In　this　article,　an　adaptive　critic　scheme　with　a　novel　performance　index　function　is　developed　to　solve　the　tracking　control　problem,　which　eliminates　the　tracking　error　and　possesses　the　adjustable　convergence　rate　in　the　offline　learning　process.　Under　some　conditions,　the　convergence　and　monotonicity　of　the　accelerated　value　function　sequence　can　be　guaranteed.　Combining　the　advantages　of　the　adjustable　and　general　value　iteration　schemes,　an　integrated　algorithm　is　proposed　with　a　fast　guaranteed　convergence,　which　involves　two　stages,　namely　the　acceleration　stage　and　the　convergence　stage.　Moreover,　an　effective　approach　is　given　to　adaptively　determine　the　acceleration　interval.　With　this　operation,　the　fast　convergence　of　the　new　value　iteration　scheme　can　be　fully　utilized.　Finally,　compared　with　the　general　value　iteration,　the　numerical　results　are　presented　to　verify　the　fast　convergence　and　the　tracking　performance　of　the　developed　adaptive　critic　design.