The First Five Years of a Phase Theory for Complex Systems and Networks

Dan Wang; Wei Chen; Li Qiu

doi:10.1109/JAS.2024.124542

Volume 11 Issue 8

Aug. 2024

IEEE/CAA Journal of Automatica Sinica

JCR Impact Factor: 15.3, Top 1 (SCI Q1)

CiteScore: 23.5, Top 2% (Q1)
Google Scholar h5-index: 77， TOP 5

Turn off MathJax

Article Contents

Article Navigation > IEEE/CAA Journal of Automatica Sinica > 2024 > 11(8): 1728-1743

D. Wang, W. Chen, and L. Qiu, “The first five years of a phase theory for complex systems and networks,” IEEE/CAA J. Autom. Sinica, vol. 11, no. 8, pp. 1728–1743, Aug. 2024. doi: 10.1109/JAS.2024.124542

Citation:

D. Wang, W. Chen, and L. Qiu, “The first five years of a phase theory for complex systems and networks,” IEEE/CAA J. Autom. Sinica, vol. 11, no. 8, pp. 1728–1743, Aug. 2024. doi: 10.1109/JAS.2024.124542

Citation:

PDF( 1410 KB)

The First Five Years of a Phase Theory for Complex Systems and Networks

doi: 10.1109/JAS.2024.124542

Dan Wang^,,
Wei Chen^{,
,},
Li Qiu^{,
,}

Funds: This work was supported in part by the National Natural Science Foundation of China (62073003, 72131001), Hong Hong Research Grants Council under GRF grants (16200619, 16201120, 16205421, 16203922), Shenzhen-Hong Kong-Macau Science and Technology Innovation Fund (SGDX20201103094600006), and Googol Technology (Shenzhen) Ltd

More Information

Author Bio:
Dan Wang (Member, IEEE) received the B.S. degree in automation from Hangzhou Dianzi University, in 2014. She received the M.Phil. and Ph.D. degrees in electronic and computer engineering from the Hong Kong University of Science and Technology, Hong Kong SAR, China, in 2016 and 2020, respectively. She is currently a Postdoc at KTH Royal Institute of Technology, Sweden. From January 2018 to July 2018, she was a Visiting Scholar at University of Illinois at Urbana Champaign, USA. Her research interests include systems and control theory, phase theory, dynamical networks, network science, power system stability, and control of epidemics

Wei Chen (Member, IEEE) received the B.S. degree in engineering and the double B.S. degree in economics from Peking University, in 2008. He received the M.Phil. and Ph.D. degrees in electronic and computer engineering from the Hong Kong University of Science and Technology, Hong Kong, China, in 2010 and 2014, respectively. He is currently an Assistant Professor in the Department of Mechanics and Engineering Science at Peking University. Prior to joining Peking University, he worked at KTH Royal Institute of Technology, Sweden and University of California at Berkeley, USA for postdoctoral research, and the Hong Kong University of Science and Technology, Hong Kong SAR, China as a Research Assistant Professor.His research interests include linear systems and control, networked control systems, phase theory, optimal control, smart grid, and network science. He was the recipient of the best student paper award at the 2012 IEEE International Conference on Information and Automation

Li Qiu (Fellow, IEEE) received the Ph.D. degree in electrical engineering from the University of Toronto in 1990. After briefly working in the Canadian Space Agency, the Fields Institute for Research in Mathematical Sciences (Waterloo), and the Institute of Mathematics and its Applications (Minneapolis), he joined the Hong Kong University of Science and Technology, Hong Kong SAR, China in 1993 and is now a Professor in the Department of Electronic and Computer Engineering. In 2023, he took a leave to work in Southern University of Science and Technology, as a Chair Professor.Prof. Qiu’s research interests include system, control, optimization theory, and mathematics for information technology, as well as their applications in manufacturing industry and energy systems. He is also interested in control education and co-authored an undergraduate textbook “Introduction to Feedback Control” which was published by Prentice-Hall in 2009. He served as an Associate Editor of the IEEE Transactions on Automatic Control and Automatica. He was the General Chair of the 7th Asian Control Conference, which was held in Hong Kong, China in 2009. He was a Distinguished Lecturer from 2007 to 2010 and was a Member of the Board of Governors in 2012 and 2017 of the IEEE Control Systems Society. He was a Vice President of the Asian Control Association and the Founding Chairperson of the Hong Kong Automatic Control Association. He is a Fellow of IEEE and a Fellow of IFAC
Corresponding author: Wei Chen, e-mail: w.chen@pku.edu.cn; Li Qiu, e-mail: eeqiu@ust.hk
Received Date: 2024-02-29
Revised Date: 2024-04-24
Accepted Date: 2024-05-12

Available Online: 2024-06-04

Abstract

Abstract

In this paper, we review the development of a phase theory for systems and networks in its first five years, represented by a trilogy: Matrix phases and their properties; The MIMO LTI system phase response, its physical interpretations, the small phase theorem, and the sectored real lemma; The synchronization of a multi-agent network using phase alignment. Towards the end, we also summarize a list of ongoing research on the phase theory and speculate what will happen in the next five years.
- Consensus and synchronization,
- matrix phases,
- multi-agent systems,
- phase theory,
- sectored real lemma,
- small phase theorem

FullText(HTML)

References(93)

References

[1]	L. Qiu and K. Zhou, Introduction to Feedback Control. Englewood Cliffs, USA: Prentice Hall, 2009.
[2]	I. Postlethwaite and A. G. J. MacFarlane, A Complex Variable Approach to the Analysis of Linear Multivariable Feedback Systems. Berlin, Germany: Springer, 1979.
[3]	I. Postlethwaite, J. Edmunds, and A. MacFarlane, “Principal gains and principal phases in the analysis of linear multivariable feedback systems,” IEEE Trans. Autom. Control, vol. 26, no. 1, pp. 32–46, Feb. 1981. doi: 10.1109/TAC.1981.1102556
[4]	D. S. Bernstein and W. M. Haddad, “Is there more to robust control theory than small gain,” in Proc. American Control Conf., Chicago, USA, 1992, pp. 83–84.
[5]	J. Chen, “Multivariable gain-phase and sensitivity integral relations and design trade-offs,” IEEE Trans. Autom. Control, vol. 43, no. 3, pp. 373–385, Mar. 1998. doi: 10.1109/9.661594
[6]	J. S. Freudenberg and D. P. Looze, Frequency Domain Properties of Scalar and Multivariable Feedback Systems. Berlin, Germany: Springer, 1988.
[7]	B. D. O. Anderson and M. Green, “Hilbert transform and gain/phase error bounds for rational functions,” IEEE Trans. Circuits Syst., vol. 35, no. 5, pp. 528–535, May 1988. doi: 10.1109/31.1780
[8]	D. H. Owens, “The numerical range: A tool for robust stability studies,” Syst. Control Lett., vol. 5, no. 3, pp. 153–158, Dec. 1984. doi: 10.1016/S0167-6911(84)80096-1
[9]	A. L. Tits, V. Balakrishnan, and L. Lee, “Robustness under bounded uncertainty with phase information,” IEEE Trans. Autom. Control, vol. 44, no. 1, pp. 50–65, Jan. 1999. doi: 10.1109/9.739067
[10]	K. Laib, A. Korniienko, M. Dinh, G. Scorletti, and F. Morel, “Hierarchical robust performance analysis of uncertain large scale systems,” IEEE Trans. Autom. Control, vol. 63, no. 7, pp. 2075–2090, Jul. 2018. doi: 10.1109/TAC.2017.2762468
[11]	A. G. J. MacFarlane and I. Postlethwaite, “The generalized Nyquist stability criterion and multivariable root loci,” Int. J. Control, vol. 25, no. 1, pp. 81–127, 1977. doi: 10.1080/00207177708922217
[12]	U. Shaked, “The angles of departure and approach of the root-loci in linear multivariable systems,” Int. J. Control, vol. 23, no. 4, pp. 445–457, 1976. doi: 10.1080/00207177608922172
[13]	P. Thompson, G. Stein, and A. Laub, “Angles of multivariable root loci,” IEEE Trans. Autom. Control, vol. 27, no. 6, pp. 1241–1243, Dec. 1982. doi: 10.1109/TAC.1982.1103117
[14]	D. Wang, W. Chen, S. Z. Khong, and L. Qiu, “On the phases of a complex matrix,” Linear Algebra Appl., vol. 593, pp. 152–179, May 2020. doi: 10.1016/j.laa.2020.01.035
[15]	D. Wang, X. Mao, W. Chen, and L. Qiu, “On the phases of a semi-sectorial matrix and the essential phase of a Laplacian,” Linear Algebra Appl., vol. 676, pp. 441–458, Nov. 2023. doi: 10.1016/j.laa.2023.07.014
[16]	W. Chen, D. Wang, S. Z. Khong, and L. Qiu, “Phase analysis of MIMO LTI systems,” in Proc. 58th IEEE Conf. Decision and Control, Nice, France, 2019, pp. 6062–6067.
[17]	L. Qiu, W. Chen, and D. Wang, “New phase of phase,” J. Syst. Sci. Complexity, vol. 34, no. 5, pp. 1821–1839, Oct. 2021. doi: 10.1007/s11424-021-1249-z
[18]	X. Mao, W. Chen, and L. Qiu, “Phases of discrete-time LTI multivariable systems,” Automatica, vol. 142, p. 110311, Aug. 2022. doi: 10.1016/j.automatica.2022.110311
[19]	W. Chen, D. Wang, S. Z. Khong, and L. Qiu, “A phase theory of multi-input multi-output linear time-invariant systems,” SIAM J. Control Optim., vol. 62, no. 2, pp. 1235–1260, Apr. 2024. doi: 10.1137/22M148968X
[20]	D. Wang, W. Chen, and L. Qiu, “Synchronization of diverse agents via phase analysis,” Automatica, vol. 159, p. 111325, Jan. 2024. doi: 10.1016/j.automatica.2023.111325
[21]	A. Horn and R. Steinberg, “Eigenvalues of the unitary part of a matrix,” Pacific J. Math., vol. 9, no. 2, pp. 541–550, Jun. 1959. doi: 10.2140/pjm.1959.9.541
[22]	S. Furtado and C. R. Johnson, “Spectral variation under congruence,” Linear Multilinear Algebra, vol. 49, no. 3, pp. 243–259, 2001. doi: 10.1080/03081080108818698
[23]	S. Furtado and C. R. Johnson, “Spectral variation under congruence for a nonsingular matrix with 0 on the boundary of its field of values,” Linear Algebra Appl., vol. 359, no. 1-3, pp. 67–78, Jan. 2003. doi: 10.1016/S0024-3795(02)00429-9
[24]	C. S. Ballantine and C. R. Johnson, “Accretive matrix products,” Linear Multilinear Algebra, vol. 3, no. 3, pp. 169–185, 1975. doi: 10.1080/03081087508817108
[25]	R. A. Horn and C. R. Johnson, Topics in Matrix Analysis. Cambridge, UK: Cambridge University Press, 1991.
[26]	T. Kato, Perturbation Theory for Linear Operators. Berlin, Germany: Springer, 1980.
[27]	G. W. Stewart and J.-G. Sun, Matrix Perturbation Theory. Amsterdam: Elsevier, 1990.
[28]	J. Liang, D. Zhao, and L. Qiu, “Feedback stability under mixed gain and phase uncertainty,” IEEE Trans. Autom. Control, arXiv preprint arXiv: 2404.05609
[29]	M. G. Safonov, “Stability margins of diagonally perturbed multivariable feedback systems,” IEE Proc. D Control Theory Appl., vol. 129, no. 6, pp. 251–256, Nov. 1982. doi: 10.1049/ip-d.1982.0054
[30]	F. L. Bauer, “Optimally scaled matrices,” Numer. Math., vol. 5, no. 1, pp. 73–87, Dec. 1963. doi: 10.1007/BF01385880
[31]	J. Doyle, “Analysis of feedback systems with structured uncertainties,” IEE Proc. D Control Theory Appl., vol. 129, no. 6, pp. 242–250, Nov. 1982. doi: 10.1049/ip-d.1982.0053
[32]	K. Zhou, J. Doyle, and K. Glover, Robust and Optimal Control. Englewood Cliffs, USA: Prentice Hall, 1996.
[33]	J. Stoer and C. Witzgall, “Transformations by diagonal matrices in a normed space,” Numer. Math., vol. 4, no. 1, pp. 158–171, Dec. 1962. doi: 10.1007/BF01386309
[34]	R. Merris, “Laplacian matrices of graphs: A survey,” Linear Algebra Appl., vol. 197-198, pp. 143–176, Jan.–Feb. 1994. doi: 10.1016/0024-3795(94)90486-3
[35]	C. Guo and S. Fan, “Winding number criterion for the origin to belong to the numerical range of a matrix on a loop of matrices,” arXiv preprint arXiv: 2311.00849, 2023.
[36]	B. Brogliato, R. Lozano, B. Maschke, and O. Egeland, Dissipative Systems Analysis and Control: Theory and Applications. 3rd ed. Cham, Switzerland: Springer, 2020.
[37]	K. Liu and Y. Yao, Robust Control: Theory and Applications. Hoboken, USA: John Wiley & Sons, 2016.
[38]	M. Vidyasagar, Input-Output Analysis of Large-Scale Interconnected Systems: Decomposition, Well-Posedness and Stability. Berlin, Germany: Springer, 1981.
[39]	A. Lanzon and I. R. Petersen, “Stability robustness of a feedback interconnection of systems with negative imaginary frequency response,” IEEE Trans. Autom. Control, vol. 53, no. 4, pp. 1042–1046, May 2008. doi: 10.1109/TAC.2008.919567
[40]	I. R. Petersen and A. Lanzon, “Feedback control of negative-imaginary systems,” IEEE Control Syst. Mag., vol. 30, no. 5, pp. 54–72, Oct. 2010. doi: 10.1109/MCS.2010.937676
[41]	M. A. Mabrok, A. G. Kallapur, I. R. Petersen, and A. Lanzon, “Generalizing negative imaginary systems theory to include free body dynamics: Control of highly resonant structures with free body motion,” IEEE Trans. Autom. Control, vol. 59, no. 10, pp. 2692–2707, Oct. 2014. doi: 10.1109/TAC.2014.2325692
[42]	M. Vidyasagar, Control System Synthesis: A Factorization Approach. Cambridge, USA: MIT Press, 1985.
[43]	Y. Shi, X. Zhu, and X. Xu, “Small phase theorem for linear time invariant systems,” 2023, submitted.
[44]	C. Chen, D. Zhao, W. Chen, S. Z. Khong, and L. Qiu, “Phase of nonlinear systems,” arXiv preprint arXiv: 2012.00692, 2021.
[45]	X. Yang, D. Wang, and W. Chen, “On a small phase theorem with necessity over phase bounded nonlinear uncertainties,” IFAC-PapersOnLine, vol. 56, no. 2, pp. 1877–1882, 2023. doi: 10.1016/j.ifacol.2023.10.1905
[46]	L. Huang, D. Wang, X. Wang, H. Xin, P. Ju, K. H. Johansson, and F. Dörfler, “Gain and phase: Decentralized stability conditions for power electronics-dominated power systems,” IEEE Trans. Power Syst., 2024. doi: 10.1109/TPWRS.2024.3380528
[47]	R. Srazhidinov, D. Zhang, and L. Qiu, “Computation of the phase and gain margins of MIMO control systems,” Automatica, vol. 149, p. 110846, Mar. 2023. doi: 10.1016/j.automatica.2022.110846
[48]	R. Srazhidinov, J. Liang, and L. Qiu, “Gain-phase margin of MIMO control systems,” IFAC-PapersOnLine, vol. 56, no. 2, pp. 1728–1735, 2023. doi: 10.1016/j.ifacol.2023.10.1881
[49]	A. Ringh, X. Mao, W. Chen, L. Qiu, and S. Z. Khong, “Gain and phase type multipliers for structured feedback robustness,” IEEE Trans. Autom. Control, arXiv preprint arXiv: 2203.11837
[50]	T. Iwasaki and S. Hara, “Generalized KYP lemma: Unified frequency domain inequalities with design applications,” IEEE Trans. Autom. Control, vol. 50, no. 1, pp. 41–59, Jan. 2005. doi: 10.1109/TAC.2004.840475
[51]	N. Kottenstette, M. J. McCourt, M. Xia, V. Gupta, and P. J. Antsaklis, “On relationships among passivity, positive realness, and dissipativity in linear systems,” Automatica, vol. 50, no. 4, pp. 1003–1016, Apr. 2014. doi: 10.1016/j.automatica.2014.02.013
[52]	K. Zhou, K. Glover, B. Bodenheimer, and J. Doyle, “Mixed H₂ and H_∞ performance objectives. I. Robust performance analysis,” IEEE Trans. Autom. Control, vol. 39, no. 8, pp. 1564–1574, Aug. 1994. doi: 10.1109/9.310030
[53]	Z. Nowomiejski, “Generalized theory of electric power,” Archiv für Elektrotechnik, vol. 63, no. 3, pp. 177–182, May 1981.
[54]	S. L. Hahn, Hilbert Transforms in Signal Processing. Boston, USA: Artech House Inc., 1996.
[55]	J. Doyle, B. Francis, and A. Tannenbaum, Feedback Control Theory. New York, USA: MacMillan, 1990.
[56]	W. Ren and R. W. Beard, Distributed Consensus in Multi-vehicle Cooperative Control. London, UK: Springer, 2008.
[57]	Z. Lin, B. Francis, and M. Maggiore, Distributed Control and Analysis of Coupled Cell Systems. Saarbrücken, Germany: VDM Verlag, 2008.
[58]	F. Bullo, J. Cortés, and S. Martínez, Distributed Control of Robotic Networks: A Mathematical Approach to Motion Coordination Algorithms. Princeton, USA: Princeton University Press, 2009.
[59]	M. Mesbahi and M. Egerstedt, Graph Theoretic Methods in Multiagent Networks. Princeton, USA: Princeton University Press, 2010.
[60]	F. L. Lewis, H. Zhang, K. Hengster-Movric, and A. Das, Cooperative Control of Multi-Agent Systems: Optimal and Adaptive Design Approaches. London, UK: Springer, 2014.
[61]	F. Bullo, Lectures on Network Systems. Charleston, USA: CreateSpace Independent Publishing Platform, 2019.
[62]	A. Jadbabaie, J. Lin, and A. S. Morse, “Coordination of groups of mobile autonomous agents using nearest neighbor rules,” IEEE Trans. Autom. Control, vol. 48, no. 6, pp. 988–1001, Jun. 2003. doi: 10.1109/TAC.2003.812781
[63]	R. Olfati-Saber and R. M. Murray, “Consensus problems in networks of agents with switching topology and time-delays,” IEEE Trans. Autom. Control, vol. 49, no. 9, pp. 1520–1533, Sep. 2004. doi: 10.1109/TAC.2004.834113
[64]	W. Ren and R. W. Beard, “Consensus seeking in multiagent systems under dynamically changing interaction topologies,” IEEE Trans. Autom. Control, vol. 50, no. 5, pp. 655–661, May 2005. doi: 10.1109/TAC.2005.846556
[65]	W. Ren, R. W. Beard, and E. M. Atkins, “A survey of consensus problems in multi-agent coordination,” in Proc. American Control Conf., Portland, USA, 2005, pp. 1859–1864.
[66]	Z. Lin, B. Francis, and M. Maggiore, “Getting mobile autonomous robots to rendezvous,” in Proc. Workshop on Control of Uncertain Systems: Modelling, Approximation, and Design, Berlin, Germany, 2006, pp. 119–137.
[67]	R. Olfati-Saber, J. A. Fax, and R. M. Murray, “Consensus and cooperation in networked multi-agent systems,” Proc. IEEE, vol. 95, no. 1, pp. 215–233, Jan. 2007. doi: 10.1109/JPROC.2006.887293
[68]	J. A. Fax and R. M. Murray, “Information flow and cooperative control of vehicle formations,” IEEE Trans. Autom. Control, vol. 49, no. 9, pp. 1465–1476, Sep. 2004. doi: 10.1109/TAC.2004.834433
[69]	C.-Q. Ma and J.-F. Zhang, “Necessary and sufficient conditions for consensusability of linear multi-agent systems,” IEEE Trans. Autom. Control, vol. 55, no. 5, pp. 1263–1268, May 2010. doi: 10.1109/TAC.2010.2042764
[70]	Z. Li, Z. Duan, G. Chen, and L. Huang, “Consensus of multiagent systems and synchronization of complex networks: A unified viewpoint,” IEEE Trans. Circuits Syst. I: Regul. Pap., vol. 57, no. 1, pp. 213–224, Jan. 2010. doi: 10.1109/TCSI.2009.2023937
[71]	K. You and L. Xie, “Network topology and communication data rate for consensusability of discrete-time multi-agent systems,” IEEE Trans. Autom. Control, vol. 56, no. 10, pp. 2262–2275, Oct. 2011. doi: 10.1109/TAC.2011.2164017
[72]	G. Gu, L. Marinovici, and F. L. Lewis, “Consensusability of discrete-time dynamic multiagent systems,” IEEE Trans. Autom. Control, vol. 57, no. 8, pp. 2085–2089, Aug. 2012. doi: 10.1109/TAC.2011.2179431
[73]	I. Lestas and G. Vinnicombe, “Heterogeneity and scalability in group agreement protocols: Beyond small gain and passivity approaches,” Automatica, vol. 46, no. 7, pp. 1141–1151, Jul. 2010. doi: 10.1016/j.automatica.2010.03.018
[74]	P. Wieland, R. Sepulchre, and F. Allgöwer, “An internal model principle is necessary and sufficient for linear output synchronization,” Automatica, vol. 47, no. 5, pp. 1068–1074, May 2011. doi: 10.1016/j.automatica.2011.01.081
[75]	M. Bürger, D. Zelazo, and F. Allgöwer, “Duality and network theory in passivity-based cooperative control,” Automatica, vol. 50, no. 8, pp. 2051–2061, Aug. 2014. doi: 10.1016/j.automatica.2014.06.002
[76]	S. Z. Khong, E. Lovisari, and A. Rantzer, “A unifying framework for robust synchronization of heterogeneous networks via integral quadratic constraints,” IEEE Trans. Autom. Control, vol. 61, no. 5, pp. 1297–1309, May 2016. doi: 10.1109/TAC.2016.2545118
[77]	M. Lu, L. Liu, and G. Feng, “Output synchronization of heterogeneous linear multi-agent systems,” in Proc. 11th Asian Control Conf., Gold Coast, Australia, 2017, pp. 156–161.
[78]	D. Wang, W. Chen, and L. Qiu, “Synchronization of heterogeneous dynamical networks via phase analysis,” IFAC-PapersOnLine, vol. 53, no. 2, pp. 3013–3018, 2020. doi: 10.1016/j.ifacol.2020.12.989
[79]	H. Jeffreys and B. Jeffreys, “Lagrange interpolation formula,” Methodsof Mathematical Physics, vol. 3, p. 260, 1988.
[80]	A. Tannenbaum, “Feedback stabilization of linear dynamical plants with uncertainty in the gain factor,” Int. J. Control, vol. 32, no. 1, pp. 1–16, Jan. 1980. doi: 10.1080/00207178008922838
[81]	R. Saeks and J. Murray, “Fractional representation, algebraic geometry, and the simultaneous stabilization problem,” IEEE Trans. Autom. Control, vol. 27, no. 4, pp. 895–903, Aug. 1982. doi: 10.1109/TAC.1982.1103005
[82]	M. Vidyasagar and N. Viswanadham, “Algebraic design techniques for reliable stabilization,” IEEE Trans. Autom. Control, vol. 27, no. 5, pp. 1085–1095, Aug. 1982. doi: 10.1109/TAC.1982.1103086
[83]	B. Ghosh and C. Byrnes, “Simultaneous stabilization and simultaneous pole-placement by nonswitching dynamic compensation,” IEEE Trans. Circuits Syst., vol. 30, no. 6, pp. 422–428, Jun. 1983. doi: 10.1109/TCS.1983.1085369
[84]	D. Zhao, A. Ringh, L. Qiu, and S. Z. Khong, “Low phase-rank approximation,” Linear Algebra Appl., vol. 639, pp. 177–204, Apr. 2022. doi: 10.1016/j.laa.2022.01.003
[85]	A. Ringh and L. Qiu, “Finsler geometries on strictly accretive matrices,” Linear Multilinear Algebra, vol. 70, no. 21, pp. 6753–6771, 2022. doi: 10.1080/03081087.2021.1968781
[86]	T. Yu, D. Zhao, and L. Qiu, “Phases of sectorial operators,” Integr. Equations Oper. Theory, vol. 95, no. 4, p. 31, Nov. 2023. doi: 10.1007/s00020-023-02752-5
[87]	Y. Liu, L. Liu, and Y. Lu, “The generalized sectorial decompositions of semi-sectorial operators,” Linear Multilinear Algebra, 2024. doi: 10.1080/03081087.2024.2311860
[88]	A. W. Marshall, I. Olkin, and B. C. Arnold, Inequalities: Theory of Majorization and its Applications. New York, USA: Academic Press, 1979.
[89]	L. Woolcock and R. Schmid, “Mixed gain/phase robustness criterion for structured perturbations with an application to power system stability,” IEEE Control Syst. Lett., vol. 7, pp. 3193–3198, Jun. 2023. doi: 10.1109/LCSYS.2023.3290183
[90]	D. Zhang, J. Wang, F. Zhang, I. Lestas, and L. Qiu, “Entangled gain and phase analysis for internet,” 2023, working paper.
[91]	G. Zames, “On the input-output stability of time-varying nonlinear feedback systems part one: Conditions derived using concepts of loop gain, conicity, and positivity,” IEEE Trans. Autom. Control, vol. 11, no. 2, pp. 228–238, Apr. 1966. doi: 10.1109/TAC.1966.1098316
[92]	J. Chen, W. Chen, and L. Qiu, “Phase analysis of N-port electrical networks under interconnections,” IFAC-PapersOnLine, vol. 56, no. 2, pp. 4295–4300, 2023. doi: 10.1016/j.ifacol.2023.10.1798
[93]	J. Chen, W. Chen, C. Chen, and L. Qiu, “Phase preservation of N-port networks under general connections,” IEEE Trans. Autom. Control, arXiv preprint arXiv: 2311.16523v1

Supplements(0)

Cited By

Proportional views

Proportional views

通讯作者: 陈斌, bchen63@163.com

1.
沈阳化工大学材料科学与工程学院沈阳 110142

Figures(8) / Tables(1)

Get Citation

PDF

XML

Article Metrics

Article views (305) PDF downloads(88)

Highlights

Matrix phases and their properties
The MIMO LTI system phase response, its physical interpretations, the small phase theorem, and the sectored real lemma
The synchronization of a multi-agent network using phase alignment

The First Five Years of a Phase Theory for Complex Systems and Networks

doi: 10.1109/JAS.2024.124542

Abstract

References

Proportional views

Catalog

通讯作者: 陈斌, bchen63@163.com

Article Metrics

Highlights

Export File

Citation

Format

Content