here !o is the operating frequency, !d is the design
frequency, ci is a factor that determines the permissible
error range of the frequency and ki is the weighting
factor for each operating frequency. Matrix K is the
positive semi-definite weighting matrix and H is the
positive definite weighting matrix. The settling time is
represented as ts, and k4 is the weighting factor for the
settling time. Since the error in the damping affects the
vibration much smaller than the error in frequency does
(Vaughan et al., 2008), the above cost function does not
consider the robustness with respect to damping.
From the above, the optimization of the multi-input
shapers can be viewed as a particle evolution in an
eight-dimensional space with respect to the cost function.
The search space can be defined by