Decompositions of Large-Scale Systems

Part of the Communications and Control Engineering book series (CCE)


From the standpoint of control theory, it is usually convenient to represent a large-scale system as a collection of interconnected subsystems. In certain cases such a decomposition can be derived directly from the physical description of the problem, which suggests a “natural” grouping of the state variables. More often than not, however, the only information that we have about the system dynamics comes from a mathematical model whose properties provide little or no insight into how the subsystems should be chosen. In order to deal with such problems in a systematic manner, one obviously needs to develop decomposition algorithms that are based exclusively on the structure of the underlying equations.

In this chapter we will consider three decompositions, all of which are designed to exploit the sparsity of large-scale state space models. The algorithms are based on graph theoretic representations, which provide the necessary mathematical framework for partitioning the system. We should also point out that each of the three decompositions corresponds to a different gain matrix structure. With that in mind, it is fair to say that the purpose of these decompositions is not only to simplify the computation, but also to help us identify the most appropriate type of control law for a given large-scale system. We will take a closer look at this aspect of the problem in Chaps. 2 and 3, which are devoted to control design with information structure constraints.


Bipartite Graph Decomposition Algorithm Diagonal Block Permutation Matrix Waveform Relaxation 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Amano, M., A. I. Zečević and D. D. Šiljak (1996). An improved block-parallel Newton method via epsilon decompositions for load-flow calculations. IEEE Transactions on Power Systems, 11, 1519–1527.CrossRefGoogle Scholar
  2. Asratian, A., T. Denley and R. Häggkvist (1998). Bipartite Graphs and Their Applications. Cambridge University Press, Cambridge.MATHGoogle Scholar
  3. Csermely, P. (2006). Weak Links: Stabilizers of Complex Systems from Proteins to Social Networks. Springer, Berlin.Google Scholar
  4. Duff, I. S. (1981). On algorithms for obtaining a maximal transversal. ACM Transactions on Mathematical Software, 7, 315–330.CrossRefGoogle Scholar
  5. Duff, I. S., A. M. Erisman and J. K. Reid (1986). Direct Methods for Sparse Matrices. Clarendon, Oxford.MATHGoogle Scholar
  6. Ferrari, R. M. G., T. Parisini and M. M. Polycarpou (2009). Distributed fault diagnosis with overlapping decompositions: An adaptive approximation approach. IEEE Transactions on Automatic Control, 54, 794–799.CrossRefMathSciNetGoogle Scholar
  7. Finney, J. D. and B. S. Heck (1996). Matrix scaling for large-scale system decomposition. Automatica, 32, 1177–1181.MATHCrossRefMathSciNetGoogle Scholar
  8. Gačić, N., A. I. Zečević and D. D. Šiljak (1998). Coherency recognition using epsilon decomposition. IEEE Transactions on Power Systems, 13, 314–319.CrossRefGoogle Scholar
  9. Gajić, Z. and X. Shen (1993). Parallel Algorithms for Optimal Control of Large Scale Linear Systems. Springer, London.MATHGoogle Scholar
  10. George, A. and J. W. H. Liu (1981). Computer Solution of Large Sparse Positive Definite Systems. Prentice-Hall, Upper Saddle River, NJ.MATHGoogle Scholar
  11. George, A. and J. W. H. Liu (1989). The evolution of the minimal degree ordering algorithm. SIAM Review, 31, 1–19.MATHCrossRefMathSciNetGoogle Scholar
  12. George, A., J. Gilbert and J. W. H. Liu (Eds.) (1993). Graph Theory and Sparse Matrix Computation. Springer, New York.MATHGoogle Scholar
  13. Gould, R. (1988). Graph Theory. Benjamin/Cummings Publishing Co., Menlo Park, CA.MATHGoogle Scholar
  14. Happ, H. H. (1971). Diakoptics and Networks. Academic, New York.Google Scholar
  15. Iftar, A. (1993). Decentralized estimation and control with overlapping input, state and output decomposition. Automatica, 29, 511–516.MATHCrossRefMathSciNetGoogle Scholar
  16. Ikeda, M. and D. D. Šiljak (1986). Overlapping decentralized control with input, state and output inclusion. Control Theory and Advanced Technology, 2, 155–172.Google Scholar
  17. Ikeda, M., D. D. Šiljak and D. E. White (1984). An inclusion principle for dynamic systems. IEEE Transactions on Automatic Control, 29, 244–249.MATHCrossRefGoogle Scholar
  18. Peponides, G. M. and P. K. Kokotovic (1983). Weak connections, time scales, and aggregation of nonlinear systems. IEEE Transactions on Automatic Control, 28, 729–735.CrossRefMathSciNetGoogle Scholar
  19. Pothen, A., H. D. Simon and K. P. Liou (1990). Partitioning sparse matrices with eigenvectors of graphs. SIAM Journal of Matrix Analysis and Applications, 11, 430–452.MATHCrossRefMathSciNetGoogle Scholar
  20. Sezer, M. E. and D. D. Šiljak (1986). Nested ε decompositions and clustering of complex systems. Automatica, 22, 321–331.MATHCrossRefGoogle Scholar
  21. Sezer, M. E. and D. D. Šiljak (1991). Nested epsilon decompositions and clustering of linear systems: weakly coupled and overlapping blocks. SIAM Journal on Matrix Analysis and Applications, 12, 521–533.MATHCrossRefMathSciNetGoogle Scholar
  22. Šiljak, D. D. (1978). Large-Scale Dynamic Systems: Stability and Structure. North Holland, New York.MATHGoogle Scholar
  23. Šiljak, D. D. (1991). Decentralized Control of Complex Systems. Academic, Cambridge, MA.Google Scholar
  24. Šiljak, D. D. and A. I. Zečević (1995). A nested decomposition algorithm for parallel computations of very large sparse systems. Mathematical Problems in Engineering, 1, 41–57.MATHCrossRefGoogle Scholar
  25. Stanković, S. S. and D. D. Šiljak (2001). Contractibility of overlapping decentralized control. Systems and Control Letters, 44, 189–199.MATHCrossRefMathSciNetGoogle Scholar
  26. Tarjan, R. (1972). Depth-first search and linear graph algorithms. SIAM Journal on Computing, 1, 146–160.MATHCrossRefMathSciNetGoogle Scholar
  27. Tinney, W. F. and J. W. Walker (1967). Direct solutions of sparse equations by optimally ordered triangular factorization. Proceedings of the IEEE, 55, 1801–1809.CrossRefGoogle Scholar
  28. Wallach, Y. (1986). Calculations and Programs for Power System Networks. Prentice-Hall, Upper Saddle River, NJ.Google Scholar
  29. Zečević, A. I. and D. D. Šiljak (1994a). A block-parallel Newton method via overlapping epsilon decompositions. SIAM Journal on Matrix Analysis and Applications, 15, 824–844.MATHCrossRefMathSciNetGoogle Scholar
  30. Zečević, A. I. and D. D. Šiljak (1994b). Balanced decompositions of sparse systems for multilevel parallel processing. IEEE Transactions on Circuits and Systems, 41, 220–233.CrossRefGoogle Scholar
  31. Zečević, A. I. and N. Gačić (1999). A partitioning algorithm for the parallel solution of differential-algebraic equations by waveform relaxation. IEEE Transactions on Circuits and Systems, 46, 421–434.MATHCrossRefGoogle Scholar
  32. Zečević, A. I. and D. D. Šiljak (1999). Parallel solutions of very large sparse Lyapunov equations by balanced BBD decompositions. IEEE Transactions on Automatic Control, 44, 612–618.MATHCrossRefGoogle Scholar
  33. Zečević, A. I. and D. D. Šiljak (2005). A decomposition-based control strategy for large, sparse dynamic systems. Mathematical Problems in Engineering, 11, 33–48.Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  • Aleksandar I. Zečević
    • 1
  • Dragoslav D. Šiljak
    • 1
  1. 1.Dept. Electrical EngineeringSanta Clara UniversitySanta ClaraUSA

Personalised recommendations