Article
An adaptive PID-like controller for vibration suppression of piezo-actuated flexible beams
Journal of Vibration and Control 1–15 ! The Author(s) 2017 Reprints and permissions: sagepub.co.uk/journalsPermissions.nav DOI: 10.1177/1077546317692160 journals.sagepub.com/home/jvc
Teerawat Sangpet1, Suwat Kuntanapreeda1 and Ru¨diger Schmidt2
Abstract Flexible structures have been increasingly utilized in many applications because of their light-weight and low production cost. However, being flexible leads to vibration problems. Vibration suppression of flexible structures is a challenging control problem because the structures are actually infinite-dimensional systems. In this paper, an adaptive control scheme is proposed for the vibration suppression of a piezo-actuated flexible beam. The controller makes use of the configuration of the prominent proportional-integral-derivative controller and is derived using an infinite-dimensional Lyapunov method. In contrast to existing schemes, the present scheme does not require any approximated finitedimensional model of the beam. Thus, the stability of the closed loop system is guaranteed for all vibration modes. Experimental results have illustrated the feasibility of the proposed control scheme.
Keywords Vibration control, adaptive control, infinite-dimensional systems, flexible structures, integral action
1. Introduction Because of their light-weight and low production cost, flexible structures have been increasingly utilized in many applications, especially in aerospace (Azadi et al., 2011; Lee and Singh, 2012; Sales et al., 2013; Sharma et al., 2015) and robotics (Dadfarnia et al., 2004; Zhang et al., 2005; Li et al., 2013; Sharifnia and Akbarzadeh, 2016). However, being flexible leads to vibration problems. Thus, vibration control is usually needed. Since governing equations of vibrating flexible structures are partial differential equations (PDEs), the structures are infinite-dimensional systems. Thus, vibration control of flexible structures, especially its stability analysis, is challenging. Piezoelectric materials have been shown to be effective actuators and sensors for the vibration control of flexible structures. For example, Wu et al. (2014) proposed an independent modal space control method for the vibration control of a flexible beam using piezoelectric actuators and sensors. The method decouples the states of the controlled plant by representing the states in terms of modal co-ordinates. The stability analysis assumes that every mode is controlled by each
negative velocity feedback controller. Experimental and numerical results indicated that by using piezo-patches as actuators the independent mode control method considering only the first three modes is very effective and good at suppressing the vibration of a flexible structure. Zhang et al. (2014) proposed a disturbance rejection control method for vibration suppression of piezoelectric laminated thin-walled structures. A generalized proportional-integral observer was utilized for the estimation of the disturbance. Stability analysis of the estimation error is based on the Lyapunov stability criterion for the linear model. The results illustrated 1 Department of Mechanical and Aerospace Engineering, Faculty of Engineering, King Mongkut’s University of Technology North Bangkok, Thailand 2 Institute of Structural Mechanics and Lightweight Design, RWTH Aachen University, Germany
Received: 10 August 2016; accepted: 11 January 2017 Corresponding author: Teerawat Sangpet, Department of Mechanical and Aerospace Engineering, Faculty of Engineering, King Mongkut’s University of Technology North Bangkok, Bangkok 10800, Thailand. Email:
[email protected]
2 that the vibrations were better suppressed by the proposed method when compared to a LQR control method. Qiu and Xu (2016) presented the vibration control of a rotating two-connected flexible beam using chatter-free sliding mode controllers and piezoelectric sensors/actuators. The controlled plant was modeled as a finite-dimensional state-space model and the stability of the closed-loop system was proven by utilizing the Lyapunov stability method. The experimental results demonstrated that the controllers can significantly improve the vibration reduction performance. When there are uncertainties in the parameters of the controlled plants, the control performance might be deteriorated. In order to overcome this problem, adaptive control is usually adopted. Azadi et al. (2011) proposed an adaptive robust control scheme for a flexible satellite. The appendages were considered as Euler– Bernoulli beams and the piezoelectric layers were used as actuators. The control design was based on an approximated ordinary differential equation (ODE). Zhang et al. (2013) presented adaptive vibration control for a cantilever beam bonded with a piezoelectric actuator. The control plant was online-modeled as a controlled autoregressive moving average (CARMA) model. The adaptive controller was designed based on a minimum variance direct control method. All of the above mentioned works employed approximated finite-dimensional models for designing the controllers, as well as proving the stability of the control systems. Thus, the stability is guaranteed for only those finite numbers of vibration modes. This approach can cause the systems to become unstable due to a spillover effect (Montazeri et al., 2011). Vibration control of flexible structures without using any approximated finite-dimensional model has been recently investigated. Dadfarnia et al. (2004) presented an infinite-dimensional Lyapunov-based control strategy for the regulation of a flexible Cartesian manipulator. Zhang et al. (2005) proposed a control design scheme for a flexible two-link manipulator. The scheme made use of passivity and Lyapunov techniques. Luemchamloey and Kuntanapreeda (2014) presented an experimental study of nonadaptive proportional-derivative (PD) vibration control of a piezoactuated flexible beam. The control law was directly designed based on a Lyapunov stability theory of infinite-dimensional systems. This work was then extended by Sangpet et al. (2014) for an adaptive version. Proportional-integral-derivative (PID) controllers have been effectively used in many applications, including vibration control of flexible structures (Jovanovic´ et al., 2013; Saad et al., 2015; Sharifnia and Akbarzadeh, 2016; Simonovic´ et al., 2016). The integrator in PID controllers is a decisive element for
Journal of Vibration and Control eliminating steady state errors in linear control systems. The integral action has been also introduced in nonlinear control to improve the steady state performance (Donaire and Junco, 2009; Kuntanapreeda and Marusak, 2012; Esbrook et al., 2014). This paper proposes an adaptive PID-like control design to suppress the vibration of a flexible beam bonded with a piezoelectric actuator and sensor. The design is motivated by the advantages of integral actions and adaptive mechanisms. Moreover, in order to guarantee the control stability for all vibration modes, the design adopts an infinite-dimensional Lyapunov stability method, without using any approximated finite-dimensional model. The work can be considered as an extension of Luemchamloey and Kuntanapreeda (2014) by imposing an integral action and an adaptive mechanism into the control law to enhance the steady state performance of the control system. The feasibility of the proposed controller is demonstrated through experiments. The rest of the paper is organized as follows. In the next section, the control system is described. Main results are presented in Section 3, where the proposed controller and stability analysis are detailed. Experimental results are given in Section 4, and concluding remarks are given in Section 5.
2. Control system 2.1. System description A schematic of the control system under research is shown in Figure 1. The system consists of a flexible cantilever beam, a piezoelectric actuator, a piezoelectric sensor, a high-voltage power amplifier, an ADC/DAC interface board, and a computer. The beam is made of aluminum. The actuator and sensor are lead zirconium titanate (PZT) and polyvinylidene fluoride (PVDF), respectively. They are bonded on the beam as a collocated pair. The output voltage signal y(t) from the sensor is fed back to the computer through the interface board. The control signal u(t) from the computer is converted to an analog signal by the interface board, and is transmitted through the power amplifier to the actuator to close the control loop. In the figure, L is the length of the beam, l1 and l2 are respectively the distances from the clamped end to the leading edge and the tailing edge of the actuator, and w(x, t) is the transverse displacement of the beam at the position x as a function of time t.
2.2. Mathematical model The governing equation of the system under research is given in this section. The effects of the sensor on the dynamics of the beam is ignored since the thickness of
Sangpet et al.
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Figure 1. Schematic diagram of flexible beam system.
the PDVF sensor is very thin, and the linear mass density and flexural rigidity are very small comparing to those of the beam and the actuator. Based on the Euler–Bernoulli theory and Hamilton’s principle, the governing PDE of the beam can be expressed as (Queiroz et al., 2000; Dadfarnia et al., 2004)
t2b =4 þ tb ta =2 þ t3a =3 , Eb and Ea are respectively the Young’s moduli of the beam and the actuator. By considering the sensor as a parallel plate capacitor, the output voltage signal y(t) across the sensor due to the bending of the beam can be expressed as (Dadfarnia et al., 2004) yðtÞ ¼
1 00 b € tÞ ¼ ðM00 ðEIðxÞw wðx, ðx, tÞÞ00 c AðxÞ
ð1Þ
0
_ tÞ Cw_ ðx, tÞÞ Bwðx, with the boundary conditions wð0, tÞ ¼ w0 ð0, tÞ ¼ 0 w00 ðL, tÞ ¼ w000 ðL, tÞ ¼ 0
ð2Þ
where the dot and prime notations respectively represent derivatives with respect to time and the variable x, M is the applied bending moment produced by the actuator, B and C are respectively the viscous and strucc tural damping coefficients, AðxÞ is the linear mass b density, and EIðxÞ is the flexural rigidity. The linear mass density and flexural rigidity are given as c AðxÞ ¼ bb b tb þ ba a ta SðxÞ
ð3Þ
b b b þ EI b a SðxÞ EIðxÞ ¼ EI
ð4Þ
and
where b and a are respectively the densities of the beam and the actuator, bb and ba are respectively the widths of the beam and the actuator, tb and ta are respectively the thicknesses of the beam and the actuator, SðxÞ ¼ Hðx l1 Þ Hðx l2 Þ, H(x) is the b b ¼ bb t3 Eb =12, EI b a ¼ ba ta Ea Heaviside function, EI b
ðw0 ðl2 , tÞ w0 ðl1 , tÞÞ
ð5Þ
where ¼ bs Es d31,s ðtb þ ts Þ=2Cs is the voltage constant of the piezoelectric sensor, bs is the width of the sensor, Es is the Young’s modulus of the sensor, d31, s is the piezoelectric constant of the sensor, ts is the thickness of the sensor, and Cs is the piezoelectric capacitance of the sensor. The equation relating the control signal u(t) applied to the power amplifier of the actuator and the bending moment M on the beam produced by the actuator is written as (Dadfarnia et al., 2004; Queiroz et al., 2000) M ¼ Ma uðtÞSðxÞ
ð6Þ
where Ma ¼ Gpa Ea d31,a ba ðtb þ ta Þ=2, Gpa is the gain of the power amplifier, Ea is the Young’s modulus of the actuator, and d31, a is the piezoelectric constant of the actuator. Note that Ma > 0 since d31, a < 0 for the actuator. From (1) to (6), by considering u(t) and y(t) respectively as the input and output, the governing equation of the system under research can be expressed as € tÞ ¼ wðx,
1 00 b ðMa uðtÞS00 ðxÞ ðEIðxÞw ðx, tÞÞ00 c AðxÞ _ tÞ Cw_ 0 ðx, tÞÞ Bwðx,
yðtÞ ¼ ðw0 ðl2 , tÞ w0 ðl1 , tÞÞ with the boundary conditions in (2).
ð7Þ
4
2.3. Experimental system Photographs of the experimental test bench on which all experiments are carried out to demonstrate the feasibility are shown in Figure 2. The beam is made of aluminum. Its material properties and dimensions are summarized in Table 1. The actuator used is a PSI-5A-S4-ENH (Piezo System, USA). Table 2 summarizes the properties and dimensions of the actuator. The sensor is a LDT1-028K (Measurement Specialities, USA). Its properties and dimensions are summarized in Table 3. The power amplifier used is a E-463 HVPZT (Physik Instrumente, Germany). Its gain (Gpa) is 150. The interface board is a NI PCIe-6321 (National Instruments, USA). Its resolution is 16 bits. Figure 3 displays an example of the dynamic response to an impulse input of the uncontrolled beam. The impulsive loading conditions were identified by finite element simulation as presented in Appendix A1.1. From the figure, the logarithmic decrement of 0.03112 and damping ratio of 0.004952 were found.
3. Main results 3.1. Proposed controller Motivated by the works in Dadfarnia et al. (2004) and Luemchamloey and Kuntanapreeda (2014) and the
Journal of Vibration and Control advantages of integral actions and adaptive mechanisms, an adaptive controller with an integral action is proposed as _ uðtÞ ¼ kp ðtÞ yðtÞ ki ðtÞ yI ðtÞ kd ðtÞyðtÞ Rt yI ðtÞ ¼ 0 yðÞd
ð8Þ
with the following proposed adaptive law _ þ 0 y2 ðtÞ kp ðtÞ k_p ðtÞ ¼ 1 yðtÞyðtÞ _ þ 0 yðtÞÞ yI ðtÞ k_i ðtÞ ¼ 2 ðyðtÞ _ k_d ðtÞ ¼ 3 0 yðtÞyðtÞ
ð9Þ
where 0 > 1, i,i¼1,2 4 0, i,i¼1,2,3 4 0, and 0 are the design parameters.
3.2. Stability analysis An infinite-dimensional Lyapunov stability method is utilized to examine the stability of the closed-loop PDE system consisting of the system (7) with the boundary conditions (2), the control law (8), and the adaptive law (9). The following lemmas and properties are used in the analysis.
Figure 2. Experimental test bench; 1-beam, 2-actuator, 3-power amplifier, and 4-sensor.
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Table 1. Material properties and dimensions of the beam.
0.8
Parameter
Value
Unit
Eb b L bb tb l1 l2 B C
Young’s modulus Density Poisson’s ratio Length Width Thickness Actuator’s leading edge pos. Actuator’s tailing edge pos. Viscous damping coeff. Structural damping coeff.
65 2700 0.35 340 15 0.8 15 60 100 0.04
GPa kg/m3 mm mm mm mm mm kg/s kg m/s
Table 2. Material properties and dimensions of the actuator. Symbol
Parameter
Value
Unit
Ea a la ba ta d31,
Young’s modulus Density Length Width Thickness Piezoelectric constant
66.67 7800 45 15 0.5 171e12
GPa kg/m3 mm mm mm m/V
a
Table 3. Material properties and dimensions of the sensor. Symbol
Parameter
Value
Unit
Es s ls bs ts Cs d31,
Young’s modulus Density Length Width Thickness Piezoelectric capacitance Piezoelectric constant
2 1780 45 12 28 1.38 23e12
GPa kg/m3 mm mm
m nF m/V
s
0.6 Output signal (V)
Symbol
0.4 0.2 0 −0.2 −0.4 −0.6 −0.8
0
1
2
3
4
5
Time (s)
Figure 3. Dynamic response of an uncontrolled beam.
Property 1. (Queiroz et al., 2000) If the kinetic energy of the system (7), which is given as
d ¼1 KEðtÞ 2
Z
L
c w_ 2 ðx, tÞdx AðxÞ
ð12Þ
0
_ tÞ, w_ 0 ðx, tÞ and w_ 00 ðx, tÞ is bounded 8t 2 ½0, 1Þ, then wðx, are bounded 8t 2 ½0, 1Þ. Property 2. (Queiroz et al., 2000) If the potential energy of the system (7), which is given as c ¼1 PEðtÞ 2
Z
L
002 b ðx, tÞdx EIðxÞw
ð13Þ
0
is bounded 8t 2 ½0, 1Þ, then w00 ðx, tÞ, w000 ðx, tÞ and w0000 ðx, tÞ are bounded 8t 2 ½0, 1Þ. For examining of the closed-loop stability, first consider the following Lyapunov function candidate VðtÞ ¼ V1 ðtÞ þ 0 V2 ðtÞ þ V3 ðtÞ þ V4 ðtÞ
Lemma 1. (Queiroz et al., 2000; Dadfarnia et al., 2004) Let p, q, 2 1. The conditions to ensure the positive definite property of V(t) are derived below. c By substituting AðxÞ ¼ b bb tb þ a ba ta SðxÞ b b b a SðxÞ EI b b into (15), it b bb tb and EIðxÞ ¼ EIb þ EI gives V1 ðtÞ
Substituting (20) and (22) into (16) yields
L
2 w2 ðx, tÞdx 0 Z L 002 w ðx, tÞ 2 þL dx 2 0
01 0 b C0 L2 EIb 2 2 0 B c 2 Amax 1 C2 0 1 2
ð26Þ
it yields V(t) is positive definite. Since satisfying cmax 1 0 and B C2 0 both conditions B4 A 4 yields the third condition in (26), the conditions in (26) can be written as
ð22Þ 20 1
B 2C0 L2 B 2 cmax EI bb 4C 4A
ð27Þ
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Then, by combining the conditions in (27) and the condition 0 > 1, a sufficient condition for 0 to ensure the positive definite property of V(t) can be written as b bB B EI 1 5 0 min , cmax 8C2 L2 8A
Integrating by parts the first term on the right-hand side of (34) and using the boundary conditions in (2) yields Z
L
0
! ð28Þ
_ tÞMa uðtÞS00 ðxÞdx wðx, L _ tÞS0 ðxÞ0 ¼ Ma uðtÞðwðx, ZL w_ 0 ðx, tÞS0 ðxÞdxÞ 0
L _ tÞS0 ðxÞjL0 w_ 0 ðx, tÞSðxÞ0 ¼ Ma uðtÞðwðx, ZL þ w_ 00 ðx, tÞSðxÞdxÞ 0 ZL ¼ Ma uðtÞ w_ 00 ðx, tÞSðxÞdx
Next, the time derivative of V(t) along the trajectories of (7) is determined. From (14) to (18), the derivative is obtained as follows _ ¼ V_ 1 ðtÞ þ 0 V_ 2 ðtÞ þ V_ 3 ðtÞ þ V_ 4 ðtÞ VðtÞ
ð29Þ
Z
0 l2
w_ 00 ðx, tÞdx
¼ Ma uðtÞ
where V_ 1 ðtÞ ¼
Z Ma þ
V_ 2 ðtÞ ¼
Z Ma
L
0
Ma
L
l1 0
¼ Ma uðtÞðw_ ðl2 , tÞ w_ 0 ðl1 , tÞÞ Ma _ uðtÞyðtÞ ¼
c wðx, _ tÞwðx, € tÞdx AðxÞ
Z
L
ð30Þ 00 b ðx, tÞw_ 00 ðx, tÞdx EIðxÞw
0
Similarly, the second and fourth terms on the righthand side of (34) can be obtained respectively as
c € AðxÞwðx,tÞ wðx,tÞdx
0
Z
ð35Þ
Z
L
Z
00 00 b _ tÞ EIðxÞw ðx, tÞ dx wðx, 0 00 b ¼ w_ 0 ðL, tÞ EIðLÞw ðL, tÞ 00 b þ w_ 0 ð0, tÞ EIð0Þw ð0, tÞ ZL 00 b þ ðx, tÞw_ 00 ðx, tÞdx EIðxÞw
L
c w_ 2 ðx,tÞdx þ B _ AðxÞ wðx,tÞwðx,tÞdx Ma 0 Ma 0 ZL ZL C C _ tÞw0 ðx,tÞdx wðx,tÞw_ 0 ðx, tÞdx þ wðx, þ Ma 0 Ma 0 ð31Þ
þ
_ þ 2 ð0 1Þ yI ðtÞ yðtÞ V_ 3 ðtÞ ¼ 1 yðtÞyðtÞ _ Þ þ 2 ðyI ðtÞ þ yðtÞÞðyðtÞ þ yðtÞ
L
ð36Þ
0
Z
ð32Þ
L
¼
00 b ðx, tÞw_ 00 ðx, tÞdx EIðxÞw
0
1 1 V_ 4 ðtÞ ¼ k_p ðtÞ kp ðtÞ kp þ k_i ðtÞ ki ðtÞ ki 1 2 1 _ þ kd ðtÞ kd ðtÞ kd 3
ð33Þ
and Z
L 0
By substituting (7) into (30), it yields
V_ 1 ðtÞ ¼
Z Ma
L 00
_ tÞMa uðtÞS ðxÞdx wðx, 0
Z
ð37Þ
Substituting (35) to (37) into (34) yields B _ V_ 1 ðtÞ ¼ uðtÞyðtÞ Ma
Z
L
w_ 2 ðx, tÞdx 0
C w_ 2 ðL, tÞ 2Ma ð38Þ
L
00 b _ tÞEIðxÞw ðx, tÞ00 dx wðx, Ma 0 ZL ZL B C 2 _ tÞw_ 0 ðx, tÞdx w_ ðx, tÞdx wðx, Ma 0 Ma 0 ZL 00 b ðx, tÞw_ 00 ðx, tÞdx EIðxÞw þ Ma 0 ð34Þ
1 _ tÞw_ 0 ðx, tÞdx ¼ Cw_ 2 ðL, tÞ Cwðx, 2
Next, by substituting the governing equation (7) into (31), it results in V_ 2 ðtÞ ¼
Z Ma Ma
L
0 ZL 0
wðx; tÞMa uðtÞS00 ðxÞdx 00 00 b wðx; tÞ EIðxÞw ðx; tÞ dx
8
Journal of Vibration and Control Z
L
þ
cmax ¼ b bb tb þ a ba ta AðxÞ, Then, by utilizing A Lemma 1 and Lemma 2, it results in
c w_ 2 ðx; tÞdx AðxÞ
Ma 0 ZL C _ tÞw0 ðx; tÞdx wðx; þ Ma 0
ð39Þ
Integrating the first term of (39) and substituting the boundary conditions in (2) yields Z
Z
ZL 002 cmax w_ 2 dx b A dx EIðxÞw Ma 0 Ma 0 Z L 2 C w_ ðx, tÞ þ 3 L2 w002 ðx, tÞ dx þ uðtÞ yðtÞ þ Ma 0 3 ð43Þ L
L
wðx, tÞMa uðtÞS00 ðxÞdx 0 L ¼ Ma uðtÞðwðx, tÞS0 ðxÞ0 ZL w0 ðx, tÞS0 ðxÞdxÞ
where 3 is a positive constant. By substituting (32), (33), (38) and (43) into (29), the following inequality is obtained
0
L ¼ Ma uðtÞðwðx, tÞS0 ðxÞjL0 w0 ðx, tÞSðxÞ0 ZL þ w00 ðx, tÞSðxÞdxÞ 0
Z Z
ð40Þ
w00 ðx, tÞSðxÞdx 0 l2
¼ Ma uðtÞ
w00 ðx, tÞdx
l1 0
¼ Ma uðtÞðw ðl2 , tÞ w0 ðl1 , tÞÞ Ma uðtÞ yðtÞ ¼
Similarly, the second terms on the right-hand side of (39) can be obtained as 00 00 b wðx, tÞ EIðxÞw ðx, tÞ dx 0 00 b ¼ w0 ðL, tÞ EIðLÞw ðL, tÞ ZL ð41Þ 002 b þ w0 ð0, tÞðEIð0Þw00 ð0, tÞÞ þ ðx, tÞdx EIðxÞw 0 ZL 002 b ðx, tÞdx EIðxÞw ¼ L
0
Substituting (40) and (41) into (39) yields V_ 2 ðtÞ ¼ uðtÞ yðtÞ Z þ
Ma
L 0
Z Ma
L
002 b dx EIðxÞw
0
c w_ 2 dx þ C AðxÞ Ma
Z
Z L cmax C0 w_ 2 ðx, tÞdx B 0 A Ma 0 3 Z L 0 b EIðxÞ C3 L2 w002 ðx, tÞdx Ma 0 C _ w_ 2 ðL, tÞ þ 1 yðtÞyðtÞ 2Ma _ Þn þ 2 ð0 1Þ yI ðtÞ yðtÞ þ 2 ðyI ðtÞ þ yðtÞÞðyðtÞ þ yðtÞ 1 1 þ k_p ðtÞ kp ðtÞ kp þ k_i ðtÞ ki ðtÞ ki 1 2 1 _ ÞuðtÞ þ k_d ðtÞ kd ðtÞ kd þ ð0 yðtÞ þ yðtÞ 3 ð44Þ
_ VðtÞ
L
¼ Ma uðtÞ
Z
V_ 2 ðtÞ
L
_ tÞw0 ðx, tÞdx wðx,
0
ð42Þ
Subsequently, substituting the control law (8) gives cmax C0 w_ 2 ðx, tÞdx B 0 A Ma 0 3 Z L 0 b EIðxÞ C3 L2 w002 ðx, tÞdx Ma 0 C w_ 2 ðL, tÞ kd ðtÞy_ 2 ðtÞ 2Ma 22 2 _ þ 0 y2 ðtÞ 2 y ðtÞ kp ðtÞ yðtÞyðtÞ 0 _ þ 0 yI ðtÞ yðtÞÞ ðki ðtÞ 2 ÞðyI ðtÞyðtÞ 1 22 _ 1 þ 2 0 yðtÞyðtÞ kd ðtÞ 0 0 1 þ k_p ðtÞ kp ðtÞ kp 1 1 1 þ k_i ðtÞ ki ðtÞ ki þ k_d ðtÞ kd ðtÞ kd 2 3 ð45Þ
_ VðtÞ
Z
L
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9
2 i ¼ 2 and kd ¼ 1 1 þ 2 22 By setting kp ¼ 2 , k 0 0 0 and substituting the adaptive law (9) into (45) results in Z L C0 2 c B0 Amax w_ ðx,tÞdx Ma 0 3 Z L 0 C 2 002 b EIðxÞC w_ 2 ðL,tÞ 3 L w ðx,tÞdx Ma 0 2Ma kd ðtÞy_ 2 ðtÞ2 y2 ðtÞ k2p ðtÞþ kp ðtÞkp 1 1 ð46Þ
Next, the conditions (48) and (49) are examined. Since b b b =CL2 satisfies the conb b 5 EIðxÞ, choosing 3 ¼ EI EI b b =CL2 into dition (49). Thus, by substituting 3 ¼ EI (48) yields
_ VðtÞ
Then, applying Lemma 1 to the last term on the righthand side yields Z L cmax C0 w_ 2 ðx,tÞdx _ VðtÞ B0 A Ma 0 3 Z L 0 C 2 002 b EIðxÞC w_ 2 ðL,tÞ 3 L w ðx,tÞdx Ma 0 2Ma ð14 Þk2p ðtÞ k2p kd ðtÞy_ 2 ðtÞ2 y2 ðtÞ þ 1 1 4 ð47Þ where 0 5 4 5 1. Thus, if kd ðtÞ 0 and the following conditions are met cmax B 0 A
C0 0 3
ð48Þ
and b EIðxÞ C3 L2 0
ð49Þ
_ can further be bounded as VðtÞ _ VðtÞ 2 y2 ðtÞ þ
0
bb b bB B BEI EI 1 5 0 min , , 2 2 cmax 8C L C2 L2 þ EI cmax b b A 8A
1 4
rffiffiffiffiffiffiffiffiffiffiffiffiffiffi 8 j yðtÞj kp 2 1 4
3 0 2 y ðtÞ y2 ð0Þ þ kd ð0Þ 2
ð50Þ
ð55Þ Thus, the conditions (53) and (55) sufficiently yield V(t) _ as negative outside of the as positive definite and VðtÞ
qffiffiffiffiffiffiffiffiffiffi set yðtÞ : j yðtÞj 2 1 4 kp . Therefore, by considering V(t) as a Lyapunov function, it yields that RL R 002 c w_ 2 ðx, tÞdx, and L EIðxÞw b ðx, tÞdx yI ðtÞ, yðtÞ, 0 AðxÞ 0 are semi-globally bounded. Hence, the y(t) can be made arbitrarily small by appropriately adjusting the controller parameters. In addition, the boundness of RL RL 00 2 c _2 b ðx, tÞdx imply that 0 AðxÞw ðx, tÞdx and 0 EIðxÞw the kinetic and potential energies of the controlled _ is also beam are bounded. Thus, from Property 1, yðtÞ bounded. Moreover, if ¼ 0 is chosen, it results in lim yðtÞ ¼ 0. The asymptotic convergence of y(t) can be proved by using Barbalat’s Lemma (Slotine, 1991; Khalil, 2002) as follows. Setting ¼ 0 in (50) yields _ VðtÞ 2 y2
ð51Þ
ð52Þ
Thus, kd ðtÞ 0 for 8t 2 ½0, 1Þ if kd ð0Þ satisfies the following condition
kd ð0Þ
!
t!1
k2p
The nonnegative property of kd ðtÞ can be achieved as follows. By solving the adaptive law _ in (9) results in k_d ðtÞ ¼ 3 0 yðtÞyðtÞ kd ðtÞ ¼
ð54Þ
which guarantees the conditions (48) and (49). Then, by combining with the condition (28) leads to a new condition for 0 as
It immediately follows that _ VðtÞ 0,
bb BEI cmax b b A C2 L2 þ EI
3 0 2 y ð0Þ 2
ð53Þ
ð56Þ
By solving (56), it results in Z
t
y2 ðÞd 0
1 Vð0Þ ðVð0Þ VðtÞÞ 2 2
ð57Þ
which implies that y2(t) is integrable. Taking the deriva_ _ are tive of y2(t) results in 2yðtÞyðtÞ. Since y(t) and y(t) bounded, the derivative of y2(t) is bounded. Thus, y2(t) is uniformly continuous. Therefore, by Barbalat’s Lemma (Slotine, 1991; Khalil, 2002), it yields that lim yðtÞ ¼ 0. t!1
Remark 1. The closed-loop stability is theoretically guaranteed for all vibration modes since the analysis was proven directly from the governing PDE model,
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Figure 4. Experimental control results using the proposed adaptive PID-like controller: (a) output signal, (b) control signal, (c) adaptive gains.
which is infinite-dimensional. Because most controllers are nowadays implemented in computers, any time needed for computing control laws creates additional time delay, which can cause instability about high-frequency modes. However, computers have become very fast. Thus, the delay will be very small and can be neglected. In addition, the viscous and structural damping existing in the beam will damp out high-frequency modes quickly.
4. Experimental studies The experimental test bench described in Section 2 was used in the experimental studies. The proposed controller (8) with the adaptive law (9) was implemented digitally on the computer with the sampling period of the control loop of 1 ms. Based on prior finite element simulation studies, the control design parameters were chosen as follows: 1 ¼ 2 ¼ 3 ¼ 0.01, kp(0) ¼ ki(0) ¼ kd(0) ¼ 0.05, and 0 ¼ 10. Note that 0 ¼ 10 and kd(0) ¼ 0.05 satisfy respectively the conditions (55) and (53), assuming yð0Þ 1. Moreover, the adaptive law (9) contains y2(t), which can cause kp(t) to grow unboundedly due
the measurement noise in y(t) if ¼ 0 is chosen. Thus, to prevent this problem, ¼ 0.0005 was chosen in the experiment. Note that the utilization of nonzero is comparable to the well-known -modification technique (Ioannou and Sun, 1996) in classical adaptive control. The output signal y(t) was measured directly from _ the sensor whereas yðtÞ and yI(t) were numerically computed from the output signal y(t). To reduce the influence of measurement noise, a recently developed robust differentiator called the Uniform Robust Exact Differentiator (URED) (Cruz–Zavala et al., 2011) was _ utilized to obtain yðtÞ. The URED for the output signal y(t) can be summarized as follows (Cruz–Zavala et al., 2011) z_1 ¼ k1 1 ðsÞ þ z2 z_2 ¼ k2 2 ðsÞ 1 ðsÞ ¼ 1 jsj1=2 signðsÞ þ 2 jsj3=2 signðsÞ
2 2 ðsÞ ¼ 21 signðsÞ þ 2 1 2 s þ 32 22 jsj2 signðsÞ
ð58Þ
_ where s ¼ z1 yðtÞ and z2 estimates yðtÞ. The parameters of the URED used in the experiment were chosen as k1 ¼ k2 ¼ 2 and 1 ¼ 2 ¼ 2.
Sangpet et al.
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Figure 5. Experimental control results using a velocity feedback controller: (a) output signal, (b) control signal.
Figure 6. Experimental control results using a PD controller: (a) output signal, (b) control signal.
The experimental results are shown in Figure 4. The controller was activated at t ¼ 1 s. The results show that the proposed controller effectively suppressed the vibration of the beam with a zero steady-state output. Notice that the adaptive gains converged to constants within about 1 second after the controller was activated. For comparison purposes, a conventional velocity feedback controller and a PD controller were also implemented. The conventional velocity feedback controller was implemented as
controller; however, they resulted in nonzero steadystate outputs.
_ uðtÞ ¼ kd yðtÞ
ð59Þ
where kd is the controller parameter. Here, kd ¼ 0.051 was chosen such that the time response of the controlled system was approximately the same as of the proposed adaptive PID-like controller. The result is shown in Figure 5. The PD controller was implemented as (Luemchamloey and Kuntanapreeda, 2014) _ uðtÞ ¼ Kð0 yðtÞ þ yðtÞÞ
ð60Þ
where K and 0 are the controller parameters. Similarly, K ¼ 20 and 0 ¼ 0.3 were chosen. The result is shown in Figure 6. As shown in Figures 5 and 6, both controllers provided similar vibration suppression rates compared to those of the proposed adaptive PID-like
Remark 2. The proposed controller was derived as a continuous-time controller, but the implementation was carried out in a computer that involved sampling. Thus, the implemented controller was actually a discrete-time controller with a zero-order hold. However, from a practitioner’s point of view, the sampling effect can be ignored since the sampling is very fast. The stability analysis taking consideration on the sampling is considered as another challenging research topic.
5. Conclusion In this paper, an adaptive PID-like controller for vibration suppression of a flexible beam bonded with a collocated piezoelectric actuator and sensor pair has been presented. A governing partial differential equation (PDE) of the beam was used in the controller design. The design made use of an integral action and an adaptive mechanism to enhance steady state error performance of the control system. The stability of the closed-loop PDE system was provided using an infinite-dimensional Lyapunov stability method. The feasibility of the proposed controller was validated by the experimental results.
12 Acknowledgements The authors would like to thank Professor PC Mu¨ller, University of Wuppertal, Germany, for his useful remarks and comments. Thanks are also due to the anonymous referees for their valuable and constructive criticism and suggestions.
Declaration of conflicting interests The author(s) declared no potential conflicts of interest with respect to research, authorship, and/or publication of this article.
Funding The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by the Royal Golden Jubilee PhD program of the Thailand Research Fund.
References Azadi M, Fazelzadeh SA, Eghtesad M, et al. (2011) Vibration suppression and adaptive-robust control of a smart flexible satellite with three axes maneuvering. Acta Astronautical 69(5): 307–322. Cruz–Zavala E, Moreno J and Fridman L (2011) Uniform robust exact differentiator. IEEE Transactions on Automatic Control 56(11): 2727–2733. Bandyopadhyay B, Manjunath TC and Umapathy M (2007) Modeling and Implementation of Smart Structure: A FEMState Space Approach. Berlin: Springer–Verlag. Dadfarnia M, Jalili N, Xian B, et al. (2004) A Lyapunovbased piezoelectric controller for flexible Cartesian robot manipulators. Journal of Dynamic Systems, Measurement, and Control 126(2): 347–358. Donaire A and Junco S (2009) On the addition of integral action to port-controlled Hamiltonian systems. Automatica 45(8): 1910–1916. Esbrook A, Tan X and Khalil HK (2014) Inversion-free stabilization and regulation of systems with hysteresis via integral action. Automatica 50(4): 1017–1025. Guo BZ and Lin FF (2013) The active disturbance rejection and sliding mode control approach to the stabilization of the Euler–Bernoulli beam equation with boundary input disturbance. Automatica 49(9): 2911–2918. Ioannou PA and Sun J (1996) Robust Adaptive Control. Englewood Cliffs: Prentice–Hall. Jovanovic´ MM, Simonovic´ AM, Zoric´ ND, et al. (2013) Experimental Studies on active vibration control of a smart composite beam using a PID controller. Smart Materials and Structures 22(11): 115038 (8 pages). Khalil HK (2002) Nonlinear Systems. Englewood Cliffs: Prentice–Hall. Kuntanapreeda S and Marusak PM (2012) Nonlinear extended output feedback control for CSTRs with van de Vusse reaction. Computers and Chemical Engineering 41(11): 10–23. Lee KW and Singh SN (2012) L1 adaptive control of flexible spacecraft despite disturbances. Acta Astronautica 80: 24–35. Li Y, Tong S and Li T (2013) Adaptive fuzzy output feedback control for a single-link flexible robot manipulator driven
Journal of Vibration and Control DC motor via backstepping. Nonlinear Analysis: Real World Applications 14(1): 483–494. Luemchamloey A and Kuntanapreeda S (2014) Active vibration control of flexible beams based on infinite-dimensional Lyapunov stability theory: an experimental study. Journal of Control, Automation and Electrical Systems 25(6): 649–656. Montazeri A, Poshtan J and Yousefi–Koma A (2011) Design and analysis of robust minimax LQG controller for an experimental beam considering spill-over effect. IEEE Transactions on Control Systems Technology 19(5): 1251–1259. Qiu Z and Xu Y (2016) Vibration control of a rotating twoconnected flexible beam using chattering-free sliding mode controllers. Proceedings of the Institution of Mechanical Engineers, Part G: Journal of Aerospace Engineering 230(3): 444–468. Queiroz MS, Dawson DM, Nagarkatti SP, et al. (2000) Lyapunov Based Control of Mechanical Systems. Boston: Birkha¨user. Sales TP, Rade DA and De Souza LCG (2013) Passive vibration control of flexible spacecraft using shunted piezoelectric transducers. Aerospace Science and Technology 29(1): 403–412. Saad MS, Jamaluddin H and Darus IZM (2015) Active vibration control of a flexible beam using system identification and controller tuning by evolutionary algorithm. Journal of Vibration and Control 21(10): 2027–2042. Sangpet T, Kuntanapreeda S and Schmidt R (2014) Adaptive vibration control of piezoactuated Euler–Bernoulli beams using infinite-dimensional Lyapunov method and highorder sliding-mode differentiation. Journal of Engineering 2014: 839128 (9 pages). Sharifnia M and Akbarzadeh A (2016) An analytical model for vibration and control of a PR-PRP parallel robot with flexible platform and prismatic joint. Journal of Vibration and Control 22(3): 632–648. Sharma A, Kumar R, Vaish R, et al. (2015) Active vibration control of space antenna reflector over wide temperature range. Composite Structures 128: 291–304. Simonovic´ AM, Jovanovic´ MM, Lukic´ NS, et al. (2016) Experimental studies on active vibration control of smart plate using a modified PID controller with optimal orientation of piezoelectric actuator. Journal of Vibration and Control 22(11): 2619–2631. Slotine JE (1991) Applied Nonlinear Control. Englewood Cliffs, NJ: Prentice–Hall. Wu D, Huang L, Pan B, et al. (2014) Experimental study and numerical simulation of active vibration control of a highly flexible beam using piezoelectric intelligent material. Aerospace Science and Technology 37: 10–19. Zhang SQ, Li HN, Schmidt R, et al. (2014) Disturbance rejection control for vibration suppression of piezoelectric laminated thin-walled structures. Journal of Sound and Vibration 333(5): 1209–1223. Zhang T, Li HG and Cai GP (2013) Hysteresis identification and adaptive vibration control for a smart cantilever beam by a piezoelectric actuator. Sensors and Actuators A: Physical 203: 168–175. Zhang X, Xu W, Nair SS, et al. (2005) PDE modeling and control of a flexible two-link manipulator. IEEE Transactions on Control Systems Technology 13(2): 301–312.
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A1.1 Numerical simulations The beam described in Section 2 was simulated using a finite element (FE) method (Bandyopadhyay et al., 2007). The sensor was not included in the FE model since the effect of the sensor on the dynamic of the beam is very small. For simplicity, the model was divided into 11 elements as shown in Figure 7. Each element has two nodes and each node has two degrees of freedom. Figure 8 displays the dynamic response of the uncontrolled beam under an impulsive load 0.2 N acting for 0.005 s at the node connecting elements 2 and 3. The result shows that the FE model represents the experimental system very well. Next, the FE model was used to test the proposed PID-like controller. The same control design parameters used in the experiment were also used here. The control results from the simulation are shown in Figure 9. The controller was activated at t ¼ 1 s. The results show that the simulation gave similar results to those of the experiment.
sented in this appendix. First, 0 was increased from 10 to 100. The results are shown in Figure 10. By comparing to the results in Figure 4, it is noticed that the convergent rate is faster; however, the control signal is larger. Next, i,i¼1,2,3 were increased from 0.01 to 0.1 whereas 0 ¼ 10. The results are shown in Figure 11. Similar results to those in Figure 4 were obtained.
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Appendix 1
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A1.2 Experimental results with other design parameters More experimental results with other values of the control design parameters 0 and i,i¼1,2,3 are pre-
Figure 7. Finite element model of the experimental system.
Figure 8. Uncontrolled dynamic responses of the beam.
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Figure 10. Experimental control results using the proposed adaptive PID-like controller with 0 ¼ 100: (a) output signal, (b) control signal, (c) adaptive gains.
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Figure 11. Experimental control results using the proposed adaptive PID-like controller with i,i¼1,2,3 ¼ 0.1: (a) output signal, (b) control signal, (c) adaptive gains.
