{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T04:01:14Z","timestamp":1760241674938,"version":"build-2065373602"},"reference-count":35,"publisher":"MDPI AG","issue":"7","license":[{"start":{"date-parts":[[2018,6,26]],"date-time":"2018-06-26T00:00:00Z","timestamp":1529971200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Algorithms"],"abstract":"<jats:p>The significant task for control performance assessment (CPA) is to review and evaluate the performance of the control system. The control system in the semiconductor industry exhibits a complex dynamic behavior, which is hard to analyze. This paper investigates the interesting crossover properties of Hurst exponent estimations and proposes a novel method for feature extraction of the nonlinear multi-input multi-output (MIMO) systems. At first, coupled data from real industry are analyzed by multifractal detrended fluctuation analysis (MFDFA) and the resultant multifractal spectrum is obtained. Secondly, the crossover points with spline fit in the scale-law curve are located and then employed to segment the entire scale-law curve into several different scaling regions, in which a single Hurst exponent can be estimated. Thirdly, to further ascertain the origin of the multifractality of control signals, the generalized Hurst exponents of the original series are compared with shuffled data. At last, non-Gaussian statistical properties, multifractal properties and Hurst exponents of the process control variables are derived and compared with different sets of tuning parameters. The results have shown that CPA of the MIMO system can be better employed with the help of fractional order signal processing (FOSP).<\/jats:p>","DOI":"10.3390\/a11070090","type":"journal-article","created":{"date-parts":[[2018,6,26]],"date-time":"2018-06-26T10:40:50Z","timestamp":1530009650000},"page":"90","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":14,"title":["A Novel Method for Control Performance Assessment with Fractional Order Signal Processing and Its Application to Semiconductor Manufacturing"],"prefix":"10.3390","volume":"11","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-9822-4544","authenticated-orcid":false,"given":"Kai","family":"Liu","sequence":"first","affiliation":[{"name":"School of Mechanical Electronic &amp; Information Engineering, China University of Mining and Technology, Beijing 100083, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-7422-5988","authenticated-orcid":false,"given":"YangQuan","family":"Chen","sequence":"additional","affiliation":[{"name":"Mechatronics, Embedded Systems and Automation Lab, University of California, Merced, CA 95343, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Pawe\u0142 D.","family":"Doma\u0144ski","sequence":"additional","affiliation":[{"name":"Institute of Control and Computation Engineering, Warsaw University of Technology, Nowowiejska 15\/19, 00-665 Warsaw, Poland"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-7574-9487","authenticated-orcid":false,"given":"Xi","family":"Zhang","sequence":"additional","affiliation":[{"name":"School of Mechanical Electronic &amp; Information Engineering, China University of Mining and Technology, Beijing 100083, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2018,6,26]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Jelali, M. (2012). Control Performance Management in Industrial Automation: Assessment, Diagnosis and Improvement of Control Loop Performance, Springer Science & Business Media.","DOI":"10.1007\/978-1-4471-4546-2"},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Sheng, H., Chen, Y., and Qiu, T. (2011). Fractional Processes and Fractional-Order Signal Processing: Techniques and Applications, Springer Science & Business Media.","DOI":"10.1007\/978-1-4471-2233-3"},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"Chen, Y., Sun, R., and Zhou, A. (2007, January 4\u20137). An overview of fractional order signal processing (FOSP) techniques. Proceedings of the International Design Engineering Technical Conferences and Computers and Information in Engineering Conference, Las Vegas, NV, USA.","DOI":"10.1115\/DETC2007-34228"},{"key":"ref_4","doi-asserted-by":"crossref","unstructured":"West, B.J. (2016). Fractional Calculus View of Complexity: Tomorrow\u2019s Science, CRC Press.","DOI":"10.1201\/b18911"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"441","DOI":"10.1016\/j.conengprac.2005.11.005","article-title":"An overview of control performance assessment technology and industrial applications","volume":"14","author":"Jelali","year":"2006","journal-title":"Control Eng. Pract."},{"key":"ref_6","doi-asserted-by":"crossref","unstructured":"Doma\u0144ski, P.D. (2017, January 24\u201326). On-line control loop assessment with non-Gaussian statistical and fractal measures. Proceedings of the American Control Conference (ACC), Seattle, WA, USA.","DOI":"10.23919\/ACC.2017.7963011"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"1750094","DOI":"10.1142\/S0218127417500948","article-title":"Multifractal Properties of Process Control Variables","volume":"27","year":"2017","journal-title":"Int. J. Bifurc. Chaos"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"773","DOI":"10.1007\/s11071-017-3484-3","article-title":"Assessment of predictive control performance using fractal measures","volume":"89","year":"2017","journal-title":"Nonlinear Dyn."},{"key":"ref_9","doi-asserted-by":"crossref","unstructured":"Mandelbrot, B.B. (1983). The Fractal Geometry of Nature, Freeman.","DOI":"10.1119\/1.13295"},{"key":"ref_10","unstructured":"Feder, J. (2013). Fractals, Springer Science & Business Media."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"R1923","DOI":"10.1152\/ajpregu.00372.2007","article-title":"Fractal scale-invariant and nonlinear properties of cardiac dynamics remain stable with advanced age: A new mechanistic picture of cardiac control in healthy elderly","volume":"293","author":"Schmitt","year":"2007","journal-title":"Am. J. Physiol. Regul. Integr. Comp. Physiol."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"31","DOI":"10.1080\/10652460310001600717","article-title":"Generalized Mittag-Leffler function and generalized fractional calculus operators","volume":"15","author":"Kilbas","year":"2004","journal-title":"Integral Transf. Spec. Funct."},{"key":"ref_13","doi-asserted-by":"crossref","unstructured":"Mainardi, F. (2010). Fractional Calculus and Waves in Linear Viscoelasticity: An Introduction to Mathematical Models, Imperial College Press.","DOI":"10.1142\/9781848163300"},{"key":"ref_14","unstructured":"Podlubny, I. (1998). Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications, Academic Press."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"422","DOI":"10.1137\/1010093","article-title":"Fractional Brownian motions, fractional noises and applications","volume":"10","author":"Mandelbrot","year":"1968","journal-title":"SIAM Rev."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"770","DOI":"10.1061\/TACEAT.0006518","article-title":"Long-term storage capacity of reservoirs","volume":"116","author":"Hurst","year":"1951","journal-title":"Trans. Am. Soc. Civ. Eng."},{"key":"ref_17","unstructured":"Samorodnitsky, G., and Taqqu, M.S. (1994). Stable Non-Gaussian Random Processes: Stochastic Models with Infinite Variance, CRC Press."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"849","DOI":"10.1049\/iet-spr.2012.0050","article-title":"Effects of trends and seasonalities on robustness of the Hurst parameter estimators","volume":"6","author":"Ye","year":"2012","journal-title":"IET Signal Process."},{"key":"ref_19","first-page":"1471","article-title":"Asymptotic behavior for partial autocorrelation functions of fractional ARIMA processes","volume":"12","author":"Inoue","year":"2002","journal-title":"Ann. Appl. Probabil."},{"key":"ref_20","doi-asserted-by":"crossref","unstructured":"Woodward, W.A., Gray, H.L., and Elliott, A.C. (2016). Applied Time Series Analysis with R, CRC Press. [2nd ed.].","DOI":"10.1201\/9781315161143"},{"key":"ref_21","doi-asserted-by":"crossref","unstructured":"Liu, K., Chen, Y., and Zhang, X. (2017). An Evaluation of ARFIMA (Autoregressive Fractional Integral Moving Average) Programs. Axioms, 6.","DOI":"10.3390\/axioms6020016"},{"key":"ref_22","first-page":"377","article-title":"Fractional calculus and stable probability distributions","volume":"50","author":"Gorenflo","year":"1998","journal-title":"Arch. Mech."},{"key":"ref_23","unstructured":"Nikias, C.L., and Shao, M. (1995). Signal Processing With Alpha-Stable Distributions and Applications, Wiley-Interscience."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"918","DOI":"10.1080\/01621459.1980.10477573","article-title":"Regression-type estimation of the parameters of stable laws","volume":"75","author":"Koutrouvelis","year":"1980","journal-title":"J. Am. Stat. Assoc."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"1109","DOI":"10.1080\/03610918608812563","article-title":"Simple consistent estimators of stable distribution parameters","volume":"15","author":"McCulloch","year":"1986","journal-title":"Commun. Stat. Simul. Comput."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"P02003","DOI":"10.1088\/1742-5468\/2006\/02\/P02003","article-title":"Multifractal detrended fluctuation analysis of sunspot time series","volume":"2006","author":"Movahed","year":"2006","journal-title":"J. Stat. Mech. Theory Exp."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"1359","DOI":"10.1016\/j.automatica.2012.04.003","article-title":"Control loop performance assessment using detrended fluctuation analysis (DFA)","volume":"48","author":"Srinivasan","year":"2012","journal-title":"Automatica"},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"4997","DOI":"10.1002\/hyp.7119","article-title":"Multifractal detrended fluctuation analysis of streamflow series of the Yangtze River basin, China","volume":"22","author":"Zhang","year":"2008","journal-title":"Hydrol. Process."},{"key":"ref_29","first-page":"P07001","article-title":"Analysis of the time dynamics in wind records by means of multifractal detrended fluctuation analysis and the Fisher\u2013Shannon information plane","volume":"7","author":"Telesca","year":"2011","journal-title":"J. Stat. Mech. Theory Exp."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"817","DOI":"10.1016\/j.physa.2010.11.002","article-title":"Analysis of the efficiency and multifractality of gold markets based on multifractal detrended fluctuation analysis","volume":"390","author":"Wang","year":"2011","journal-title":"Phys. A Stat. Mech. Appl."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"82","DOI":"10.1016\/j.chaos.2006.06.019","article-title":"Detecting long-range correlations of traffic time series with multifractal detrended fluctuation analysis","volume":"36","author":"Shang","year":"2008","journal-title":"Chaos Solitons Fractals"},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"515","DOI":"10.1016\/j.ymssp.2012.12.014","article-title":"Fault diagnosis of rolling bearings based on multifractal detrended fluctuation analysis and Mahalanobis distance criterion","volume":"38","author":"Lin","year":"2013","journal-title":"Mech. Syst. Signal Process."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"82","DOI":"10.1063\/1.166141","article-title":"Quantification of scaling exponents and crossover phenomena in nonstationary heartbeat time series","volume":"5","author":"Peng","year":"1995","journal-title":"Chaos"},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"87","DOI":"10.1016\/S0378-4371(02)01383-3","article-title":"Multifractal detrended fluctuation analysis of nonstationary time series","volume":"316","author":"Kantelhardt","year":"2002","journal-title":"Phys. A Stat. Mech. Appl."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"141","DOI":"10.3389\/fphys.2012.00141","article-title":"Introduction to multifractal detrended fluctuation analysis in Matlab","volume":"3","author":"Ihlen","year":"2012","journal-title":"Front. Physiol."}],"container-title":["Algorithms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1999-4893\/11\/7\/90\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T15:10:13Z","timestamp":1760195413000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1999-4893\/11\/7\/90"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2018,6,26]]},"references-count":35,"journal-issue":{"issue":"7","published-online":{"date-parts":[[2018,7]]}},"alternative-id":["a11070090"],"URL":"https:\/\/doi.org\/10.3390\/a11070090","relation":{},"ISSN":["1999-4893"],"issn-type":[{"type":"electronic","value":"1999-4893"}],"subject":[],"published":{"date-parts":[[2018,6,26]]}}}