Document Zbl 1316.93074

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The generic hypersonic flight vehicle is a multi-input multi-output system whose longitudinal model is high-order, highly nonlinear, tight coupling and most of all includes big uncertainties. Interval type-2 fuzzy sets with Gaussian membership functions are used in antecedent and consequent parts of fuzzy rules. The IT2-FNN directly outputs elevator deflection and throttle setting which make the GHFV track the altitude command signal and meanwhile maintain its velocity. The parameter adaptive law of IT2-FNN is derived using backpropagation method. The deviation of the control signal from the nominal dynamic inversion control signal is used as the reference output signal of IT2-FNN. The tracking errors of velocity and altitude are used as inputs of IT2-FNN. Tracking differentiator is designed to form an Arranged Transition Process (ATP) of the command signal as well as ATP’s high-order derivatives. Nonlinear state observer is designed to get the approximations of velocity, altitude as well as their high-order derivatives. Simulation results validate the effectiveness and robustness of the proposed controller especially under big uncertainties.</div> <div class="clear"></div> <br> <div class="citations"><div class="clear"><a href="/?q=rf%3A6464801">Cited in <strong>4</strong> Documents</a></div></div> <div class="classification"> <h3>MSC:</h3> <table><tr> <td> <a class="mono" href="/classification/?q=cc%3A93C42" title="MSC2020">93C42</a> </td> <td class="space"> Fuzzy control/observation systems </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A93C40" title="MSC2020">93C40</a> </td> <td class="space"> Adaptive control/observation systems </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A92B20" title="MSC2020">92B20</a> </td> <td class="space"> Neural networks for/in biological studies, artificial life and related topics </td> </tr></table> </div><div class="keywords"> <h3>Keywords:</h3><a href="/?q=ut%3Atype-2+fuzzy+logic">type-2 fuzzy logic</a>; <a href="/?q=ut%3Afuzzy+neural+network">fuzzy neural network</a>; <a href="/?q=ut%3Ahypersonic">hypersonic</a>; <a href="/?q=ut%3Adirect+adaptive+control">direct adaptive control</a>; <a href="/?q=ut%3Auncertainty">uncertainty</a></div>  <div class="modal fade" id="metadataModal" tabindex="-1" role="dialog" aria-labelledby="myModalLabel"> <div class="modal-dialog" role="document"> <div class="modal-content"> <div class="modal-header"> <button type="button" class="close" data-dismiss="modal" aria-label="Close"><span aria-hidden="true">×</span></button> <h4 class="modal-title" id="myModalLabel">Cite</h4> </div> <div class="modal-body"> <div class="form-group"> <label for="select-output" class="control-label">Format</label> <select id="select-output" class="form-control" aria-label="Select Metadata format"></select> </div> <div class="form-group"> <label for="metadataText" class="control-label">Result</label> <textarea class="form-control" id="metadataText" rows="10" style="min-width: 100%;max-width: 100%"></textarea> </div> <div id="metadata-alert" class="alert alert-danger" role="alert" style="display: none;">  </div> </div> <div class="modal-footer"> <button type="button" class="btn btn-primary" onclick="copyMetadata()">Copy to clipboard</button> <button type="button" class="btn btn-default" data-dismiss="modal">Close</button> </div> </div> </div> </div> <div class="functions clearfix"> <div class="function">  <a type="button" class="btn btn-default btn-xs pdf" data-toggle="modal" data-target="#metadataModal" data-itemtype="Zbl" data-itemname="Zbl 1316.93074" data-ciurl="/ci/06464801" data-biburl="/bibtex/06464801.bib" data-amsurl="/amsrefs/06464801.bib" data-xmlurl="/xml/06464801.xml" > Cite </a> <a class="btn btn-default btn-xs pdf" data-container="body" type="button" href="/pdf/06464801.pdf" title="Zbl 1316.93074 as PDF">Review PDF</a> </div> <div class="fulltexts"> <span class="fulltext">Full Text:</span> <a class="btn btn-default btn-xs" type="button" href="https://doi.org/10.1007/s00500-013-1123-6" aria-label="DOI for “Direct adaptive type-2 fuzzy neural network control for a generic hypersonic flight vehicle”" title="10.1007/s00500-013-1123-6">DOI</a> </div> <div class="sfx" style="float: right;"> </div> </div> <div class="references"> <h3>References:</h3> <table><tr> <td>[1]</td> <td class="space">Castillo O, Melin P (2008) Type-2 fuzzy logic theory and applications. Springer, Berlin · <a href="/1171.93002" class="nowrap">Zbl 1171.93002</a></td> </tr><tr> <td>[2]</td> <td class="space">Chen X (2012) Intelligent control method research for attitude control of mav. PhD thesis, Changchun Institute of Optics, Fine Mechanics and Physics, Chinese Academy of Science</td> </tr><tr> <td>[3]</td> <td class="space">Fazel Zarandi MH, Gamasaee R (2012) Type-2 fuzzy hybrid expert system for prediction of tardiness in scheduling of steel continuous casting process. Soft Comput 16(8):1287-1302. doi:10.1007/s00500-012-0812-x</td> </tr><tr> <td>[4]</td> <td class="space">Han J (2009) Active disturbance rejection control technique-the technique for estimating and compensating the uncertainties. National Defense Industry Press, Beijing</td> </tr><tr> <td>[5]</td> <td class="space">Huang H, Wan H, Han J (2001) Arranging the transient process is an effective method improved the “robustness, adaptability and stability” of closded-loop system. Control Theory Appl 18(Supply):89-94 · <a href="/0995.93502" class="nowrap">Zbl 0995.93502</a></td> </tr><tr> <td>[6]</td> <td class="space">Karnik NN, Mendel JM (1998) Type-2 fuzzy logic systems: type-reduction. In: IEEE international conference on systems, man, and cybernetics, pp 2046-2051</td> </tr><tr> <td>[7]</td> <td class="space">Karnik NN, Mendel JM, Liang Q (1999) Type-2 fuzzy logic systems. IEEE Trans Fuzzy Syst 7(6):643-658 · <a href="https://doi.org/10.1109/91.811231" class="nowrap">doi:10.1109/91.811231</a></td> </tr><tr> <td>[8]</td> <td class="space">Keshmiri S, Colgren R, Mirmirani M (2006) Six-dof modeling and simulation of a generic hypersonic vehicle for control and navigation purposes. In: AIAA guidance, navigation, and control conference and exhibit, AIAA</td> </tr><tr> <td>[9]</td> <td class="space">Khalil H (2001) Nonlinear system, 3rd edn. Prentice Hall, NJ</td> </tr><tr> <td>[10]</td> <td class="space">Li C, Yi J, Wang M, Guiqing Z (2012) Monotonic type-2 fuzzy neural network and its application to thermal comfort prediction. Neural Comput Appl. doi:10.1007/s00521-012-1140-x</td> </tr><tr> <td>[11]</td> <td class="space">Li C, Zhang G, Yi J, Wang M (2013) Uncertainty degree and modeling of interval type-2 fuzzy sets: definition, method and application. Comput Math Appl. doi:10.1016/j.camwa.2013.07.021 · <a href="/1381.68288" class="nowrap">Zbl 1381.68288</a></td> </tr><tr> <td>[12]</td> <td class="space">Liang Q, Mendel JM (2000) Interval type-2 fuzzy logic systems: theory and design. IEEE Trans Fuzzy Syst 8(5):535-550 · <a href="https://doi.org/10.1109/91.873577" class="nowrap">doi:10.1109/91.873577</a></td> </tr><tr> <td>[13]</td> <td class="space">Lin TC, Liu HL, Kuo MJ (2009) Direct adaptive interval type-2 fuzzy control of multivariable nonlinear systems. Eng Appl Artif Intell 22(3):420-430 · <a href="https://doi.org/10.1016/j.engappai.2008.10.024" class="nowrap">doi:10.1016/j.engappai.2008.10.024</a></td> </tr><tr> <td>[14]</td> <td class="space">Liu Y, Lu Y (2009a) Nonlinear adaptive inversion control with neural network compensation for a longitudinal hypersonic vehicle model. Int Conf Intell Comput Intell Syst, IEEE, pp 264-268</td> </tr><tr> <td>[15]</td> <td class="space">Liu Y, Lu Y (2009b) Nonlinear fuzzy robust adaptive control of a longitudinal hypersonic aircraft model. In: International conference on artificial intelligence and, computational intelligence, pp 31-35</td> </tr><tr> <td>[16]</td> <td class="space">Mendel JM (2007) Type-2 fuzzy sets and systems: an overiew. IEEE Comput Intell Mag 2(1):20-29 · <a href="https://doi.org/10.1109/MCI.2007.380672" class="nowrap">doi:10.1109/MCI.2007.380672</a></td> </tr><tr> <td>[17]</td> <td class="space">Mendel JM, John RI (2002) Type-2 fuzzy sets made simple. IEEE Trans Fuzzy Syst 10(2):117-127</td> </tr><tr> <td>[18]</td> <td class="space">Nurmaini S, Zaiton S, Norhayati D (2009) An embedded interval type-2 neuro-fuzzy controller for mobile robot navigation. In: IEEE international conference on systems, man, and cybernetics, pp 4315-4321</td> </tr><tr> <td>[19]</td> <td class="space">Ougli AE, Lagrat I, Boumhidi I (2008) A type-2 fuzzy adaptive controller of a class of nonlinear system. Int J Inf Math Sci 4(4):282-288</td> </tr><tr> <td>[20]</td> <td class="space">Pan Y, Huang D, Sun Z (2011) Overview of type-2 fuzzy logic control. Control Theory Appl 28(1):13-23 · <a href="/1240.93220" class="nowrap">Zbl 1240.93220</a></td> </tr><tr> <td>[21]</td> <td class="space">Parker JT, Serrani A, Yurkovich S, Bolender MA, Doman DB (2007) Control-oriented modeling of an air-breathing hypersonic vehicle. J Guid Control Dyn 30(3):846-869 · <a href="https://doi.org/10.2514/1.27830" class="nowrap">doi:10.2514/1.27830</a></td> </tr><tr> <td>[22]</td> <td class="space">Rehman OU, Fidan B, Petersen IR (2009) Robust minimax optimal control of nonlinear uncertain systems using feedback linearization with application to hypersonic flight vehicles. In: Joint 48th IEEE conference on decision and control and 28th Chinese control conference, IEEE, pp 720-726</td> </tr><tr> <td>[23]</td> <td class="space">Shakiba M, Serrani A (2011) Control oriented modeling of 6-dof hypersonic vehicle dynamics. In: AIAA guidance, navigation, and control conference, AIAA, pp 1-27</td> </tr><tr> <td>[24]</td> <td class="space">Shao L, Liao X, Xia Y, Han J (2008) Stability analysis and synthesis of third order discrete extended state observer. Inf Control 37(2):135-139</td> </tr><tr> <td>[25]</td> <td class="space">Shaughnessy JD, Pinckney SZ, McMinn JD, Cruz CI, Kelley ML (1990) Hypersonic vehicle simulation model: winged-cone configuration. Technical report, NASA Langley Research Center</td> </tr><tr> <td>[26]</td> <td class="space">Singh M, Srivastava S, Gupta JRP, Hanmandlu M (2004) A type-2 fuzzy neural model based control of a nonlinear system. In: IEEE cybernetics and intelligent systems, pp 1352-1356</td> </tr><tr> <td>[27]</td> <td class="space">Wang LX (1996) A course in fuzzy systems & control. Prentice Hall, NJ</td> </tr><tr> <td>[28]</td> <td class="space">Wu H, Mendel JM (2002) Uncertainty bounds and their use in the design of interval type-2 fuzzy logic systems. IEEE Trans Fuzzy Syst 10(5):622-639 · <a href="https://doi.org/10.1109/TFUZZ.2002.803496" class="nowrap">doi:10.1109/TFUZZ.2002.803496</a></td> </tr><tr> <td>[29]</td> <td class="space">Xu H, Mirmirani M, Ioannou P (2004) Adaptive sliding mode control design for a hypersonic flight vehicle. AIAA JGCD 27(5):829-838</td> </tr><tr> <td>[30]</td> <td class="space">Yang F, Yi J, Tan X, Yuan R (2012) Parameter-optimized high-order active disturbance rejection controllers for a generic hypersonic flight vehicle. Commun Inf Sci Manag Eng 2(10):6-12</td> </tr><tr> <td>[31]</td> <td class="space">Yang F, Tan X, Yi J (2013a) A type-2 adaptive fuzzy logic controller for a generic hypersonic flight vehicle. ICIC Express Lett 7(5):1583-1588</td> </tr><tr> <td>[32]</td> <td class="space">Yang F, Yuan R, Yi J, Fan G, Tan X (2013b) Backstepping based type-2 adaptive fuzzy control for a generic hypersonic flight vehicle. In: Chinese intelligent automation conference, pp 180-188</td> </tr><tr> <td>[33]</td> <td class="space">Zaheer SA, Kim JH (2011) Type-2 fuzzy airplane altitude control: a comparative study. In: IEEE international conference on fuzzy systems, pp 2170-2176</td> </tr></table> <div class="reference_disclaimer"> This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. 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