Control input to reduce residual vibration for a nonlinear time-varying system was developed. The nonlinear equation is first solved with nominal input then linearized by nominal response which can be defined with some equilibrium point. An additional input can be found by solving the linearized equation that should satisfy the boundary conditions which is obtained by applying the nominal input to original nonlinear equation.
Residual vibration can be reduced by applying nominal and additional input together but still have some amount of vibration. So newly updated additional input should be required to further reduce the residual vibration. The new additional input can be obtained by solving the above linearized equation with the new boundary conditions which is the sum of two results (response by nominal input + response by nominal and additional input). By repeating this process an optimized additional input can be defined which reduces the residual vibration to the extent as we need. Coordinate selection should be done to describe the nominal response about some equilibrium point. For the case of the system which moves in three-dimensional space, only transitional coordinates can properly define the nominal motion.
