As　an　alternative　to　bang-bang　control,　a　time　optimal　control　(TOC)　algorithm　for　discrete-time　systems　was　first　reported　by　Han　(1).　This　algorithm　not　only　acts　as　a　noise-tolerant　tracking　differentiator　(TD)　to　avoid　setpoint　jumps　in　control　processes,　but　also　has　wide　applications　in　the　design　of　controllers　and　observers.　However,　determination　of　the　real-time　state　position　on　the　phase　plane　involves　complex　boundary　transformations,　which　renders　this　algorithm　impractical　for　some　engineering　applications.　This　paper　proposes　a　methodology　for　discrete-time　optimal　control　(DTOC)　of　double　integrators　with　disturbances.　The　closed-form　solution　with　lower　computational　burden　can　be　easily　extended　to　general　second-order　systems.　Further,　in　consideration　of　the　inevitable　disturbances　in　the　systems,　a　rigorous　and　full-convergence　proof　is　presented　for　the　proposed　algorithm.　The　results　show　finite-time　and　fast　convergence　as　well　as　provide　the　ultimate　stable　attraction　regions　for　the　system　states.　Examples　and　experiments　are　also　presented　to　demonstrate　the　effectiveness　of　the　proposed　algorithm　for　solving　a　signal　processing　problem　in　a　maglev　train.　©　2022　The　Institute　of　Electrical　Engineers　of　Japan.