{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,30]],"date-time":"2026-04-30T08:32:50Z","timestamp":1777537970579,"version":"3.51.4"},"reference-count":38,"publisher":"SAGE Publications","issue":"12","license":[{"start":{"date-parts":[[2015,12,1]],"date-time":"2015-12-01T00:00:00Z","timestamp":1448928000000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/journals.sagepub.com\/page\/policies\/text-and-data-mining-license"}],"content-domain":{"domain":["journals.sagepub.com"],"crossmark-restriction":true},"short-container-title":["Advances in Mechanical Engineering"],"published-print":{"date-parts":[[2015,12,1]]},"abstract":"<jats:p>This article presents a novel method for visualizing the control systems behavior. The proposed scheme uses the tools of fractional calculus and computes the signals propagating within the system structure as a time\/frequency-space wave. Linear and nonlinear closed-loop control systems are analyzed, for both the time and frequency responses, under the action of a reference step input signal. Several nonlinearities, namely, Coulomb friction and backlash, are also tested. The numerical experiments demonstrate the feasibility of the proposed methodology as a visualization tool and motivate its extension for other systems and classes of nonlinearities.<\/jats:p>","DOI":"10.1177\/1687814015619831","type":"journal-article","created":{"date-parts":[[2015,12,8]],"date-time":"2015-12-08T20:29:09Z","timestamp":1449606549000},"update-policy":"https:\/\/doi.org\/10.1177\/sage-journals-update-policy","source":"Crossref","is-referenced-by-count":3,"title":["Visualizing control systems performance: A fractional perspective"],"prefix":"10.1177","volume":"7","author":[{"given":"Ant\u00f3nio M","family":"Lopes","sequence":"first","affiliation":[{"name":"UISPA\u2013LAETA\/INEGI, Faculty of Engineering, University of Porto, Porto, Portugal"}]},{"given":"Jos\u00e9 A","family":"Tenreiro Machado","sequence":"additional","affiliation":[{"name":"Department of Electrical Engineering, Institute of Engineering, Polytechnic of Porto, Porto, Portugal"}]}],"member":"179","published-online":{"date-parts":[[2015,12,7]]},"reference":[{"key":"bibr1-1687814015619831","volume-title":"Modern control systems","author":"Dorf RC","year":"1998"},{"key":"bibr2-1687814015619831","volume-title":"Discrete-time control systems","volume":"2","author":"Ogata K","year":"1995"},{"key":"bibr3-1687814015619831","volume-title":"Digital control of dynamic systems","volume":"3","author":"Franklin GF","year":"1998"},{"key":"bibr4-1687814015619831","volume-title":"Multivariable feedback control: analysis and design","volume":"2","author":"Skogestad S","year":"2007"},{"key":"bibr5-1687814015619831","volume-title":"Nonlinear dynamical control systems","author":"Nijmeijer H","year":"2013"},{"key":"bibr6-1687814015619831","volume-title":"Applied nonlinear control","volume":"199","author":"Slotine JJE","year":"1991"},{"key":"bibr7-1687814015619831","volume-title":"Nonlinear systems","volume":"3","author":"Khalil HK","year":"1996"},{"key":"bibr8-1687814015619831","volume-title":"Robust nonlinear control design: state-space and Lyapunov techniques","author":"Freeman RA","year":"2008"},{"key":"bibr9-1687814015619831","doi-asserted-by":"publisher","DOI":"10.1007\/s11071-008-9405-8"},{"key":"bibr10-1687814015619831","doi-asserted-by":"publisher","DOI":"10.1007\/s11071-008-9410-y"},{"key":"bibr11-1687814015619831","doi-asserted-by":"publisher","DOI":"10.1016\/j.sigpro.2014.05.012"},{"key":"bibr12-1687814015619831","volume-title":"Models and numerical methods","volume":"3","author":"Baleanu D","year":"2012"},{"key":"bibr13-1687814015619831","volume-title":"An introduction to the fractional calculus and fractional differential equations","author":"Kenneth M","year":"1993"},{"key":"bibr14-1687814015619831","doi-asserted-by":"publisher","DOI":"10.1142\/p614"},{"key":"bibr15-1687814015619831","doi-asserted-by":"publisher","DOI":"10.1002\/9781118387726"},{"key":"bibr16-1687814015619831","volume-title":"Fractional processes and fractional-order signal processing: techniques and applications","author":"Sheng H","year":"2011"},{"key":"bibr17-1687814015619831","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4471-5388-7"},{"key":"bibr18-1687814015619831","doi-asserted-by":"publisher","DOI":"10.1177\/1077546312473323"},{"key":"bibr19-1687814015619831","doi-asserted-by":"publisher","DOI":"10.1007\/s11071-004-3770-8"},{"key":"bibr20-1687814015619831","doi-asserted-by":"publisher","DOI":"10.1016\/j.camwa.2013.02.003"},{"key":"bibr21-1687814015619831","doi-asserted-by":"publisher","DOI":"10.1155\/2014\/238459"},{"key":"bibr22-1687814015619831","doi-asserted-by":"publisher","DOI":"10.1016\/j.cnsns.2010.05.027"},{"key":"bibr23-1687814015619831","doi-asserted-by":"publisher","DOI":"10.1140\/epjst\/e2013-01967-y"},{"key":"bibr24-1687814015619831","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-642-18101-6"},{"key":"bibr25-1687814015619831","doi-asserted-by":"publisher","DOI":"10.1155\/2008\/528934"},{"key":"bibr26-1687814015619831","doi-asserted-by":"publisher","DOI":"10.1155\/2013\/102068"},{"key":"bibr27-1687814015619831","doi-asserted-by":"publisher","DOI":"10.1016\/S0005-1098(00)00011-X"},{"key":"bibr28-1687814015619831","unstructured":"Dorcak L. 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