Optimized Active Control of a Smart Cantilever Beam Using Genetic Algorithm

Abstract

Vibration is one of the most dangerous phenomena that happens to a structure. It leads to premature fatigue and eventually failure, with potentially fatal consequences. A smart structure is an excellent solution to this problem; it adds an actuator, a sensor, and an appropriate control law to the system to reduce/eliminate the vibration. This study developed a complete analytical model for a cantilever beam with a collocated PZT sensor/actuator pair. First, we used a coupling of a collocated PZT sensor and an actuator to measure and control vibration levels based on a PID control law considering the physical constraints associated with PZT operation as the voltage level of the actuator. Next, the damping coefficient of the structure was determined by using genetic algorithms best fit to satisfy specific vibration conditions. Finally, we conducted a complete optimization for sensor/actuator position and PID parameters, using genetic algorithms. Thus, this paper gives a thorough understanding of the potential vibration control of the cantilever beam.

1. Introduction

Even though vibration can be helpful in some applications, and we can harvest energy from it, as shown by Jamadar et al. in Reference [1], vibration is, in most cases, harmful to structures. It can lead to high cycle fatigue failure; internal rubs, especially in rotating machines; decreased productivity; and fastener loss, which is a significant concern in high-performance machinery [2]. Crawley et al. [3] and Bailey et al. [4] introduced vibration control of structures by using a piezoelectric material in the mid-1980s. Subsequently, a lot of research addressed this subject. The concept of vibration control using Piezoelectric material transforms the structure into a smart one which usually consists of four major components: an actuator, a sensor, a controller, and the structure itself [5]. Piezoelectric material is widely used for its fast response time, low weight, relatively low cost, low power consumption, easy fabrication, design flexibility, and generation of no magnetic field while converting electrical energy to mechanical one [5,6,7]. Moreover, these materials can be used as actuators/sensors in several forms: surface bonded patches, embedded patches, distributed layer, cylindrical stacks, screen printed layer, etc. [6]. A crucial criterion to consider for vibration control is the choice of the control law and the placement of the actuators/sensors. In the literature, a lot of research has been conducted around the first criterion, and several control laws were studied, such as direct proportional feedback control, constant negative velocity feedback, Linear Quadratic Gaussian (LQG), Linear Quadratic Regulator (LQR), Proportional Integral Derivative (PID), sliding mode, and H₂, as detailed in Reference [8]. Kumar et al. [9] studied the direct proportional position feedback control law, the constant gain negative velocity feedback control, and the LQR optimal control. He shows that the constant gain negative velocity feedback control is more effective than the direct proportional position feedback control. Chandrashekhara et al. [10] reached similar conclusions.

Moreover, Kumar has shown that the LQR control offers an effective control by using lower actuator voltages than the negative velocity feedback. Qui et al. [11] showed that a combined position feedback control (PPF) with a proportional derivative (PD) control can be effective in vibration suppression, especially for small-vibration amplitude. Yang et al. [7] showed the effectiveness of the velocity feedback control in vibration suppression of a composite laminated plate with different fiber orientations, using an embedded PZT actuator/sensor.

In addition, He et al. [12] proved the effectiveness of the constant velocity feedback control law in attenuating the vibration amplitude of functionally graded material (FGM) plates. Waghulde et al. [5] showed that the LQG control law can reduce 30% of a smart cantilever beam’s first mode vibration response. LQR is widely used [13,14,15,16] because of its capability to stabilize the closed-loop system and allow the user to define weights on inputs and states [13]. Zhao et al. [14] tested a hybrid control consisting of a passive shunt circuit with an LQR controller and has shown better vibration attenuation. Lee et al. [17] applied the LQR controller to a system of laminated composite stiffener with PZT layers at the top and bottom. They searched the optimal position of the stiffener. Abdelrahman et al. [16] studied the effect of the size and material of PZT on vibration control. Using an LQR controller gave considerable vibration reduction in settling time and the actuation force. PID controller has also been widely used, and its effectiveness in vibration reduction has been proven [11,18,19]. Khot et al. [20] used the PID controller based on output feedback to compare the full and reduced model of a beam, and a good agreement was shown.

On the other hand, Qui et al. [21] proposed a sliding mode controller based on acceleration sensor control strategy. They have shown that this controller, along with proportional feedback control, can effectively reduce the first and second bending modes of vibration. Furthermore, Rodriguez et al. [22] confirmed the effectiveness and robustness of sliding control law to uncertainties and noise. Finally, Caruso et al. [23] proposed using multiple pairs of actuators/sensors along with an appropriate H₂ control strategy designed on a model with three vibration modes. The controller has shown high performance in vibration reduction of the clamped plate taken into consideration.

The other criterion mentioned above, the placement of the actuator/sensor, has been widely studied in the literature. Actuators should be placed in appropriate locations to excite the desired modes to control effectively [3], as performed in a study by Botta et al. [24] wherein the positions of the actuators were optimized to control each of the first five modes of vibration and combined modes of vibration. A bad positioning for the actuator and sensors leads to non-effective vibration control, and in some cases, more excitation of the spillover effect. The spillover effect comes from the fact that only the first few low-frequency modes are considered when modeling a system for vibration, and the high-frequency modes (residual modes) are treated as uncertainties. In this case, the state feedback based on the reduced model may destabilize the residual modes, as many controlled structures are lightly damped [25]. Gupta et al. [6] presented five criteria for the optimal placement of piezoelectric sensors and actuators: maximizing modal forces/moments applied to the structure by the PZT actuators, maximizing the deflection of the structure, minimizing the control energy utilized by the actuators, maximizing the degree of controllability, maximizing the degree of observability, and minimizing the effects of spillover.

Several papers have studied the optimal placement of the actuator/sensor. For example, Qui et al. [11] used an H₂ norm to maximize the plate’s controllability and observability. As a result, the actuator/sensor pair’s best placement for suppressing the bending modal vibration turns out to be the root of the plate; instead, for suppressing torsional vibration, the pair should be placed at the tip of the plate. The genetic algorithm was also used to study the optimal placement in References [15,26,27]. The main objective was to maximize controllability and observability and reduce the spillover effect.

In the subsequent sections, we use a couple of collocated PZT sensor and actuator to measure and control vibration levels of the damped beam based on a PID control law. In this model, we consider the physical constraints associated with PZT operation as the voltage level of the actuator. Next, the damping coefficient of the structure is determined by using genetic algorithms best fit to satisfy specific vibration conditions. Finally, we conducted a complete optimization for sensor/actuator position and PID parameters, using genetic algorithms.

The novelty of this paper resides in a thorough analysis of the beam’s vibration control and the insertion of the actuator’s physical constraints in the derivation of motion’s analytical equation, which is very important to achieve a more realistic system representation. In addition, it can improve the vibration control of the structure, since it allows us to take better advantage of the actuator by applying the best voltage without damaging it. Moreover, the use of the genetic algorithm helps optimize the vibration control by determining the actuator’s best position and the PID controller parameters.

2. Analytical Model for Damped Cantilevered Beam Controlled with PZT Actuator

Obtaining a dynamic system model is very important during structural vibration control systems analysis. There are three standard techniques for procuring models of flexible structures [28]: The first is the finite element analysis, which is an approximate method giving a high-order spatially discrete system. This method was widely used in the literature to obtain a model for a cantilevered structure with a PZT actuator/sensor. The second technique is system identification, which uses experimental data to identify composite structural and piezoelectric model parameters. The third one is the analytical model, which uses the modal approach to obtain and solve the partial differential equation of the system.

Figure 1 shows a schematic of the controlled damped cantilever beam used to build the analytical model. As shown in Figure 1, we have a beam with a pair of PZT patches fixed in position x₁. One acts as an actuator at the beam’s top, and the second, fixed at the bottom, acts as a sensor. Moreover, an initial load, P, is applied to the tip of the beam.

In this study, we did not consider the beam’s mass-induced deformation. The partial differential equation describing the system of a cantilever beam with the pair of sensor/actuator is as follows [28]:

$$E_b{I}_b\frac{{\partial }^4\omega \left(x,t\right)}{\partial x^4}+{\rho }_b{A}_b\frac{{\partial }^2\omega \left(x,t\right)}{\partial t^2}+c_b\frac{\partial \omega \left(x,t\right)}{\partial t}=M_a\left({\frac{\partial \delta }{\partial x}}_{\left(x-x_2\right)}-{\frac{\partial \delta }{\partial x}}_{\left(x-x_1\right)}\right)$$

(1)

where $M_a$ is the moment applied by the PZT actuator, $\delta$ is the Dirac function, $c_b$ is the damping coefficient of the beam, and $x_1$ and $x_2$ are the coordinates of the PZT actuator.

Appendix A explains, in detail, the different terms of the partial differential equation of the system (1).

The moment applied by the PZT actuator can be defined as follows [13]:

$$M_a=-E_a·b_a·d_{31}·\left(\frac{h_b+h_a}{2}\right)·\frac{I_b}{I_{eq}}{V}_a=K^{*}·\frac{I_b}{I_{eq}}·V_a=K_a·V_a$$

(2)

where $E_a, b_a, d_{31}, h_b, h_a, V_a, I_b, \mathrm{and} I_{eq}$ represent, respectively, Young’s modulus of the PZT actuator, the width of the actuator, the piezoelectric strain constant, beam height, PZT actuator height, applied actuator voltage, the moment of inertia of the beam, and the equivalent moment of inertia of the beam with the PZT pair of actuator/sensor.

In this study, since the pair of actuator/sensor has the exact dimensions; thus, $h_a=h_s$ and $b_a=b_s$, where $h_s$ and $b_s$ are the height and the width of the sensor.

$I_{eq}$ can be defined as follows [13]:

$$I_{eq}=\frac{b_b{h}_b{}^3}{12}+\frac{b_{a eq}{h}_a^3}{12}+b_a{h}_a{\left(\frac{h_b}{2}+\frac{h_a}{2}\right)}^2+\frac{b_{s eq}{h}_s^3}{12}+b_s{h}_s{\left(\frac{h_b}{2}+\frac{h_s}{2}\right)}^2$$

$$I_{eq}=\frac{b_b{h}_b{}^3}{12}+2\frac{b_{a eq}{h}_a^3}{12}+2b_a{h}_a{\left(\frac{h_b}{2}+\frac{h_a}{2}\right)}^2$$

and $b_{a eq}=b_{s eq}=b_a·\frac{E_a}{E_b}$ is the equivalent PZT width.

As we can see in Equation (1), the choice of the control law will affect all the subsequent model calculations. This study used a Proportional Integral Derivative (PID) controller based on the sensor voltage. The system’s movement dictates the sensor voltage signal. This controller aims to reduce the system vibration, reaching stable sensor voltage. In this case, the voltage fed to the actuator becomes the following:

$$V_a=K_P·(V_s-V_{desired})+K_D·\left({\dot{V}}_s-{\dot{V}}_{desired}\right)+K_I·\int (V_s-V_{desired})dt$$

(3)

where $K_P, K_D, \mathrm{and} K_I$ are the parameters of the controller, and $V_{desired}$ represents the desired sensor voltage. In our case, the desired sensor voltage is zero, as we try to reduce and eventually stop the system’s vibration. $V_s$ is the measured sensor voltage:

$$V_s=-E_s·b_s·\frac{g_{31}}{2C}\left(\frac{h_b}{2}\right)·{\frac{\partial \omega \left(x,t\right)}{\partial x}|}_{x_1}^{x_2}=K_s·{\frac{\partial \omega \left(x,t\right)}{\partial x}|}_{x_1}^{x_2}$$

where $E_s, b_s, g_{31}, \mathrm{and} C$ represent, respectively, Young’s modulus of the PZT sensor, the width of the sensor, the piezoelectric voltage coefficient, and the capacitance of the PZT sensor.

Thus, we have the following:

$$V_a=K_P·V_s+K_D·{\dot{V}}_s+K_I·\int V_{s}dt$$

(4)

Figure 2 shows the schematic of the PID controller. The PID controller uses the sensor voltage to calculate the appropriate voltage applied to the actuator.

By substituting all the variables mentioned above in Equation (1), we obtain the general equation:

$$\begin{array}{cc}\hfill E_b{I}_b\frac{{\partial }^4\omega \left(x,t\right)}{\partial x^4}& +{\rho }_b{A}_b\frac{{\partial }^2\omega \left(x,t\right)}{\partial t^2}+c_b\frac{\partial \omega \left(x,t\right)}{\partial t}\hfill \\ & =K_a·K_s·(K_P·{\frac{\partial \omega \left(x,t\right)}{\partial x}|}_{x_1}^{x_2}+K_I·\int {\frac{\partial \omega \left(x,t\right)}{\partial x}|}_{x_1}^{x_2}dt\hfill \\ & +K_D·\frac{\partial }{\partial t}{\frac{\partial \omega \left(x,t\right)}{\partial x}|}_{x_1}^{x_2})·({\frac{\partial \delta }{\partial x}}_{\left(x-x_2\right)}-{\frac{\partial \delta }{\partial x}}_{\left(x-x_1\right)})\hfill \end{array}$$

(5)

The displacement $\omega \left(x,t\right)$ can be separated in space and time by using the modal analysis approach [29]:

$$\omega \left(x,t\right)=\sum _{i=1}^{\infty }{\varphi }_i\left(x\right)f_i\left(t\right)$$

(6)

Here, ${\varphi }_i\left(x\right)$ represents the eigenfunction and is known as the mode-shape. In our case, we assume that the actuator’s impacts on mode shapes are insignificant, which is true if the actuator’s dimensions are tiny compared to the beam’s [13]. Thus, the mode-shape equation for a cantilevered beam is as follows [30,31]:

$${\varphi }_i\left(x\right)=\frac{1}{2}\left[\cos \left({\gamma }_{i}L\right)-\cos h\left({\gamma }_{i}L\right)+\left(\frac{-\cos \left({\gamma }_{i}L\right)-\cos h\left({\gamma }_{i}L\right)}{\sin \left({\gamma }_{i}L\right)+\sin h\left({\gamma }_{i}L\right)}\right)\left(\sin \left({\gamma }_{i}L\right)-\sin h\left({\gamma }_{i}L\right)\right)\right]$$

(7)

where ${\gamma }_i{}^4=\frac{{\omega }_i{}^2}{c^2}$.

Moreover, $c=\sqrt{\frac{E_b{I}_b}{{\rho }_b{A}_b}}$. Substituting Equation (6) in (5), multiplying by ${\varphi }_j\left(x\right)$, and integrating over all the length of the beam gives the following:

$$\begin{array}{l}{\displaystyle \sum _{i=1}^n}{\int }_0^L(E_b{I}_b·f_i\left(t\right){\varphi }_j\left(x\right)·\frac{{\partial }^4{\varphi }_i\left(x\right)}{\partial x^4}+{\rho }_b{A}_b·{\varphi }_j\left(x\right)·{\varphi }_i\left(x\right)\frac{{\partial }^2{f}_i\left(t\right)}{\partial t^2}\\ +c_b·{\varphi }_j\left(x\right)·{\varphi }_i\left(x\right)\frac{\partial f_i\left(t\right)}{\partial t})dx\\ ={\int }_0^L\{K_a·K_s{\displaystyle \sum _{i=1}^n}(K_P·f_i\left(t\right)·{\varphi }_j\left(x\right){\frac{\partial {\varphi }_i\left(x\right)}{\partial x}|}_{x_1}^{x_2}\\ =K_I·{\varphi }_j\left(x\right){\frac{\partial {\varphi }_i\left(x\right)}{\partial x}|}_{x_1}^{x_2}·\int f_i\left(t\right)dt\\ +K_D·\frac{\partial f_i\left(t\right)}{\partial t}·{\varphi }_j\left(x\right){\frac{\partial {\varphi }_i\left(x\right)}{\partial x}|}_{x_1}^{x_2})·({\frac{\partial \delta }{\partial x}}_{\left(x-x_2\right)}\\ -{\frac{\partial \delta }{\partial x}}_{\left(x-x_1\right)})\}dx\end{array}$$

(8)

Using the property of orthogonality of the mode-shapes results in the following [29]:

$${\int }_0^L{\varphi }_j\left(x\right){\varphi }_i\left(x\right)dx={\delta }_{ij}$$

where ${\delta }_{ij}=\{\begin{array}{c}1 if i=j\\ 0 otherwise\end{array}$

In addition, using the property of the Dirac Delta function, we achieve the following:

$${\int }_{-\infty }^{\infty }{\delta }^{\left(n\right)}(x-\theta )\varphi \left(x\right)dx={(-1)}^n{\varphi }^{\left(n\right)}(\theta )$$

The Equation (8) can be reduced to the following:

$$\begin{array}{l}{E}_b{I}_b·f_i(t){\int }_0^L{\varphi }_i\left(x\right)·\frac{{\partial }^4{\varphi }_i\left(x\right)}{\partial x^4}dx+{\rho }_b{A}_b·\frac{{\partial }^2{f}_i\left(t\right)}{\partial t^2}+c_b·\frac{\partial f_i\left(t\right)}{\partial t}\\ =K_a·K_s·{\left({\frac{\partial {\varphi }_i\left(x\right)}{\partial x}|}_{x_1}^{x_2}\right)}^2·[K_P·f_i\left(t\right)+K_I·\int f_i\left(t\right)dt\\ +K_D·\frac{\partial f_i\left(t\right)}{\partial t}]\end{array}$$

(9)

Regrouping terms give the following:

$${\ddot{f}}_i\left(t\right)+\frac{c_b}{{\rho }_b{A}_b}{\dot{f}}_i\left(t\right)+{\Omega }^2{f}_i\left(t\right)=K_{Ti}·\left(K_P·f_i\left(t\right)+K_D{\dot{f}}_i\left(t\right)+K_I·\int f_i\left(t\right)dt\right)$$

(10)

where ${\Omega }^2=\frac{E_b{I}_b}{{\rho }_b{A}_b}{\int }_0^L{\varphi }_i\left(x\right)·\frac{{\partial }^4{\varphi }_i\left(x\right)}{\partial x^4}dx$

$$K_{Ti}=K_a·K_s·{\left({\frac{\partial {\varphi }_i\left(x\right)}{\partial x}|}_{x_1}^{x_2}\right)}^2$$

To solve Equation (10), the integral term of the PID was integrated by iterations of solving the following equation:

$$\begin{array}{l}{\ddot{f}}_i\left(t\right)+\frac{c_b}{{\rho }_b{A}_b}{\dot{f}}_i\left(t\right)+{\Omega }^2{f}_i\left(t\right)\\ =K_{Ti}·(K_P·f_i(t)+K_D{\dot{f}}_i(t)\\ +K_I(f_i(t)+f_i(t-1)+\cdots f_i(0)))\end{array}$$

(11)

At time, t, all the precedent $f_i\left(t-1\right)$, $f_i\left(t-2\right),\dots f_i\left(0\right)$ are already calculated and can be inserted as coefficients, $a_i$.

Thus Equation (11) can be organized as follows:

$${\ddot{f}}_i\left(t\right)+\left(\frac{c_b}{{\rho }_b{A}_b}+K_i·K_D\right){\dot{f}}_i\left(t\right)+\left({\Omega }^2+K_{Ti}{K}_P+K_{Ti}·K_I\right)f_i\left(t\right)+a_i=0$$

(12)

Equation (12) is for the ith vibration mode. If we truncate to the first n vibration modes, the dynamics of the system can be represented by the state-space equation:

$$\dot{Z}\left(t\right)-D·Z\left(t\right)=A$$

(13)

where $Z\left(t\right)={\left[\begin{array}{cccccccc}{f}_1\left(t\right),& f_2\left(t\right),& \dots & f_n\left(t\right),& {\dot{f}}_1\left(t\right),& {\dot{f}}_2\left(t\right),& \dots & {\dot{f}}_n\left(t\right)\end{array}\right]}^T$ represents the state vector.

In this study, we chose $n$ equal three, considering that the first three modes have the greatest impact on the vibration response.

The other terms in Equation (13) are as follows:

$$D=\left[\begin{array}{cc}{0}_{nxn}& I_{nxn}\\ {\Omega }^2+K_{Ti}{K}_P+K_{Ti}·K_I& \frac{c_b}{{\rho }_b{A}_b}+K_i·K_D\end{array}\right]$$

$$A=\left[\begin{array}{c}\begin{array}{c}{0}_{nx1}\\ a_1\end{array}\\ a_2\\ \begin{array}{c}⋮\\ a_n\end{array}\end{array}\right]$$

At this level, we here introduce the voltage limit of the PZT actuator. A PZT actuator can hold $\left|V_a\right|\le V_{limit}$. Otherwise, the actuator will crack as the generated tensile stress in PZT exceeds the fracture tensile strength of the material [32].

The methodology used in our code to solve this system is to check at every iteration of time, t, the actuator voltage, and if this voltage is higher than the limit voltage, the voltage attributed to the actuator becomes $\pm V_{limit}$ based on the sign of $V_a$. Equation (5) then becomes as follows:

$$\begin{array}{cc}\hfill E_b{I}_b\frac{{\partial }^4\omega \left(x,t\right)}{\partial x^4}& +{\rho }_b{A}_b\frac{{\partial }^2\omega \left(x,t\right)}{\partial t^2}+c_b\frac{\partial \omega \left(x,t\right)}{\partial t}\hfill \\ & =K_a·\pm V_{limit}·\left({\frac{\partial \delta }{\partial x}}_{\left(x-x_2\right)}-{\frac{\partial \delta }{\partial x}}_{\left(x-x_1\right)}\right)\hfill \end{array}$$

(14)

After performing all of the above mentioned subsequent steps, the final equation to solve is as follows:

$${\ddot{f}}_i\left(t\right)+\frac{c_b}{{\rho }_b{A}_b}{\dot{f}}_i\left(t\right)+{\Omega }^2{f}_i\left(t\right)=K_a·\pm V_{limit}·{\frac{\partial {\varphi }_i\left(x\right)}{\partial x}|}_{x_1}^{x_2}$$

(15)

As we can see, the right-handed side of the equation is time-independent, and it can be replaced by a coefficient, $a_i{}^{*}$, and the state-space equation is as follows:

$$\dot{Z}\left(t\right)-D^{*}·Z\left(t\right)=A^{*}$$

(16)

where

$$D^{*}=\left[\begin{array}{cc}{0}_{nxn}& I_{nxn}\\ {\Omega }^2& \frac{c_b}{{\rho }_b{A}_b}\end{array}\right]$$

$$A^{*}=\left[\begin{array}{c}\begin{array}{c}{0}_{nx1}\\ a_1{}^{*}\end{array}\\ a_2{}^{*}\\ \begin{array}{c}⋮\\ a_n{}^{*}\end{array}\end{array}\right]$$

Figure 3 shows the methodology used in our MATLAB code to solve the equation of motion of the beam controlled by the PZT actuator, with a PID voltage controller and considering the voltage limit that the actuator can withstand. After calculating $Z\left(t\right)$ in Equations (13) and (16), as in Figure 3, the final solution of $\omega \left(x,t\right)$ can be calculated by using Equation (6).

3. Optimization Using Genetic Algorithm

The genetic algorithm (GA) is one of the earliest population-based stochastic algorithms proposed in history [33]. This research method that was developed by John Holland back in 1975 [34] is inspired by natural Darwinian evolution [35]. In GA, every individual is a candidate (chromosome) whose fitness is evaluated every iteration, using an objective function known as the fitness function. Best individuals are chosen randomly by using a selection mechanism, and these candidates are used to improve other solutions, using the crossover between individuals. The crossover consists of exchanging genetic material of two parents at a random position in the chromosomes to produce a new child exploiting the area between the parents, as shown in Figure 4. The parents’ chromosomes, blue and red, are exchanged at a specific position (single point or double point, as shown in the figure) to obtain children with alternating blue and red chromosomes. GA also randomly adds new genetic information to the so-called “mutation” search process. This process adds diversity to the population and helps avoid local optimal solutions [35]. Figure 5 shows an example of how the mutation method processes. The genetic information of the randomly chosen chromosomes for the children is changed from zero to one, as shown in the figure.

It must be noted that the evaluation process is based on the fitness function; $p_c$ is the parameter representing the probability of crossover, usually between 0.5 and 1; and $p_m$ is the probability of mutation, and it is usually very low, i.e., less than 0.05 [36].

We first used GA as a search technique to determine the structural damping coefficient of the beam with a prescribed vibration attenuation. After that, we used it to optimize the pair of PZT sensor/actuator placement. Finally, we used it to determine the optimal PID parameters.

3.1. Finding the Damping Coefficient of the Structure

The damping coefficient is very difficult to determine in any structure. In our case, we assume that the structural damping coefficient is such that the tip displacement is reduced by 60% after 10 s. Therefore, we need to find the appropriate damping coefficient that helps reach this attenuation. We used the genetic algorithm to determine the value that respects the desired conditions. Here, the objective was to determine the damping coefficient “$c_b$” that gives an attenuation of 60% of the tip displacement after $t_{finale}=$ 10 s of free vibration. The fitness function chosen minimizes the error between data “${\omega }_s\left(L,t\right)$” from the searched $c_b$, using the GA and the desired tip displacement “${\omega }_d\left(L,t\right)$” (free undamped tip displacement with 60% attenuation after 10 s). The fitness function is defined as follows:

$$J=\underset{0}{\overset{t_{finale}}{{\displaystyle \int }}}{\left[{\omega }_s\left(L,t\right)-{\omega }_d\left(L,t\right)\right]}^{2}dt$$

The parameter of the aluminum beam used and the PZT-5H actuator/sensor are summarized in

Table 1. Summary of characteristics of the beam and PZT actuator/sensor pair.

Parameter	Value
Length of the beam L (m)	1
Width of the beam $w_b$ (m)	0.3
Thickness of the beam $h_b$ (m)	0.004
Beam Young’s modulus $E_b$ (Pa)	69 × 109
Density of the beam ${\rho }_b$ (kg/m3)	2705
Damping coefficient c	0.5079
Initial load applied to beam’s tip P (N)	7.038
Length of PZT $L_p$ (m)	0.1
Width of PZT $w_p$ (m)	0.1
Thickness of PZT $h_p$ (m)	0.002
PZT Young’s modulus $E_p$ (Pa)	6.4 × 1010
Density of PZT ${\rho }_p$ (kg/m3)	7500
Piezoelectric constant $d_{31}$ (C/N)	−274 × 10−12
Piezoelectric voltage coefficient $g_{31}$ (V/N)	−0.15099
Free dielectric constant $K_3^T$	3400
PZT capacitance $C_p$ [37]	0.3008

.

The obtained “$c_b$” was 0.5079. Figure 6 shows a good agreement between the tip displacement of the beam, with “$c_b$” obtained by using the GA and the desired tip displacement.

3.2. Optimization of the Actuator Position

As mentioned above, the actuator position has a crucial role in controlling the structure. In our case, GA was used to obtain the actuator’s best position to attenuate the vibration of the beam’s tip. This study aimed to reduce the tip displacement as much as possible with the control of the first three vibration modes, as mentioned above. Thus, the fitness function (objective function) is used to minimize the tip displacement:

$$J=\underset{0}{\overset{t_{final}}{{\displaystyle \int }}}{\left[{\omega }_s\left(L,t\right)-{\omega }_d\left(L,t\right)\right]}^{2}dt$$

where $t_{final}$ = 10 s, ${\omega }_s\left(L,t\right)$ is the tip displacement based on parameters defined in the GA optimization process and ${\omega }_d\left(L,t\right)$ is the desired tip displacement (equal to zero).

Using the GA, the optimal $x_1$ found was zero near the fixed end of the beam. To test this result, the controller parameters ($K_P, K_D, \mathrm{and} K_I$) were fixed as follows:

$$\begin{gathered} K_P=-\mathrm{1,000,000} \\ {K}_I=0 \\ {K}_D=-\mathrm{1,000,000} \end{gathered}$$

and several positions of the actuator were tested. As shown in Figure 7, a good agreement with the GA results is obtained. The actuator becomes less and less efficient when moving away from the fixed end. The controller itself introduces an instability (vibration increase) for x₁ = 0.3.

3.3. Optimization of Controller Parameter Using GA

Considering the complexity to determine the optimal parameters of the PID controller explained above, the genetic algorithm was used to find the best values of $K_P, K_D, \mathrm{and} K_I$. The objective function is again to minimize the beam’s tip displacement:

$$J=\underset{0}{\overset{t_{final}}{{\displaystyle \int }}}{\left[{\omega }_s\left(L,t\right)-{\omega }_d\left(L,t\right)\right]}^{2}dt$$

where $t_{final}$ = 10 s, ${\omega }_s\left(L,t\right)$ is the tip displacement based on parameters defined in the GA optimization process and ${\omega }_d\left(L,t\right)$ is the desired tip displacement (equal to zero). The beam’s damping coefficient is ‘c = 0.5079’, as obtained above (Section 3.1), and the position of the actuator/sensor pair is given by the previous optimization (Section 3.2), ‘x₁ = 0’. $V_{limit}$ is 150 V.

The best PID parameters obtained by using the GA were as follows:

$$\begin{gathered} K_P=-\mathrm{500,227,077.54} \\ {K}_I=0.329 \\ {K}_D=-\mathrm{69,151,875.29} \end{gathered}$$

As shown in Figure 8, the PID controller with optimized parameters using the GA helps eliminate the structure’s vibration in less than 0.4 s. Without control, the beam takes over 20 s until vibration stops.

As shown in Figure 9, the voltage limit of 150 V helps reach the stability of the structure more rapidly, as it exploits the capacity of the actuator more efficiently.

To better understand the integration of voltage limit in the model, a test compares the model with voltage limit with a model without voltage limit. In fact, in the case of the model without a voltage limit, one can never know precisely the voltage of the actuator, and it can mount to unrealistic values. We took a case when the PID controller parameters give an actuator voltage near 150 V without exceeding it. Figure 10 shows that, in this case, the beam needs a lot more time to stabilize than the model with a voltage limit. Figure 11 shows the exploitation of the two models for the actuator voltage. As we can see, the model with a voltage limit gives a more stable response and fast vibration suppression.

4. Conclusions

This paper presented the analytical vibration models of a damped cantilevered beam controlled with a PID controller. The model considered the damping coefficient of the beam and the integration of the voltage limit for the PZT actuator. This model can be used to add multiple actuators. This paper presents the step-by-step method of resolution of this model. Furthermore, the genetic algorithm was used to find the damping coefficient, optimize the PZT actuator position, and determine the optimal parameters of the PID controller. Analytical results show a good efficiency of the PID controller in suppressing the vibration.

Moreover, integrating the actuator voltage limit leads to fast vibration elimination. This method is different from traditional control. It helps the controller take advantage of the capacity of the actuators without damaging its physical structure and eventually avoid its premature fatigue and failure. The results in this paper can be used in oncoming research to validate experimental or numerical methods subsequently applied for more complex geometries for which analytical models are not available.

The proposed system has multiple applications. It can control the wind turbine blade vibration by adding an appropriate number of PZT actuators. In aviation, the PZT system is particularly advantageous, as the lightweight of the vibration control system is crucial. It can be used in most applications having vibration problems. For example, we have implemented this system to control an approximate model of a wind turbine blade. The results are encouraging and will be submitted soon for publication.