This paper presents an adaptive control scheme to suppress vibration of flexible beams using a collocated piezoelectric actuator-sensor configuration. A governing equation of the beams is modelled by a partial differential equation based on Euler-Bernoulli theory. Thus, the beams are infinite-dimensional systems. Whereas conventional control design techniques for infinite-dimensional systems make use of approximated finite-dimensional models, the present adaptive control law is derived based on the infinite-dimensional Lyapunov method, without using any approximated finite-dimension model. Thus, the stability of the control system is guaranteed for all vibration modes. The implementation of the control law requires a derivative of the sensor output for feedback. A high-order sliding mode differentiation technique is used to estimate the derivative. The technique features robust exact differentiation with finite-time convergence. Numerical simulation and experimental results illustrate the effectiveness of the controller.
1. Introduction
Flexible structures have attracted interest because of their lighter weight compared to traditional structures. They have been widely used in aerospace applications and robotics [1–3]. However, the flexibility leads to vibration problems. Therefore, vibration control is needed. Over the past few decades, active vibration control has drawn more interests from researchers since it can effectively suppress the vibration [4–7].
Piezoelectric actuators and sensors provide an effective means of vibration suppression of flexible structures [8]. The advantages of using piezoelectric actuators and piezoelectric sensors include nanometer scale resolution, high stiffness, and fast response. Many researchers have studied the vibration suppression of flexible structures using piezoelectric actuators and piezoelectric sensors. In [9], Tavakolpour et al. proposed a self-learning vibration control strategy for flexible plate structures. A control algorithm is based on a P-type iterative learning with displacement feedback. Wang et al. [10] presented a simple control law for reducing the vibration of the flexible structure. Linear feedback control was derived using a linear matrix inequality method. Qiu et al. [11] proposed a neural network controller based on PD control with collocated piezoelectric actuator and sensor. The back-propagation algorithm was utilized for adapting the controller parameters. In [12], Sangpet et al. utilized a fractional-order control approach to improve the delay margin in the control system of a piezoactuated flexible beam. The controller parameters were tuned experimentally. An et al. [13] presented a time-delay acceleration feedback controller for vibration suppression of cantilever beams. Stability boundaries of the closed-loop system were determined by a Hurwitz stability criterion. In [14], Takács et al. presented an adaptive-predictive vibration control system using extended Kalman filter. The viability of the control method was experimentally evaluated. In [15], Sangpet et al. proposed an observer-based hysteresis compensation scheme for tracking control of a piezoactuated flexible beam. The observer based on a PI observer and a Kalman-filter algorithm was employed for estimation of the hysteresis in the beam. The scheme was successfully implemented in experimental tests.
The governing equation of vibrating flexible structures is a partial differential equation (PDE). Thus, the systems are distributed-parameter systems and have an infinite number of vibration modes. Most of control design techniques exploit an approximated finite-dimensional model by ignoring the higher frequency modes. However, this approach can cause the control system to become unstable due to a spillover effect [16, 17].
Infinite-dimensional control for the systems modeled by PDEs has been recently investigated. Extensions of finite-dimensional techniques to infinite-dimensional systems as well as innovative infinite-dimensional specific control design approaches have been proposed; see [18] and references therein. The Lyapunov-based control method has played a major role in nonlinear control of finite-dimensional complex systems [19–22]. Infinite-dimensional Lyapunov-based control is its extension to infinite-dimensional systems, which has been successfully applied to many engineering problems [23–28].
In [23], Dadfarnia et al. proposed infinite-dimensional Lyapunov-based control for a flexible Cartesian robot with a piezoelectric patch actuator attachment. The robot was modeled as a flexible cantilever beam with a translational base support and a tip mass. The controller was designed to exponentially suppress the vibration of the beam and regulate the base motion. Coron et al. [24] presented a strict Lyapunov function for hyperbolic systems of conservation laws. The time derivative of this Lyapunov function can be made strictly negative definite by an appropriate choice of the boundary conditions. The method was illustrated by a hydraulic application. Cheng et al. [25] proposed sliding-mode boundary control of a one-dimensional unstable heat conduction system modeled by parabolic PDE systems with parameter variations and boundary uncertainties. An infinite-dimensional sliding manifold was constructed via Lyapunov’s direct method. Simulation studies were conducted to verify the effectiveness of the sliding mode control law. In [26], Shang and Xu proposed a new control strategy for a one-dimensional cantilever Euler-Bernoulli beam with an input delay. The control system is either exponentially or asymptotically stable, depending on the time delay. In [27], Guo and Jin presented an active disturbance rejection and sliding mode control approach for a stabilization problem of Euler-Bernoulli beams with boundary input disturbance. The controller design is based on a PDE model. In [28], Luemchamloey and Kuntanapreeda presented an experimental study of active vibration control of flexible beams. The flexible beam was modeled by PDE. The controller was designed based on the infinite-dimensional Lyapunov method. He et al. [29] considered vibration control of a flexible string system. With Lyapunov’s direct method, adaptive boundary control was developed to suppress the string’s vibration and the adaptive law was designed to compensate for the system parametric uncertainties. Numerical simulations were carried out to verify the effectiveness of the controller.
In addition, high-order sliding-mode differentiation [30, 31] has been attracting interest. It is an alternative differentiation that features robust exact differentiation with finite-time convergence. It has been successfully used in control laws, replacing the use of derivative measurements [32, 33].
This paper presents an adaptive feedback control law to suppress vibration of Euler-Bernoulli beams with a collocated piezoelectric actuator/sensor pair. The actuator and sensor are lead zirconate titanate (PZT). The filtered signal from the sensor is used to provide the feedback signal. The governing equation of the beams is a PDE. Thus, it has an infinite number of vibration modes. The controller is designed based on an infinite-dimensional Lyapunov method, which does not involve any approximated finite-dimensional models. Thus, the stability of the control system including the filter dynamic is guaranteed for all vibration modes. The high-order differentiation is used to estimate the derivative of the sensor signal and then used in the control law. The rest of the paper is organized as follows. Section 2 provides some preliminaries. Section 3 describes the flexible beam that is used as an experimental test bench. Section 4 presents a finite element model of the experimental beam. The proposed controller design is given in Section 5. Simulation and experimental results are presented in Section 6. The last section concludes the paper.
2. Preliminaries2.1. System Modeling
Consider a flexible cantilever beam with piezoelectric actuator and sensor patches. The patches are bonded on the top and bottom surfaces of the beam as shown in Figure 1. A governing equation of the beam can be derived based on an Euler-Bernoulli beam equation and Hamilton’s principle [19] as
(1)w¨x,t=1ρ(x)MpoutS′′x-∂2EIxw′′x,t∂x2hhhhhhhh-∂2EIxw′′x,t∂x2Bw˙x,t-Cw˙′x,t,
with the boundary conditions
(2)w(0,t)=w˙(0,t)=w′(0,t)=0,w′′(L,t)=w′′′(L,t)=0,
where the dot and prime notations represent derivatives with respect to time and the variable x, respectively, w(x,t) is the transverse displacement of the beam, u(t) is the input voltage applied to the actuator, L is the length of the beam, ρ(x)=bbρbtb+bpρptpS(x), ρb, ρp are the densities of the beam and the piezoelectric patch actuator, respectively, bb, bp are the widths of the beam and the actuator, tb, tp are the thicknesses of the beam and the actuator, S(x)=H(x-l1)-H(x-l2), H(x) is the Heaviside function, l1,l2 are the distances from the clamped end to the leading edge and the tailing edge of the actuator, respectively, Mpo=-bpEpd31tb+tp is the moment constant of the actuator, EI(x)=EIb+EIpS(x) is the bending stiffness, EIp=2bpEptptb2/4+tbtp/2+tp3/3EIb=Ebbbtb3/12, Eb, Ep are the Young’s moduli of the beam and the actuator, and B, C are the viscous and structural damping coefficients, respectively.
Beam with a piezoelectric actuator and a piezoelectric sensor.
The equation relating the applied voltage u and the axial strain εxx produced by the piezoelectric patch actuator is written as [8, 11]:
(3)εxx=d31utp,
where d31 and tp are the dielectric constant and the thickness of the actuator, respectively. The normal stress σxx and the bending moment Ma on the beam produced by the actuator are
(4)σxx=Epd31utp,(5)Ma=∫tb/2tb/2+tpσxxzdA=Epd31bpz-u,
where Ep is Young’s modulus of the actuator, bp is the width of the actuator, z-=(tb+tp)/2, and tb, tp are the thicknesses of the beam and the actuator.
By considering the piezoelectric patch sensor as a parallel plate capacitor, the voltage vs(t) across the two layers produced by the sensor is given as [23]:
(6)vst=ψw′l2,t-w′l1,t,
where ψ=bp,sEp,sd31,s(tb+tp,s)/2Cp,s is the piezoelectric constant, bp,s is the width of the sensor, Ep,s is Young’s modulus of the sensor, d31,s is the dielectric constant of the sensor, tp,s is the thickness of the sensor, and Cp,s is the piezoelectric capacitance.
2.2. Finite Element Equation
Consider a finite element (FE) model of a cantilever beam shown in Figure 2. It is assumed that the beam is divided into n elements. Each element has two nodes and each node has two degrees of freedom: transverse direction (w1,w2) and slope (w1′,w2′).
Finite element model.
The transverse displacement at any x position on the element can be expressed as [8, 30]:
(7)w(x)=ΦTq,
where qT=w1w1′w2w2′ is the nodal displacement vector and Φ=ϕ1ϕ2ϕ3ϕ4 is the mode-shape vector function. By using the Lagrangian method, the differential equation of motion to be solved can be expressed as [34, 35]:
(8)Mq¨+Cq˙+Kq=F,
where M=ρA∫0LΦTΦdx is the mass matrix, K=EI∫0LΦ′′TΦ′′dx is the stiffness matrix, C is the damping matrix, and F is the force matrix. For Rayleigh proportional damping, C=αM+βK, where α,β are the damping constants. The force matrix F for the element without the piezoelectric actuator is F=0000T and is, for the element bonded with the actuator, patch F=0Epd31bpz-0-Epd31bpz-Tu, which is directly derived from (5). Note that the piezoelectric sensor is not included in the FE model since the bonded sensor in the experimental system is very thin and light compared to those of the beam and the actuator.
2.3. High-Order Sliding-Mode Differentiation
In this subsection, a high-order sliding-mode differentiation method is shortly described. To explain the method, let f(t) be a function from which we want to determine its derivative. Consider the following auxiliary system:
(9)z˙=η.
The main idea is to find η to keep z-f(t)=0, resulting in η=f˙(t). Based on a second-order sliding-mode algorithm, one obtains [30]
(10)η=η1-μ1z-f(t)1/2signz-f(t),η˙1=-μ2signz-ft,
where μ1,μ2>0 and sign(x) is a signum function. A general form of nth-order differentiators can be written as
(11)η˙0=κ0,κ0=-μ0Γ1/n+1η0-ftn/n+1η˙0=κ0,κ0=×signη0-ft+η1,η˙1=κ1,κ1=-μ1Γ1/nη1-κ0n-1/nsignη0-κ0+η2,⋮η˙n-1=κn-1,κn-1=-μn-1Γ1/2ηn-1-κn-21/2η˙n-1=κn-1,κn-1=×signηn-1-κn-2+ηn,η˙n=-μnΓsignηn-κn-1,
where ηn is an approximated value of the nth derivative of the function f(t), Γ>0 is a Lipschitz constant of the function f(t), and μi(i=0,1,…,n) are constant parameters. The reader is referred to Levant [30, 31] for full details of the method. In this paper, this method will be used for estimating the first derivative of the signal from the piezoelectric sensor.
3. Experimental System
A schematic and a photograph of the experimental system used in this paper are shown in Figure 3. The system consists of a flexible cantilever beam, a piezoelectric patch actuator, a piezoelectric patch sensor, a high-voltage power amplifier, a 16-bit ADC/DAC interface board, and a PC. The beam is made of aluminum. The dimensions and properties of the beam are summarized in Table 1. The piezoelectric actuator and sensor are lead zirconate titanate (PZT). They are bonded on the beam at the distance x=15mm from the clamped point. The dimensions and properties of the actuator and sensor are summarized in Table 2. The sensor measures the axial strain of the beam. The system is also equipped with a laser displacement sensor for measuring the beam’s tip deflection; however, the displacement sensor is not used in the control loop. The signals from the sensors are fed back to the PC through the interface board. The control signal from the computer is converted to an analogue signal by the interface board and is transmitted through the high-voltage power amplifier to the actuator to close the control loop. The gain of the amplifier is 150. Figure 4 displays the pulse response and the frequency response functions of the beam where the signal from the piezoelectric sensor is considered as the output. The frequency response shows that the first two natural frequencies of the beam are 39.28 rad/s and 224.7 rad/s.
Beam’s dimensions and properties.
Dimensions/properties
Symbol
Value
Unit
Young’s modulus
Eb
70
GPa
Density
ρb
2700
kg/m^{3}
Poisson ratio
ν
0.35
—
Length
L
340
mm
Width
bb
15
mm
Thickness
tb
0.8
mm
Piezoelectric material’s dimensions and properties.
Dimensions/properties
Symbol
Value
Unit
Actuator
Young’s modulus
Ep
66.67
GPa
Density
ρp
7800
kg/m^{3}
Length
lp
45
mm
Width
bp
15
mm
Thickness
tp
0.5
mm
Strain coefficient
d31
−171e^{−12}
m/V
Piezoelectric position
l1
15
mm
l2
60
mm
Sensor
Young’s modulus
Ep,s
2.5
GPa
Length
lp,s
45
mm
Width
bp,s
15
mm
Thickness
tp,s
0.5
mm
Piezoelectric capacitance
Cp,s
379e^{−12}
Farad
Strain coefficient
d31,s
23e^{−12}
m/V
Experimental system: (a) schematic and (b) photograph.
Experimental system’s responses: (a) pulse response and (b) frequency response function.
4. Finite Element Model
A finite element (FE) model of the experimental beam was developed to be used in the control design process. For simplicity, n=11 was chosen. The pulse response and the frequency response functions obtained from the developed FE model are shown in Figure 5. The responses agree with those of the experimental system.
System’s responses based on FE model: (a) pulse response and (b) frequency response function.
5. Controller Design
The proposed control design is given in this section. The control objective is to suppress the vibration of the beam by using only the signal from the piezoelectric sensor. The schematic for the control loop is shown in Figure 6.
Controlled system.
Instead of directly using vs and v˙s for feedback, let vs and v˙s pass through a low-pass filter to obtain a new feedback signal y(t) as follows:
(12)y˙t=1τa1vst+a2v˙st-yt,
where τ>0 is the time constant of the filter and a1,a2≥0(a1+a2≠0) are weighting parameters. Note that including a low-pass filter in the control loop is usually done in practice to attenuate the effect of noise in the feedback signal. In the following, the dynamic of the filter is also included in the controller design.
Define a Lyapunov function candidate as
(13)V1=12Mpo∫0Lρ(x)w˙2(x,t)dx+12Mpo∫0LEIxw′′2x,tdx+λ2y2,
where λ is considered as a controller parameter. By taking the time derivative of (13) and substituting (1), (2), (6), and (12) into the derivative result, it yields
(14)V˙1=-BMpo∫0Lw˙2x,tdx-C2Mpow˙2L,t+u(t)(w˙′l2,t-w˙′(l1,t))+λyτa1ψw′l2,t-w′l1,thhhhhhll+a2ψw˙′l2,t-w˙′l1,t-y=-BMpo∫0Lw˙2x,tdx-C2Mpow˙2L,t-λy2τ+u(t)(w˙′l2,t-w˙′(l1,t))+λyτa1ψw′l2,t-w′l1,thhhhlhhh+a2ψw˙′l2,t-w˙′l1,t.
By choosing the following control law:
(15)ut=-λa2ψyτ-λa1ψyτw′l2,t-w′l1,t(w˙′l2,t-w˙′(l1,t)),
it yields
(16)V˙1=-BMpo∫0Lw˙2(x,t)dx-C2Mpow˙2(L,t)-λy2τ≤-λy2τ≤0.
By Barbalat’s lemma [36, 37], y→0 is achieved. Note that w′(l2,t)-w′(l1,t) can be obtained directly from the piezoelectric sensor (6); that is, vs(t)=ψ(w′(l2,t)-w′(l1,t)), and w˙′l2,t-w˙′(l1,t) can be obtained from v˙s(t). In our experiment, we used the high-order sliding-mode differentiation method described in Section 2.3 to determine v˙s(t).
Note that control law (15) has a singularity at w˙′l2,t-w˙′l1,t=0. Thus, implementations of the control law must empirically avoid this singularity. For simplicity, we chose a1=0 and a2=1 in the experiment to avoid the singularity.
Moreover, the parameter ψ is usually uncertain or even unknown. Thus, control law (15) with a1=0 and a2=1 is modified as
(17)u=-λktyτ,
where k(t) is an adaptive variable replacing ψ. Define e(t)=k(t)-ψ. Take
(18)V=V1+12e2=12Mpo∫0Lρxw˙2x,tdx+12Mpo∫0LEI(x)w′′2(x,t)dx+λ2y2
as a new Lyapunov function candidate. Taking its derivative with respect to time and substituting (17) into the result yields
(19)V˙=-BMpo∫0Lw˙2x,tdx-C2Mpow˙2L,t+(-λktyτ)(w˙′l2,t-w˙′l1,t)+λyτψw˙′l2,t-w˙′l1,t-y+kt-ψk˙(t)=-BMpo∫0Lw˙2x,tdx-C2Mpow˙2L,t-λy2τ+k˙t-λyw˙′l2,t-w˙′l1,tτkt-ψ.
By choosing the following adaptive law:
(20)k˙t=λyw˙′l2,t-w˙′l1,tτ,
it results in
(21)V˙=-BMpo∫0Lw˙2(x,t)dx-C2Mpow˙2(L,t)-λy2τ≤-λy2τ≤0.
Applying Barbalat’s lemma [36, 37] results in y→0.
6. Results6.1. Simulation Study
A simulation study was conducted using the FE model described in Section 2.2 to tune the control parameter λ. It was observed that the convergence rate of the controlled response increases with λ. Here, the control gain of λ=0.052 was chosen. Simulation results are shown in Figure 7. The controller was activated at t=1 sec. The results show that the controller is able to suppress the vibration effectively. Note that k(t) converged to a finite value.
FEM simulation result (the controller was activated at t=1 sec): (a) feedback signal, (b) control signal, (c) adaptive variable, and (d) tip deflection.
6.2. Experimental Results
Controller (17) was implemented digitally on the PC with the sampling period of the control loop of 1 msec. The high-order sliding differentiation technique with Γ=700, μ0=μ1=1, was used. The experimental results are shown in Figure 8. The controller provides satisfactory responses similar to those obtained by the simulation. Note that the controller was activated at t=1 sec.
Experimental result (the controller was activated at t=1 sec): (a) feedback signal, (b) control signal, (c) adaptive variable, and (d) tip deflection.
Note that we set the initial conditions of the FEM simulation and experiment roughly the same for a comparison purpose. It was found that there are slight differences in the FEM simulation and experimental results. This is mainly due to the measurement noise and the mismatch in the initial conditions. The difference between the maximum control inputs is about 12.08%. The difference of the steady state adaptive gains is about 3.35%. In summary, both results are practically comparable.
7. Conclusions
In this paper we proposed an adaptive controller for vibration suppression of flexible structures. The structure under consideration is a cantilever beam with a collocated piezoelectric actuator and sensor pair. The governing equation of the beam is a partial differential equation. Since the beam is an infinite-dimensional system, we derived the control law using the infinite-dimensional Lyapunov method, which does not require any approximated finite-dimension model. Thus, the stability of the control system is guaranteed for all vibration modes. We also employed a high-order sliding mode differentiation technique to obtain the derivative of the sensor’s signal for being used in the control law. The technique features robust exact differentiation with finite-time convergence. Effectiveness of the controller is illustrated both in finite-element model and experiment, with results that show good agreement.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
This work was supported by the Royal Golden Jubilee Ph.D. program of the Thailand Research Fund. The authors also would like to thank anonymous referees for their valuable comments.
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