We　consider　a　general　nonlinear　time-delay　system　in　which　the　input　signal　is　piecewise-constant.　Such　systems　arise　in　a　wide　range　of　industrial　applications,　including　evaporation　and　purification　processes　and　chromatography.　We　assume　that　the　time-delays-one　involving　the　state　variables　and　the　other　involving　the　input　variables-are　unknown　and　need　to　be　estimated　using　experimental　data.　We　formulate　the　problem　of　estimating　the　unknown　delays　as　a　nonlinear　optimization　problem　in　which　the　cost　function　measures　the　least-squares　error　between　predicted　and　measured　system　output.　The　main　difficulty　with　this　problem　is　that　the　delays　are　decision　variables　to　be　optimized,　rather　than　fixed　values.　Thus,　conventional　optimization　techniques　are　not　directly　applicable.　We　propose　anew　computational　approach　based　on　a　novel　algorithm　for　computing　the　cost　function＇s　gradient.　We　then　apply　this　approach　to　estimate　the　time-delays　in　two　industrial　chemical　processes:　a　zinc　sulphate　purification　process　and　a　sodium　aluminate　evaporation　process.　(C)　2013　Elsevier　Inc.　All　rights　reserved.