Optimization of Piezoelectric Sensor-Actuator for Plate Vibration Control Using Evolutionary Computation: Modeling, Simulation and Experimentation

The development of lightweight, stronger, and more flexible structures has received the utmost interest from many researchers. For this reason, piezoelectric materials, with their inherent electromechanical coupling, have been widely incorporated in the development of such structures to attenuate their vibrations. However, one of the main challenges is to find the optimal control and sensor-actuator placement. This paper presents an active vibration control for flexible structures, whereby a simply supported plate is taken as the benchmark model. A feedback controller with a collocated sensor-actuator configuration is used. Both disturbance and control signal acting on the plate is created by using piezoelectric (PZT) patches. The analytical model is derived based on the Euler-Bernoulli model. The Optimal location of the collocated sensor-actuator, as well as PID controller gains, are determined using Ant Colony Optimization (ACO) technique, then compared with the Genetic Algorithm (GA) and enumerative method (EM). Optimization in this paper is based on minimizing frequency average energy. The optimal performance value of piezoelectric patch sensor-actuator position and PID controller gains are verified experimentally. It was found that PID controller gains and collocated sensor-actuator location optimizations using ACO, GA and enumerative methods give similar results, which implies the effectiveness of ACO as an optimization technique. More than 20 % of attenuation achieved using the available hardware setup.


I. INTRODUCTION
Lighter, stronger, and more flexible structures are said to be the ideal forms of structures in vibration suppression with minimum power consumption [1]. In recent years, the development of lightweight yet stronger structures that can withstand vibrational loads has gained the utmost attention by researchers. Such structures can be developed when piezoelectric patches (sensor-actuator) are integrated with an optimized active vibration control (AVC) strategy [2]. In the advancement of such smart structures, piezoelectrics have The associate editor coordinating the review of this manuscript and approving it for publication was Yingxiang Liu . shown great promise. Piezoelectric materials can generate an electrical voltage when subjected to mechanical strains and stresses, and vice versa. Under the application of electrical voltage, these materials can generate mechanical strain in response to the applied voltage. Hence, allowing them to be utilized as both sensors and actuators when placed on flexible structures. Piezoelectric actuators have shown significant benefits in vibration control due to their ability to excite only the structures' elastic modes without exciting the rigid body modes. It is crucial since it is often required to control the structures' elastic motions [3].
The continuous nature of the structures has provided flexibility to the placement of the piezoelectric patches on VOLUME 9, 2021 This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ the structures. To achieve optimal performance, one must ensure the optimal location of the piezoelectric sensoractuator. Many researchers have investigated the optimal placement of the piezoelectric sensors and actuators on structures. Caruso et al. [4] investigated the optimal placement of the actuators and sensors for collocated flexible plates, and the optimal locations were achieved by maximizing the modal controllability and observability of the structure. Hongwei et al. [5] have investigated the actuators' optimal locations in a smart structure using optimal control theory. Using the input energy correlation, it was concluded that size, location and control gain were independent of the initial conditions. Bruant and Proslier [6] considered the optimal location of actuators and sensors in the AVC of structures while considering the residual modes to limit spillover effects. By minimizing the linear quadratic regulator cost, Kumar and Narayanan [7] were able to optimize the piezoelectric actuators' location on steel plates. Darivandi et al. [8] applied a subgradient-based algorithm to optimize the location of piezoelectric actuators on flexible structures that have been much more efficient and accurate than genetic algorithm optimization. Liu et al. [9] have investigated the optimal placements of piezoelectric actuators on structures and demonstrated that particle swarm optimization's computational efficiency is higher than that of the genetic algorithm. Prion et al. [10] developed a criterion concerning the optimal placement of a collocated sensor-actuator pair of AVC structures based on the maximization of the pole-zero distance in open-loop, which correlates with the maximum achievable damping ratio in closed-loop. Contero-Chinchilla et al. [11] optimized the sensor/actuator placement for structural health monitoring using an objective function that combines parameters uncertainty, expected information entropy and the cost of both sensors and actuators. Optimal sensor-actuator positions and gains used in the controller are vital to enhancing the performance of a control system and numerous works using conventional optimization techniques such as H (infinity) norm [12], linear quadratic regulator (LQR) [13], and H2 norms [14], were proposed to cope with such problem. Various conventional optimization techniques have been employed for this purpose. However, biologically inspired optimization techniques have shown great potential in optimizing the sensor/ actuator location and controller gains of AVC systems Optimization of sensor and/or actuator location and controller gains in AVC system were successfully carried out through biologically-inspired optimization techniques, including particle swarm optimization (PSO) [14], artificial neural network (ANN) [16] and genetic algorithm (GA) [17]- [19]. In 1992, another bio-inspired optimization method based on swarm intelligence, namely Ant Colony Optimization (ACO), was founded. Since then, it has been successfully implemented [20]. The ant colony optimization ACO technique is another bio-inspired method that can be successfully implemented. Furthermore, ACO has shown great potential as an optimization technique due to its simplicity and efficiency. Nevertheless, even with its simplicity, it is shown that attempts using ACO in optimizing the location of the sensor-actuator and controllers for the AVC system is still considerably low. ACO was used to optimally tune the Proportional-Integral-Derivative (PID) controller [21], but these were tested only on arbitrary plants. Hardware-inthe-loop (HIL) optimization with the help of a continuous ant colony system (TCACS) suggested by Nobahari et al. [22], to optimize controller and observer parameters for a beam was conducted successfully. Mohamad et al. [23], proposed SISO control of a flexible beam using continuous ACO (ACO R ).
This paper presents theoretical and experimental works on AVC strategy on a simply supported thin plate using a piezoelectric patch for modal overlap factor (MOF) less than 1. The individual vibration modes distinct, which is equivalent to a frequency range of 0 to 200 Hz. Optimization sensor-actuator location and controller gains are carried out utilizing the ACO and later verified using GA. Using ACO offers simplicity, less complexity and yields a similar level of accuracy. Although many algorithms can be used, this study focused on ACO and GA algorithms. These bio-inspired computing optimizing algorithm has gained interest as it is a promising approach, that can be further developed for new and robust competing techniques. The values obtained from ACO and GA are compared with the enumerative method (the actual value) with objective function as total energy reduction. The purpose of this study is to demonstrate the effectiveness of ACO as an optimization technique. This paper is organized as follows. Section 2 presents the equation of motion (EOM) of the system. Section 3 explains the optimization algorithms used in this research, i.e., ACO and GA. Section 4 discusses the simulation results for optimal controller gains and sensor-actuator location. Section 5 presents the experimental verification of the optimal parameters found in Section 4.

II. PLANT MODEL
In this section, the system's equation of motion of the simply supported thin rectangular plate with attached piezoelectric patches for excitation and control is derived. The time-averaged plate energy EP is then derived and used in the objective function used for the optimization. The mathematical model of the simply supported plate with piezoelectric patches is derived using the Euler-Bernoulli model; which limits the simulation results to thin plates only. Lagrange's method is employed to derive the equation of motion of a simply supported plate with attached piezoelectric patches.
Consider a simply-supported thin rectangular plate of length a, width b and thickness h is taken for this study. Piezoelectric patches were used for excitation and control forces. For linear isotropic material, the constitutive equation of piezoelectric as the actuator is [24]: where σ is the stress vector, e is the dielectric permittivity matrix, c E is the matrix of elastic coefficients under constant electric field, ε is the strain vector, E = V (t)/t p is the electric field vector, V (t) is the voltage supplied to the piezoelectric actuator, t p is the piezoelectric patch thickness. The spatial deflection of the simply supported plate during vibration can be represented as the summation of modes in double series [25]: where W mn is the magnitude, and m and n refer to the half-wave number in the x and y directions, respectively. The natural frequency (m, n) th of the plate can be written as: where D is the flexural rigidity, ρ is the plate density, L x and L y refer to the dimensions in the x and y directions. The potential energy U and kinetic energy T are given by [26]: S (x, y) is the product of Heaviside function for a patch on the plate and can be obtained by: The flexural rigidity of the plate is: The flexural rigidity of the plate under the patch area is: (8) and the flexural rigidity of the piezoelectric patch is obtained by: and, where ρ and ρ pz are the plate and piezoelectric density, respectively, and A is the plate area. d 31  Substitution of both potential and kinetic energies into Lagrange's Equation produce an equation of motion of the system as: where ω is the angular frequency, M is the mass matrix. By substituting both the derived potential and kinetic energies into the Lagrange's equation, it produces the equation of motion of the system, where the term (jη + 1)K is substituted by K c for simplicity. Where K is the stiffness matrix, and j is an imaginary point. η is the modal loss factor, W mn is the vector representing modal amplitude, and f e and f a are the modal force vector for exciter and actuator patches, respectively. PID controller involves feedback signal, its derivative, and it's integral, i.e. the feedback signal is equivalent to velocity term. The mass, stiffness, and damping matrices due to PID controller is denoted by M PID , K PID and C PID respectively. The EOM of the system with a PID controller is written as [27]:  (13) Thus, the frequency-averaged plate enrgy, EP, measured from an initial frequency f in to final frequency f fi is:

III. OPTIMIZATION FUNCTION
The objective function used for optimization is defined as: where EP c and EP nc are the average plate energy with and without control, respectively. This objective function aims to find five optimal parameters: the x-and y-coordinates of sensor-actuator location (x sa , y sa ) and the K p , K i and K d of the PID controller. Optimization of five variables simultaneously is difficult and time-consuming; thus, the process is divided into two [18]. 1. Optimize x sa and y sa while considering velocity feedback control with 1000 Ns/m, approximately 100 times driving point impedance, as an initial gain. 2. Optimize K p , K i and K d , using optimal sensor-actuator location found above. Vibration mode shapes of a flexible structure have nodes and anti-nodes. It is difficult to control the vibration by placing the actuator on a node even with a large actuating force. The location of the sensor-actuator patches is more critical than the PID gains, hence, the PID gains are optimized after finding the optimal sensor-actuator location.

A. ANT COLONY OPTIMIZATION
In this paper, we utilized Simple-ACO (SACO) algorithm for optimization purpose. Inspired by a colony of ants looking for a food source, the algorithm finds optimal parameter values of any objective function based on the probability equation given by: where τ is the pheromone trail mimicking trail made by an ant while searching for food, t is time, α is the relative importance of the pheromone values and T k does the ant k effectuate the path at a given time. The magnitude of pheromone disposed of by an ant during a tour is represented by: where Q is a constant and L k is the cost of tour by ant k or the objective function. In our case, L k is the inverse of energy reduction calculated using Eq. (13). The next ants follow the trails with more potent pheromones and reinforce it with their pheromone. On the other hand, the weaker pheromone trails will eventually evaporate. Mathematically, this can be represented by: Forgetfulness of the bad choices are introduced by σ , the pheromone decay 0 < σ < 1, and NA is the number of ants. ACO parameters used in this paper are given in Table 1.

B. GENETIC ALGORITHM
GA, a well-established algorithm, is used in this paper to ascertain the results found by ACO method. It is inspired by the natural evolutionary process mechanics and had successfully solved problems involving optimization of actuator location and controller gains [28]- [33]. The GA parameters used in this study are listed in Table 2, and other settings are left to their default value. Flowcharts of the SACO algorithm is given in Figure 1, and the optimization strategy is illustrated in Figure 2.
The optimal values from both ACO and GA are compared against the enumerative method since it gives the actual optimal point. As the purpose is to reduce modal energy, the exciter should be placed at a single location that can excite the required modes.

IV. SIMULATION RESULTS
Parameters of the simulation model are given in Table 3. For simulation purpose, the exciter patch is assumed to be located at (0.15 m, 0.15 m), while the control patch with collocated sensor-actuator configuration will be found through optimization. The frequency range of interest is between 0 to 200 Hz, and the resonant frequencies within this range are shown in Table 4.

A. OPTIMAL SENSOR-ACTUATOR LOCATION
To make the results as precise as possible, 10000 possible values of x sa and 10000 possible values of y sa are used, making the distances between consecutive one-two nodes along the x-axis and y-axis as 4.5005 × 10 −5 m and 5.8006 × 10 −5 m, respectively. Table 5 and Figures 3(a) and 3(b) show optimization results.    Table 5, the results from both optimization methods agree to each other. The effect of a negative velocity feedback control to the plate energy is clearly shown in Figure 3(a), whereby the controlled energy plot experiences a significant reduction of amplitude without any natural frequency shifting. The best location to place the sensor-actuator patch is near the exciter patch with more than  10% frequency-averaged energy reduction. This is because the highest energy reduction can be achieved by directly locking the disturbance force. Nevertheless, the actuator's placement close to the disturbance point is not a practical solution since we normally do not know the location that causes vibration and its corresponding magnitude. Hence, optimal location is searched again by setting at least 0.1 m gabs, along x and y axes, between excitation and sensor-actuator locations. Table 6 shows the result and illustrated in Figures. 4(a)  and 4(b). The revised optimal sensor-actuator locations found are at (0.3511 m, 0.4550 m) and (0.3472 m, 0.4592 m) using ACO and GA, respectively.

As shown in
An enumerative method is used to verify results from GA and ACO. It is a conservative technique in which the sensor-actuator is systematically placed to all grid points of the plate, and its corresponding energy reduction is measured. The accuracy of the result depends on the number of grid points taken for simulation. Even with coarse grid points, this method can predict the optimal location. A 3-D plot describing energy reduction concerning the location of the sensor-actuator is shown in Figure 5. The optimal sensor-actuator locations for first and second highest energy reduction are (0.160 m, 0.155 m) and (0.358 m, 0.4547 m).

B. OPTIMAL PID CONTROLLER GAINS
The optimal value of K p , K i and K d were found when the sensor-actuator was at the optimal location i.e. (x sa = 0.3511, y sa = 0.4550). The range of values for K p , K i and K d considered here are 0 to 1000, 0 to 500 and 0 to 50, respectively.  Table 7 shows Kp's optimal values, K i and K d obtained using ACO and GA and its corresponding energy reduction. Figure 6(a) shows the PID controller had altered the system's natural frequencies. This was due to the change in mass-stiffness-damping matrices as given in Eq. (12) above. Figure 6(b) shows energy reduction for each mode. It can be seen that the energy of the mode whose frequency is around 110 Hz is increased. However, This is the trade-off done when controlling broadband vibration using a single actuator. The location of the sensor-actuator is chosen so that frequency-average energy is reduced. As such, the optimization algorithm will identify an actuator that optimally applies force on many modes within the bandwidth; this, in turn, may undesirably add energy to some modes. The situation would be different if the aim is to control individual mode, as different actuator location is needed for every mode. When using a piezoelectric patch as an actuator, changing its location is impossible since it is bonded strongly/permanently on the plate. Another option is to use multiple actuators in different locations to target various modes. The current work aims to use a single piezoelectric actuator to target and control many modes optimally.

V. EXPERIMENTAL RESULTS
The experimental setup comprised of a thin stainless-steel plate with a dimension of 0.6 m × 0.5 m × 0.001 m lying on a frame to mimic a simply supported boundary condition. Vibration excitation and suppression are generated by 2 MIDE piezoelectric patches and each is connected to ACX power amplifier model EL-1225 with 20-time amplification factor. Eight units of PCB accelerometer model 352C33 are used for energy estimation, while another unit located on top of the actuator patch meant for feedback signal purpose. Only one actuator is used to implement the broadband vibration control optimally. The generation of the excitation signal and the calculation of energy are performed via LABVIEW program with the help of Ni-cDAQ model NI-9174, two input modules model NI-9263 and one output module model NI-9234. A NI-cRIO controller model NI-9004 with one NI-9263 input module and one NI-9234 output module embedded with an FPGA-coded controller program is used to actively control the plate's vibration. Figures. 7(a) and 7(b) show experimental setup and interconnection diagram, respectively.

A. OPTIMAL SENSOR-ACTUATOR LOCATION
This section aims to experimentally gauge the performance of the optimal parameters found in Section 4. Recall that the optimal parameters found theoretically are x sa = 0.3511, y sa = 0.4550, K p = 990, k i = 210 and k d = 19.88. Fifteen natural frequencies are detected for less than 200 Hz, as shown in Table 8, compared to twelve natural frequencies found in the simulation. This is due to the placement of accelerometers on the plate, which consequently affected the plate's mass distribution and created another three natural frequencies, i.e., at 31.85 Hz, 57.5 Hz and 99.78 Hz. A significant reduction in acceleration measured by each accelerometer can be seen in Figure 8 and Table 9.
The following steps are taken to estimate the plate energy: i Divide the plate into 8 divisions and place an accelerometer at the center of each division. ii Calculate each plate division's kinetic energy separately by considering the mass of each division with the sensor and velocity sensed by each accelerometer. The total energy is the summation of energy for each division. Figure. 8 shows the comparison of energy generated between uncontrolled and controlled system while the corresponding reduction levels measured at each mode is given in Figures 9(a) and 9(b).   The frequency-averaged energy reduction achieved is 19.52%, which is slightly lower compared to the simulation result. The frequency response of a continuous structure vibration is generally grouped into low, mid and high frequencies, based on the modal overlapping factor (MOF). Low-frequency response is deterministic, which exhibits distinct individual modes and is not sensitive to structural uncertainties (i.e., a small variation in mass or stiffness matrices) [34], [35].
In this current article, the work is performed within the low-frequency regions. The total mass of the accelerometers and portion of cables on the plate are considered small, and their effect on the low-frequency modes is insignificant.

VI. CONCLUSION
In this work, the ACO technique was utilized to optimise collocated sensor-actuator location on the flexible structure and obtain the optimal PID controller gains. The optimal values obtained from ACO are found comparable to the results from GA and the enumerative method. Based on the results, the PID controller gains and collocated sensor-actuator location optimizations using ACO, GA and enumerative methods give similar results. This implies the effectiveness of ACO for such optimization problems. The best location to place a collocated sensor-actuator to achieve the highest reduction of plate energy (or reduction of displacement for the whole plate) is near the exciter patch with more than 10% frequency-averaged energy reduction. This is because the highest energy reduction can be achieved by directly locking the disturbance force. However, in reality, the information about the disturbance such as magnitude and location is inaccessible. Since the location that causes vibration, and its corresponding magnitude can be difficult to identify. Hence, optimal location is investigated by setting at least 0.1 m gabs, along x and y axes, between excitation and sensor-actuator locations. As for the PID gains, with the PID controller gains been capped due to the output module's limitation (NI-9263) to produce voltage with the range of -10V to 10V only, about 20% of frequency-averaged energy reduction is achieved. A much better reduction would be accomplished if higher gains, or a better amplifier were used.