Numerical Simulation and Verification

The numerical simulation was also conducted using the numerical model studied by Fan et al.

in 2009. Again, those ten earthquakes with 400 gal PGA were used to evaluate the control performance of different control algorithms. To optimize the control performance, the SQP was used to minimize the objective functions iteratively with randomized initial values and various constraints.

For the traditional control algorithms such as the LQR and the SMC, the state cost matrix, 𝐐, and the sliding surface matrix, 𝐏, were assumed to be diagonal matrices and the last four elements in these matrices were assigned to be 1. The input cost matrix, 𝐑, and switching parameter matrix, 𝛅, were reduced to two positive scales, 𝑅 and 𝛿 , because of only one MR damper in this numerical simulation. For the DSMC, only the relative responses of the equipment isolation system were assumed to be measured. Hence, three parameters, 𝑃, 𝛿, and 𝜂, were needed for the DSMC-SEIS, two parameters, 𝑃 and 𝛿 , were used for the DSMC-SOF, two parameters, 𝑃 and 𝜐 , was prespecified for the DSMC-OCC, and only one parameter, 𝑃, was required for the DSMC-OC. For other control algorithms, the maximum energy dissipation, no parameter was involved. For the WDSMC, two sliding surfaces, 𝑃_𝑒 and 𝑃_𝑠, two switching parameters, 𝛿_𝑒 and 𝛿_𝑠, and two adjusting parameters, 𝜌_𝑒 and 𝜌_𝑠, were optimized for the DSMC-WASOF and the DSMC-WASEIS. Another two parameters, 𝜂_𝑒 and 𝜂_𝑠, were also predefined for the DSMC-WASEIS. All these control parameters for different control strategies are shown in Table 5.6, Table 5.7, and Table 5.8. Moreover, the control performance for different control strategies is shown in Table 5.9, Table 5.10, and Table 5.11.

First of all, the best results for the passive control were provided constant command voltages with the maximum and minimum values in the control strategy (1) and the control strategy (2), respectively. Once the balance between the responses of equipment and the structural responses was focused, the constant command voltage needed to be 0.33 volts for the optimized result. Furthermore, the three objective functions were evaluated as 90.94%, 104.98%, and 87.13% if the constant command voltage was 0 volt, as 75.31%, 113.6%, and 77.98% if the constant command voltage was 0.33 volts, and as 52.12%, 127.40%, and 83.65% if the constant command voltage was 1.2 volts. This observation revealed the trade-off between the equipment and the structural system. The absolute

doi:10.6342/NTU201901242 92

acceleration responses of equipment gained the largest reduction because the maximum command voltage gave the largest damper capacity and dissipated most vibration energy within the equipment isolation system; however, some of the energy transmitted to the structural system by increasing the inter-story drift ratios, and vice versa.

The same trend can be observed from the results produced by the DSMC-OC (which is equivalent to the control based on Lyapunov stability theory) and DSMC-OCC. For example, the parameter, 𝜐, that used to limit the maximum command voltage in the DSMC-OCC was optimized as 1 in the control strategy (1), as 0 in the control strategy (2), and as 0.3555 in the control strategy (3). Besides, the control performance and the sliding surface of the DSMC-OC and the DSMC-OCC were very similar in the control strategy (1) because 𝜐 = 1, but the DSMC-OC performed worse than the DSMC-OCC in the control strategy (2) and the control strategy (3) as a larger control force was always generated if there is no constraint on the command voltage. This phenomenon is the main reason why the DSMC-OCC stands out from the DSMC-OC.

Being the only one control algorithm with no control parameters, the maximum energy dissipation tried the best to reduce the energy stored in the equipment isolation system. The relative vibration energy in Equation (5.46) eventually forced this control algorithm to have a decentralized control architecture as the 𝐇₁ matrix in Equation (5.63) had only two non-zero elements. As a consequence, it had a control performance very similar to the passive control, the DSMC-OC, and the DSMC-OCC in the control strategy (1). Unfortunately, the limited information provided by the measured responses in the vicinity of the control device’s location was not enough to control the whole structures, and the results in the control strategy (2) and the control strategy (3) were then almost the same with the DSMC-OC. Actually, a combination of the maximum energy dissipation and the semi-active control devices is some kind of trick because the devices are essentially designed to dissipate the relative vibration energy. The results are always similar to the one that keeps the largest hysteresis loops, for example, the MR damper continuously fed with the maximum command voltage.

Figure 5.26, Figure 5.27, and Figure 5.28 illustrate the values of objective functions for different control algorithms in the control strategy (1), the control strategy (2), and the control strategy (3), respectively. In the control strategy (1), the control parameters were optimized based on the first objective function, 𝐽₁, and therefore, all the control algorithms performed very well. The LQR, the SMC, the DSMC-SOF, the DSMC-SEIS, the DSMC-WASOF, and the DSMC-WASEIS reduced more than 50% in absolute acceleration responses compared to the one only used the equipment

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isolation system. The results were even better than the best one produced by the passive control, even on the other objective functions, 𝐽₂ and 𝐽₃; actually, the passive control had the worst control performance in this control strategy.

However, in the control strategy (2), several control algorithms can effectively reduce the structural responses, such as the LQR, the SMC, the OCC, the WASOF, the DSMC-WASEIS, and the passive control. To be honest, the results provided by the DSMC-OCC and the passive control were trivial ones because these two control algorithms forced the command voltage to be zero throughout the simulation to avoid high structural responses. The DSMC-WASOF, and the DSMC-WASEIS had good feedback controllers and were performed slightly better than the above two control algorithms. Most importantly, they still had a comparable performance although the LQR and the SMC were supposed to be much more informed since the number of feedback signals in these control algorithms was quite higher. The results in Figure 5.27 again reveal the trade-off between the equipment and the structural system; most of the control algorithms increased the first objective function, 𝐽₁, up to 80%.

A similar observation can be examined in the control strategy (3); the proposed control algorithms combined the DSMC and the real-time wavelet analysis provided the excellent results with very limited feedback signals due to the decentralized control architecture. The LQR and the SMC had the responses lower than the one only used the equipment isolation system, 67.07% and 66.66%, respectively. The DSMC-WASOF and the DSMC-WASEIS were comparable with them and left 69.46% and 69.35% responses, respectively. Nevertheless, the WDSMC seemed to pay more attention to the responses of equipment because of the local measurements, and therefore the LQR and the SMC favored the reduction of the inter-story drift ratios. The other control algorithms, even the DSMC-SOF and the DSMC-SEIS, can barely decrease 30% responses.

doi:10.6342/NTU201901242 94

doi:10.6342/NTU201901242 95