1. F. Dörfler and F. Bullo. On the Critical Coupling for Kuramoto Oscillators. SIAM Journal on Applied Dynamical Systems, 10(3):1070-1099, 2011. Keyword(s): Power Networks, Complex Oscillator Networks. Abstract:

   The celebrated Kuramoto model captures various synchronization phenomena in biological and man-made dynamical systems of coupled oscillators. It is well known that there exists a critical coupling strength among the oscillators at which a phase transition from incoherency to synchronization occurs. This paper features four contributions. First, we characterize and distinguish the different notions of synchronization used throughout the literature and formally introduce the concept of phase cohesiveness as an analysis tool and performance index for synchronization. Second, we review the vast literature providing necessary, sufficient, implicit, and explicit estimates of the critical coupling strength in the finite- and infinite-dimensional cases and for both first-order and second-order Kuramoto models. Third, we present the first explicit necessary and sufficient condition on the critical coupling strength to achieve synchronization in the finite-dimensional Kuramoto model for an arbitrary distribution of the natural frequencies. The multiplicative gap in the synchronization condition yields a practical stability result determining the admissible initial and the guaranteed ultimate phase cohesiveness as well as the guaranteed asymptotic magnitude of the order parameter. For supplementary results, we provide a statistical comparison of our synchronization condition with other conditions proposed in the literature, and we show that our results also hold for switching and smoothly time-varying natural frequencies. Fourth and finally, we extend our analysis to multirate Kuramoto models consisting of second-order Kuramoto oscillators with inertia and viscous damping together with first-order Kuramoto oscillators with multiple time constants. We prove that such a heterogeneous network is locally topologically conjugate to a first-order Kuramoto model with scaled natural frequencies. Finally, we present necessary and sufficient conditions for almost global phase synchronization and local frequency synchronization in the multirate Kuramoto model. Interestingly, our provably correct synchronization conditions do not depend on the inertial coefficients which contradicts prior observations on the role of inertial effects in synchronization of second-order Kuramoto oscillators.

   
   

   @article{FD-FB:10w,
   abstract = {The celebrated Kuramoto model captures various synchronization phenomena in biological and man-made dynamical systems of coupled oscillators. It is well known that there exists a critical coupling strength among the oscillators at which a phase transition from incoherency to synchronization occurs. This paper features four contributions. First, we characterize and distinguish the different notions of synchronization used throughout the literature and formally introduce the concept of phase cohesiveness as an analysis tool and performance index for synchronization. Second, we review the vast literature providing necessary, sufficient, implicit, and explicit estimates of the critical coupling strength in the finite- and infinite-dimensional cases and for both first-order and second-order Kuramoto models. Third, we present the first explicit necessary and sufficient condition on the critical coupling strength to achieve synchronization in the finite-dimensional Kuramoto model for an arbitrary distribution of the natural frequencies. The multiplicative gap in the synchronization condition yields a practical stability result determining the admissible initial and the guaranteed ultimate phase cohesiveness as well as the guaranteed asymptotic magnitude of the order parameter. For supplementary results, we provide a statistical comparison of our synchronization condition with other conditions proposed in the literature, and we show that our results also hold for switching and smoothly time-varying natural frequencies. Fourth and finally, we extend our analysis to multirate Kuramoto models consisting of second-order Kuramoto oscillators with inertia and viscous damping together with first-order Kuramoto oscillators with multiple time constants. We prove that such a heterogeneous network is locally topologically conjugate to a first-order Kuramoto model with scaled natural frequencies. Finally, we present necessary and sufficient conditions for almost global phase synchronization and local frequency synchronization in the multirate Kuramoto model. Interestingly, our provably correct synchronization conditions do not depend on the inertial coefficients which contradicts prior observations on the role of inertial effects in synchronization of second-order Kuramoto oscillators. },
   author = {F. D{ö}rfler and F. Bullo},
   date-added = {2012-07-02 15:41:45 -0600},
   date-modified = {2013-10-01 04:26:16 +0000},
   funding = {IIS-0904501, CNS-0834446, CPS-1135819},
   journal = {SIAM Journal on Applied Dynamical Systems},
   keywords = {Power Networks, Complex Oscillator Networks},
   number = 3,
   pages = {1070-1099},
   pdf = {http://epubs.siam.org/doi/pdf/10.1137/10081530X},
   title = {On the Critical Coupling for {K}uramoto Oscillators},
   url = {http://epubs.siam.org/action/showAbstract?page=1070&volume=10&issue=3&journalCode=sjaday&isAuthorized=no},
   volume = 10,
   year = 2011,
   bdsk-url-1 = {http://epubs.siam.org/action/showAbstract?page=1070&volume=10&issue=3&journalCode=sjaday&isAuthorized=no} 
   }
   

1. F. Dörfler and F. Bullo. On the critical coupling strength for Kuramoto oscillators. In American Control Conference, San Francisco, CA, USA, pages 3239-3244, June 2011. Keyword(s): Power Networks, Complex Oscillator Networks.
   

   @inproceedings{FD-FB:11-a,
   address = {San Francisco, CA, USA},
   author = {F. Dörfler and F. Bullo},
   booktitle = {{A}merican {C}ontrol {C}onference},
   date-added = {2012-01-19 21:32:43 -0800},
   date-modified = {2013-10-01 05:32:44 +0000},
   keyword = {Power Networks},
   keywords = {Complex Oscillator Networks},
   month = jun,
   pages = {3239--3244},
   pdf = {http://motion.me.ucsb.edu/pdf/2010t-db.pdf},
   title = {On the critical coupling strength for {K}uramoto oscillators},
   url = {http://ieeexplore.ieee.org/xpl/freeabs_all.jsp?arnumber=5991303},
   year = 2011,
   bdsk-url-1 = {http://ieeexplore.ieee.org/xpl/freeabs_all.jsp?arnumber=5991303} 
   }
   




2. F. Dörfler and F. Bullo. Topological equivalence of a structure-preserving power network model and a non-uniform Kuramoto model of coupled oscillators. In IEEE Conf. on Decision and Control and European Control Conference, Orlando, FL, USA, pages 7099-7104, December 2011. Keyword(s): Power Networks, Complex Oscillator Networks.
   

   @inproceedings{FD-FB:11-b,
   address = {Orlando, FL, USA},
   author = {F. Dörfler and F. Bullo},
   booktitle = {{IEEE} Conf.\ on Decision and Control and European Control Conference},
   date-added = {2012-01-19 21:33:49 -0800},
   date-modified = {2013-10-01 05:39:06 +0000},
   keywords = {Power Networks, Complex Oscillator Networks},
   month = dec,
   pages = {7099-7104},
   pdf = {http://motion.me.ucsb.edu/pdf/2011j-db.pdf},
   title = {Topological equivalence of a structure-preserving power network model and a non-uniform {K}uramoto model of coupled oscillators},
   url = {http://ieeexplore.ieee.org/xpl/login.jsp?tp=&arnumber=6160337&url=http%3A%2F%2Fieeexplore.ieee.org%2Fxpls%2Fabs_all.jsp%3Farnumber%3D6160337},
   year = 2011,
   bdsk-url-1 = {http://ieeexplore.ieee.org/xpl/login.jsp?tp=&arnumber=6160337&url=http%3A%2F%2Fieeexplore.ieee.org%2Fxpls%2Fabs_all.jsp%3Farnumber%3D6160337} 
   }
   




3. F. Dörfler, F. Pasqualetti, and F. Bullo. Distributed detection of cyber-physical attacks in power networks: A waveform relaxation approach. In Allerton Conf. on Communications, Control and Computing, pages 1486-1491, September 2011. Keyword(s): Linear Control Design, Distributed Control Algorithms, Power Networks, Secure Control Systems.
   

   @inproceedings{FD-FP-FB:11-c,
   author = {F. Dörfler and F. Pasqualetti and F. Bullo},
   booktitle = {Allerton Conf.\ on Communications, Control and Computing},
   date-added = {2012-01-19 21:34:27 -0800},
   date-modified = {2013-10-01 05:26:19 +0000},
   keywords = {Linear Control Design, Distributed Control Algorithms, Power Networks, Secure Control Systems},
   month = sep,
   pages = {1486-1491},
   pdf = {http://motion.me.ucsb.edu/pdf/2011t-dpb.pdf},
   title = {Distributed detection of cyber-physical attacks in power networks: {A} waveform relaxation approach},
   url = {http://ieeexplore.ieee.org/xpls/abs_all.jsp?arnumber=6120343},
   year = 2011,
   bdsk-url-1 = {http://ieeexplore.ieee.org/xpls/abs_all.jsp?arnumber=6120343} 
   }
   




4. F. Pasqualetti, F. Dörfler, and F. Bullo. Cyber-physical attacks in power networks: Models, fundamental limitations and monitor design. In IEEE Conf. on Decision and Control and European Control Conference, Orlando, FL, USA, pages 2195-2201, December 2011. Keyword(s): Power Networks, Secure Control Systems, Linear Control Design, Distributed Control Algorithms.
   

   @inproceedings{FP-FD-FB:11i,
   address = {Orlando, FL, USA},
   author = {F. Pasqualetti and F. D{ö}rfler and F. Bullo},
   booktitle = {{IEEE} Conf.\ on Decision and Control and European Control Conference},
   date-added = {2012-07-02 15:55:24 -0600},
   date-modified = {2013-10-01 05:24:32 +0000},
   funding = {ARO-DAAD19-03-D-0004, NSF-CPS-1135819},
   keywords = {Power Networks, Secure Control Systems, Linear Control Design, Distributed Control Algorithms},
   month = dec,
   pages = {2195-2201},
   pdf = {http://motion.me.ucsb.edu/pdf/2011i-pdb.pdf},
   title = {Cyber-physical attacks in power networks: {M}odels, fundamental limitations and monitor design},
   url = {http://ieeexplore.ieee.org/xpl/login.jsp?tp=&arnumber=6160641&url=http%3A%2F%2Fieeexplore.ieee.org%2Fxpls%2Fabs_all.jsp%3Farnumber%3D6160641},
   year = 2011,
   bdsk-url-1 = {http://ieeexplore.ieee.org/xpl/login.jsp?tp=&arnumber=6160641&url=http%3A%2F%2Fieeexplore.ieee.org%2Fxpls%2Fabs_all.jsp%3Farnumber%3D6160641} 
   }
   

1. F. Dörfler and F. Bullo. Kron Reduction of Graphs with Applications to Electrical Networks, February 2011. Note: Available at http://arxiv.org/abs/1102.2950. Keyword(s): Power Networks, Algebraic Graph Theory, Power Networks.
   

   @misc{FD-FB:11d-arxiv,
   author = {F. D{ö}rfler and F. Bullo},
   date-added = {2012-11-29 16:31:36 +0000},
   date-modified = {2013-10-01 04:55:43 +0000},
   funding = {IIS-0904501, CPS-1135819},
   keyword = {Power Networks},
   keywords = {Algebraic Graph Theory, Power Networks},
   month = feb,
   note = {{Available at {http://arxiv.org/abs/1102.2950}}},
   pdf = {http://arxiv.org/pdf/1102.2950v1.pdf},
   title = {{K}ron Reduction of Graphs with Applications to Electrical Networks},
   year = 2011 
   }
   




2. F. Pasqualetti, F. Dörfler, and F. Bullo. Cyber-Physical Attacks in Power Networks: Models, Fundamental Limitations and Monitor Design, September 2011. Note: Available at http://arxiv.org/abs/1103.2795. Keyword(s): Secure Control Systems, Distributed Control Algorithms, Power Networks, Linear Control Design.
   

   @misc{FP-FD-FB:11-arxiv,
   author = {Pasqualetti, F. and D{ö}rfler, F. and Bullo, F.},
   date-added = {2012-10-17 18:24:08 +0000},
   date-modified = {2013-10-01 05:07:27 +0000},
   keywords = {Secure Control Systems, Distributed Control Algorithms, Power Networks, Linear Control Design},
   month = sep,
   note = {Available at {http://arxiv.org/abs/1103.2795}},
   pdf = {http://arxiv.org/pdf/1103.2795v1.pdf},
   title = {Cyber-Physical Attacks in Power Networks: Models, Fundamental Limitations and Monitor Design},
   year = 2011,
   bdsk-url-1 = {http://arxiv.org/pdf/1103.2795v1.pdf} 
   }
   

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