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Set inversion
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<h2 id="example">Example</h2> <p>The set <i>X</i> = <i>f</i> −1([4,9]) where <i>f</i> (<i>x</i>1, <i>x</i>2) = <i>x</i>21 + <i>x</i>22 is represented on the figure. </p><p>For instance, since [−2,1]2 + [4,5]2 = [0,4] + [16,25] = [16,29] does not <a href="/facts/Intersection_(set_theory)/RPfUNJh9">intersect</a> the interval [4,9], we conclude that the box [−2,1] × [4,5] is outside <i>X</i>. Since [−1,1]2 + [2,√5]2 = [0,1] + [4,5] = [4,6] is inside [4,9], we conclude that the whole box [−1,1] × [2,√5] is inside <i>X</i>. </p> <h2 id="application">Application</h2> <p>Set inversion is mainly used for <a href="/facts/Motion_planning/DoSPv4dB">path planning</a>, for nonlinear parameter <a href="/facts/Set_estimation/I6jlICRp">set estimation</a>,<a class="footnote-ref" id="fnref:3" href="#fn:3"><sup>3</sup></a><a class="footnote-ref" id="fnref:4" href="#fn:4"><sup>4</sup></a> for localization<a class="footnote-ref" id="fnref:5" href="#fn:5"><sup>5</sup></a><a class="footnote-ref" id="fnref:6" href="#fn:6"><sup>6</sup></a> or for the characterization of stability domains of <a href="/facts/Linear_dynamical_system/tYnoVnKs">linear dynamical systems</a>.<a class="footnote-ref" id="fnref:7" href="#fn:7"><sup>7</sup></a> </p> <h2 id="references">References</h2> <ol> <li id="fn:1"><p>Jaulin, L.; Walter, E. (1993). "Set inversion via interval analysis for nonlinear bounded-error estimation" (PDF). Automatica. 29 (4): 1053–1064. doi:10.1016/0005-1098(93)90106-4. <a href="http://www.ensta-bretagne.fr/jaulin/paper_automatica93.pdf" target="_blank">http://www.ensta-bretagne.fr/jaulin/paper_automatica93.pdf</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li> <li id="fn:2"><p>Jaulin, L.; Kieffer, M.; Didrit, O.; Walter, E. (2001). Applied Interval Analysis. Berlin: Springer. ISBN 1-85233-219-0. <a href="1-85233-219-0" target="_blank">1-85233-219-0</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li> <li id="fn:3"><p>Jaulin, L.; Godet, J.L; Walter, E.; Elliasmine, A.; Leduff, Y. (1997). "Light scattering data analysis via set inversion" (PDF). Journal of Physics A: Mathematical and General. 30 (22): 7733–7738. Bibcode:1997JPhA...30.7733J. doi:10.1088/0305-4470/30/22/012. <a href="http://www.ensta-bretagne.fr/jaulin/simcf4.pdf" target="_blank">http://www.ensta-bretagne.fr/jaulin/simcf4.pdf</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li> <li id="fn:4"><p>Braems, I.; Berthier, F.; Jaulin, L.; Kieffer, M.; Walter, E. (2001). "Guaranteed Estimation of Electrochemical Parameters by Set Inversion Using Interval Analysis" (PDF). Journal of Electroanalytical Chemistry. 495 (1). <a href="http://www.ensta-bretagne.fr/jaulin/electrochimie.pdf" target="_blank">http://www.ensta-bretagne.fr/jaulin/electrochimie.pdf</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li> <li id="fn:5"><p>Colle, E.; Galerne, S. (2013). "Mobile robot localization by multiangulation using set inversion". Robotics and Autonomous Systems. 66 (1): 39–48. doi:10.1016/j.robot.2012.09.006. <a href="/wiki/Doi_(identifier)" target="_blank">/wiki/Doi_(identifier)</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></p></li> <li id="fn:6"><p>Drevelle, V.; Bonnifait, Ph. (2011). "A set-membership approach for high integrity height-aided satellite positioning" (PDF). GPS Solutions. 15 (4): 357–368. Bibcode:2011GPSS...15..357D. doi:10.1007/s10291-010-0195-3. S2CID 121728552. <a href="https://hal.science/hal-00608133/file/Drevelle_rev3_HAL.pdf" target="_blank">https://hal.science/hal-00608133/file/Drevelle_rev3_HAL.pdf</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></p></li> <li id="fn:7"><p>Walter, E.; Jaulin, L. (1994). "Guaranteed characterization of stability domains via set inversion" (PDF). IEEE Trans. Autom. Control. 39 (4): 886–889. doi:10.1109/9.286277. <a href="http://www.ensta-bretagne.fr/jaulin/IEEEStab.pdf" target="_blank">http://www.ensta-bretagne.fr/jaulin/IEEEStab.pdf</a> <a href="#fnref:7" class="footnote-back-ref">↩</a></p></li> </ol>