*7.2. Modal Identification*

A chirp signal is utilized to determine the natural frequencies of the piezoelectric smart plate. Actuator *a* was excited by a chirp signal with an amplitude of 100V, while the output voltages of the Sensor *a* were stored after filtering. The starting frequency was 0.5 Hz, and the stopping frequency was 50 Hz. The sweep time was 100 s. Then, the fast Fourier transform (FFT) of the time response was calculated. Figure 18b depicts the frequency responses of sensor *a* when applying FFT to the time-domain signal plotted in Figure 18a. The experimental natural frequencies obtained from the FFT plot are given in the same table (Table 4) for comparison. The biggest difference, 13.9%, occurs in the second natural frequency.

**Figure 18.** Measured vibration responses excited by actuator *a*: (**a**) Time-domain responses of sensor *a*, (**b**) frequency responses of sensor *a*.

Conditions that may lead to the difference of modal frequencies between the numerical solutions and experimental solutions are considered as follows: (1) The clamp-free boundary condition in the simulations is ideal. While in the experiments, the boundary of the plate may not be totally clamped. (2) The geometric parameters and material properties of the piezoelectric smart plate cannot be known accurately enough. The parameters used in the numerical simulations are not precisely consistent with those of the plate applied in the experimental material. (3) The mass of both the glue and the connected signal wire of the piezoelectric patches are not considered in the simulations. The modal analysis in simulations is used to provide an approximate solution for verifying the feasibility of the control algorithms, the difference of modal frequencies between the numerical results and the experimental results is acceptable. It is clear that the FE model sufficiently predicts the natural frequencies of the piezoelectric smart plate. Filters were designed and utilized in the experiments to deal with high frequency noise. For investigating the control algorithms, the designed low-pass filters were applied. The cutoff frequency of the low-pass filters was specified at 30 Hz.

#### *7.3. Experiments Results and Discussions*

During the experiments, the robust MFA-IL control and P-type IL control algorithms were used to suppress the structural vibration of the cantilevered plate. The first mode control was tested in this section, where the parameters of the P-type IL controller are chosen as follows: The fixed learning gain Φ = 0.54. The maximum number of iterations was set to 1500 as to avoid control spillover or system instability. The parameters of the robust MFA-IL controller are listed in Table 7. The stopping criteria in the experiment are the same with those in Section 6.2. The sensor output signals (shown in Figure 19a,b) were moved forward to compensate for the phase lag effect. Five seconds after harmonic excitation driving, the proposed control algorithms were implemented to suppress the structural vibration.


**Table 7.** The control parameters.

**Figure 19.** The time-domain response of actuators/sensors: (**a**) Actuators *<sup>a</sup>*/*b*, (**b**) actuator *c*, (**c**) sensors *<sup>a</sup>*/*b*, (**d**) sensor *c*.

The measured signals of sensors *a*/*b* and sensor *c*, and the corresponding actuation voltages obtained from the robust MFA-IL controller and the P-type IL controller are shown in Figure 19. The sensor *a*/*b* and sensor *c* output signals used to calculate the RMS values were recorded after learning termination. The RMS values (shown in Table 6) were used to quantitatively evaluate the control performance of the various methods. From the RMS values, it can be seen that the P-type IL control provides a 31.9% reduction and the robust MFA-IL control provides a 33.73% reduction. It is clear that the robust MFA-IL control is more effective at reducing the vibration amplitude of the flexible plate than the P-type IL control. It can be concluded that the proposed control method exhibits excellent performance when integrating MFA control and SM control. The learning processes of the feedback gains are shown in Figure 20. Under the control of the robust MFA-IL method, the structural vibration can be suppressed faster when compared with the robust MFA-IL algorithm.

**Figure 20.** The learning processes of feedback gains: (**a**) Actuators *<sup>a</sup>*/*b*, (**b**) actuator *c*.

The percentages of contributed feedback gain from the MFA and SM methods are shown in Figure 21, where it can be seen that SM control has a larger impact on the convergence speed of feedback gain than MFA control. The real-time monitoring curves of the fused BPA results and the BPAs from each information source are shown in Figure 22. By applying evidence theory to the design of the stopping criteria, the learning processes of all controllers can be evaluated in real-time. The decision-making based on the appropriate stopping criteria makes all controllers learn sufficiently, so that a better control performance is obtained.

**Figure 21.** The contribution percentages of various methods to feedback gain: (**a**) Actuators *<sup>a</sup>*/*b*, (**b**) actuator *c*.

**Figure 22.** The BPAs curves.

Comparing the experimental results with the numerical results, it can be seen that the vibration suppression trends performed similarly. Due to the difference of the sample period and the unknown nature of the experimental system, the parameters selected in the simulations and experiments are different.

#### **8. Conclusions and Outlooks**

A robust MFA-IL control method was developed for the active vibration control of piezoelectric smart structures. Considering the uncertainty of interaction among all actuators in the control process, the MFA control was incorporated into P-type IL control for the adjustment of learning gain. In order to achieve a fast control response and enhance the stability of the system, SM control was adopted to ensure a fast dynamic response and compensate for the influence of uncertain noise. The multi-source information fusion method based on the evidence theory was adopted to design the stopping criteria of the robust MFA-IL method. The vibration control equations of piezoelectric smart structures were derived from the dynamic FE equations of a linear elastic system. The dynamic linear method was applied to transfer the vibration control equations for the design of the robust MFA-IL controller. The simulation and experimental results were presented and compared with the corresponding results using the P-type IL control approach.

As long as the locations and sizes of actuators and sensors are chosen appropriately, both P-type IL control and robust MFA-IL control can effectively suppress structural vibration when the piezoelectric smart plate is excited by its first natural frequency. Furthermore, the whole structure presents good controllability, rather than small portions bonded with piezoelectric sensors, using both the P-type IL method and the robust MFA-IL method.

The robust MFA-IL control method accelerates the learning speed of the controller. Based on the comprehensive comparative analysis of the numerical and experimental results, the proposed control method can achieve better control performance and is more robust to external disturbances when compared with the P-type IL control. In other words, the proposed control method overcomes the inherent drawbacks of P-type IL control and achieves the desired control performance.

Although the robust MFA-IL control method was applied for a plate structure in this paper, considering its advantages presented above, this approach can be applied for other structures, like beam structures, extending the possibilities of engineering and research applications. Dynamic linearization was an effective method in developing the proposed method for nonlinear systems. Dynamic linearization mainly introduces optimal technology as a tool for the controller design and analysis. In future work, it will be expected that the dynamic linearization method will be combined with an online model identification technique to deal with much more practical problems, those typically encountered in industrial applications. This may result in a simpler controller structure and obtain a better degree of control precision.

**Author Contributions:** L.B. conceived and designed the experiments; X.X. performed the experiment; Y.F. and N.L. contributed the analysis tools; and L.B. analyzed the data and wrote the paper.

**Acknowledgments:** The work was supported by National Natural Science Foundation of China under Grant 10577015.

**Conflicts of Interest:** The authors declare no conflict of interest.
