Understanding the fundamental principles and limitations of controlling complex networks is of paramount importance across natural, social, and engineering sciences. The classic notion of controllability does not capture the effort needed to control dynamical networks, and quantitative measures of controllability have been proposed to remedy this problem. This article presents an introductory overview of the practical (i.e., energy-related) aspects of controlling networks governed by linear dynamics. First, we introduce a class of energy-aware controllability metrics and discuss their properties. Then, we establish bounds on these metrics, which allow us to understand how the structure of the network impacts the control energy. Finally, we examine the problem of optimally selecting a set of control nodes so as to minimize the control effort, and compare the performance of some simple strategies to approximately solve this problem. Throughout the article, we include examples of structured and random networks to illustrate our results.


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Literature Cited

  1. 1. 
    Newman M, Barabasi AL, Watts DJ. 2011. The Structure and Dynamics of Networks Princeton, NJ: Princeton Univ. Press
  2. 2. 
    Strogatz SH, Stewart I. 1993. Coupled oscillators and biological synchronization. Sci. Am. 269:6102–9
    [Google Scholar]
  3. 3. 
    Mesbahi M, Egerstedt M. 2010. Graph Theoretic Methods in Multiagent Networks Princeton, NJ: Princeton Univ. Press
  4. 4. 
    Cornelius SP, Kath WL, Motter AE 2013. Realistic control of network dynamics. Nat. Commun. 4:1942
    [Google Scholar]
  5. 5. 
    Cui R, Ge SS, How BVE, Choo YS. 2010. Leader–follower formation control of underactuated autonomous underwater vehicles. Ocean Eng. 37:1491–502
    [Google Scholar]
  6. 6. 
    Papageorgiou M, Diakaki C, Dinopoulou V, Kotsialos A, Wang Y 2003. Review of road traffic control strategies. Proc. IEEE 91:2043–67
    [Google Scholar]
  7. 7. 
    Liu Y, Slotine J, Barabási AL. 2011. Controllability of complex networks. Nature 473:167–73
    [Google Scholar]
  8. 8. 
    Liu YY, Barabási AL. 2016. Control principles of complex systems. Rev. Mod. Phys. 88:035006
    [Google Scholar]
  9. 9. 
    Kailath T. 1980. Linear Systems Englewood Cliffs, NJ: Prentice Hall
  10. 10. 
    Reinschke KJ. 1988. Multivariable Control: A Graph-Theoretic Approach Berlin: Springer
  11. 11. 
    Ramos G, Aguiar AP, Pequito S. 2020. Structural systems theory: an overview of the last 15 years. arXiv:2008.11223 [math.OC]
  12. 12. 
    Pasqualetti F, Zampieri S, Bullo F 2014. Controllability metrics, limitations and algorithms for complex networks. IEEE Trans. Control Netw. Syst. 1:40–52
    [Google Scholar]
  13. 13. 
    Whalen AJ, Brennan SN, Sauer TD, Schiff SJ. 2015. Observability and controllability of nonlinear networks: the role of symmetry. Phys. Rev. X 5:011005
    [Google Scholar]
  14. 14. 
    Zhao Y, Cortés J. 2016. Gramian-based reachability metrics for bilinear networks. IEEE Trans. Control Netw. Syst. 4:620–31
    [Google Scholar]
  15. 15. 
    Georges D. 1995. The use of observability and controllability Gramians or functions for optimal sensor and actuator location in finite-dimensional systems. Proceedings of the 1995 34th IEEE Conference on Decision and Control 43319–24 Piscataway, NJ: IEEE
    [Google Scholar]
  16. 16. 
    Van De Wal M, De Jager B. 2001. A review of methods for input/output selection. Automatica 37:487–510
    [Google Scholar]
  17. 17. 
    Gu S, Pasqualetti F, Cieslak M, Telesford QK, Yu AB et al. 2015. Controllability of structural brain networks. Nat. Commun. 6:8414
    [Google Scholar]
  18. 18. 
    Yuill W, Edwards A, Chowdhury S, Chowdhury SP 2011. Optimal PMU placement: a comprehensive literature review. 2011 IEEE Power and Energy Society General Meeting Piscataway, NJ: IEEE https://doi.org/10.1109/PES.2011.6039376
    [Crossref] [Google Scholar]
  19. 19. 
    Müller P, Weber H. 1972. Analysis and optimization of certain qualities of controllability and observability for linear dynamical systems. Automatica 8:237–46
    [Google Scholar]
  20. 20. 
    Wicks M, DeCarlo R. 1988. An energy approach to controllability. Proceedings of the 27th IEEE Conference on Decision and Control2072–77 Piscataway, NJ: IEEE
  21. 21. 
    Antoulas AC. 2005. Approximation of Large-Scale Dynamical Systems Philadelphia: Soc. Ind. Appl. Math.
  22. 22. 
    Yan G, Ren J, Lai YC, Lai CH, Li B. 2012. Controlling complex networks: How much energy is needed?. Phys. Rev. Lett. 108:218703
    [Google Scholar]
  23. 23. 
    Olshevsky A. 2016. Eigenvalue clustering, control energy, and logarithmic capacity. Syst. Control Lett. 96:45–50
    [Google Scholar]
  24. 24. 
    Yan G, Tsekenis G, Barzel B, Slotine JJ, Liu YY, Barabási AL. 2015. Spectrum of controlling and observing complex networks. Nat. Phys. 11:779–86
    [Google Scholar]
  25. 25. 
    Bof N, Baggio G, Zampieri S. 2016. On the role of network centrality in the controllability of complex networks. IEEE Trans. Control Netw. Syst. 4:643–53
    [Google Scholar]
  26. 26. 
    Klickstein I, Sorrentino F. 2020. The controllability Gramian of lattice graphs. Automatica 114:108833
    [Google Scholar]
  27. 27. 
    Hou B. 2021. Relevance of network characteristics to controllability degree. IEEE Trans. Autom. Control 66:5436–43
    [Google Scholar]
  28. 28. 
    She B, Mehta SS, Doucette E, Ton C, Kan Z. 2021. Characterizing energy-related controllability of composite complex networks via graph product. IEEE Trans. Autom. Control 66:3205–12
    [Google Scholar]
  29. 29. 
    Lindmark G. 2020. Controllability of complex networks at minimum cost PhD Thesis Linköping Univ. Linköping, Sweden:
  30. 30. 
    Baggio G, Zampieri S. 2021. Non-normality improves information transmission performance of network systems. IEEE Trans. Control Netw. Syst. 8:1846–58
    [Google Scholar]
  31. 31. 
    Trefethen LN, Embree M. 2005. Spectra and Pseudospectra: The Behavior of Nonnormal Matrices and Operators Princeton, NJ: Princeton Univ. Press
  32. 32. 
    Sun J, Motter AE. 2013. Controllability transition and nonlocality in network control. Phys. Rev. Lett. 110:208701
    [Google Scholar]
  33. 33. 
    Van Loan C. 1977. The sensitivity of the matrix exponential. SIAM J. Numer. Anal. 14:971–81
    [Google Scholar]
  34. 34. 
    Bianchin G, Pasqualetti F, Zampieri S 2015. The role of diameter in the controllability of complex networks. 2015 54th IEEE Conference on Decision and Control980–85 Piscataway, NJ: IEEE
  35. 35. 
    Zhao S, Pasqualetti F. 2019. Networks with diagonal controllability Gramian: analysis, graphical conditions, and design algorithms. Automatica 102:10–18
    [Google Scholar]
  36. 36. 
    Summers TH, Cortesi FL, Lygeros J. 2015. On submodularity and controllability in complex dynamical networks. IEEE Trans. Control Netw. Syst. 3:91–101
    [Google Scholar]
  37. 37. 
    Li G, Hu W, Xiao G, Deng L, Tang P et al. 2015. Minimum-cost control of complex networks. New J. Phys. 18:013012
    [Google Scholar]
  38. 38. 
    Tzoumas V, Rahimian M, Pappas G, Jadbabaie A. 2016. Minimal actuator placement with bounds on control effort. IEEE Trans. Control Netw. Syst. 3:67–78
    [Google Scholar]
  39. 39. 
    Chanekar PV, Chopra N, Azarm S 2017. Optimal actuator placement for linear systems with limited number of actuators. 2017 American Control Conference334–39 Piscataway, NJ: IEEE
    [Google Scholar]
  40. 40. 
    Olshevsky A. 2017. On (non)supermodularity of average control energy. IEEE Trans. Control Netw. Syst. 5:1177–81
    [Google Scholar]
  41. 41. 
    Nozari E, Pasqualetti F, Cortés J 2019. Heterogeneity of central nodes explains the benefits of time-varying control scheduling in complex dynamical networks. J. Complex Netw. 7:659–801
    [Google Scholar]
  42. 42. 
    Lindmark G, Altafini C. 2017. A driver node selection strategy for minimizing the control energy in complex networks. IFAC-PapersOnLine 50:18309–14
    [Google Scholar]
  43. 43. 
    Lindmark G, Altafini C. 2019. Combining centrality measures for control energy reduction in network controllability problems. 2019 18th European Control Conference1518–23 Piscataway, NJ: IEEE
  44. 44. 
    Olshevsky A. 2020. On a relaxation of time-varying actuator placement. IEEE Control Syst. Lett. 4:656–61
    [Google Scholar]
  45. 45. 
    Taha A, Gatsis N, Summers T, Nugroho S. 2019. Time-varying sensor and actuator selection for uncertain cyber-physical systems. IEEE Trans. Control Netw. Syst. 6:750–62
    [Google Scholar]
  46. 46. 
    Baggio G, Zampieri S, Scherer CW 2019. Gramian optimization with input-power constraints. 2019 IEEE 58th Conference on Decision and Control5686–91 Piscataway, NJ: IEEE
  47. 47. 
    Guo B, Karaca O, Summers T, Kamgarpour M. 2021. Actuator placement under structural controllability using forward and reverse greedy algorithms. IEEE Trans. Autom. Control 66:5845–60
    [Google Scholar]
  48. 48. 
    Siami M, Olshevsky A, Jadbabaie A. 2021. Deterministic and randomized actuator scheduling with guaranteed performance bounds. IEEE Trans. Autom. Control 66:1686–701
    [Google Scholar]
  49. 49. 
    Klickstein I, Sorrentino F. 2020. Selecting energy efficient inputs using graph structure. arXiv:2008.12940 [math.OC]
  50. 50. 
    Lindmark G, Altafini C. 2021. Centrality measures and the role of non-normality for network control energy reduction. IEEE Control Syst. Lett. 5:1013–18
    [Google Scholar]
  51. 51. 
    Dhingra NK, Jovanović MR, Luo ZQ. 2014. An ADMM algorithm for optimal sensor and actuator selection. 53rd IEEE Conference on Decision and Control4039–44 Piscataway, NJ: IEEE
  52. 52. 
    Münz U, Pfister M, Wolfrum P. 2014. Sensor and actuator placement for linear systems based on and optimization. IEEE Trans. Autom. Control 59:2984–89
    [Google Scholar]
  53. 53. 
    Joshi S, Boyd S. 2008. Sensor selection via convex optimization. IEEE Trans. Signal Process. 57:451–62
    [Google Scholar]
  54. 54. 
    Silva VL, Chamon LF, Ribeiro A. 2019. Model predictive selection: a receding horizon scheme for actuator selection. 2019 American Control Conference347–53 Piscataway, NJ: IEEE
  55. 55. 
    Summers T, Kamgarpour M. 2019. Performance guarantees for greedy maximization of non-submodular controllability metrics. 2019 European Control Conference2796–801 Piscataway, NJ: IEEE
    [Google Scholar]
  56. 56. 
    Golub GH, Van Loan CF. 2013. Matrix Computations Baltimore, MD: Johns Hopkins Univ. Press, 4th ed..
  57. 57. 
    Zhao S, Pasqualetti F. 2018. Controllability degree of directed line networks: nodal energy and asymptotic bounds. 2018 European Control Conference1857–62 Piscataway, NJ: IEEE
    [Google Scholar]

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