state-observer-based approach for synchronization in complex dynamical networks,” IEEE Trans. on Circuits &Systems-I, vol. 53, pp. 2739-2745, 2006

18. NanJing University of Posts & Telecommunications

19. Outer synchronization: (State-coupling Model) The model of a dynamical complex network: state variables of the node outer coupling variable inner coupling matrix (network topology)

20. NanJing University of Posts & Telecommunications Outer synchronization: The response network: Hypothesis1 (H1): Hypothesis2 (H2): Hypothesis3 (H3): Fan C-X, Jiang G-P, Jiang F-H. Synchronization between two complex networks using scalar signals under pinning control [J]. IEEE Transaction on Circuits and Systems-I: Regular Papers, vol. 57, no. 11, 2010

21. NanJing University of Posts & Telecommunications Outer synchronization: The error system: The synchronization criteria : Suppose that (H1)-(H4) hold. If there exists a constant k such that the following inequality hold then the error system is asymptotically stable.

22. NanJing University of Posts & Telecommunications Simulation result The network is consisting of 10 nodes, where node dynamical is Lorenz system K=-10, A=[1 0 0; 0 1 0; 0 0 1], B=[4 5 6], H=[1 1 1] C=[-5 1 1 1 0 0 1 0 0 1 ; 1 -2 0 1 0 0 0 0 0 0;[-5 1 1 1 0 0 1 0 0 1 ; 1 -2 0 1 0 0 0 0 0 0; 1 0 -4 0 0 1 1 1 0 0; 1 1 0 -4 0 0 1 0 0 1; 0 0 0 0 -4 0 1 1 1 1; 0 0 1 0 0 -2 0 0 1 0; 1 0 1 1 1 0 -5 0 0 1; 0 0 1 0 1 0 0 -2 0 0; 0 0 0 0 1 1 0 0 -2 0 ; 1 0 0 1 1 0 1 0 0 -4]

23. NanJing University of Posts & Telecommunications Outer synchronization: (output-coupling model) Another model of a dynamical complex network: state variables of the node outer coupling variable inner coupling matrix (network topology)

24. NanJing University of Posts & Telecommunications Outer synchronization: The response network: Hypothesis1 (H1): Hypothesis2 (H2): Hypothesis3 (H3):

25. NanJing University of Posts & Telecommunications Outer synchronization: The error system: The synchronization criteria : Suppose that (H1)-(H3) hold. If there exists a constant k such that the following inequality hold then the error system is asymptotically stable.

26. NanJing University of Posts & Telecommunications Simulation result The network is consisting of 10 nodes, where node dynamics is Lorenz system. K=-10, L=[ 1 2 3]’, H=[1 1 1], B=[ 4 5 6]’ C=[-5 1 1 1 0 0 1 0 0 1 ; 1 -2 0 1 0 0 0 0 0 0;[-5 1 1 1 0 0 1 0 0 1 ; 1 -2 0 1 0 0 0 0 0 0; 1 0 -4 0 0 1 1 1 0 0; 1 1 0 -4 0 0 1 0 0 1; 0 0 0 0 -4 0 1 1 1 1; 0 0 1 0 0 -2 0 0 1 0; 1 0 1 1 1 0 -5 0 0 1; 0 0 1 0 1 0 0 -2 0 0; 0 0 0 0 1 1 0 0 -2 0 ; 1 0 0 1 1 0 1 0 0 -4]

27. NanJing University of Posts & Telecommunications Topological identification and fault diagnosis based on outer synchronization using output variable

28. NanJing University of Posts & Telecommunications The model of a dynamical complex network: state variables of the node: outer coupling variable: inner coupling matrix: (connected) (network topology) (otherwise) observer gain matrix: [1] G. –P. Jiang, W. K. -S. Tang, G. Chen, “A state-observer-based approach for synchronization in complex dynamical networks,” IEEE Trans. on Circuits &Systems-I, vol. 53, pp. 2739-2745, 2006

29. NanJing University of Posts & Telecommunications Observer design: We can design an observer as follows: The error system can be written as: Where,,, is estimation of, is the re-state vector, [1] H. Liu, G. –P. Jiang, C..-X Fan “State-observer-based approach for identification and monitoring of complex dynamical networks,” IEEE Asia-Pacific Conference on Circuits and Systems, 2008, Macao, China. [2] Liu H, Song Y-R, Fan C-X, Jiang G-P. Fault diagnosis of time-delay complex dynamical networks using output signals [J]. Chinese Physics B, 2010, 19 (7):070508 [1] H. Liu, G. –P. Jiang, C..-X Fan “State-observer-based approach for identification and monitoring of complex dynamical networks,” IEEE Asia-Pacific Conference on Circuits and Systems, 2008, Macao, China. [2] Liu H, Song Y-R, Fan C-X, Jiang G-P. Fault diagnosis of time-delay complex dynamical networks using output signals [J]. Chinese Physics B, 2010, 19 (7):070508

30. NanJing University of Posts & Telecommunications Lyapunov function: Consider a positive Lyapnov function as: Assuming that

31. NanJing University of Posts & Telecommunications We have the differential coefficient of as: Where

32. NanJing University of Posts & Telecommunications Theorem 1: If a suitable is chosen such that, then one gets and

33. NanJing University of Posts & Telecommunications Time-delay case: The network with time-delay can be modelled as: where is time-varying delay. Liu H, Song Y-R, Fan C-X, Jiang G-P. Fault diagnosis of time-delay complex dynamical networks using output signals [J]. Chinese Physics B, 2010, 19 (7):070508 （ SCI ）

34. NanJing University of Posts & Telecommunications First we induce some assumptions and a lemma. Assumption 1: Suppose that there exists a positive constant satisfying: where are time-varying vectors, represents 2-norm. Assumption 2: is a differential function with: Lemma 1: For any vectors and positive define matrix, the following matrix inequality holds

35. NanJing University of Posts & Telecommunications We can design the observer as follows: The error system can be written as:

36. NanJing University of Posts & Telecommunications Consider a positive Lyapnov function as: According to Assumption 1, we get:

37. NanJing University of Posts & Telecommunications We have the differential coefficient of as:

38. NanJing University of Posts & Telecommunications Using the assumption 2 and lemma 1, one gets: Where

39. NanJing University of Posts & Telecommunications So, the matrix is negative definite if we get the suitable. Therefore, base on the Lyapnov stability theorem, one gets and converges to a constant when, so the topology can be approximately identified and duly monitored.

40. NanJing University of Posts & Telecommunications Simulation results: In the simulation, a network of 7 nodes is constructed with each node being a Lorenz model. The Lorenz model can be described as When a=16,b=4,c=45, the first state variable is depicted in Fig. 1.

41. NanJing University of Posts & Telecommunications Fig.1 Lorenz model

42. NanJing University of Posts & Telecommunications A dynamical network of seven nodes is constructed, as shown in Fig.2 Fig.

43. NanJing University of Posts & Telecommunications The initial values are given as:

44. NanJing University of Posts & Telecommunications

45. Fig.4. Estimation of C 12 and the derivative of error. The connection between nodes 1and 2 is broken at t=50s.

46. NanJing University of Posts & Telecommunications Fig.5. The synchronization errors

47. NanJing University of Posts & Telecommunications Thank You for Listening!