**6. Numerical Examples**

#### *6.1. FE Model Validation*

The purpose of this subsection is to examine the accuracy of the dynamic FE model established by ANSYS, by comparison of the numerical and analytical results in the open literature.

Considering the square laminate plate, which composes of three layers of graphite-epoxy (GE, carbon-fibre reinforced) composite material (0/90/0) covered by PZT-4 piezoelectric layers poled in the z-direction (through-thickness). Then, a five-layer laminate plate (PZT-4/GE 0◦/GE 90◦/GE 0◦/PZT-4) was formed. The geometrical data of the square laminated plate are given in Figure 2. The material properties of the GE composite material and the piezoelectric material are given in Table 1, and the densities of all materials considered are to be considered unitary (*ρS* = 1 kg/m3) for the purpose of comparison. The assumed boundary conditions are simply supported. The SOLID46 3-D solid element, which can be used to model the laminated composite beam or plate type structures with various layer orientations, is used for simulating the GE composite plate and the SOLID5 3-D solid element (which has a 3D piezoelectric and structural capability between the fields and is applied for simulating piezoelectric layers).

**Figure 2.** Square laminated plate.

**Table 1.** Properties of the graphite-epoxy (GE) composite material and the piezoelectric material.


The first five natural frequencies *<sup>x</sup>*,*<sup>y</sup>* can be non-dimensionalized by the expression *<sup>λ</sup><sup>x</sup>*,*<sup>y</sup>* = *<sup>x</sup>*,*<sup>y</sup>LS*2/*HS*√*ρ<sup>S</sup>* × 103. The dynamic FE numerical analysis results in this paper are compared to different approaches: (a) The finite element solution FPS (an equivalent single-layer approach and a layerwise representation of the electric potential, using the finite element method with four nodes and five degrees of freedom) by W. Larbi et al. [48]; (b) the finite element solution Q9-HSDT (higher shear deformation theory, using the finite element method with nine nodes and eleven degree of freedom) by Victor M. Franco Correia et al. [50]; (c) the finite element solutions TDST (third order shear deformation theory, using the finite element method with four nodes and seven degrees of freedom) by Tatiane Corrêa de Godoy et al. [49]; and (d) two-dimensional analytical solutions (layerwise first-order shear deformation theory and quadratic electric potential) by Ayech Benjeddou et al. [51]. The present results and their percent errors are relative to the published results (Δ%) and are shown in Table 2. It was found that a reasonably good approximation to FPS and Q9-HSDT theories was obtained, although the relative errors for some natural frequencies were greater than 5%.

**Table 2.** First five frequencies parameters.


#### *6.2. Modeling and Choice of Controller Parameters*

In this section, a piezoelectric smart plate is considered for vibration control simulations. The piezoelectric smart plate is made of a host composite plate and six piezoelectric patches bonded in pairs on both sides of the plate. From Figure 3, there are three piezoelectric actuator-sensor pairs marked with *a*, *b* and *c*, respectively. The upper piezoelectric patches work as actuators to control structural vibration, while the lower ones work as sensors to obtain vibration information. The dimensions of the composite plate and the piezoelectric patches are 414 mm × 120 mm × 1 mm and 60 mm × 24 mm × 1 mm, and the configuration of the structure is shown in Figure 3. The localization of the actuator-sensor pairs is selected by referencing to [52]. The host composite plate is made of a GE composite material with five substrate layers, the stacking sequence of which is symmetric angle-ply [0/90/0/90/0]. The properties of the GE composite material and the piezoelectric material are listed in Table 3. The locations of points A, B and C are shown in Figure 3. A clamped boundary condition is assigned in the root of the piezoelectric smart plate.

**Figure 3.** The piezoelectric smart plate.

**Table 3.** Properties of the GE composite material and the piezoelectric material.


In this study, the layered solid element SOLID46 was used to model the composite plate, while SOLID5 was used to model the piezoelectric patches. The composite plate was meshed with 69 × 20 × 1 elements, and each piezoelectric patch was meshed with 10 × 4 × 1 elements. It was assumed that the same magnitude but the opposite electric field direction was applied to the upper and lower piezoelectric patches. The degrees of electric freedom for the nodes at the top and bottom surfaces of the piezoelectric patches were coupled by using the ANSYS command *CP*. Model analysis was implemented to find out the natural frequencies of the plate and determine the sampling period for the closed-loop system simulations [28]. The numerical and experimental results of the first three natural frequencies are given in Table 4. The sampling period was defined as *T* = 1/(20 *f*1), where *f*1 is the first natural frequency. The Rayleigh damping coefficients were considered as *α* = 0.003 and *β* = 0.0015.

**Table 4.** First three natural frequencies of the piezoelectric smart plate.


The initial value of feedback gain *G*(*k*) in Equation (15) was assumed to be zero. The parameters of the robust MFA-IL controller are given in Table 5. There are three actuator-sensor pairs in the system, and the corresponding controllers offer measurements. The frame of discernment for system state diagnosis was constructed with two types of states, namely the normal learning state and the learning stopping state. The feedback gains of the controllers were chosen as the feature parameters. Let *r* = 2 in Equation (39) and the distance converges to the Euclidean distance. The BPAs can be calculated using the method based on information sources in Section 5.2. The stopping criteria of the robust MFA-IL control algorithm are defined as follows: (a) According to Dempster's rule of combination, each row vector of the BPA matrix *P* can be fused. The threshold value was set 0.985. If the fused BPA is higher than the threshold value, the learning processes of all controllers should be stopped. (b) Controllers connected to the actuators at different locations may lead to distinct learning speeds. To make all controllers learn sufficiently, the threshold value for the BPAs of each controller should also be proposed. If the BPAs from a certain controller are higher than 0.800, the learning process of the corresponding controller should be stopped. If one of the above conditions is met, the learning process of the controller is then terminated. Otherwise, the controller is in normal learning state. Considering the vibrations generated by various external disturbances, different simulations were investigated to evaluate the effectiveness of the proposed method. In the design of the P-type IL controllers, the maximum number of iterations was limited to 500 as the stopping criterion, and the fixed learning gains were selected as Φ1 = 0.063 and Φ2 = 0.100 for various cases.

**Table 5.** The control parameters.


## *6.3. Harmonic Excitation*

In this case, the first mode control was tested by applying the harmonic force *f*(*t*) = 6 cos(*<sup>ω</sup>*1*<sup>t</sup>*)*<sup>N</sup>* at the point C, where *ω*1 = 17.083 rad/s (5.4377 Hz). Due to the symmetry of the piezoelectric smart structure and the excitation location, sensors *a* and *b* have the same control feedback signals in the process of vibration.

The displacement responses of points A and B are displayed in Figure 4a,b, where it can be seen that both the robust MFA-IL control and the P-type IL control can effectively suppress the first mode vibration. The control effectiveness of the actuators is obvious at the positions with sensors (e.g., point A) and without sensors (e.g., point B). However, it is noteworthy that these results are different from Salehs [13], which claimed that the P-type IL control was able to compensate for the unwanted vibration at the observation point, while not being effective at other points. In addition, it was also pointed out that the P-type IL method cannot effectively attenuate the amplitude as long as the smart structure is excited by its first natural frequency.

**Figure 4.** Displacement responses: (**a**) Point A, (**b**) point B.

An effective control system has the ability to suppress the structural vibration of the whole plate rather than a small portion of the plate area. The control strategy plays an important role in designing a vibration control system for obtaining a desired performance. Besides, the locations and sizes of the piezoelectric actuators and sensors should also be seriously considered [53]. The areas of the structure where the mechanical strain is highest are always the best locations for actuators and sensors. To guarantee the actuators generate the desired control forces to suppress structural vibration, the dimensions of the actuators should also be designed appropriately. The sizes of sensors should also be chosen properly, so that accurate information of structural deformation can be obtained. A misread or incorrect sensor measurement signal may lead to unreasonable measurements and inappropriate control force generation, which will deteriorate the dynamic behavior. As long as the locations and the sizes of the actuators and sensors are selected appropriately, the P-type IL control presents good performance on first mode control. Furthermore, both the locations with sensors and the positions without sensors on the smart plate can provide good controllability of structural vibration.

The displacement responses of points A and B are given in Figure 4a,b. The output voltages of sensor *a*/*b* and sensor *c* are shown in Figure 5c,d. By comparing with the P-type IL approach, a smaller amplitude can be obtained when the system is controlled by the robust MFA-IL method. In this paper, the root mean square (RMS) values of the amplitude at points A and B and output electric potential are used to quantitatively evaluate the control performance of the robust MFA-IL control and the P-type IL control. The data used to calculate the RMSs were recorded after all the controllers stopped learning, and the RMSs of amplitude are given in Table 6.

**Figure 5.** The time-domain responses of actuators/sensors: (**a**) Actuator *<sup>a</sup>*/*b*, (**b**) actuator *c*, (**c**) sensor *<sup>a</sup>*/*b*, (**d**) sensor *c*.



The control voltages applied to actuators *a*/*b* and actuator *c* are shown in Figure 5a,b. In the process of iterative learning, the actuation voltages changed sharply under the control of the P-type IL algorithm at 4.3 s. After the learning was terminated, the amplitude recovered to a smooth value. The feedback gain of the controllers at different locations had a distinct convergence rate, which may cause the control force produced by controllers to mismatch among each other. If an actuator fails to perform as expected, the performance of its neighboring actuators will be negatively affected. In order to avoid this problem, more iterations are necessary to reach satisfying level of control stability. A smaller iterative number may directly cause control spillover or even system instability. The controllers connected with actuators *a*/*b* have the same learning processes, as shown in Figure 6a. Figure 6b presents the learning processes of the feedback gains in actuator *e*. Observing the results of Figures 5 and 6, the robust MFA-IL control has a faster convergence rate when compared with the P-type IL method, and the input voltages of the actuators change smoothly. Therefore, it can be seen that the proposed method enables the system to enhance control effectiveness and maintain superior stability.

**Figure 6.** The learning processes of feedback gains: (**a**) Actuator *<sup>a</sup>*/*b*, (**b**) actuator *c*.

The main contribution of the robust MFA-IL control is the improvement of learning speed. Two parts influence the learning speed of the proposed algorithm: The MFA control and the SM control. Both of these two methods accelerate the convergence speed of the feedback gains in the learning processes. Their contribution percentages to the feedback gains are shown in Figure 7, where it can be seen that the SM control plays a more important role than the MFA control.

**Figure 7.** The contribution percentages of various methods to the feedback gains: (**a**) Actuators *<sup>a</sup>*/*b*, (**b**) actuator *c*.

Real-time monitoring for the fused BPA results and the BPA from each controller was implemented and is shown in Figure 8. The BPAs update at each period and the monitoring curve moves forward over time. First, the BPAs obtained from actuator *c* meet the stopping criteria. After a short period, the controller connected with actuators *a*/*b* stops learning. All controllers can learn sufficiently based on the evidence theory, and thus preferable control performance can be obtained.

**Figure 8.** The basic probability assignment (BPA) curves.

## *6.4. Random Excitation*

In the last simulation, a random excitation (shown in Figure 9) was applied to point C to drive the piezoelectric smart plate.

**Figure 9.** The random excitation.

Figure 10a,b presents the dynamic displacement responses of points A and B, respectively. The control voltages applied on actuators *a*/*b* and actuator *c* are depicted in Figure 11a,b. The output signals of the corresponding sensors are shown in Figure 11c,d. By comparing with the robust MFA-IL method, the actuation voltage amplitudes controlled by the P-type IL method are smaller in the initial stage of simulation. Under the control of the P-type IL algorithm, the actuators cannot work effectively to consume the energy of the vibration system in the initial period. According to Figures 10 and 11, it can be seen that the P-type IL control cannot suppress structural vibration in the short term. Since several hundreds of iterations lead to convergence rate of feedback gain slow. However, the learning speed of the robust MFA-IL algorithm is faster than that of the P-type IL method which has fixed gain. The time-varying learning gain is updated using I/O data, which can reflect the system dynamic behavior in real-time.

**Figure 10.** Displacement responses: (**a**) Point A, (**b**) point B.

**Figure 11.** The time-domain responses of actuators/sensors: (**a**) Actuators *<sup>a</sup>*/*b*, (**b**) actuator *c*, (**c**) sensors *<sup>a</sup>*/*b*, (**d**) sensor *c*.

The learning processes for feedback gain for the controllers connected actuators *a*/*b* and actuator *c* are shown in Figure 12. The percentages contributed by the MFA control and the SM control are shown in Figure 13, where it can be seen that the SM control has a larger impact on the convergence rate of feedback gain than the MFA control.

**Figure 12.** The learning processes of feedback gains: (**a**) Actuators *<sup>a</sup>*/*b*, (**b**) actuator *c*.

**Figure 13.** The contribution percentages of various methods to feedback gain: (**a**) Actuators *<sup>a</sup>*/*b*, (**b**) actuator *c*.

The RMSs of amplitude were calculated and are shown in Table 6. A similar control effect to the previous simulation can be observed. The robust MFA-IL control in this simulation presents a better control performance and makes the learning speed of the controller faster by comparing with the P-type IL control. The real-time monitoring results of the fused BPAs and the BPAs from each controller are depicted in Figure 14.

**Figure 14.** The BPAs curves.

In order to test the robustness of the proposed method, the harmonic noise signal *f*(*t*) = cos(<sup>152</sup> · *t*)*N* was added to the external excitation at point C. The controller's parameters were set up the same as mentioned above. Figure 15 shows the sensor output signals. The added noise resulted in a decrease of system performance and divergence as long as the system was controlled by the P-type IL method, while the robust MFA-IL control could maintain the stability of the control system. These comparative results validate that the proposed control method possesses excellent control performance and robustness to the noise from external excitation.

**Figure 15.** The time-domain responses of sensors: (**a**) Sensors *<sup>a</sup>*/*b*, (**b**) sensor *c*.
