Stability Analysis and Hadamard Synergic Control for a Class of Dynamical Networks

ABSTRACT

Hadamard synergic control is a new kind of control problem which is achieved via a composite strategy of the state feedback control and the direct regulation of the part of connection coefficients of system state variables. Such a control is actually used very often in the practical areas. In this paper, we discuss Hadamard synergic stabilization problem for a class of dynamical networks. We analyze three cases: 1) Synergic stabilization problem for the general twonodenetwork. 2) Synergic stabilization problem for a special kind of networks. 3) Synergic stabilization problem for special kind of networks with communication timedelays. The mechanism of the synergic action between two control strategies: feedback control and the connection coefficients regulations are presented.

Hadamard synergic control is a new kind of control problem which is achieved via a composite strategy of the state feedback control and the direct regulation of the part of connection coefficients of system state variables. Such a control is actually used very often in the practical areas. In this paper, we discuss Hadamard synergic stabilization problem for a class of dynamical networks. We analyze three cases: 1) Synergic stabilization problem for the general twonodenetwork. 2) Synergic stabilization problem for a special kind of networks. 3) Synergic stabilization problem for special kind of networks with communication timedelays. The mechanism of the synergic action between two control strategies: feedback control and the connection coefficients regulations are presented.

KEYWORDS

Hadamard Synergic Control, Algebraically Graph Theory, Decentralized Feedback Control, Connection Coefficient Gain Matrix

Hadamard Synergic Control, Algebraically Graph Theory, Decentralized Feedback Control, Connection Coefficient Gain Matrix

Cite this paper

nullX. Liu and Y. Zou, "Stability Analysis and Hadamard Synergic Control for a Class of Dynamical Networks,"*Intelligent Control and Automation*, Vol. 1 No. 1, 2010, pp. 36-47. doi: 10.4236/ica.2010.11005.

nullX. Liu and Y. Zou, "Stability Analysis and Hadamard Synergic Control for a Class of Dynamical Networks,"

References

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[13] X. F. Yan, Y. Zou and J. Li, “Optimal Quarantine and Isolation Strategies in Epidemics Control,” World Journal of Modeling and Simulation, Vol. 3, No. 3, 2007, pp. 202211.

[14] X. F. Yan and Y. Zou, “Optimal Control for Internet Worm,” ETRI Journal, Vol. 30, No. 1, 2008, pp. 8188.

[15] X. F. Yan and Y. Zou, “Isolation Treatment Strategy for Emergency Control,” Journal of systems Engineering, Vol. 24, No. 2, 2009, pp. 129135.

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[17] K. J. Astrom, “Limitations on Control System Performance,” European Journal of Control, Vol. 4, No. 1, 2000, pp. 220.

[18] R. M. Murray, K. J. Astrom, S. P. Boyd, et al., “Future Directions in Control in an InformationRich World,” IEEE Control Systems Magazine, Vol. 23, No. 2, 2003, pp. 2033.

[19] L. Guo, “Exploring the Maximum Capability of Adaptive Feedback,” International Journal of Adaptive Control and Signal Processing, Vol. 16, No. 1, 2002, pp. 341 354.

[20] J. L. Chen and X. H. Chen, “Special Matrices,” Tsinghua Publisher, Beijing, 2001.

[21] X. J. Liu and Y. Zou, “On the Hadamard Synergic Stabilization Problem: The Stabilization Via a Composite Strategy of ConnectionRegulation and Feedback Control: Matrix Inequality Approach,” International Journal of Innovative Computing, Information and Control, Vol. 6, No. 5, 2010, pp. 23832392.

[22] Z. S. Duan, J. Z. Wang, G. R. Chen and L. Huang, “Stability Analysis and Decentralized Control of a Class of Complex Dynamical Networks,” Automatica, Vol. 44, No. 4, 2008, pp. 10281035.

[23] A. B. Gumel, S. Ruan, T. Day, et al., “Modeling Strategies for Controlling SARS Outbreaks,” Proceedings of the Royal Society, Vol. 49, No. 9, 2004, pp. 14651476.

[24] X. Zhang, X. Hu and Z. Bao, “Blind Source Separation Based on Grading Learning,” Science in ChinaSeries E, Vol. 32, No. 5, 2002, pp. 693793.

[25] S. Chen, “A Lower Bound for the Minimum Eigenvalue of the Hadamard Product of Matrices,” Linear Algebra and its Applications, Vol. 37, No. 8, 2004, pp. 159166.

[26] H. Li, T. Huang, S. Shen, et al., “Lower Bounds for the Minimum Eigenvalue of Hadamard Product of an M matrix and its Inverse,” Linear Algebra and its Applications, Vol. 4, No. 20, 2007, pp. 235247.

[27] G. Visick, “A Quantitative Version of the Observation that the Hadamard Product is a Principal Submatrix of the Kronecker Product,” Linear Algebra and its Applications, Vol. 30, No. 4, 2000, pp. 4568.

[28] M. Neumann, “A Conjecture Concerning the Hadamard Product of Inverses of Mmatrices,” Linear Algebra and its Applications, Vol. 28, No. 5, 1998, pp. 277290.

[29] L. Qiu and X. Zhan, “On the Span of Hadamard Products of Vectors,” Linear Algebra and its Applications, Vol. 422, No. 1, 2006, pp. 304307.

[30] J. A. Fax and R. M. Murray, “Information Flow and Cooperative Control of Vehicle Formations,” IEEE Transactions on Automatic Control, Vol. 49, No. 9, 2004, pp. 14651476.

[31] R. O. Saber and M. Murray, “Consensus Protocols for Networks of Dynamic Agents,” Proceedings of the American Control Conference, Denver, 2003, pp. 951956.

[32] Z. S. Duan, L. Huang, L. Wang and J. Z. Wang, “Some Applications of Small Gain Theorem to Interconnected Systems,” Systems and Control Letters, Vol. 52, No. 34, 2004, pp. 263273.

[33] P. Gahinet, A. Nemirovski and A. J. Laub, “LMI Control Toolbox,” TheMath Works, Inc., Natick, 1995.

[34] R. O. Saber and M. Murray, “Consensus Problems in Networks of Agents with Switching Topology and Time Delays,” IEEE Transactions on Automatic Control, Vol. 49, No. 9, 2004, pp. 15201533.

[1] X. F. Wang, “Complex Networks: Topology, Dynamics, and Synchronization,” International Journal of Bifurcation and Chaos, Vol. 5, No. 12, 2002, pp. 885916.

[2] D. J. Watts and S. H. Strogatz, “Collective Dynamics of ‘SmallWorld’ Networks,” Nature, Vol. 393, No. 6684, 1998, pp. 440442.

[3] X. Li, X. F. Wang and G. R. Chen., “Pinning a Complex Dynamical Network to its Equilibrium,” IEEE Transactions on Circuits and SystemsI, Vol. 51, No. 10, 2004, pp. 20742085.

[4] V. Lesser, C. L. Ortiz and M. Tamb, “Distributed Sensor Networks: A Multiagent Perspective,” Kluwer Academic Publishers, Boston, 2003.

[5] S. S. Mascolo, “Congestion Control in HighSpeed Communication Networks Using the Smith Principle,” Automatica, Vol. 35, No. 11, 1999, pp. 19211935.

[6] L. Huang and Z. S. Duan, “Complexity in Control Science,” Acta Automatica Sinica, Vol. 29, No. 5, 2003, pp. 748753.

[7] Z. S. Duan, L. Huang, J. Z. Wang and L. Wang, “Harmonic Control between Two Systems,” Acta Automatica Sinica, Vol. l29, No. 1, 2003, pp. 1421.

[8] Z. S. Duan, J. Z. Wang and L.Huang, “Special Decentralized Control Problems and Effectiveness of ParameterDepend Lyaounov Function Method,” Proceedings of the American Control Conference, Boston, 2005, pp. 16971702.

[9] Y. Zou, M. H. Yin and H. D. Chiang, “Theoretical Foundation of Controlling U.E.P Method in Network Structure Preserving Power System Model,” IEEE Transactions on Circuits and Systems I, Vol. 50, No. 10, 2003, pp. 1324 1336.

[10] K. Sun, D. Z. Zheng and Q. Lu, “A Simulation Study of OBDDBased Proper Splitting Strategies for Power Systems, under Consideration of Transient Stability,” IEEE Transaction on Power Systems, Vol. l20, No. 1, 2005, pp. 389399.

[11] K. Sun, D. Z. Zheng and Q. Lu, “Splitting Strategies for Islanding Operation of LargeScale Power Systems Using OBDDBased Methods,” IEEE Transactions on Power Systems, Vol. 18, No. 2, 2003, pp. 912923.

[12] Q. C. Zhao, K. Sun, D. Z. Zheng, J. Ma and Q. Lu, “A Study of System Splitting Strategies for Island Operation of Power System: A TwoPhase Method Based on OBDDs,” IEEE Transactions on Power Systems, Vol. 18, No. 4, 2003, pp. 15561565.

[13] X. F. Yan, Y. Zou and J. Li, “Optimal Quarantine and Isolation Strategies in Epidemics Control,” World Journal of Modeling and Simulation, Vol. 3, No. 3, 2007, pp. 202211.

[14] X. F. Yan and Y. Zou, “Optimal Control for Internet Worm,” ETRI Journal, Vol. 30, No. 1, 2008, pp. 8188.

[15] X. F. Yan and Y. Zou, “Isolation Treatment Strategy for Emergency Control,” Journal of systems Engineering, Vol. 24, No. 2, 2009, pp. 129135.

[16] Y. Zou, “Investigation on Mechanism and Extension of Feedback: An Extended Closedloop Coordinate Control of Interconnectionregulation and Feedback,” Journal of Nanjing University of Science and Technology, Vol. 34, No. 1, 2010, pp. 17.

[17] K. J. Astrom, “Limitations on Control System Performance,” European Journal of Control, Vol. 4, No. 1, 2000, pp. 220.

[18] R. M. Murray, K. J. Astrom, S. P. Boyd, et al., “Future Directions in Control in an InformationRich World,” IEEE Control Systems Magazine, Vol. 23, No. 2, 2003, pp. 2033.

[19] L. Guo, “Exploring the Maximum Capability of Adaptive Feedback,” International Journal of Adaptive Control and Signal Processing, Vol. 16, No. 1, 2002, pp. 341 354.

[20] J. L. Chen and X. H. Chen, “Special Matrices,” Tsinghua Publisher, Beijing, 2001.

[21] X. J. Liu and Y. Zou, “On the Hadamard Synergic Stabilization Problem: The Stabilization Via a Composite Strategy of ConnectionRegulation and Feedback Control: Matrix Inequality Approach,” International Journal of Innovative Computing, Information and Control, Vol. 6, No. 5, 2010, pp. 23832392.

[22] Z. S. Duan, J. Z. Wang, G. R. Chen and L. Huang, “Stability Analysis and Decentralized Control of a Class of Complex Dynamical Networks,” Automatica, Vol. 44, No. 4, 2008, pp. 10281035.

[23] A. B. Gumel, S. Ruan, T. Day, et al., “Modeling Strategies for Controlling SARS Outbreaks,” Proceedings of the Royal Society, Vol. 49, No. 9, 2004, pp. 14651476.

[24] X. Zhang, X. Hu and Z. Bao, “Blind Source Separation Based on Grading Learning,” Science in ChinaSeries E, Vol. 32, No. 5, 2002, pp. 693793.

[25] S. Chen, “A Lower Bound for the Minimum Eigenvalue of the Hadamard Product of Matrices,” Linear Algebra and its Applications, Vol. 37, No. 8, 2004, pp. 159166.

[26] H. Li, T. Huang, S. Shen, et al., “Lower Bounds for the Minimum Eigenvalue of Hadamard Product of an M matrix and its Inverse,” Linear Algebra and its Applications, Vol. 4, No. 20, 2007, pp. 235247.

[27] G. Visick, “A Quantitative Version of the Observation that the Hadamard Product is a Principal Submatrix of the Kronecker Product,” Linear Algebra and its Applications, Vol. 30, No. 4, 2000, pp. 4568.

[28] M. Neumann, “A Conjecture Concerning the Hadamard Product of Inverses of Mmatrices,” Linear Algebra and its Applications, Vol. 28, No. 5, 1998, pp. 277290.

[29] L. Qiu and X. Zhan, “On the Span of Hadamard Products of Vectors,” Linear Algebra and its Applications, Vol. 422, No. 1, 2006, pp. 304307.

[30] J. A. Fax and R. M. Murray, “Information Flow and Cooperative Control of Vehicle Formations,” IEEE Transactions on Automatic Control, Vol. 49, No. 9, 2004, pp. 14651476.

[31] R. O. Saber and M. Murray, “Consensus Protocols for Networks of Dynamic Agents,” Proceedings of the American Control Conference, Denver, 2003, pp. 951956.

[32] Z. S. Duan, L. Huang, L. Wang and J. Z. Wang, “Some Applications of Small Gain Theorem to Interconnected Systems,” Systems and Control Letters, Vol. 52, No. 34, 2004, pp. 263273.

[33] P. Gahinet, A. Nemirovski and A. J. Laub, “LMI Control Toolbox,” TheMath Works, Inc., Natick, 1995.

[34] R. O. Saber and M. Murray, “Consensus Problems in Networks of Agents with Switching Topology and Time Delays,” IEEE Transactions on Automatic Control, Vol. 49, No. 9, 2004, pp. 15201533.