# Influence of Hysteresis on the Vibration Control of a Smart Beam with a Piezoelectric Actuator by the Bouc–Wen Model

Authors: Ting Zhang
Journal: Shock and Vibration
 (2020)

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## Abstract

The hysteresis property in a smart structure has attracted much attention from researchers for several decades. Hysteresis not only affects the response precision of the smart structure but also threatens the stability of the system. This paper focuses on how the hysteresis property influences the control effect of vibration suppression for a smart beam. Furthermore, the Bouc–Wen model is adopted to describe the hysteresis property of a smart beam and the hysteresis parameters of the hysteresis model are identified with a genetic algorithm. Based on the identification results, the hysteresis model is validated to represent the hysteresis property of the smart beam. Based on the hysteresis model, model reference adaptive control is designed to explore the influence of hysteresis on the vibration control of the smart beam. With some simulations and experiments, it is found that the vibration control effect is influenced when the hysteresis item changes. The vibration control effect will be improved when the hysteresis coefficient in the Bouc–Wen model, as the expected objective model of the adaptive reference model, is within a proper numerical range where the control system is stable. Furthermore, when the time delay is considered in the closed-loop control system, the principle of the hysteresis influence is different. The results indicate that the hysteresis property affects not only the control effect but also the stability of the control system for a smart cantilever beam.


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## Body

## 1. Introduction

Hysteresis is a common nonlinearity of piezoelectric materials [1–3]. Hysteresis is a lagging phenomenon where a physical effect on the smart material lags behind its cause. When a smart material acts as an actuator or a sensor of space structure, the response of the smart structure lags behind the input excitation of the system [4, 5]. The nonlinearity not only affects the control precision of the smart structure with a control law but also threatens the stability of the control system [6]. Therefore, it is an urgent challenge to address how the hysteresis property exactly influences the control precision and the stability of the control system.Some researchers have not considered the hysteresis property in their studies when performing active control investigations of smart structures [7]. Furthermore, some linear control laws are commonly designed based on a linear model for the smart structure. For example, PID control was implemented to damp both the free and forced vibrations of a thin-walled structure and compared with LQR optimal control [8], a new intelligent methodology was introduced to mitigate the vibration response of flexible cantilever plates [9], and a minimax-linear quadratic Gaussian controller was designed based on an uncertain system model for the positioning of a piezoelectric tube scanner used in an atomic force microscope [10]. These linear control approaches are limited in terms of their applications to linear system models. Once the controlled system is requested to have high precision and a fast response, these linear control methods cannot meet the requirements. Therefore, modeling the hysteresis nonlinearities of smart structures is important for designing a controller to explore the influence of hysteresis on the control of the smart structures.At present, the hysteresis modeling approaches include Preisach model [11], Prandtl–Ishlinskii (PI) model [12–14], and Bouc–Wen model [15]. The Preisach model and Prandtl–Ishlinskii model are rate-independent hysteresis models and are also phenomenological hysteresis models [16]. Based on the two types of models, the main control laws were adopted with only their inverse hysteresis model as open-loop control to compensate for the hysteresis property of the smart structure [17, 18]. It is difficult to design closed-loop control strategies on account of the two models. However, the Bouc–Wen model is a rate-dependent model and a physics-based hysteresis model. The hysteresis parameters of the Bouc–Wen model are easily identified [19–21]. In addition, feedback or feedforward control may be easily realized with its inverse model and a certain control law. The control laws applied to suppress vibration mainly include PID control [22], adaptive control [6, 23], robust control [24], and intelligent control [25, 26]. These control laws have been used to compensate for the hysteresis property or track the expected trajectory. However, few studies have analyzed how to affect the control result by the hysteresis of a smart structure. Furthermore, compared with other control methods, the adaptive control has strongly robust, adaptive, and anti-interference properties. Because of these advantages, the model reference adaptive control [23, 27] is adopted to explore the influence of hysteresis on the control effect of vibration suppression of smart structures with the Bouc–Wen model in this paper.The rest of this paper is organized as follows. A dynamic model for a smart flexible beam, dynamic model validations, and stability analyses are presented in Section2. Model reference adaptive control with the Bouc–Wen model as an expected reference model is introduced in Section 3. Simulations and experiments on vibration suppression of the smart beam are discussed in Section 4. Finally, conclusions are presented in Section 5.


## 2. Dynamic Modeling of a Smart Cantilever Beam

### 2.1. A Model of a Smart Beam with the Bouc–Wen Equation

A flexible beam bonded with a piezoelectric actuator is shown in Figure1. The smart beam is regarded as an Euler–Bernoulli beam, and its bending vibration is assumed to be ωx,t. Thus, the dynamic partial differential equation of the flexible beam is given as(1)mx∂2ωx,t∂t2+cx∂ωx,t∂t+∂2∂x2EIx∂2ωx,t∂x2=Qx,t,where m is the linear mass, c is the damping coefficient, E is the elasticity modulus, I is the inertia moment, and Q is a line distributed force.Figure 1
Smart beam.The constitutive equation of the piezoelectric material is described as(2)s1=s11ES1+d31E3,D3=d31S1+ε33XE3,where s1 and S1 are the strain and stress, respectively, E3 is the electric field, and D3 is the electric displacement. Moreover, s11E, d31, and ε33X are the elastic compliance, piezoelectric strain, and dielectric constants, respectively.When there is no external stress and the electric field isE3=U/tp (U is the external applied voltage and tp is the thickness of the piezoelectric actuator), the strain of the piezoelectric actuator is derived as(3)s=d31tpUx,t.The stress is obtained as(4)S=Eps,where Ep is the elasticity modulus of the piezoelectric actuator.Therefore, the momentM of force is expressed as(5)Mx,t=∫ASydAp,where Ap is the cross-sectional area of the piezoelectric actuator.Substituting equations (3) and (4) into equation (5), the moment is obtained to be as follows:(6)Mx,t=∫tb/2tb/2+tpEpd31tpyhpdyUx,t=KpUx,t,where tb and hp are the thickness of the beam and the height of the piezoelectric actuator, respectively, and Kp=1/2hpd31Eptb+tp.Furthermore, the input voltageU is expressed as(7)Ux,t=VtHx,where Vt is a time function and Hx is the Heaviside function.The moment of force and line distributed force are related as follows:(8)Qx,t=d2Mdx2.Substituting equations (6) and (8) into equation (1) yields(9)mx∂2ωx,t∂t2+cx∂ωx,t∂t+∂2∂x2EIx∂2ωx,t∂x2=∂2∂x2KpVtHx−x1−Hx−x0.The bending vibrationωx,t is obtained by a superposition of vibration modes as(10)ωx,t=∑i=1∞ϕixqit,where ϕix is a mode and qit is the generalized coordinate:(11)ϕix=coshβix−cosβix−sinhβilb−sinβilbcoshβilb+cosβilbsinhβix−sinβix.where β1lb=1.875, β2lb=4.694, β3lb=7.855,…, and lb is the length of the beam.Substituting equation (10) into equation (9) yields(12)∑i=1∞mxϕixq¨it+∑i=1∞cxϕixq˙it+∑i=1∞d2dx2EIxd2ϕixdx2qit=∂2∂x2KpVtHx−x1−Hx−x0.Because of the mode shape orthogonality,(13)∫ϕixϕjxdx=0,i≠j.It is assumed thatEIxd4ϕix/dx4=mΩi2ϕix. The mode ϕjx is multiplied by both sides of equation (12) ; then, equation (12) is integrated with respect to x as follows:(14)∫0lbϕjxmxϕjxdxq¨jt+∫0lbϕjxcxϕjxdxq˙jt+∫0lbϕjxEIxβj4ϕjxdxqjt=KpVtϕj′x1−ϕj′x0.The natural frequency is obtained as(15)Ωj=βj2∫0lbEbIbϕj2xdx+∫x0x1EpIpϕj2xdx∫0lbmbϕj2xdx+∫x0x1mbϕj2xdx,where mband mp are the masses of the cantilever beam and the piezoelectric patch, respectively, Ib and Ip are the moments of inertia of the cantilever beam, and Ep is the elastic modulus of the cantilever beam.Then, assumingMj=∫0lbϕjxmϕjxdx and Cj=∫0lbϕjxcϕjxdx, equation (15) is expressed as(16)Mjq¨jt+Cjq˙jt+MjΩj2qjt=KpVtϕj′x1−ϕj′x0.Both sides of equation (17) are multiplied by 1/Mj, yielding(17)q¨jt+2ζjΩjq˙jt+Ωj2qjt=1MjKpVtϕj′x1−ϕj′x0,where ζj is the j-th order damping coefficient.Some corresponding physical parameters of the smart cantilever beam are given in Table1. However, the smart structure has a nonlinear property, namely, the hysteresis phenomenon. Therefore, when the first order of the vibration modes of the beam with a piezoelectric actuator is considered with the Bouc–Wen equation, a dynamic model with hysteresis of the smart beam is derived from equation (16) as(18)q¨1t+2ζ1Ωq˙1t+Ω12q1t+Azzt=1M1KpVtϕ1′x1−ϕ1′x0,z˙t=αq˙1t−βq˙1tztn−γq˙1tztn−1zt,where zt denotes the hysteresis item and the parameters Az, n, α, β, and γ are used to obtain the form of the hysteresis property.Table 1
Corresponding parameters.
Beam
Piezoelectric
Others
l
b
=
0.4770
m
l
p
=
0.0280
m
x
11
=
2.4
cm
h
b
=
0.0305
m
h
p
=
0.0140
m
x
12
=
5.2
cm
t
b
=
0.7
×
10
−
3
m
t
p
=
0.3
×
10
−
3
m
ζ
1
=
0.042
E
b
=
71
GPa
E
p
=
15.857
GPa
ρ
b
=
2.77
×
10
3
kg
/
m
3
ρ
p
=
5.44
×
10
3
kg
/
m
3
d
31
=
−
1.7
×
10
−
10
C
/
N


### 2.2. Model Identification by a Genetic Algorithm

For the above hysteresis parameters,Az, α, β, and γ, a genetic identification algorithm [28, 29] is adopted to identify the hysteresis phenomenon of the smart beam. The parameters Az, α, β, and γ are estimated by the input voltage and output response (bending deflection ωx0,t at x0=0.115m) of the smart beam in the experiment. These estimated parameters are substituted in the original model to identify the output response. Therefore, an objective function is first defined as(19)fs=∑k=1NXk−X¯2N−1,where fs is the standard deviation, Xk is the k-th error between the identified data and experimental response, k is the sampling point, X¯ is the average error, and N is the number of identified data points.Before the genetic algorithm is implemented, some initial conditions are set. The maximum number of generations is MAXGEN = 100, the number of individuals is NIND = 100, the crossover probability and mutation probability are 0.7 and 0.02, respectively, and the generation gap is GGAP = 0.8. The default ranges of the identified parameters in Table2 are very important for the convergence rate of the iterations for the genetic algorithm. Moreover, the experimental response is collected with a laser displacement sensor when a sinusoidal voltage V=150×sin2π×2.5t is applied to the piezoelectric actuator. Therefore, with the input voltage and output response data from the experiment, the identified parameters Az, α, β, and γ are obtained in the default ranges, as shown in Table 2.Table 2
The identified parameters.
Parameters
Default ranges
Identified values
A
z
[0, 0.5]
0.110
α
[0, 0.5]
0.4971
β
[0, 0.5]
0.2636
γ
[0.5, 1]
0.8282Figure2 shows the convergence of the genetic algorithm when the hysteresis parameters Az, α, β, and γ are identified. The minimum standard deviation converges to a stable state after a generation number of approximately 20. Moreover, when the identified parameters in the third column of Table 2 at the last generation number 100 are substituted into equation (18), the identified displacement is obtained and compared with the experimental displacement in Figure 3(a), and a part of Figure 3(a) from 26 s to 28 s is magnified in Figure 3(b). Then, to display the detailed hysteresis property, a period of experimental and identified data is excerpted from Figure 3(a) and is shown Figure 4. The red identified discrete data are consistent with the blue experimental solid curve. As a result, the dynamic model equation (18) is valid for describing the hysteresis property of the smart cantilever beam with the genetic identification algorithm.Figure 2
Convergence of the genetic algorithm.Figure 3
Comparison of the experimental displacement with the identified result. (a) Identified and experimental displacement responses. (b) Magnification of a part of Figure3(a) from 26 s to 28 s.
(a)
(b)Figure 4
Hysteresis loop of the experiment and identification.


### 2.3. Stability Analysis for the Bouc–Wen Model

For the constructed dynamic model, the stability of the identified system is analyzed. Based on the above identified parameters, equation (18) is first transferred as(20)u˙1u˙2u˙3=u3αq˙1t−βq˙1tztn−γq˙1tztn−1zt−2ζ1Ωq˙1t−Ω12q1t−Azzt+00Kpϕ1′x1−ϕ1′x0M1Vt,where u1=q1, u2=z, and u3=q˙1.Then, ifn=1, a further equation simplification is made and equation (20) is expressed as(21)u˙1u˙2u˙3=u3αu3−δ1β+δ2γu3u2−2ζ1Ωu3−Ω12u1−Azu2+00DVVt,where δ1=1 when u2>0, δ1=−1 and δ2=1 when u3>0, and δ2=−1. DV=Kpϕ1′x1−ϕ1′x0/M1.Next, equation (21) is given in vector form as(22)u˙=Pu+DVVt,where(23)u=u1u2u3T,Pu=u3αu3−δ1β+δ2γu3u2−2ζ1Ωu3−Ω12u1−Azu2,DV=00DVT.IfPus=0, us is the equilibrium point in the phase plane. By the nonlinear function Pu, us=us1us20T and us2=−Ω12/Azus1. Therefore, the stability analyses of equation (22) are explored near the equilibrium point us. In addition, equation (22) without external excitation voltage is written as(24)u˙=DPus∗u−us+Ou−us2,where(25)DPu=0010−δ1β+δ2γu3α−δ1β+δ2γus2−Ω12−Az−2ζ1Ωdu1du2du3,and O⋅ is the minimum value of approximately u−us2.Therefore, equation (23) is simplified as(26)u˙=Aλu−us,where(27)Aλ=0010−δ1β+δ2γu3α−δ1β+δ2γu2−Ω12−Az−2ζ1Ωu=us=00100α−δ1β+δ2γus2−Ω12−Az−2ζ1Ω.Furthermore, the eigenvaluesλ are calculated by the following equation:(28)λI−Aλ=0,where I is the unit matrix.By equation (28), the characteristic equation is derived as(29)λλ2+2ξΩ1λ+Azα−δ1β+δ2γus2+Ω12=0.If the system is stabilized, the real parts of its eigenvalues are negative. As a result, the third item plus the fourth item is greater than zero and is given as(30)0<Azα−δ1β+δ2γus2+Ω12.Finally, for the stabilized model of smart beam, the relation between the identified parametersAz, α, β, and γ is presented as(31)Azδ1β+δ2γus2<αAz+Ω12.Ifus2=0, then(32)−Ω12<αAz.According to the identified parameter values in Table2, assuming Az=0.110az, the smart beam system is stable at az>−4.5044 from equation (32).


## 2.1. A Model of a Smart Beam with the Bouc–Wen Equation

A flexible beam bonded with a piezoelectric actuator is shown in Figure1. The smart beam is regarded as an Euler–Bernoulli beam, and its bending vibration is assumed to be ωx,t. Thus, the dynamic partial differential equation of the flexible beam is given as(1)mx∂2ωx,t∂t2+cx∂ωx,t∂t+∂2∂x2EIx∂2ωx,t∂x2=Qx,t,where m is the linear mass, c is the damping coefficient, E is the elasticity modulus, I is the inertia moment, and Q is a line distributed force.Figure 1
Smart beam.The constitutive equation of the piezoelectric material is described as(2)s1=s11ES1+d31E3,D3=d31S1+ε33XE3,where s1 and S1 are the strain and stress, respectively, E3 is the electric field, and D3 is the electric displacement. Moreover, s11E, d31, and ε33X are the elastic compliance, piezoelectric strain, and dielectric constants, respectively.When there is no external stress and the electric field isE3=U/tp (U is the external applied voltage and tp is the thickness of the piezoelectric actuator), the strain of the piezoelectric actuator is derived as(3)s=d31tpUx,t.The stress is obtained as(4)S=Eps,where Ep is the elasticity modulus of the piezoelectric actuator.Therefore, the momentM of force is expressed as(5)Mx,t=∫ASydAp,where Ap is the cross-sectional area of the piezoelectric actuator.Substituting equations (3) and (4) into equation (5), the moment is obtained to be as follows:(6)Mx,t=∫tb/2tb/2+tpEpd31tpyhpdyUx,t=KpUx,t,where tb and hp are the thickness of the beam and the height of the piezoelectric actuator, respectively, and Kp=1/2hpd31Eptb+tp.Furthermore, the input voltageU is expressed as(7)Ux,t=VtHx,where Vt is a time function and Hx is the Heaviside function.The moment of force and line distributed force are related as follows:(8)Qx,t=d2Mdx2.Substituting equations (6) and (8) into equation (1) yields(9)mx∂2ωx,t∂t2+cx∂ωx,t∂t+∂2∂x2EIx∂2ωx,t∂x2=∂2∂x2KpVtHx−x1−Hx−x0.The bending vibrationωx,t is obtained by a superposition of vibration modes as(10)ωx,t=∑i=1∞ϕixqit,where ϕix is a mode and qit is the generalized coordinate:(11)ϕix=coshβix−cosβix−sinhβilb−sinβilbcoshβilb+cosβilbsinhβix−sinβix.where β1lb=1.875, β2lb=4.694, β3lb=7.855,…, and lb is the length of the beam.Substituting equation (10) into equation (9) yields(12)∑i=1∞mxϕixq¨it+∑i=1∞cxϕixq˙it+∑i=1∞d2dx2EIxd2ϕixdx2qit=∂2∂x2KpVtHx−x1−Hx−x0.Because of the mode shape orthogonality,(13)∫ϕixϕjxdx=0,i≠j.It is assumed thatEIxd4ϕix/dx4=mΩi2ϕix. The mode ϕjx is multiplied by both sides of equation (12) ; then, equation (12) is integrated with respect to x as follows:(14)∫0lbϕjxmxϕjxdxq¨jt+∫0lbϕjxcxϕjxdxq˙jt+∫0lbϕjxEIxβj4ϕjxdxqjt=KpVtϕj′x1−ϕj′x0.The natural frequency is obtained as(15)Ωj=βj2∫0lbEbIbϕj2xdx+∫x0x1EpIpϕj2xdx∫0lbmbϕj2xdx+∫x0x1mbϕj2xdx,where mband mp are the masses of the cantilever beam and the piezoelectric patch, respectively, Ib and Ip are the moments of inertia of the cantilever beam, and Ep is the elastic modulus of the cantilever beam.Then, assumingMj=∫0lbϕjxmϕjxdx and Cj=∫0lbϕjxcϕjxdx, equation (15) is expressed as(16)Mjq¨jt+Cjq˙jt+MjΩj2qjt=KpVtϕj′x1−ϕj′x0.Both sides of equation (17) are multiplied by 1/Mj, yielding(17)q¨jt+2ζjΩjq˙jt+Ωj2qjt=1MjKpVtϕj′x1−ϕj′x0,where ζj is the j-th order damping coefficient.Some corresponding physical parameters of the smart cantilever beam are given in Table1. However, the smart structure has a nonlinear property, namely, the hysteresis phenomenon. Therefore, when the first order of the vibration modes of the beam with a piezoelectric actuator is considered with the Bouc–Wen equation, a dynamic model with hysteresis of the smart beam is derived from equation (16) as(18)q¨1t+2ζ1Ωq˙1t+Ω12q1t+Azzt=1M1KpVtϕ1′x1−ϕ1′x0,z˙t=αq˙1t−βq˙1tztn−γq˙1tztn−1zt,where zt denotes the hysteresis item and the parameters Az, n, α, β, and γ are used to obtain the form of the hysteresis property.Table 1
Corresponding parameters.
Beam
Piezoelectric
Others
l
b
=
0.4770
m
l
p
=
0.0280
m
x
11
=
2.4
cm
h
b
=
0.0305
m
h
p
=
0.0140
m
x
12
=
5.2
cm
t
b
=
0.7
×
10
−
3
m
t
p
=
0.3
×
10
−
3
m
ζ
1
=
0.042
E
b
=
71
GPa
E
p
=
15.857
GPa
ρ
b
=
2.77
×
10
3
kg
/
m
3
ρ
p
=
5.44
×
10
3
kg
/
m
3
d
31
=
−
1.7
×
10
−
10
C
/
N


## 2.2. Model Identification by a Genetic Algorithm

For the above hysteresis parameters,Az, α, β, and γ, a genetic identification algorithm [28, 29] is adopted to identify the hysteresis phenomenon of the smart beam. The parameters Az, α, β, and γ are estimated by the input voltage and output response (bending deflection ωx0,t at x0=0.115m) of the smart beam in the experiment. These estimated parameters are substituted in the original model to identify the output response. Therefore, an objective function is first defined as(19)fs=∑k=1NXk−X¯2N−1,where fs is the standard deviation, Xk is the k-th error between the identified data and experimental response, k is the sampling point, X¯ is the average error, and N is the number of identified data points.Before the genetic algorithm is implemented, some initial conditions are set. The maximum number of generations is MAXGEN = 100, the number of individuals is NIND = 100, the crossover probability and mutation probability are 0.7 and 0.02, respectively, and the generation gap is GGAP = 0.8. The default ranges of the identified parameters in Table2 are very important for the convergence rate of the iterations for the genetic algorithm. Moreover, the experimental response is collected with a laser displacement sensor when a sinusoidal voltage V=150×sin2π×2.5t is applied to the piezoelectric actuator. Therefore, with the input voltage and output response data from the experiment, the identified parameters Az, α, β, and γ are obtained in the default ranges, as shown in Table 2.Table 2
The identified parameters.
Parameters
Default ranges
Identified values
A
z
[0, 0.5]
0.110
α
[0, 0.5]
0.4971
β
[0, 0.5]
0.2636
γ
[0.5, 1]
0.8282Figure2 shows the convergence of the genetic algorithm when the hysteresis parameters Az, α, β, and γ are identified. The minimum standard deviation converges to a stable state after a generation number of approximately 20. Moreover, when the identified parameters in the third column of Table 2 at the last generation number 100 are substituted into equation (18), the identified displacement is obtained and compared with the experimental displacement in Figure 3(a), and a part of Figure 3(a) from 26 s to 28 s is magnified in Figure 3(b). Then, to display the detailed hysteresis property, a period of experimental and identified data is excerpted from Figure 3(a) and is shown Figure 4. The red identified discrete data are consistent with the blue experimental solid curve. As a result, the dynamic model equation (18) is valid for describing the hysteresis property of the smart cantilever beam with the genetic identification algorithm.Figure 2
Convergence of the genetic algorithm.Figure 3
Comparison of the experimental displacement with the identified result. (a) Identified and experimental displacement responses. (b) Magnification of a part of Figure3(a) from 26 s to 28 s.
(a)
(b)Figure 4
Hysteresis loop of the experiment and identification.


## 2.3. Stability Analysis for the Bouc–Wen Model

For the constructed dynamic model, the stability of the identified system is analyzed. Based on the above identified parameters, equation (18) is first transferred as(20)u˙1u˙2u˙3=u3αq˙1t−βq˙1tztn−γq˙1tztn−1zt−2ζ1Ωq˙1t−Ω12q1t−Azzt+00Kpϕ1′x1−ϕ1′x0M1Vt,where u1=q1, u2=z, and u3=q˙1.Then, ifn=1, a further equation simplification is made and equation (20) is expressed as(21)u˙1u˙2u˙3=u3αu3−δ1β+δ2γu3u2−2ζ1Ωu3−Ω12u1−Azu2+00DVVt,where δ1=1 when u2>0, δ1=−1 and δ2=1 when u3>0, and δ2=−1. DV=Kpϕ1′x1−ϕ1′x0/M1.Next, equation (21) is given in vector form as(22)u˙=Pu+DVVt,where(23)u=u1u2u3T,Pu=u3αu3−δ1β+δ2γu3u2−2ζ1Ωu3−Ω12u1−Azu2,DV=00DVT.IfPus=0, us is the equilibrium point in the phase plane. By the nonlinear function Pu, us=us1us20T and us2=−Ω12/Azus1. Therefore, the stability analyses of equation (22) are explored near the equilibrium point us. In addition, equation (22) without external excitation voltage is written as(24)u˙=DPus∗u−us+Ou−us2,where(25)DPu=0010−δ1β+δ2γu3α−δ1β+δ2γus2−Ω12−Az−2ζ1Ωdu1du2du3,and O⋅ is the minimum value of approximately u−us2.Therefore, equation (23) is simplified as(26)u˙=Aλu−us,where(27)Aλ=0010−δ1β+δ2γu3α−δ1β+δ2γu2−Ω12−Az−2ζ1Ωu=us=00100α−δ1β+δ2γus2−Ω12−Az−2ζ1Ω.Furthermore, the eigenvaluesλ are calculated by the following equation:(28)λI−Aλ=0,where I is the unit matrix.By equation (28), the characteristic equation is derived as(29)λλ2+2ξΩ1λ+Azα−δ1β+δ2γus2+Ω12=0.If the system is stabilized, the real parts of its eigenvalues are negative. As a result, the third item plus the fourth item is greater than zero and is given as(30)0<Azα−δ1β+δ2γus2+Ω12.Finally, for the stabilized model of smart beam, the relation between the identified parametersAz, α, β, and γ is presented as(31)Azδ1β+δ2γus2<αAz+Ω12.Ifus2=0, then(32)−Ω12<αAz.According to the identified parameter values in Table2, assuming Az=0.110az, the smart beam system is stable at az>−4.5044 from equation (32).


## 3. Model Reference Adaptive Control with the Bouc–Wen Model

### 3.1. Adaptive Control Law

In this paper, the vibration suppression problem of a smart beam considering the hysteresis property is explored by using a model reference adaptive control (MRAC) [22, 25]. For adaptive control, a reference model is assumed as an expected objective model. According to the error between the output states of the smart beam and the expected objective hysteresis model, the adaptive control law is designed online with gains Kr and Ku, as shown in Figure 5.Figure 5
Model reference adaptive control.First, the expected objective hysteresis model is given as(33)u¨eo=−2ζeoΩ1u˙eo−Ω12ueo−azAzz+DVur,where ζeo is set as 0.707, az is a hysteresis coefficient, and azAz satisfies equation (32).(34)Vc=Krur+Kuu.Substituting equation (34) into equation (18), the identified model is derived as(35)u¨=−2ζ1Ω1u˙−Ω12u−Azz+DVKrur+DVKuu.Then, the error between the outputs of the expected objective model and the smart beam is expressed as(36)e=ueo−u.Furthermore, equation (36) is transformed into a state-space equation with equations (33) and (37), which is displayed as(37)e˙e¨=Aeoee˙+Aeo−Au−BuKuuu˙−Beo−BuKrur+0az−1Azz,where(38)Aeo=01−Ω12−2ξeoΩ1,Au=01−Ω12−2ξ1Ω1,Beo=0DVeo,Bu=0DV.If the second item and the third item on the right of equation (37) are equal to zero, the error e of equation (36) is convergent and limt⟶∞et=0. The fourth item of equation (36) is only considered when the influence of hysteresis on the control effect is studied. Therefore, the control gains Krt⟶∞=Kr∞ and Kut⟶∞=Ku∞ must satisfy the following equations:(39)Aeo−Au−BuKu∞=0,Beo−BuKr∞=0,or(40)Au=Aeo−BuKu∞−1,Bu=BeoKr∞−1.Moreover, substituting equation (40) into equation (37) yields(41)e˙=Aeoe+BeoKr−1K¯uu+BeoKr−1K¯rur+az−1Azz,where e=ee˙T, K¯u=Ku∞−Ku, and K¯r=Kr∞−Kr.Furthermore, for the stability analyses of the control system, a Lyapunov function is constructed as(42)V=eTPe+traceK¯uTPu−1K¯u+traceK¯rTPr−1K¯r,where P, Pu, and Pr are positive definite matrices. In addition, trace (·) calculates the trace of a matrix.For equation (42), the derivative is obtained as(43)V˙=e˙TPe+eTPe˙+traceK¯˙uTPu−1K¯u+K¯uTPu−1K¯˙u+traceK¯˙rTPr−1K¯r+K¯rTPr−1K¯˙r.Substituting equation (41) into equation (43) yields(44)V˙=eTAeoTP+PAeoe+2traceK¯˙uTPu−1K¯u+ueTPBeoKu∞−1K¯u+2traceK¯˙rTPr−1K¯r+ureTPBeoKr∞−1K¯r.If the first item of equation (44) is negative, namely, AeoTP+PAeo=−Q, Q is a positive definite matrix. Therefore, the second and third items are equal to zero. With K¯u=Ku∞−Ku and K¯r=Kr∞−Kr, the control gain matrices Ku and Kr are expressed as(45)K˙u=R1BeoPeuT,K˙r=R2BeoPeurT,where R1=PuKu∞−1T and R2=PrKu∞−1T.If the actual control gains are realized in discrete form, equation (45) is also represented as(46)Kuk=Kuk−1+h×R1BeoPeuT,Krk=Krk−1+h×R2BeoPeurT,where k is the sampling point and h is the sampling interval.


## 3.1. Adaptive Control Law

In this paper, the vibration suppression problem of a smart beam considering the hysteresis property is explored by using a model reference adaptive control (MRAC) [22, 25]. For adaptive control, a reference model is assumed as an expected objective model. According to the error between the output states of the smart beam and the expected objective hysteresis model, the adaptive control law is designed online with gains Kr and Ku, as shown in Figure 5.Figure 5
Model reference adaptive control.First, the expected objective hysteresis model is given as(33)u¨eo=−2ζeoΩ1u˙eo−Ω12ueo−azAzz+DVur,where ζeo is set as 0.707, az is a hysteresis coefficient, and azAz satisfies equation (32).(34)Vc=Krur+Kuu.Substituting equation (34) into equation (18), the identified model is derived as(35)u¨=−2ζ1Ω1u˙−Ω12u−Azz+DVKrur+DVKuu.Then, the error between the outputs of the expected objective model and the smart beam is expressed as(36)e=ueo−u.Furthermore, equation (36) is transformed into a state-space equation with equations (33) and (37), which is displayed as(37)e˙e¨=Aeoee˙+Aeo−Au−BuKuuu˙−Beo−BuKrur+0az−1Azz,where(38)Aeo=01−Ω12−2ξeoΩ1,Au=01−Ω12−2ξ1Ω1,Beo=0DVeo,Bu=0DV.If the second item and the third item on the right of equation (37) are equal to zero, the error e of equation (36) is convergent and limt⟶∞et=0. The fourth item of equation (36) is only considered when the influence of hysteresis on the control effect is studied. Therefore, the control gains Krt⟶∞=Kr∞ and Kut⟶∞=Ku∞ must satisfy the following equations:(39)Aeo−Au−BuKu∞=0,Beo−BuKr∞=0,or(40)Au=Aeo−BuKu∞−1,Bu=BeoKr∞−1.Moreover, substituting equation (40) into equation (37) yields(41)e˙=Aeoe+BeoKr−1K¯uu+BeoKr−1K¯rur+az−1Azz,where e=ee˙T, K¯u=Ku∞−Ku, and K¯r=Kr∞−Kr.Furthermore, for the stability analyses of the control system, a Lyapunov function is constructed as(42)V=eTPe+traceK¯uTPu−1K¯u+traceK¯rTPr−1K¯r,where P, Pu, and Pr are positive definite matrices. In addition, trace (·) calculates the trace of a matrix.For equation (42), the derivative is obtained as(43)V˙=e˙TPe+eTPe˙+traceK¯˙uTPu−1K¯u+K¯uTPu−1K¯˙u+traceK¯˙rTPr−1K¯r+K¯rTPr−1K¯˙r.Substituting equation (41) into equation (43) yields(44)V˙=eTAeoTP+PAeoe+2traceK¯˙uTPu−1K¯u+ueTPBeoKu∞−1K¯u+2traceK¯˙rTPr−1K¯r+ureTPBeoKr∞−1K¯r.If the first item of equation (44) is negative, namely, AeoTP+PAeo=−Q, Q is a positive definite matrix. Therefore, the second and third items are equal to zero. With K¯u=Ku∞−Ku and K¯r=Kr∞−Kr, the control gain matrices Ku and Kr are expressed as(45)K˙u=R1BeoPeuT,K˙r=R2BeoPeurT,where R1=PuKu∞−1T and R2=PrKu∞−1T.If the actual control gains are realized in discrete form, equation (45) is also represented as(46)Kuk=Kuk−1+h×R1BeoPeuT,Krk=Krk−1+h×R2BeoPeurT,where k is the sampling point and h is the sampling interval.


## 4. Simulation and Experimental Verifications of Model Reference Adaptive Control

### 4.1. Simulation Verifications

After a hysteresis model is constructed and an adaptive control law is designed, a numerical simulation is implemented in MATLAB/Simulink. The adaptive control law with the hysteresis model in MATLAB/Simulink is solved with the Runge–Kutta method (ODE45). How the hysteresis property affects the adaptive control effect for the smart beam is explored with numerical simulation analyses.Figure6 shows the control result with no hysteresis in the expected objective hysteresis model of equation (33). Figure 6(a) shows the control voltage. Moreover, Figure 6(b) displays the free vibration of the cantilever beam with adaptive control at az=0 compared with the result with no control. If the hysteresis coefficient az changes, the control result for the free vibration of the smart beam is different. When the hysteresis coefficient az is set to −4, −3,…, 8,…, it is found that the control effect at az=8 is the best.Figure 6
Control result ataz=0. (a) Control voltage (b) Free vibration.
(a)
(b)Figure7 shows the control result considering the hysteresis property in the expected objective hysteresis model at az=8. Figure 7(a) shows the control voltage and Figure 7(b) presents the free vibration of the smart beam at az=8 compared with that at az=0. Obviously, the vibration amplitude at az=8 is less than that at az=0. Figure 8 displays the frequency response of the curves in Figure 7(b) obtained by a fast Fourier transform (FFT). The amplitude peak value of the frequency response at az=8 is 0.03083 mm, the amplitude peak value of the frequency response at az=0 is 0.05875 mm, and the amplitude peak value of the frequency response with no control is 0.2567 mm. The control effect considering the hysteresis property in the expected objective hysteresis model at az=8 is superior to that with a linear expected objective model at az=0.Figure 7
Control result ataz=8. (a) Control voltage (b) Free vibration.
(a)
(b)Figure 8
Amplitude-frequency characteristic of the free vibration.When the hysteresis coefficientaz changes from −4 to 15, the amplitude peak values of the frequency response are given in Figure 9(a). If the practical time delay of the sensor and the processing time spent in the computer are not considered in the simulation, the amplitude peak value of the frequency response tends to be constant under a limited control voltage applied by the power equipment with hysteresis item az gradually increasing. Moreover, when the hysteresis coefficient az is less than −4, the control system of the smart beam is not stable. However, when the time delay of the sensing data and the processing time of the PC are added in the closed-loop control in the simulation, the control system of the smart beam is not stable at az<5 and az>12. The amplitude peak value of the frequency response at approximately az=−2 is the minimum of the dotted curve in Figure 9(a). Figure 9(b) shows peak of amplitude-frequency characteristic curves with different time delays (0.02 s, 0.05 s, 0.056 s, 0.058 s, 0.059 s, and 0.06 s), and the limited peak value is the minimum at az = −5, −6, and −7, when the time delay are set to 0.06 s, 0.059 s, and 0.058 s, respectively. Moreover, different time delays will affect the stability of the control system. Therefore, adding the time delay in the closed-loop control system will produce a different control result.Figure 9
Peak of the amplitude-frequency characteristic curve. (a) Curves with and without time delay. (b) Curves with different time delays.
(a)
(b)


### 4.2. Experimental Verifications

An experimental platform is set up to verify the vibration control effect of the smart beam by the adaptive model reference control with a Bouc–Wen model as the expected objective model. The experimental equipment is shown in Figure10. The adaptive control law is realized in LabVIEW. The control voltage is transported to the data acquisition unit. A power supply is used to amplify the control voltage from the data acquisition unit. Then, the control voltage is applied to the piezoelectric patch to suppress the vibration of the cantilever beam when the beam suffers from an initial disturbance. A laser displacement sensor is used to sense the bending deflection of the smart beam.Figure 10
Experimental equipment.Figure11 shows the vibration control result at az=0 for the smart beam. Figure 11(a) shows the control voltage, while Figure 11(b) displays the free vibration of the beam when the initial displacement at the free end of the beam is approximately 2 cm. It is observed easily that the vibration amplitude with control at az=0 is 0.15 mm at 0.45 s. Furthermore, Figure 12 shows the control result at az=−6. Figure 12(a) shows the control voltage, and Figure 12(b) displays the free vibration of the cantilever beam with control at az=−6. Compared with the vibration amplitude at az=0, the vibration amplitude az=−6 is 0.09 mm at 0.4 s. Notably, the control effect at az=−6 is better than that at az=0.Figure 11
Control result ataz=0.Figure 12
Control result ataz=−6.Similarly, the frequency response of the three curves in Figure12(b) is shown in Figure 13. The amplitude peak of the frequency response at az=−6 is 0.01157 mm at a frequency of 5.2490 Hz, the amplitude peak of the frequency response at az=0 is 0.02431 mm at a frequency of 8.8060 Hz, and the amplitude peak of the frequency response with no control is 0.2063 mm at a frequency of 2.5740 Hz. It is clearly indicated that the vibration control effect az=−6 for the cantilever beam is better than that at az=0.Figure 13
Amplitude-frequency characteristic of the free vibration.When the hysteresis coefficientaz changes, the peak value in the experiment exhibits the same changing trend in contrast to the result of the simulation considering the time delay in closed-loop control system, as shown in Figure 14. The solid curve is a polyfit of the curve from the experiment. The dashed curve is the amplitude peak value of the frequency response from the simulation considering the time delay. Then, both the solid curve and the dashed curve have their own minimum peak values. However, the hysteresis coefficients az of the two minimum peak values are different. One value is −2 and the other value is −6. The deviation between the experiment and simulation is caused by the time delay. In the experimental system, the time delay includes not only the sensing delay of the sensor but also the processing time of the PC. The sensing delay of the sensor is obtained from the user manual of the sensor. However, the processing time of the PC in the simulation is not calculated accurately. When too many application programs are running on the computer, the processing velocity of the PC slows down and the processing time of PC is large, and vice versa.Figure 14
Peak of the amplitude-frequency characteristic curve.Through the simulation and experimental verifications, adaptive model reference control with a Bouc–Wen model as an expected objective model is proven to be feasible for suppressing the vibration of the cantilever beam when it suffers from an initial displacement at the free end. It is illustrated that the hysteresis property will affect the stability and control effect of the control system when considering the time delay.


## 4.1. Simulation Verifications

After a hysteresis model is constructed and an adaptive control law is designed, a numerical simulation is implemented in MATLAB/Simulink. The adaptive control law with the hysteresis model in MATLAB/Simulink is solved with the Runge–Kutta method (ODE45). How the hysteresis property affects the adaptive control effect for the smart beam is explored with numerical simulation analyses.Figure6 shows the control result with no hysteresis in the expected objective hysteresis model of equation (33). Figure 6(a) shows the control voltage. Moreover, Figure 6(b) displays the free vibration of the cantilever beam with adaptive control at az=0 compared with the result with no control. If the hysteresis coefficient az changes, the control result for the free vibration of the smart beam is different. When the hysteresis coefficient az is set to −4, −3,…, 8,…, it is found that the control effect at az=8 is the best.Figure 6
Control result ataz=0. (a) Control voltage (b) Free vibration.
(a)
(b)Figure7 shows the control result considering the hysteresis property in the expected objective hysteresis model at az=8. Figure 7(a) shows the control voltage and Figure 7(b) presents the free vibration of the smart beam at az=8 compared with that at az=0. Obviously, the vibration amplitude at az=8 is less than that at az=0. Figure 8 displays the frequency response of the curves in Figure 7(b) obtained by a fast Fourier transform (FFT). The amplitude peak value of the frequency response at az=8 is 0.03083 mm, the amplitude peak value of the frequency response at az=0 is 0.05875 mm, and the amplitude peak value of the frequency response with no control is 0.2567 mm. The control effect considering the hysteresis property in the expected objective hysteresis model at az=8 is superior to that with a linear expected objective model at az=0.Figure 7
Control result ataz=8. (a) Control voltage (b) Free vibration.
(a)
(b)Figure 8
Amplitude-frequency characteristic of the free vibration.When the hysteresis coefficientaz changes from −4 to 15, the amplitude peak values of the frequency response are given in Figure 9(a). If the practical time delay of the sensor and the processing time spent in the computer are not considered in the simulation, the amplitude peak value of the frequency response tends to be constant under a limited control voltage applied by the power equipment with hysteresis item az gradually increasing. Moreover, when the hysteresis coefficient az is less than −4, the control system of the smart beam is not stable. However, when the time delay of the sensing data and the processing time of the PC are added in the closed-loop control in the simulation, the control system of the smart beam is not stable at az<5 and az>12. The amplitude peak value of the frequency response at approximately az=−2 is the minimum of the dotted curve in Figure 9(a). Figure 9(b) shows peak of amplitude-frequency characteristic curves with different time delays (0.02 s, 0.05 s, 0.056 s, 0.058 s, 0.059 s, and 0.06 s), and the limited peak value is the minimum at az = −5, −6, and −7, when the time delay are set to 0.06 s, 0.059 s, and 0.058 s, respectively. Moreover, different time delays will affect the stability of the control system. Therefore, adding the time delay in the closed-loop control system will produce a different control result.Figure 9
Peak of the amplitude-frequency characteristic curve. (a) Curves with and without time delay. (b) Curves with different time delays.
(a)
(b)


## 4.2. Experimental Verifications

An experimental platform is set up to verify the vibration control effect of the smart beam by the adaptive model reference control with a Bouc–Wen model as the expected objective model. The experimental equipment is shown in Figure10. The adaptive control law is realized in LabVIEW. The control voltage is transported to the data acquisition unit. A power supply is used to amplify the control voltage from the data acquisition unit. Then, the control voltage is applied to the piezoelectric patch to suppress the vibration of the cantilever beam when the beam suffers from an initial disturbance. A laser displacement sensor is used to sense the bending deflection of the smart beam.Figure 10
Experimental equipment.Figure11 shows the vibration control result at az=0 for the smart beam. Figure 11(a) shows the control voltage, while Figure 11(b) displays the free vibration of the beam when the initial displacement at the free end of the beam is approximately 2 cm. It is observed easily that the vibration amplitude with control at az=0 is 0.15 mm at 0.45 s. Furthermore, Figure 12 shows the control result at az=−6. Figure 12(a) shows the control voltage, and Figure 12(b) displays the free vibration of the cantilever beam with control at az=−6. Compared with the vibration amplitude at az=0, the vibration amplitude az=−6 is 0.09 mm at 0.4 s. Notably, the control effect at az=−6 is better than that at az=0.Figure 11
Control result ataz=0.Figure 12
Control result ataz=−6.Similarly, the frequency response of the three curves in Figure12(b) is shown in Figure 13. The amplitude peak of the frequency response at az=−6 is 0.01157 mm at a frequency of 5.2490 Hz, the amplitude peak of the frequency response at az=0 is 0.02431 mm at a frequency of 8.8060 Hz, and the amplitude peak of the frequency response with no control is 0.2063 mm at a frequency of 2.5740 Hz. It is clearly indicated that the vibration control effect az=−6 for the cantilever beam is better than that at az=0.Figure 13
Amplitude-frequency characteristic of the free vibration.When the hysteresis coefficientaz changes, the peak value in the experiment exhibits the same changing trend in contrast to the result of the simulation considering the time delay in closed-loop control system, as shown in Figure 14. The solid curve is a polyfit of the curve from the experiment. The dashed curve is the amplitude peak value of the frequency response from the simulation considering the time delay. Then, both the solid curve and the dashed curve have their own minimum peak values. However, the hysteresis coefficients az of the two minimum peak values are different. One value is −2 and the other value is −6. The deviation between the experiment and simulation is caused by the time delay. In the experimental system, the time delay includes not only the sensing delay of the sensor but also the processing time of the PC. The sensing delay of the sensor is obtained from the user manual of the sensor. However, the processing time of the PC in the simulation is not calculated accurately. When too many application programs are running on the computer, the processing velocity of the PC slows down and the processing time of PC is large, and vice versa.Figure 14
Peak of the amplitude-frequency characteristic curve.Through the simulation and experimental verifications, adaptive model reference control with a Bouc–Wen model as an expected objective model is proven to be feasible for suppressing the vibration of the cantilever beam when it suffers from an initial displacement at the free end. It is illustrated that the hysteresis property will affect the stability and control effect of the control system when considering the time delay.


## 5. Conclusion

In this paper, a Bouc–Wen model is constructed with a genetic identification algorithm to describe the hysteresis property of a smart beam with a piezoelectric actuator. With the identification result, the identified hysteresis loop can be consistent with the experimental hysteresis loop. Based on the hysteresis model as an expected objective model, a stability analysis of the smart system is presented, and adaptive model reference control is designed to explore the influence of the hysteresis characteristic on the control effect. By the simulation and experimental verifications, adaptive model reference control is demonstrated to be feasible for suppressing the vibration of the cantilever beam and the hysteresis item in the expected objective model will affect the stability and control effect of adaptive control for the smart beam with a time delay.



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*Source: 8138726-2020-03-09.xml*