Multi-mode optimal fuzzy active vibration control of composite beams laminated with photostrictive actuators

Mncng JIA , Shijie ZHENG ,*, Rongbo HE ,b

a Institute of Smart Materials and Structures, State Key Laboratory of Mechanics and Control of Mechanical Structures,Nanjing University of Aeronautics and Astronautics, Nanjing 210016, China

b School of Electrical Information, Anhui University of Technology, Maanshan 243002, China

KEYWORDS Active control;Finite element method;Multiple modes;Optimal fuzzy control algorithm;Photostrictive actuators

Abstract Firstly, a multi-f ield coupling f inite element formulation of composited beam laminated with the photostrictive actuators is developed in this paper.Moreover,an optimal fuzzy active control algorithm is also proposed on the basis of the combination of optimal control and fuzzy one.This method opens a new avenue to resolve the contradiction between the linear system control method and nonlinear actuating characteristics of photostrictive actuators. The desired control for suppressing multi-modal vibration of photoelectric laminated beam is f irstly obtained through optimal control and then the fuzzy control is used to approach the desired mechanical strain induced by photostrictive actuators. Thus, the multi-mode vibration control of beam is realized.In the design process of optimal fuzzy controller,the design of fuzzy control is independent of optimal control. The simulation results demonstrate that the proposed control method can effectively realize multi-modal vibration control of photoelectric laminated beams, and the control effect of optimal fuzzy control is better than that of optimal state feedback control.

1. Introduction

The newly developed intelligent material, lanthanum-modif ied lead zirconate titanate (PLZT), which has good photostrictive effect,can directly transform light energy to mechanical deformation. Thus it can be used for the optical control system without electromagnetic noise. Since the advent of PLZT, a great deal of theoretical and experimental investigations1-20have focused on knowing photostrictive effect and its application in wireless control.As a result of ultraviolet light illumination to the PLZT actuator, high voltage can be generated between two electrode surfaces of PLZT actuator in the direction of polarization. Then the strain will be generated in the direction of polarization because of the converse piezoelectric effect. This is photostrictive effect. Hence the PLZT actuator can be used to achieve active control of vibration. The photostrictive actuator does not require additional high voltage and strong magnetic power equipment, so the actuator can avoid the impact of electromagnetic disturbances. It also has the advantages of light weight and small size. So it can be used for structural remote control, especially for active vibration control and signal transmission in vacuum environment. Tzou and Chou1did the pioneering work of using the photostrictive actuator for active control of smart structure,and proposed a two-dimensional distributed photostrictive actuator model with multiple degrees of freedom. Liu and Tzou2presented a kind of variable light intensity control method to realize the independent modal vibration suppression of a simply supported plate structure by using the PLZT actuator.For this algorithm,the light intensity is proportional to the speed,and the direction of the light is changed according to the direction of the current modal velocity. Shih et al.3derived the modal control equation of an open cylindrical shell,and investigated the relationship between the PLZT actuator position and the modal control factor. Shih et al.4proposed a theoretical model for active vibration control and discussed the inf luence of PLZT actuator layout on the controllability of a simply supported plate.Shih and his collaborators5,6also respectively established the modal control equation for hemispherical shell and parabolic shell laminated with PLZT actuator, and analyzed the effect of shell curvature on the membrane/bending control actions.Wang et al.9discussed the main differences and similarities between the velocity feedback control and the Lyapunov control as well as the uniform irradiation and the alternative irradiation.Jiang et al.10used a hybrid PLZT/PVDF actuation mechanism to realize vibration suppression of a simply supported cylindrical shell. Sun and Tong11presented a research on the wireless vibration control of thin beam with PLZT actuator.It is worth noting that control methods in the above mentioned investigations can be classif ied into Lyapunov control, namely constant light intensity and velocity proportional feedback control, namely variable light intensity. In order to overcome the limitations of these two kinds of control methods, He and Zheng7,8introduced the fuzzy control method to the wireless active vibration control of plate and shell structures and realized independent modal variable structure fuzzy active vibration suppression.Their investigations show that the vibration suppression characteristics of fuzzy logic are signif icantly better than those of velocity feedback control and Lyapunov control algorithm.

Note that most of the aforementioned literatures are mainly concerned with single-mode vibration control. So far, only a few works were reported for PLZT-based vibration control with multiple modes. Zheng et al.12suppressed the f irst two transverse bending modes of an isotropic beam under simply supported, simply supported-clamped and clamped-clamped boundary conditions by applying the genetic algorithm to optimize the placement of PLZT actuators,but this method is not straightforward to be extended to control much more modes.Recently, He et al.13,14proposed fuzzy neural network and self-organizing fuzzy sliding mode control methods to achieve multi-mode vibration suppression of isotropic cylindrical shells. A literature survey1-14demonstrated that the above mentioned studies on the active vibration control using PLZT actuators are restricted to isotropic material and simple boundary conditions for which the theoretical solution exists.Zheng21made the f irst attempt to establish a f inite element formulation to simulate single modal wireless vibration suppression, in which the universal assumption of regular structure,simple boundary condition, mechanically isotropic as well as ignoring the effective stiffness and inertial mass of actuators are abandoned, but this investigation is limited to singlemode vibration control. As seen, there are very few investigations studying wireless vibration active control for orthotropic composite laminated structures,let alone f inite element simulation. In this study, in order to f ill this gap, f inite element simulations of multi-mode vibration active control for composite laminated structures are studied.

On the other hand, modern control technologies represented by fuzzy logic7,8,22,23, fuzzy neural network vibration control,13and self-organizing fuzzy sliding mode14have been applied to active vibration control of photoelectric laminated structure. Despite many efforts to advance the optimal Linear Quadratic Regulator (LQR) algorithm, which is well-known to be insensitive to systemic errors and widely applied in the piezoelectric based vibration active control,24-26very little work has been done in developing LQR scheme for PLZT based wireless active vibration control. The main reason for this phenomenon is that LQR is an optimal control method focusing on the linear system and it is not straightforward to be applied for a nonlinear one. Different from the piezoelectric actuator, in which the induced strain is linearly proportional to the applied electrical voltage,27,28the responses of PLZT actuators are time-varying and vary nonlinearly with the incident light intensity. This unique property is not only an opportunity but also a challenge for wireless vibration active control. Wang et al.29approximately regarded relationship between the actuating strain and the incident light intensity as a linear process and applied LQR and PLZT actuators to realize active vibration control of open spherical shell. Chen et al.30regarded the mechanical strain S-(t) of PLZT actuator as a function in the form of the incident light intensities at two neighboring time steps and expressed the nonlinear system equation as a linear state-space representation. However, the above cited linear methodologies30deserve elaboration, and indeed, these methods cannot get a standard linear state-space representation in a real sense. It is worth noting that the modal control force is linearly proportional to the photo-induced strain; hence a standard linear statespace representation written in the photo-induced strain rather than light intensity is established in this paper. This approach can resolve the contradiction between the linear system control method and nonlinear actuating characteristics of PLZT actuators, but the determination of the light intensity to induce the desired photo-induced strain will come to us. Thus, an optimal fuzzy active control algorithm is proposed in this paper on the basis of the combination of optimal control and fuzzy logic. In view of the nonlinear driving characteristics of photostrictive actuators, the desired control for suppressing multi-mode vibration of photoelectric laminated beam is f irstly obtained through optimal control.And then the fuzzy control is used to approach the desired mechanical strain induced by photostrictive actuators. Thus,the multi-mode vibration control of beam is realized. In the design process of optimal fuzzy controller, the design of fuzzy control is independent of optimal control.

The outline of this paper is organized as follows. A novel one-dimensional two-node f inite element formulation of composite beam laminated with the photostrictive actuators is developed in Section 2 and subsequently an optimal fuzzy control algorithm is proposed for multi-mode vibration control in Section 3. Sections 4 and 5 present the verif ication of the proposed f inite element formulation and numerical simulation for vibration control, respectively. Finally, some conclusions are drawn in Section 6.

2. Finite element formula and modal equations

As shown in Fig.1,a pair of PLZT actuators are pasted on the upper and lower surfaces of a composite beam. The polarity direction of PLZT actuators is in the x-direction. Under the illumination of ultraviolet light,the actuator generates a strain along the x-axis direction, resulting in bending moment control force. The location of the actuators are def ined by edge coordinate,x1and x2.And this bending moment plays a significant role in the active control of the transverse vibration of the beam.

The photo-induced strain generated by the PLZT actuator under the action of the multi-f ield coupling can be expressed as4

where E t( )is the total electric f ield produced by the light,T t( )is the temperature of the actuator structure at current moment,and the specif ic expressions of E t( ) and T t( ) are given in Appendix A; Yadenotes the elastic modulus of the actuator;d33and λ represent the piezoelectric-strain and thermal stress constant, respectively.

From the equations in Appendix A, it is discovered that both the temperature and electric f ield have time-varying nonlinear relation with light intensity.Then Eq.(1)shows that the strain has complex nonlinear relation with the illumination.

The relationship between the bending strain εb,shear strain, and displacement is

where w and θ denote the transverse deformation and the rotation of the beam element in the mid-plane, respectively.

This paper adopts the following Timoshenko's composite beam functions31,32to describe transverse def lection w and rotation θ at any point in the middle-plane of the beam,where ueis the interior displacement vector of the element,q is the vector of nodal displacement, and

Fig. 1 Model of photoelectric laminated beam.

and L1=1-x/L, L2=x/L, μe=1/(1 +12λe) andx is the coordinate along the length direction, L is the length of element, Dbbis the element bending stiffness, Dssis the element transverse shear stiffness, and Dbband Dsscan be expressed as

where biis width of the i th layer, Eiis the elastic modulus of the i th layer, Giis the shear modulus of the i th layer, ziis the lower surface coordinate of the i th layer,κ is the shear correction factor, n is the total number of layer.

Substitute Eq.(3)into Eq.(2),and Eq.(2)can be rewritten as

where Bbis the bending strain matrix,and Bsis the shear strain matrix.

Ignoring the system damping, the principle of virtual displacements for the dynamic case can be expressed as

Based on Eq. (6), motion equation could be derived as

If the external force is absent, Eq. (7) can be rewritten as

According to the modal analysis based on the f inite element program, the n-order natural frequency of the structure(ω1, ω2, ..., ωn) and corresponding normalized modes[φ1φ2··· φn] can be obtained. The transverse dynamic response q x,t( ) can be written by the modal superposition of the f irst r retained modes as

where ψ= [φ1φ2··· φr] denotes the truncated modal matrix with r retained modes, and η= [η1η2··· ηr]Tthe modal coordinate vector.

Using the modal expansion, i.e. substituting Eq. (9) into Eq. (8) and imposing the modal orthogonality result in the n-th transverse modal equation of the beam,

where fnS is the n-order modal control force,and fnis the control factor.

3. Optimal fuzzy controller design

In this thesis,the beam structure under various boundary conditions is taken as the controlled object, the f irst three modes are the controlled ones, and the active control simulation are carried out.

The f irst three modal control equations in Eq.(10)are written in the form of state-space.

Because of the anomalous photovoltaic effect, PLZT ceramics irradiated by ultraviolet light behaves with lightthermal-elastic-electric coupling effect, and the relationship between light intensity and strain is complex and not linear,which is shown in Appendix A. If the light intensity is chosen as the input of the system,it is diff icult to express the nonlinear system equation as a linear state-space representation.But the system will be a simple linear one with the photo-induced strain as the input, and then the LQR control method can be used to design the optimal control law to realize the multimode vibration suppression of the system.The control method adopted in this paper is the optimal fuzzy control algorithm,which combines the merits of both LQR control and fuzzy control.Its general procedure is shown in Fig.2.LQR optimal controller can get the optimal state feedback gain matrix K,and then the desired photo-induced strain of the current time can be obtained based on u=-K η-.The fuzzy control module adjusts the light intensity with the optimal control law u and its changing rate ˙u as input, then obtains the appropriate light intensity I(t) to make the real actuator strain S(t) of the actuator to approximate the optimal control law|u|.The direction of the light depends on the sign of u (when the sign of u changes, reverse the ultraviolet light illumination in the opposite direction). After a certain number of iterations, the multimode vibration suppression of the beam structure can be realized.

Fig. 2 Optimal fuzzy active vibration control system.

3.1. LQR optimal controller

For the vibration control problem, there are several requirements for LQR controller design: the closed-loop system is stable, and the feedback controller K should minimize the following quadratic performance target J,

where Q is symmetric positive semi-def inite weighting matrices for state vectorand R is positive weight coeff icient for input variable u.

Minimizing the quadratic performance target J,the optimal control law can be obtained as follows:

P satisf ies the algebraic Riccati equation

The values of Q and R can be determined according to the actual response curve after repeated adjustments.In this paper,Q has the following form:

3.2. Fuzzy controller

The present paper chooses the optimal control law u and its changing rate ˙u as the input of the fuzzy controller, and the control output is the light intensity I. The illumination direction of light depends on the sign of u. The input u and ˙u could be transformed into their respective domain by multiplying the quantization factors Kuand K˙u. The input domain is divided into 7 fuzzy subsets {NB, NM, NS, ZO, PS, PM,PB}, which refer to negative big, negative medium, negative small,zero,positive small as well as positive medium and positive big. The inputs are def ined on the universe of discourse[-3, 3]. The intensity must be positive, def ining the output on the universe of discourse [0, 3]. The output is divided into 4 fuzzy subsets{ZO,PS,PM,PB}.The triangular membership functions shown in Fig.3 are used,μ is the membership of the variable,U, ˙U and O are the universe of discourse of the input and output.

According to the driving characteristics of the photostrictive actuator, the establishment of fuzzy control rules should follow the two rules: (A) when the input u is a larger positive value and the changing rate ˙u is positive,the output light intensity should be a larger positive value to approximate the optimal control law u as soon as possible;(B)when the input u is a larger positive value and the changing rate ˙u is negative, the output light intensity should be a smaller positive value or zero. Each input contains seven fuzzy subsets, so there are 49 fuzzy control rules showed in Table 1.

The inference algorithm of the fuzzy controller is Mamdani method. The fuzzy value can be obtained. Then the value should be converted to a f inal crisp output. The Center of Gravity(COG)method is used.The crisp value on the universe is

where ζ is a point in the universe Z of the resulting fuzzy set,and μζthe membership of the resulting conclusion set. The intensity of illumination I can be obtained as the product of O and the quantization factor KI.

Fig. 3 Membership function of inputs and output.

4. Verif ication of presented f inite element model

To assess the accuracy of the presented f inite element program,which is written by FORTRAN, the static analysis and modal analysis of a four-ply composite cantilever were carried out.

A lumped mass P=4.536 kg is applied at the free end of the cantilever beam. The lamination under consideration is[0°/90°/90°/0°], elastic modulus E1=E3=206.8 GPa,E2=83.74 GPa, Poisson's ratio υ12=υ13=υ23=0.12, shear elastic modulus G12=G13=G23=48.27 GPa, density ρ=7.8×103kg/m3, length l=254 mm, width b=25.4 mm, and thickness of each layer are h1=h4=2.54 mm, h2=h3=3.81 mm.

The computational results of the presented formulation are compared with those of ANSYS. From the comparison of Tables 2 and 3, this f inite element program is feasible and effective for the mechanical analysis of the laminated beam structure, so it can be used to analyze the mechanical properties of the photoelectric laminated beam.

5. Numerical simulation

Having validated the presented f inite element formulation,the multi-mode LQR vibration control of beams under cantilever,simply supported and clamped-clamped boundary conditions are simulated in this section. The geometry of the four-ply composite beam is 1 m×0.05 m×0.005 m, the thickness of each layer are the same, and the lamination under consideration is [0°/90°/90°/0°]. The material parameters of the beam are taken as E2=E3=6.9×109Pa, E1=25.0E2,G12=G23=0.5E2, G23=0.2E2, υ12=υ13=υ23=0.25, and the density is 7800 kg/m3.The PLZT actuator is isotropic with elastic modulus Ea=6.3×1010N/m2, and its dimensions are 0.10 m by 0.05 m by 0.0004 m. The density is taken as ρa(bǔ)=7.6×103kg/m3. Other parameters of PLZT actuator are shown in Table 4.In the following investigation,on behalf of demonstrating clearly the effects of the presented control strategy, it is assumed that the structural damping is absent.It is also assumed that the values of LQR weighting factors R and k are respectively f ixed to be 1×104and 10-2unless otherwise stated.In this paper,the Newmark-β method is used to estimate discrete time responses and the iteration step of the simulation is f ixed to be 0.0001 s.

Table 2 Comparison of modal analysis result.

Table 3 Comparison of static analysis result.

Table 1 Fuzzy control rules.

5.1. Multi-mode vibration control of a simply supported beam

The origin of the coordinate system (xz) is located at the far left end of the simply supported beam and the position of the actuator is x1=0.20 m,x2=0.30 m.A concentrated load of F=0.2sin(65t)+0.5sin(250t)+sin(565t)N is acted at the point A 0.20 m away from the left-most end of the simply supported beam for a duration of 1.0 s, and the controller isswitched on after another 1 s. The f irst three natural frequencies of the beam are obtained as 10.01, 40.07 and 89.70 Hz.

Table 4 Parameters of PLZT actuator.

Fig.4 depicts the modal amplitude of the f irst three modes.The time history of control light intensity is also shown in Fig. 5. Fig. 6 shows the frequency spectrum with different LQR parameter k. It is concluded that the higher the value of LQR parameter k is,the faster the reduction in the vibration amplitude is. Fig. 7 shows transverse displacement at point A of the beam versus time with and without control.

In order to show clearly the necessity of incorporating the fuzzy control to overcome the non-linearity of the actuator,the fuzzy part in the controller is removed.The optimal control law obtained by LQR is used to control the light intensity directly. The intensity of light I=KIuu=KIuKx｜ ｜, where KIuis the quantization factor from u to I, and the direction of the light depends on the sign of u too. The simulation results are shown in Fig.8.Compared with Fig.4,a worse amplitude control is achieved in the case of the optimal state feedback control scheme without fuzzy logic. It is also concluded from these two f igures that the fuzzy control strategy can effectively overcome the non-linearity of the PLZT actuator.

5.2. Multi-mode vibration control of a cantilever beam

A pair of PLZT actuator is pasted on the upper and lower surfaces of the f ixed end of the cantilever beam and its right side is 0.9 m away from the free end. A concentrated load of F=0.03sin(25t)+0.3sin(140t)+0.6sin(400t) N is acted at the free tip of the cantilever beam for a duration of 1.0 s,and the controller is switched on after another 1.0 s.This cantilevered composite beam is modeled with the presented element formulation. The f irst three natural frequencies of the beam are obtained as 3.68, 22.82 and 63.26 Hz.

Fig. 9 shows transverse displacement at the free tip versus time with and without control. And Fig. 10 depicts the amplitude frequency responses of the cantilever beam for the uncontrolled and controlled cases. These two f igures once again demonstrate the effectiveness of the presented controlling strategy.

Fig. 4 Displacement response curves of the f irst three modes of simply supported beam.

Fig. 5 Light intensity of upper and lower actuators.

Fig. 6 Frequency response of different LQR parameters.

5.3. Multi-mode vibration control of a clamped-clamped beam

Fig. 7 Displacement response curves of simply supported beam with and without control.

For this case, the PLZT actuator is pasted at the position x1=0.00 m, x2=0.10 m of the clamped-clamped beam. A concentrated load of F = 1.2sin(143t) + 0.6sin(395t) +0.7sin(770t) N is acted at the point A 0.20 m away from the far left end of the clamped-clamped beam for a duration of 2.0 s, and the controller is switched on after another 1.0 s.The f irst three natural frequencies of the clamped-clamped beam are obtained as 23.15, 63.08 and 122.40 Hz. Fig. 11 shows transverse displacement at point A of the beam versus time with and without control. And Fig. 12 depicts the amplitude frequency variations of the clamped-clamped beam for the uncontrolled and controlled cases. It can be observed that the presented control scheme is effective in obtaining good reduction in the vibration amplitude.

Fig. 8 Displacement response curves without fuzzy control.

Fig. 9 Displacement response curves of cantilever beam with and without control.

Fig. 10 Amplitude frequency response curves for uncontrolled case and controlled case.

Fig. 11 Displacement response curves of clamped-clamped beam with and without control.

Fig. 12 Amplitude frequency response curves with and without control.

6. Conclusions

A f inite element formulation of composite beam laminated with the photostrictive actuators is developed in this paper.Moreover, an optimal fuzzy active control algorithm is also proposed on the basis of the combination of optimal control and fuzzy one. Numerical simulations verify the correctness of the f inite element formulation and the effectiveness of the optimal fuzzy active control strategy. For the f inite element simulation, the hysteresis problem between the photoelectric f ield and the photo-induced strain of PLZT is not considered,and it can be studied in subsequent work. The present investigations suggest the following conclusions:

(1) This paper opens a new avenue to resolve the contradiction between the linear system control method and nonlinear actuating characteristics of PLZT actuators.Numerical simulations show, for the photoelectric laminated beam structure under various different boundary conditions, the optimal fuzzy active control algorithm could effectively realize multi-mode vibration control, and the optimal fuzzy control provides better controllability than optimal state feedback control.

(2) It is also observed that the higher the value of LQR parameter k is, the faster the reduction in the vibration amplitude is.

(3) The optimal fuzzy active control algorithm can also be extended for multi-mode active vibration suppression of plate and shell structures bonded with PLZT actuators. Corresponding work is in progress.

(4) The optimal fuzzy active control algorithm can also provide reference and help for optimal state feedback control of other nonlinear systems.

Acknowledgement

This research was supported by the National Natural Science Foundation of China, China (No. 11572151).

Appendix A. Constitutive relations of PLZT material

It is easily found that the illumination intensity and the strain of PLZT actuator have complex time-varying nonlinear relationship according to the constitutive relations. In this paper, fuzzy control is used to overcome the non-linearity of the actuator.

Under the illumination of ultraviolet light on the actuator,light-induced electric f ield EI(t)could be generated between the electrode surfaces in the direction of polarization due to the photovoltaic effect,

where Esdenotes the saturated photovoltaic f ield,asand α represent the aspect ratio and the optical actuator constant,respectively, I(tj) is the illumination intensity at the time instant tj, and β denotes the voltage leakage constant.

The body temperature T(t) also rises with the light,

where H and P represent the heat capacity and the power of absorbed heat, respectively, γ is the heat transfer rate.

Because of the pyroelectric effect,electric f ield ETproduced by temperature can be estimated by

where Pxand ε represent the pyroelectric constant and the permittivity, respectively. The total electric f ield E is