{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,7,7]],"date-time":"2026-07-07T15:39:29Z","timestamp":1783438769364,"version":"3.54.6"},"reference-count":23,"publisher":"MDPI AG","issue":"7","license":[{"start":{"date-parts":[[2011,6,27]],"date-time":"2011-06-27T00:00:00Z","timestamp":1309132800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/3.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Sensors"],"abstract":"<jats:p>The Phase-Locked Loop (PLL) is a key component of modern electronic communication and control systems. PLL is designed to extract signals from transmission channels. It plays an important role in systems where it is required to estimate the phase of a received signal, such as carrier tracking from global positioning system satellites. In order to robustly provide centimeter-level accuracy, it is crucial for the PLL to estimate the instantaneous phase of an incoming signal which is usually buried in random noise or some type of interference. This paper presents an approach that utilizes the recent development in the semi-definite programming and sum-of-squares field. A Lyapunov function will be searched as the certificate of the pull-in range of the PLL system. Moreover, a polynomial design procedure is proposed to further refine the controller parameters for system response away from the equilibrium point. Several simulation results as well as an experiment result are provided to show the effectiveness of this approach.<\/jats:p>","DOI":"10.3390\/s110706575","type":"journal-article","created":{"date-parts":[[2011,6,27]],"date-time":"2011-06-27T22:44:03Z","timestamp":1309214643000},"page":"6575-6592","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":16,"title":["Polynomial Method for PLL Controller Optimization"],"prefix":"10.3390","volume":"11","author":[{"given":"Ta-Chung","family":"Wang","sequence":"first","affiliation":[{"name":"Institute of Civil Aviation, National Cheng-Kung University, No.1 University Road, Tainan 701, Taiwan"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Sanjay","family":"Lall","sequence":"additional","affiliation":[{"name":"Department of Aeronautics and Astronautics, Stanford University, Stanford, CA 94305, USA"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Tsung-Yu","family":"Chiou","sequence":"additional","affiliation":[{"name":"MediaTek Inc., No.1 Dusing Road. 1, Hsinchu Science Park, Hsinchu 30078, Taiwan"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2011,6,27]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"627","DOI":"10.1016\/0005-1098(71)90027-6","article-title":"A computational method for determining quadratic Lyapunov functions for non-linear systems","volume":"7","author":"Davison","year":"1971","journal-title":"Automatica"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"2471","DOI":"10.1109\/9.362845","article-title":"An analytical method of estimating the domain of attraction for polynomial differential equations","volume":"39","author":"Levin","year":"1994","journal-title":"IEEE Trans. Autom. Control"},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"Chesi, G, Tesi, A, and Vicino, A (2002, January 12\u201316). On Optimal Quadratic Lyapunov Functions for Polynomial Systems. South Bend, IN, USA.","DOI":"10.3182\/20020721-6-ES-1901.00257"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"307","DOI":"10.3166\/ejc.9.307-321","article-title":"Semidefinite programming relaxations and algebraic optimization in control","volume":"9","author":"Parrilo","year":"2003","journal-title":"Eur. J. Control"},{"key":"ref_5","unstructured":"Papachristodoulou, A, and Prajna, S (, January December). On the Construction of Lyapunov Functions Using the sum of Squares Decomposition. Las Vegas, NV, USA."},{"key":"ref_6","unstructured":"Prajna, S, Papachristodoulou, A, and Parrilo, P (, January December). Introducing SOSTOOLS: A General Purpose Sum of Squares Programming Solver. Las Vegas, NV, USA."},{"key":"ref_7","unstructured":"Parrilo, P (2000). California Institute of Technology. PhD Thesis."},{"key":"ref_8","unstructured":"Jarvis-Wloszek, ZW (2003). University of California. PhD Thesis."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"565","DOI":"10.1109\/TAC.2007.914221","article-title":"Stability region analysis using polynomial and composite polynomial lyapunov functions and sum-of-squares programming","volume":"53","author":"Tan","year":"2008","journal-title":"IEEE Trans. Autom. Control"},{"key":"ref_10","unstructured":"Wang, T, Lall, S, and West, M (2005, January 28\u201330). Polynomial Level-Set Methods for Nonlinear Dynamical Systems Analysis. Monticello, IL, USA."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"987","DOI":"10.1109\/31.1845","article-title":"Chaos from phase-locked loops","volume":"35","author":"Endo","year":"1988","journal-title":"IEEE Trans. Circuits Syst"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"255","DOI":"10.1109\/31.20202","article-title":"Chaos from phase-locked loops\u2014Part II: Non-Hamiltonian case","volume":"36","author":"Endo","year":"1988","journal-title":"IEEE Trans. Circuits Syst"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"1580","DOI":"10.1109\/31.108517","article-title":"Synchronization of chaos in phase-locked loops","volume":"38","author":"Endo","year":"1991","journal-title":"IEEE Trans. Circuits Syst"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"808","DOI":"10.1109\/81.251819","article-title":"Using chaos to broaden the capture range of a phase-locked loop","volume":"40","author":"Bradley","year":"1993","journal-title":"IEEE Trans. Circuits Syst"},{"key":"ref_15","unstructured":"Liu, ZF, Man, KF, Tang, KS, and Zhong, GQ (Sepember, January 31). Controlling Chaos in Phase-Locked Loop. Aachen, Germany."},{"key":"ref_16","unstructured":"Abramovitch, DY (1998, January 18\u201321). Analysis and Design of a Third order Phase-Lock Loop. Boston, MA, USA."},{"key":"ref_17","doi-asserted-by":"crossref","unstructured":"Gardner, FM (2005). Phaselock Techniques, Wiley-Interscience. [3rd ed].","DOI":"10.1002\/0471732699"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"203","DOI":"10.1007\/BF01446568","article-title":"The K-moment problem for compact semi-algebraic sets","volume":"289","year":"1991","journal-title":"Math. Ann"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"969","DOI":"10.1512\/iumj.1993.42.42045","article-title":"Positive polynomials on compact semi-algebraic sets","volume":"42","author":"Putinar","year":"1993","journal-title":"Indiana Univ. Math. J"},{"key":"ref_20","first-page":"223","article-title":"Distinguished representations of strictly positive polynomials","volume":"532","author":"Jacobi","year":"2001","journal-title":"J. Reine Angew. Math"},{"key":"ref_21","unstructured":"Guckenheimer, J, and Holmes, P (2002). Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Springer-Verlag."},{"key":"ref_22","unstructured":"Khalil, H (2000). Nonlinear Systems, Prentice-Hall Inc. [3rd ed]."},{"key":"ref_23","doi-asserted-by":"crossref","unstructured":"Wang, TC, Lall, S, and Chiou, TY (2008, January 25\u201327). Nonlinear Phase-Locked Loop Design using Semidefinite Programming. Ajaccio-Corsica, France.","DOI":"10.1109\/MED.2008.4602142"}],"container-title":["Sensors"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1424-8220\/11\/7\/6575\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T21:56:33Z","timestamp":1760219793000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1424-8220\/11\/7\/6575"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2011,6,27]]},"references-count":23,"journal-issue":{"issue":"7","published-online":{"date-parts":[[2011,7]]}},"alternative-id":["s110706575"],"URL":"https:\/\/doi.org\/10.3390\/s110706575","relation":{},"ISSN":["1424-8220"],"issn-type":[{"value":"1424-8220","type":"electronic"}],"subject":[],"published":{"date-parts":[[2011,6,27]]}}}