A negative input shaped command is presented for flexible systems to reduce the residual oscillation under unequal acceleration and braking delays of actuators that are common issues in industrial applications. Against this nonlinearity, a compensated unit magnitude zero vibration (UMZV) shaper is analytically developed with a phasor vector diagram and a ramp-step function to approximate the dynamic response of the unequal acceleration and braking delays of actuators. A closed-form solution is presented with a benchmark system without sacrificing the generality and simplicity for industrial applications. The robustness and control performance of the exact solution are numerically evaluated and compared with those of an existing negative input shaper in terms of the switch-on time, command interference, and effects of the shaper parameters. The proposed negative input shaped commands are experimentally validated with a mini-bridge crane.

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