This　paper　extends　the　existing　Proximate　Time-optimal　Servomechanism　(PTOS)　control　methodology　to　the　more　typical　second-order　servo　systems　with　a　damping　element.　A　parameterized　design　of　expanded　PTOS　control　law　with　a　speed-dependent　linear　region　is　presented　for　rapid　and　smooth　set-point　tracking　using　a　bounded　input　signal.　The　control　scheme　uses　the　time-optimal　bang-bang　control　law　to　accomplish　maximum　acceleration　or　braking　whenever　appropriate　and　then　smoothly　switches　into　a　linear　control　law　to　achieve　a　bumpless　settling.　The　control　scheme　is　then　applied　to　the　position-velocity　control　loop　in　a　permanent　magnet　synchronous　motor　servo　system　for　set-point　position　regulation.　Numerical　simulation　has　been　conducted,　followed　by　experimental　verification　based　on　a　TMS320F2812　digital　signal　controller　board.　The　results　confirm　that　the　servo　system　can　track　a　wide　range　of　target　references　with　superior　transient　performance　and　steady-state　accuracy.