The displacement responses of the first mass with and without FMPC are given in Figure 2, in which the results with FMPC and without FMPC are denoted by solid and dashed lines, respectively.
The results with FMPC and without FMPC are denoted by triangles and circles, respectively.
The proposed FMPC algorithm is based on the FPIM, whereas the general MPC algorithm is based on a Pade approximation with scaling and squaring  (the method used in the expm function in the programming environment MATLAB).
Table 1 illustrates that the CPU times change with the number of prediction points N and the number of degrees of freedom n for both FMPC and general MPC.
The maximum state response deviations and the maximum control input deviations are given in Table 2, in which x and u are the state response and control input given by FMPC and [x.sup.*] and [u.sup.*] are the state response and control input given by general MPC.
Tables 1 and 2 show that FMPC gives nearly the same computational precision as general MPC using much less CPU time.
There are many factors that influence the FMPC algorithm regarding system performance such as the weighting matrices Q and R, the prediction horizon T, and the number of prediction points N.
Furthermore, as discussed in Section 3, the important difference between the proposed FMPC and general MPC is the computation of the matrix exponential for a large-scale structural dynamic system.
Figure 11 gives the distribution of the nonzero elements of the matrix exponential for both FMPC and general MPC.
However, errors in the system parameters in actual applications can affect the performance of the FMPC method, so the robustness of the FMPC algorithm must be evaluated.
The displacement responses and control inputs for the first mass with FMPC are given in Figure 13, in which the results with the nominal parameters, 10% stochastic error and 50% stochastic error, are denoted by the solid, dotted, and dashed lines, respectively.
The displacement responses and control inputs for the first mass with FMPC are given in Figure 14, in which the results with the linear spring model and the cubic nonlinear spring stiffness with a = 1 and a = 10 are denoted by the solid, dotted, and dashed lines, respectively.