{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,26]],"date-time":"2026-02-26T00:19:33Z","timestamp":1772065173732,"version":"3.50.1"},"reference-count":45,"publisher":"Springer Science and Business Media LLC","issue":"2","license":[{"start":{"date-parts":[[2023,4,1]],"date-time":"2023-04-01T00:00:00Z","timestamp":1680307200000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"},{"start":{"date-parts":[[2023,4,1]],"date-time":"2023-04-01T00:00:00Z","timestamp":1680307200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["J Syst Sci Complex"],"published-print":{"date-parts":[[2023,4]]},"DOI":"10.1007\/s11424-023-1350-6","type":"journal-article","created":{"date-parts":[[2023,4,19]],"date-time":"2023-04-19T22:02:26Z","timestamp":1681941746000},"page":"613-631","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":7,"title":["Distributed Solving Linear Algebraic Equations with Switched Fractional Order Dynamics"],"prefix":"10.1007","volume":"36","author":[{"given":"Wenqiang","family":"Yu","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Songsong","family":"Cheng","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Shuping","family":"He","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2023,4,19]]},"reference":[{"issue":"10","key":"1350_CR1","doi-asserted-by":"publisher","first-page":"2465","DOI":"10.1109\/TAC.2009.2029397","volume":"54","author":"Q Hui","year":"2009","unstructured":"Hui Q, Haddad W M, and Bhat S P, Semistability, finite-time stability, differential inclusions, and discontinuous dynamical systems having a continuum of equilibria, IEEE Transactions on Automatic Control, 2009, 54(10): 2465\u20132470.","journal-title":"IEEE Transactions on Automatic Control"},{"issue":"5","key":"1350_CR2","doi-asserted-by":"publisher","first-page":"36","DOI":"10.1109\/MCS.1987.1105378","volume":"7","author":"C T Chen","year":"1987","unstructured":"Chen C T, Introduction to the linear algebraic method for control system design, IEEE Control Systems Magazine, 1987, 7(5): 36\u201342.","journal-title":"IEEE Control Systems Magazine"},{"issue":"13","key":"1350_CR3","doi-asserted-by":"publisher","first-page":"3459","DOI":"10.1109\/TSP.2018.2830304","volume":"66","author":"S Y Zhu","year":"2018","unstructured":"Zhu S Y, Chen C L, Xu J M, et al., Mitigating quantization effects on distributed sensor fusion: A least squares approach, IEEE Transactions on Signal Processing, 2018, 66(13): 3459\u20133474.","journal-title":"IEEE Transactions on Signal Processing"},{"issue":"18","key":"1350_CR4","doi-asserted-by":"publisher","first-page":"184001","DOI":"10.1088\/1751-8121\/ab3065","volume":"53","author":"A Braunstein","year":"2020","unstructured":"Braunstein A, Muntoni A P, Pagnani A, et al., Compressed sensing reconstruction using expectation propagation, Journal of Physics A: Mathematical and Theoretical, 2020, 53(18): 184001.","journal-title":"Journal of Physics A: Mathematical and Theoretical"},{"issue":"17","key":"1350_CR5","first-page":"819","volume":"3","author":"F N Dafchahi","year":"2008","unstructured":"Dafchahi F N, A new refinement of Jacobi method for solution of linear system equations Ax = b, International Journal of Contemporary Mathematical Sciences, 2008, 3(17): 819\u2013827.","journal-title":"International Journal of Contemporary Mathematical Sciences"},{"issue":"3","key":"1350_CR6","first-page":"109","volume":"6","author":"V Vatti","year":"2011","unstructured":"Vatti V and Gonfa G G, Refinement of generalized Jacobi (RGJ) method for solving system of linear equations, International Journal of Contemporary Mathematical Sciences, 2011, 6(3): 109\u2013116.","journal-title":"International Journal of Contemporary Mathematical Sciences"},{"issue":"4","key":"1350_CR7","doi-asserted-by":"publisher","first-page":"582","DOI":"10.1137\/0704053","volume":"4","author":"J Hurt","year":"1967","unstructured":"Hurt J, Some stability theorems for ordinary difference equations, SIAM Journal on Numerical Analysis, 1967, 4(4): 582\u2013596.","journal-title":"SIAM Journal on Numerical Analysis"},{"issue":"3","key":"1350_CR8","doi-asserted-by":"publisher","first-page":"497","DOI":"10.1137\/0703043","volume":"3","author":"J M Ortega","year":"1966","unstructured":"Ortega J M and Rockoff M L, Nonlinear difference equations and Gauss-Seidel type iterative methods, SIAM Journal on Numerical Analysis, 1966, 3(3): 497\u2013513.","journal-title":"SIAM Journal on Numerical Analysis"},{"issue":"4","key":"1350_CR9","doi-asserted-by":"publisher","first-page":"333","DOI":"10.1007\/s12532-012-0044-1","volume":"4","author":"Z W Wen","year":"2012","unstructured":"Wen Z W, Yin W T, and Zhang Y, Solving a low-rank factorization model for matrix completion by a nonlinear successive over-relaxation algorithm, Mathematical Programming Computation, 2012, 4(4): 333\u2013361.","journal-title":"Mathematical Programming Computation"},{"issue":"1","key":"1350_CR10","doi-asserted-by":"publisher","first-page":"189","DOI":"10.1016\/j.amc.2003.12.085","volume":"162","author":"T Allahviranloo","year":"2005","unstructured":"Allahviranloo T, Successive over relaxation iterative method for fuzzy system of linear equations, Applied Mathematics and Computation, 2005, 162(1): 189\u2013196.","journal-title":"Applied Mathematics and Computation"},{"issue":"10","key":"1350_CR11","first-page":"1547","volume":"46","author":"P Yi","year":"2016","unstructured":"Yi P and Hong Y G, Distributed cooperative optimization and its applications, Scientia Sinica Mathematica, 2016, 46(10): 1547\u20131564.","journal-title":"Scientia Sinica Mathematica"},{"key":"1350_CR12","doi-asserted-by":"publisher","first-page":"278","DOI":"10.1016\/j.arcontrol.2019.05.006","volume":"47","author":"T Yang","year":"2019","unstructured":"Yang T, Yi X L, Wu J F, et al., A survey of distributed optimization, Annual Reviews in Control, 2019, 47: 278\u2013305.","journal-title":"Annual Reviews in Control"},{"issue":"2","key":"1350_CR13","doi-asserted-by":"publisher","first-page":"944","DOI":"10.1137\/14096668X","volume":"20","author":"W Shi","year":"2015","unstructured":"Shi W, Ling Q, Wu G, et al., Extra: An exact first-order algorithm for decentralized consensus optimization, SIAM Journal on Optimization, 2015, 20(2): 944\u2013966.","journal-title":"SIAM Journal on Optimization"},{"issue":"2","key":"1350_CR14","doi-asserted-by":"publisher","first-page":"249","DOI":"10.1007\/s11768-021-00044-0","volume":"19","author":"S S Cheng","year":"2021","unstructured":"Cheng S S, Liang S, and Fan Y, Distributed solving Sylvester equations with fractional order dynamics, Control Theory and Technology, 2021, 19(2): 249\u2013259.","journal-title":"Control Theory and Technology"},{"issue":"6","key":"1350_CR15","doi-asserted-by":"publisher","first-page":"1914","DOI":"10.1007\/s11424-020-9021-3","volume":"33","author":"L P Mo","year":"2020","unstructured":"Mo L P, Liu X D, Cao X B, et al., Distributed second-order continuous-time optimization via adaptive algorithm with nonuniform gradient gains, Journal of Systems Science and Complexity, 2020, 33(6): 1914\u20131932.","journal-title":"Journal of Systems Science and Complexity"},{"issue":"4","key":"1350_CR16","doi-asserted-by":"publisher","first-page":"1438","DOI":"10.1007\/s11424-021-9355-5","volume":"34","author":"J X Wang","year":"2021","unstructured":"Wang J X, Fu K L, Gu Y, et al., Convergence of distributed gradient-tracking-based optimization algorithms with random graphs, Journal of Systems Science and Complexity, 2021, 34(4): 1438\u20131453.","journal-title":"Journal of Systems Science and Complexity"},{"issue":"1","key":"1350_CR17","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1007\/s11424-021-0600-8","volume":"34","author":"P Yi","year":"2021","unstructured":"Yi P and Li L, Distributed nonsmooth convex optimization over Markovian switching random networks with two step-sizes, Journal of Systems Science and Complexity, 2021, 34(1): 1\u201321.","journal-title":"Journal of Systems Science and Complexity"},{"issue":"6","key":"1350_CR18","doi-asserted-by":"publisher","first-page":"1753","DOI":"10.1109\/TAC.2017.2752001","volume":"63","author":"S Liang","year":"2017","unstructured":"Liang S, Zeng X L, and Hong Y G, Distributed nonsmooth optimization with coupled inequality constraints via modified Lagrangian function, IEEE Transactions on Automatic Control, 2017, 63(6): 1753\u20131759.","journal-title":"IEEE Transactions on Automatic Control"},{"issue":"10","key":"1350_CR19","doi-asserted-by":"publisher","first-page":"5227","DOI":"10.1109\/TAC.2016.2628807","volume":"62","author":"X L Zeng","year":"2016","unstructured":"Zeng X L, Yi P, and Hong Y G, Distributed continuous-time algorithm for constrained convex optimizations via nonsmooth analysis approach, IEEE Transactions on Automatic Control, 2016, 62(10): 5227\u20135233.","journal-title":"IEEE Transactions on Automatic Control"},{"key":"1350_CR20","doi-asserted-by":"publisher","first-page":"196","DOI":"10.1016\/j.automatica.2017.12.053","volume":"90","author":"D M Yuan","year":"2018","unstructured":"Yuan D M, Hong Y G, Daniel W C H, et al., Optimal distributed stochastic mirror descent for strongly convex optimization, Automatica, 2018, 90: 196\u2013203.","journal-title":"Automatica"},{"key":"1350_CR21","doi-asserted-by":"crossref","unstructured":"Wang Y H, Zhao W X, Hong Y G, et al., Distributed subgradient-free stochastic optimization algorithm for nonsmooth convex functions over time-varying networks, SIAM Journal on Control and Optimization, 57(4): 2821\u20132842.","DOI":"10.1137\/18M119046X"},{"key":"1350_CR22","doi-asserted-by":"publisher","first-page":"55","DOI":"10.1016\/j.sysconle.2014.02.005","volume":"67","author":"P Benner","year":"2014","unstructured":"Benner P and Breiten T, On optimality of approximate low rank solutions of large-scale matrix equations, Systems & Control Letters, 2014, 67: 55\u201364.","journal-title":"Systems & Control Letters"},{"issue":"11","key":"1350_CR23","doi-asserted-by":"publisher","first-page":"2863","DOI":"10.1109\/TAC.2015.2414771","volume":"63","author":"S S Mou","year":"2015","unstructured":"Mou S S, Liu J, and Morse A S, A distributed algorithm for solving a linear algebraic equation, IEEE Transactions on Automatic Control, 2015, 63(11): 2863\u20132878.","journal-title":"IEEE Transactions on Automatic Control"},{"issue":"2","key":"1350_CR24","doi-asserted-by":"publisher","first-page":"372","DOI":"10.1109\/TAC.2017.2714645","volume":"63","author":"J Liu","year":"2017","unstructured":"Liu J, Mou S S, and Morse A S, Asynchronous distributed algorithms for solving linear algebraic equations, IEEE Transactions on Automatic Control, 2017, 63(2): 372\u2013385.","journal-title":"IEEE Transactions on Automatic Control"},{"issue":"2","key":"1350_CR25","doi-asserted-by":"publisher","first-page":"1191","DOI":"10.1137\/19M1258864","volume":"30","author":"J L Lei","year":"2020","unstructured":"Lei J L, Yi P, Shi G D, et al., Distributed algorithms with finite data rates that solve linear equations, SIAM Journal on Optimization, 2020, 30(2): 1191\u20131222.","journal-title":"SIAM Journal on Optimization"},{"issue":"6","key":"1350_CR26","doi-asserted-by":"publisher","first-page":"2659","DOI":"10.1109\/TAC.2016.2612819","volume":"62","author":"G D Shi","year":"2017","unstructured":"Shi G D, Anderson B D O, and Helmke U, Network flows that solve linear equations, IEEE Transactions on Automatic Control, 2017, 62(6): 2659\u20132674.","journal-title":"IEEE Transactions on Automatic Control"},{"issue":"2","key":"1350_CR27","doi-asserted-by":"publisher","first-page":"414","DOI":"10.1109\/LCSYS.2019.2942711","volume":"4","author":"W Deng","year":"2019","unstructured":"Deng W, Zeng X L, and Hong Y G, Distributed computation for solving the Sylvester equation based on optimization, IEEE Control Systems Letters, 2019, 4(2): 414\u2013419.","journal-title":"IEEE Control Systems Letters"},{"issue":"5","key":"1350_CR28","doi-asserted-by":"publisher","first-page":"1858","DOI":"10.1109\/TAC.2018.2847603","volume":"64","author":"X L Zeng","year":"2018","unstructured":"Zeng X L, Liang S, Hong Y G, et al., Distributed computation of linear matrix equations: An optimization perspective, IEEE Transactions on Automatic Control, 2018, 64(5): 1858\u20131873.","journal-title":"IEEE Transactions on Automatic Control"},{"issue":"1","key":"1350_CR29","doi-asserted-by":"publisher","first-page":"65","DOI":"10.1016\/S0893-6080(03)00170-9","volume":"17","author":"A Bhaya","year":"2004","unstructured":"Bhaya A and Kaszkurewicz E, Steepest descent with momentum for quadratic functions is a version of the conjugate gradient method, Neural Networks, 2004, 17(1): 65\u201371.","journal-title":"Neural Networks"},{"key":"1350_CR30","unstructured":"Muehlebach M and Jordan M, A dynamical systems perspective on Nesterov acceleration, International Conference on Machine Learning, PMLR, 2019."},{"key":"1350_CR31","first-page":"1","volume":"17","author":"W J Su","year":"2016","unstructured":"Su W J, Boyd S, and Cand\u00e8s E J, A differential equation for modeling Nesterov\u2019s accelerated gradient method: Theory and insights, Journal of Machine Learning Research, 2016, 17: 1\u201343.","journal-title":"Journal of Machine Learning Research"},{"issue":"4","key":"1350_CR32","doi-asserted-by":"publisher","first-page":"1911","DOI":"10.1002\/asjc.2402","volume":"23","author":"Y H Wei","year":"2020","unstructured":"Wei Y H, Gao Q, Cheng S S, et al., Description and analysis of the time domain response of nabla discrete fractional order systems, Asian Journal of Control, 2020, 23(4): 1911\u20131922.","journal-title":"Asian Journal of Control"},{"key":"1350_CR33","doi-asserted-by":"crossref","unstructured":"Wang Y and Liang S, Two-DOF lifted LMI conditions for robust D-stability of polynomial matrix polytopes, International Journal of Control, Automation and Systems, 11(3): 636\u2013642.","DOI":"10.1007\/s12555-012-0471-9"},{"key":"1350_CR34","doi-asserted-by":"publisher","first-page":"79","DOI":"10.1016\/j.isatra.2015.08.011","volume":"59","author":"Y L Li","year":"2015","unstructured":"Li Y L, Meng X, Zheng B C, et al., Parameter identification of fractional order linear system based on Haar wavelet operational matrix, ISA Transactions, 2015, 59: 79\u201384.","journal-title":"ISA Transactions"},{"key":"1350_CR35","doi-asserted-by":"publisher","unstructured":"Wei Y H and Chen Y Q, Converse Lyapunov theorem for nabla asymptotic stability without conservativeness, IEEE Transactions on Systems, Man, and Cybernetics: Systems, DOI: https:\/\/doi.org\/10.1109\/TSMC.2021.3051639, 2021.","DOI":"10.1109\/TSMC.2021.3051639"},{"key":"1350_CR36","doi-asserted-by":"publisher","unstructured":"Wei Y H, Lyapunov stability theory for nonlinear nabla fractional order systems, IEEE Transactions on Circuits and Systems II: Express Briefs, DOI: https:\/\/doi.org\/10.1109\/TCSII.2021.3063914, 2021.","DOI":"10.1109\/TCSII.2021.3063914"},{"issue":"6","key":"1350_CR37","doi-asserted-by":"publisher","first-page":"1294","DOI":"10.1109\/TAC.2009.2013056","volume":"54","author":"J G Lu","year":"2009","unstructured":"Lu J G and Chen G R, Robust stability and stabilization of fractional order interval systems: An LMI approach, IEEE Transactions on Automatic Control, 2009, 54(6): 1294\u20131299.","journal-title":"IEEE Transactions on Automatic Control"},{"issue":"4","key":"1350_CR38","doi-asserted-by":"publisher","first-page":"633","DOI":"10.1007\/s11071-013-1091-5","volume":"75","author":"L P Chen","year":"2014","unstructured":"Chen L P, He Y G, Chai Y, et al., New results on stability and stabilization of a class of nonlinear fractional order systems, Nonlinear Dynamics, 2014, 75(4): 633\u2013641.","journal-title":"Nonlinear Dynamics"},{"key":"1350_CR39","doi-asserted-by":"publisher","first-page":"109679","DOI":"10.1016\/j.automatica.2021.109679","volume":"130","author":"Y Q Wei","year":"2021","unstructured":"Wei Y Q, Liu D Y, Boutat D, et al., Modulating functions based model-free fractional order differentiators using a sliding integration window, Automatica, 2021, 130: 109679.","journal-title":"Automatica"},{"key":"1350_CR40","volume-title":"Fractional Processes and Fractional-order Signal Processing: Techniques and Applications","author":"H Sheng","year":"2011","unstructured":"Sheng H, Chen Y Q, and Qiu T S, Fractional Processes and Fractional-order Signal Processing: Techniques and Applications, Springer Science & Business Media, Berlin, 2011."},{"issue":"11","key":"1350_CR41","doi-asserted-by":"publisher","first-page":"320","DOI":"10.1109\/97.873570","volume":"7","author":"T Alieva","year":"2000","unstructured":"Alieva T and Bastiaans M J, On fractional Fourier transform moments, IEEE Signal Processing Letters, 2000, 7(11): 320\u2013323.","journal-title":"IEEE Signal Processing Letters"},{"issue":"2","key":"1350_CR42","doi-asserted-by":"publisher","first-page":"516","DOI":"10.1109\/TCST.2009.2019120","volume":"18","author":"H S Li","year":"2009","unstructured":"Li H S, Luo Y, and Chen Y Q, A fractional order proportional and derivative (FOPD) motion controller: Tuning rule and experiments, IEEE Transactions on Control Systems Technology, 2009, 18(2): 516\u2013520.","journal-title":"IEEE Transactions on Control Systems Technology"},{"key":"1350_CR43","doi-asserted-by":"publisher","first-page":"67","DOI":"10.1016\/j.isatra.2016.11.019","volume":"67","author":"S S Cheng","year":"2017","unstructured":"Cheng S S, Wei Y H, Chen Y Q, et al., A universal modified LMS algorithm with iteration order hybrid switching, ISA Transactions, 2017, 67: 67\u201375.","journal-title":"ISA Transactions"},{"key":"1350_CR44","volume-title":"Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications","author":"I Podlubny","year":"1998","unstructured":"Podlubny I, Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications, Elsevier, Academic Press, San Diego, 1998."},{"issue":"1","key":"1350_CR45","doi-asserted-by":"publisher","first-page":"155","DOI":"10.1007\/s11071-004-3752-x","volume":"38","author":"Y Q Chen","year":"2004","unstructured":"Chen Y Q, Vinagre B M, and Podlubny I, Continued fraction expansion approaches to discretizing fractional order derivatives-an expository review, Nonlinear Dynamics, 2004, 38(1): 155\u2013170.","journal-title":"Nonlinear Dynamics"}],"container-title":["Journal of Systems Science and Complexity"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s11424-023-1350-6.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s11424-023-1350-6\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s11424-023-1350-6.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,4,19]],"date-time":"2023-04-19T22:04:17Z","timestamp":1681941857000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s11424-023-1350-6"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,4]]},"references-count":45,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2023,4]]}},"alternative-id":["1350"],"URL":"https:\/\/doi.org\/10.1007\/s11424-023-1350-6","relation":{},"ISSN":["1009-6124","1559-7067"],"issn-type":[{"value":"1009-6124","type":"print"},{"value":"1559-7067","type":"electronic"}],"subject":[],"published":{"date-parts":[[2023,4]]},"assertion":[{"value":"15 September 2021","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"31 October 2021","order":2,"name":"revised","label":"Revised","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"19 April 2023","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}]}}