{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T13:26:48Z","timestamp":1787318808100,"version":"build-2736575974"},"reference-count":19,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"5","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Control Optim."],"published-print":{"date-parts":[[2010,1]]},"abstract":"<jats:p>For m-input, m-output, finite-dimensional, linear systems satisfying the classical assumptions of adaptive control (i.e., (i) minimum phase, (ii) relative degree one, and (iii) positive high-frequency gain), the well-known funnel controller $k(t)=\\frac{\\varphi(t)}{1-\\varphi(t)\\|e(t)\\|}$, $u(t)=-k(t)e(t)$ achieves output regulation in the following sense: all states of the closed-loop system are bounded, and, most importantly, transient behavior of the tracking error $e=y-y_{\\mathrm{ref}}$ is ensured such that the evolution of $e(t)$ remains in a performance funnel with prespecified boundary $1\/\\varphi(t)$, where $y_{\\mathrm{ref}}$ denotes a reference signal bounded with an essentially bounded derivative. As opposed to classical adaptive high-gain output feedback, neither system identification nor the internal model is invoked and the gain $k(\\cdot)$ is not monotone. Invoking the conceptual framework of the nonlinear gap metric, we show that the funnel controller is robust in the following sense: the funnel controller copes with bounded input and output disturbances, and, more importantly, it may even be applied to a system not satisfying any of the classical conditions (i)\u2013(iii) as long as the initial conditions and the disturbances are \u201csmall\u201d and the system is \u201cclose\u201d (in terms of a \u201csmall\u201d gap) to a system satisfying (i)\u2013(iii).<\/jats:p>","DOI":"10.1137\/090758891","type":"journal-article","created":{"date-parts":[[2009,12,23]],"date-time":"2009-12-23T18:06:06Z","timestamp":1261591566000},"page":"3169-3190","source":"Crossref","is-referenced-by-count":8,"title":["Robustness of Funnel Control in the Gap Metric"],"prefix":"10.1137","volume":"48","author":[{"given":"Achim","family":"Ilchmann","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Markus","family":"Mueller","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2009,12,23]]},"reference":[{"key":"R1","doi-asserted-by":"crossref","unstructured":"C. I. Byrnes and J. C. Willems,\n                      Adaptive stabilization of multivariable linear systems\n                      , in Proceedings of the 23rd IEEE Conference on Decision and Control, IEEE Control Systems Society, Piscataway, NJ, 1984, pp. 1574\u20131577.","DOI":"10.1109\/CDC.1984.272346"},{"key":"R2","doi-asserted-by":"publisher","DOI":"10.1109\/TAC.2008.916659"},{"key":"R3","doi-asserted-by":"publisher","DOI":"10.1137\/080726070"},{"key":"R4","doi-asserted-by":"publisher","DOI":"10.1137\/0506336371"},{"key":"R5","doi-asserted-by":"publisher","DOI":"10.1109\/9.623082"},{"key":"R6","unstructured":"N. Hopfe, A. Ilchmann, and E. P. Ryan,\n                      Funnel control with saturation: Linear MIMO systems\n                      , IEEE Trans. Automat. Control, to appear."},{"key":"R7","doi-asserted-by":"publisher","DOI":"10.1137\/07070142X"},{"key":"R8","doi-asserted-by":"publisher","DOI":"10.1002\/gamm.200890000"},{"key":"R9","doi-asserted-by":"publisher","DOI":"10.1051\/cocv:2002064"},{"key":"R10","doi-asserted-by":"publisher","DOI":"10.1137\/050641946"},{"key":"R11","doi-asserted-by":"publisher","DOI":"10.1109\/TAC.2009.2013013"},{"key":"R12","doi-asserted-by":"crossref","unstructured":"A. Isidori,\n                      Nonlinear Control Systems\n                      , 3rd ed., Comm. Control Engrg. Ser., Springer-Verlag, Berlin, 1995.","DOI":"10.1007\/978-1-84628-615-5"},{"key":"R13","doi-asserted-by":"publisher","DOI":"10.1016\/S0167-6911(84)80045-6"},{"key":"R14","doi-asserted-by":"publisher","DOI":"10.1016\/0167-6911(85)90002-7"},{"key":"R15","doi-asserted-by":"publisher","DOI":"10.1109\/9.62269"},{"key":"R16","unstructured":"A. S. Morse,\n                      Recent problems in parameter adaptive control\n                      , in Outils et Mod\u00e8les Math\u00e9matiques pour l'Automatique, l'Analyse de Syst\u00e8mes et le Traitment du Signal, Vol. 3, I. D. Landau, ed. \u00c9ditions du Centre National de la Recherche Scientifique (CNRS), Paris, 1983, pp. 733\u2013740."},{"key":"R17","unstructured":"M. Mueller,\n                      Output Feedback Control and Robustness in the Gap Metric\n                      , Ph.D. thesis, Univ. Verl. Ilmenau, Ilmenau, Germany, 2009."},{"key":"R18","doi-asserted-by":"crossref","unstructured":"W. Walter,\n                      Ordinary Differential Equations\n                      , Springer-Verlag, New York, 1998.","DOI":"10.1007\/978-1-4612-0601-9"},{"key":"R19","doi-asserted-by":"crossref","unstructured":"J. C. Willems and C. I. Byrnes,\n                      Global adaptive stabilization in the absence of information on the sign of the high frequency gain\n                      , in Analysis and Optimization of Systems, Proceedings of the 6th INRIA Conference, Nice, France, Lecture Notes in Control and Inform. Sci. 62, A. Bensoussan and J.L. Lions, eds., Springer-Verlag, Berlin, 1984, pp. 49\u201357.","DOI":"10.1007\/BFb0004944"}],"container-title":["SIAM Journal on Control and Optimization"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/090758891","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T12:40:43Z","timestamp":1787316043000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/090758891"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2009,12,23]]},"references-count":19,"journal-issue":{"issue":"5","published-print":{"date-parts":[[2010,1]]}},"alternative-id":["10.1137\/090758891"],"URL":"https:\/\/doi.org\/10.1137\/090758891","relation":{},"ISSN":["0363-0129","1095-7138"],"issn-type":[{"value":"0363-0129","type":"print"},{"value":"1095-7138","type":"electronic"}],"subject":[],"published":{"date-parts":[[2009,12,23]]}}}