Numerical Simulation and Verification of WHSMC

In the numerical simulation of the traditional control algorithms and the wavelet-based control algorithms, all the structural responses and the control forces were evaluated with a sampling rate of 200 Hz using MATLAB (2010). For simplicity, 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 which represent the relative velocity responses were assigned to be 1. The input cost matrix, 𝐑, and the switching parameter matrix, 𝛅, were reduced to two positive scales, 𝑅 and 𝛿, because only one equipment isolation system and one actuator were involved in this numerical simulation. After seismic events, the four performance indices were first calculated throughout all DOFs, and the three objective functions were then calculated based on the equations in Table 5.3. For each control algorithm, the SQP was adopted to optimize the control performance and the iterative results were presented using the fmincon routine. Note that the initial values were randomly assigned, and the constraints for the control parameters were also applied.

Comparison of Traditional and Hybrid Control Algorithms

Figure 5.15, Figure 5.16, and Figure 5.17 show the responses of equipment, the structural

doi:10.6342/NTU201901242 86

responses, and the control forces for different control strategies under the El Centro earthquake in NS direction. Generally, both the LQR and the SMC were capable of having the efficacious control gains which satisfy the desired strategies despite the fact that the control performance is slightly different from each other. Hence, the best results may be obtained by taking both the absolute acceleration and the inter-story drift ratio into account and cautiously considering the balance between these two different responses, as the results of the control strategy (3) shown in Figure 5.17. In this control strategy, using actuator with both the LQR and the SMC can reduce more responses than using the equipment isolation system only.

Figure 5.18 illustrates the values of objective functions for different control strategies under the El Centro earthquake in both two directions. First of all, the LQR and the SMC achieved the objectives via very distinct ways since the control forces generated by these two traditional control algorithms worked differently in each strategy, even the control performance was comparable.

Clearly, both the SMC and the HSMC can provide a better result compared to the LQR in the control strategy (1); however, the LQR and the HSMC had great improvement on the reduction in the inter-story drift ratio compared to the SMC, as the results in the control strategy (2). In the LQR, the state-feedback control gain, 𝐆, is the optimal solution of the quadratic cost function with the state cost matrix, 𝐐, that only describes the relationship between the states consisted of the relative velocity and displacement responses. Although the state cost term, 𝐳^𝑇(𝑡)𝐐𝐳(𝑡) in Equation (5.6), can be replaced by the output cost term, 𝐲^𝑇(𝑡)𝐐𝐲(𝑡), that includes the relative acceleration responses, the absolute acceleration responses considered in the control strategy (1) still cannot be directly correlated with the optimal solution. The assumption that the state cost matrix was diagonalized in this study may be the second reason that the results came from the LQR was slightly worse than the other control algorithms because this assumption shrinks the reachable space and degrades the reachability (or controllability, depending on the definition) of the whole control system. On the other hand, the LQR is competent to mitigate the inter-story drift ratio by using the predefined state cost matrix and input cost matrix, as the results in the control strategy (2). Moreover, because of the wide space projected by the sliding surface matrix, 𝐏, the results came from the SMC and the HSMC were similar to the one from the LQR in the control strategy (3) and even performed better in the control strategy (1).

Most importantly, the HSMC matched the control performance against the SMC in the control strategy (2) and against the LQR in the control strategy (1) and (3). Consequently, the numerical simulation obviously demonstrated the advantage of the HSMC compared to the SMC and the LQR.

Comparison of Hybrid and Wavelet-based Control Algorithms

To extract the local information from the earthquake excitation, the Biorthogonal 6.8 was

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selected as the wavelet function, and the reference level in DWT was assigned to be 7 so that the frequency band in each decomposed component was approximately 0.781 Hz. The appended data length for the moving window technique was 40 points and, in each window, the earthquake excitation in the latest 256 points (the latest 1.28 seconds) was decomposed using DWT. According to Equation (2.30) and Equation (2.31), the energy entropy of the details, 𝐷₅(𝑡), 𝐷₆(𝑡), and 𝐷₇(𝑡), was summed up to calculate the energy locally stored in the frequency band ranged from 0.781 Hz to 6.25 Hz which corresponds to the four modal frequencies of the structural system. In the proposed WHSMC, the control gains were switched if the energy entropy was equal to or larger than 40.

Figure 5.19 shows the calculated energy entropy and the applied control gains, and Figure 5.20 shows the responses of equipment, the structural responses, and the control forces in the control strategy (3) under the El Centro earthquake in NS direction. The control forces generated by the HSMC and the WHSMC were quite similar throughout the earthquake duration except the short period (from 4.8 seconds to 5.8 seconds) during the peak ground acceleration occurred. It can be observed that the control gains temporarily switched to the second one and the second control gain immediately enhance the control forces; as a consequence of enhancement, not only the structural responses but also the responses of equipment were reduced. Apparently, by using the real-time wavelet, the WHSMC can provide an adaptability and a better control performance under the non-stationary excitations. To investigate the efficiency of the proposed control algof;rithm, the consumed energy is defined as the cumulative squared forces, and therefore the energy spent by the SMC, the HSMC, and the WSHM were compared with each other, as shown in Figure 5.21. The results show that the WHSMC only used 81.3% and 81.2% energy compared to the SMC and the HSMC; in addition, the control performance was 3.84% and 0.54% better than the SMC and the HSMC, respectively.

In conclusion, the WHSMC first combines the control gains from the LQR and the SMC to take advantage of these traditional control algorithms. Then, it utilizes DWT to consider the non-stationary nature of earthquake excitations and schedules the control gains accordingly. Through the numerical simulation, the control performance of various control algorithms was evaluated and compared, and furthermore, the WHSMC performed better than the LQR, the SMC, and the HSMC on the reduction in the responses as well as the consumed energy. Besides, these results clearly demonstrated that the proposed control algorithm can combine the ability of the SMC and the LQR, and it is a feasible and efficient control algorithm for the equipment isolation system integrated with the actuator. To further validate the control algorithms, a comprehensive study incorporated with semi-active control devices is investigated in the next sections.

doi:10.6342/NTU201901242 88