Vibration Control of Smart Structure Using Sliding Mode Control with Observer

Transcription

1 JOURNAL OF COMPUERS, VOL. 7, NO., FEBRUARY Vibration Control of Smart Struture Using Sliding Mode Control with Observer Junfeng Hu Shool of Mehanial & Eletrial Engineering, Jiangxi University of Siene and ehnology, Ganzhou, 3, China Dahang Zhu Shool of Mehanial & Eletrial Engineering, Jiangxi University of Siene and ehnology, Ganzhou, 3, China Abstrat his paper studies the appliation of the sliding mode ontrol method to redue the vibration of flexible struture with piezoeletri atuators and strain gage transduer in pratial omplex environment. he statespae dynami model of the system was derived by using finite element method and experimental modal test. he struture is subjeted to arbitrary, unmeasurable disturbane fores. aking into aount the unertain random disturbane and measurement noise, Kalman filter is hosen as the state estimator to obtain the modal oordinates and modal veloities for the modal spae ontrol. A sliding mode ontroller is adopted due to its distinguished robustness property of insensitiveness to parameter unertainties and external disturbanes. he sliding surfae is determined by using optimization method, and the sliding ontroller is designed by applying Lyapunov diret method. hat is, along the swithing surfae, the ost funtion of the states is minimized. A real-time ontrol system was built using dspace DS3 platform, and vibration ontrol tests were performed to experimentally verify the performanes of the proposed ontroller. he results of experiment show the ontroller an effetively attenuate elasti vibration of the struture. Index erms ative vibration ontrol, smart struture, sliding mode ontrol, experimental modal test, Lyapunov diret method I. INRODUCION In preision and aerospae industry, many researhes on lightweight and miniaturized strutures have been arried out to improve strutural performanes. Among the researhes, passive strutures using omposite material are typially known as one of the effetive methods. However, the traditional passive strutures are very sensitive to hange of internal load ondition and external environment ondition whih an even ause sudden destrution of strutures. herefore, in order to satisfy stringent requirements for preision ontrol and Manusript reeived Nov., ; revised Jan. 5, ; aepted Jan.,. lightweight miniaturization, smart materials suh as shape memory alloys, piezoeramis, eletroorheologial fluids and magnetorheologial fluids are frequently adopted for smart strutures. he performane requirements of future spae strutures, jet fighters and onept automobiles have brought muh interest to the area of smart strutures. A smart struture an be defined as a struture with bonded or embedded sensors and atuators as well as an assoiated ontrol system, whih enable the struture to respond simultaneously to external stimuli exerted on it and then suppresses undesired effets or enhane desired effets. Among various smart strutures, those with piezoeletri pathes have reeived muh attention in reent years, due to the fat that piezoeletri materials have simple mehanial properties, small volume, light weight, large useful bandwidth, effiient onversion between eletrial and mehanial energy, good ability to perform vibration ontrol and ease of integration into metalli and omposite strutures [-3]. Different ontrol tehniques have been investigated in the ontrol of smart struture. Abreu onduted experimental work for the vibration ontrol of flexible beam by using piezoeletri sensors and atuators with Linear Quadrati Gaussian (LQG) ontroller []. here are many lassial strategies that an be used when the mathematial model is available, for instane pole alloation and optimal ontrol. However, if the model has unertainties these methods are not indiated. here are many robust tehniques in strutural ontrol literature. Li investigated two ontrol strategies for robust vibration ontrol of parameter unertain systems [5]. Mayhan ombined intelligent ontrol and smart materials to produed an adaptive and robust ontroller to dampen the fundamental vibration mode of the system in the presene of modeling unertainties [6]. Zhang et al. studied the ative vibration ontrol problem for the high-speed flexible mehanisms all of whose members were onsidered as flexible by using omplex mode method and robust H ontrol sheme [7-8]. Kawabe utilized neural networks (NN) theory for ative ontrol in a longitudinal antilevered-beam system by simulation and experiment. It is found that fairly satisfatory ative ACADEMY PUBLISHER doi:.3/jp.7..-8

2 JOURNAL OF COMPUERS, VOL. 7, NO., FEBRUARY damping effet using the NN ontroller is obtained [9]. But the random disturbane and measurement noise of the atual system were not onsidered by these urrently proposed vibration ontrol strategies. he issue of robustness against external disturbanes was not addressed, and therefore the proposed vibration ontrollers annot be effetively applied to the smart struture under the random unertain disturbanes. Beause sliding mode ontrol has inherent robustness to system parameter variation and external disturbanes, it is meaningful to investigate its appliation in vibration ontrol of smart struture. We aim here to deal with the ative vibration redution problem in flexible struture with unertainties through designing reasonable sliding mode ontroller. he developed ontrol strategy integrates the sliding mode ontrol strategy and Kalman filter tehnique. In this paper, the vibration ontrol of a flexible beam is investigated by using sliding mode ontrol with observer and experimental modal test method, and taking into aount the random disturbane unertainty, modal parameter unertainty and measurement noise. he paper is organized as follows. In setion, a dynami model of a flexible beam with piezoeletri atuators and strain gauge sensors is onstruted by using finite element method. In setion 3, the sliding mode ontroller with observer is proposed, and sliding surfae is determined by using optimization method, and the sliding ontroller is designed by applying Lyapunov diret method. In setion, experimental identifiation of the flexible antilever beam is performed to obtain its modal parameters. And the experimental validation test is performed based on the dspace DS3 platform. he onlusions are given in setion 6. II. DYNAMIC MODELING OF SMAR SRUCUE he modeling of smart struture with piezoeletri atuators and sensors has been a subjet of intense researh for a long time and is only briefly desribed here. y u 3 u A u u L e u 6 B u 7 u 5 Figure. Planar beam element showing nodal degrees of freedom and oordinate systems he flexible struture is modeled by using a two-node beam element. he beam element is shown in Fig., whih has two nodes with four degrees of freedom at eah node; namely u, u 5, the longitudinal displaement, and u, u 6,the transverse displaement, and u 3, u 7, the slope, and u, u 8, the urvature. L e is the length of element. he nodal displaement vetor u with respet to referene frame A xy is expressed as u = [ u ] u u3 u u5 u6 u7 u8 u 8 x he transverse and longitudinal displaement fields of two-node beam element are onstruted using the quinti hermite and linear interpolation polynomials, respetively. V, W denotes longitudinal and transverse elasti displaement of arbitrary point, respetively. Subsript denotes matrix transpose. hey an be written by the following form V ( x, t) N ( x) = u W ( x, t) N ( x) where N ( x ) = ( ) ( ) N x N x is shape funtion, u is the nodal displaement vetor. he system dynami equations an be obtained by using finite element method MU + CU + KU = P () where M, C, K are the systemati mass, damping, stiffness, respetively. U, U, U are the generalized displaement, veloity, and aeleration vetors of the system, respetively. P is the systemati generalized disturbane fore vetor orresponding to vetor U. he piezoeletri pathes work as atuators are perfetly bonded on the upper and lower surfaes of the beam at the same loation. For the modeling of PZ atuators, literature [, ] provides a detailed derivation of oupling of PZ atuators and a host beam. hese bending moments indued by the atuators is given by A = B = d3epbp ( tp + ta ) V in () where d 3, E p, b p, tp is piezoeletri onstant, elasti module, width, and thikness of PZ path, respetively. t a is the thikness of the beam element, V in is the vetor of input voltage to the piezoeletri atuators. he moments are assembled as a part of the external moments exerted on node. Strain is the amount of deformation of a struture due to an applied fore. Strain gauge is the most ommon sensor for measuring strain. A strain gauge s eletrial resistane varies in proportion to the amount of strain plaed on it. he deformations mainly inlude ompression or tension and bending deformation for the flexible beam. he strainε in x diretion is given by ε = ε + ε (3) where ε L is ompression or tension strain and ε B are longitudinal strain due to bending deformation, respetively. hey an be given by ε L = N u= N BU, () ε = hn u= hn BU B where h is the distane between the neutral axis of the beam and the outer surfae of the beam, N, N are the first-order and the seond-order differential of the shape funtion N and N, respetively, B denotes the transformation matrix. hus, the exogenous perturbation and the ontrol inputs have no diret effet on the measured outputs. L B ACADEMY PUBLISHER

3 JOURNAL OF COMPUERS, VOL. 7, NO., FEBRUARY 3 here are a piezoeletri atuators and s strain gage sensors on the flexible struture. Combining Eq. () ~ Eq. (), the dynami equations of the struture equipped with piezoeletri atuators and strain gauge transduer an be expressed as MU + CU + KU = P + DaVin (5) y = DU s where D a is the systemati ontrol matrix related to onfiguration of atuators, D s is the systemati output matries determined by onfiguration of sensors, a a -by- vetor, and y is the strain from the sensors, a s -by- vetor. It was shown that the dynami response of the flexible struture is omposed mainly of the lower modes. In order to ontrol the lower modes, the physial-oordinate equations must be first transformed into modal oordinates. Here, we hoose the first order modes as ontrol modes. Applying the modal theory, the normalized modal transformation is introdued by U = ψη (6) where ψ is the ontrolled normalized modal matrix, η is ontrolled modal oordinate vetors. Substituting Eq. (6) into Eq. (5), the system dynami equations are rewritten as η + Cη + Kη = N + DaVin (7) y = Dsη where N = ψ P, Da = ψ Da, Ds = ψ D C s, K is diagram matrix, whih is determined by system natural frequeny and damping ratio. For ontrol synthesis, the system must be written as a system of first-order ordinary differential equations (ODEs). We an define the ontrolled state variables by the following form X [ ] = η η η η Due to the ontrolled mode number, the number of ontrolled state variables is. aking into aount measurement noise ν, the state-spae model for the system an be written as X = AX + BV in + N (8) y = CX + ν I a where A = K C, B = Da, C = [ Ds s], I is unit matrix, a, s are a, s zero matrix, respetively. III. CONROL SYSEM DESIGN Sliding mode ontroller (SMC) is a funtion of more than two strutures and gives some desirable losed-loop properties. he desirable features inlude invariane, order redution, and robustness against parameter variations and disturbanes. he stability and robustness of SMC is guaranteed using the onept of swithing funtion and Lyapunov stability theory. Sliding mode ontrol is to design a ontroller suh that the motion of the system tends to sliding mode surfae. herefore, the design of SMC inludes the determination of sliding surfae and ontroller design. he blok diagram of the ontrol system is shown in Fig.. In the figure, X ˆ is the estimated value of state vetor X by using Kalman filter. And the measurement noise ν is onsidered during the ontroller design. ˆX Kalman Estimator Sliding Mode Controller y N Vin Flexible Struture Figure. Blok diagram of the ontrol system A. Sliding Mode Surfae Design he problem now is to determine a sliding surfae whih guarantees stable sliding mode motion on the surfae itself. he sliding surfae S an be expressed as S = HX = (9) where S = [ S S S a ], Si is the sliding variable, H is a a matrix. In order to determine the matrix H, the state of the system is to make the following linear transformation. Z = ΓX () where Γ is the transformation matrix, it an be given by I BB B Γ =, B = I B where I and I are ( a) a and a a unit matrix, B and B are ( a) a and a a matrix, respetively, and B is nonsingular matrix. Substituting Eq. () into Eq. (8), the state equation and sliding surfae an be rewritten as the following form expressed by Z. Z = ΓAΓ Z + ΓBVin = AZ + BV in () S = HZ = where A = ΓA Γ, B, B = H = HΓ ν + y. he Eq. () an be partitioned to yield Z A A Z =, A =, H = H H () Z A A where Z and Z are a and a dimensional vetor, respetively, the dimension of other matrix are deided by the dimension of Z and Z. hus, ombining Eq. () and Eq. (), the following equations an be obtained ACADEMY PUBLISHER

4 JOURNAL OF COMPUERS, VOL. 7, NO., FEBRUARY Z = A Z + A Z (3) S = H Z + H Z = In order to simplify the design of sliding surfae, we an assume H = I () where I is unit matrix. herefore, ombining Eq. (3) and Eq. (), the following equations an be obtained where Z is onsidered as state variables, Z is output variables. Z = ( A A H ) Z (5) Z = H Z Using the optimal method, the matrix H is determined by minimizing the following performane index J. J = X t RX dt (6) where R is positive define weighting matrix. Considering the physial meaning of state variable X, the weighting matrix an be seleted by the following expression. K M R = M M where M, K is the mass matrix and stiffness matrix orresponding to the first ontrol modes, respetively. Substituting Eq. () into Eq. (6), the performane index funtion an be written by the form of state variable Z J Z = t Z Z dt (7) Z where = ( Γ ) RΓ = Combing Eq. (5) and Eq. (7), the solution of the performane funtion is a linear quadrati optimization problem. Using the Linear Quadrati (LQ) algorithm, the optimization problem an be solved and the following expression an be obtained Z = KoZ (8) Comparing Eq. (8) and Eq. (5), we an obtain H = K,H = [ Ko I] Γ (9) Substituting Eq. (9) into Eq. (9), the sliding surfae an be obtained. B. Sliding Mode Controller Lyapunov diret method is applied to design the sliding mode ontroller suh that the response of the system an tend to the sliding surfae expressed in Eq. (9) by determining the ontrol input. Suppose the Lyapunov funtion is defined as v = S S = X H HX () Aording to Lyapunov asymptoti stability ondition and Eq. (), we an obtain v = S S () where S = HX. Substituting Eq. (8) into Eq. (), the above equation an be rewritten as v = S HX = S H ( AX + BVin + N ) = χ ( Vin G) where χ = S HB, G = ( HB) ( HAX + HN ). In order to satisfy the Lyapunov stable ondition v, the ontrol voltages an be seleted as V = in G δχ () where δ is diagonal matrix whose diagonal elements δi. hus, the expression v = χδχ an be satisfied. Meanwhile, taking aount into the bound of ontrol voltage, thus, the ontrol inputs are set as Vi Vi Vm Vi = (3) Vmsgn( Vi ) Vi > Vm where subsript i denotes the i th ontrol input voltage, V i is the i th voltage omputed by the ontroller, Vi is the atual voltage applied to the atuator, V m is the bound of the ontrol voltage, sgn( ) represents sign funtion. C. Kalman state estimator Sine the modal positions and veloities of the system are not diretly measurable, a state observer may be employed to obtain the estimated states. he Kalman filter an provide an effiient way to estimate the state of a system, in a way that minimizes the mean of the squared error. Considering external disturbane and measurement noise, we an design a state estimator given the state-spae model of the system shown in Eq. (8) by using Kalman tehnology. Here, suppose the exogenous disturbane and measurement noise satisfy the following onditions E( N ) = E( ν) = () E( NN ) = Qd, E( νν ) = Rn where E() denotes the expeted value of a variable, Q d, R n are ovariane of the external disturbane and measurement noise, respetively. We an onstrut a state estimate X ˆ that minimizes the steady-state error ovariane P = E X Xˆ X X ˆ ({ }{ } ) lim t he optimal solution is the Kalman filter with the following state equations ˆ X = AXˆ + BV in + L( y CXˆ ) yˆ = C ˆ X where the filter gain matrix L is determined by solving orresponding algebrai Riati equation, ŷ are estimated value of y. he estimator uses the known inputs V in and the measurements y to generate the output and state estimates as shown in Fig.. ACADEMY PUBLISHER

5 JOURNAL OF COMPUERS, VOL. 7, NO., FEBRUARY 5 VI. EXPERIMENAL INVESIGAEION In this setion, we shall experimentally evaluate the effetiveness of the proposed ontrol method in the vibration ontrol of a flexible beam. A. Experimental setup N N S S N N 3 N5 N N N E E3 E5 E7 A E A E A E 3 6 Figure 3. Configuration of the elements and nodes, atuator and sensor, N, E, A, S denote node, atuator, sensor, respetively (h) (i) (j) (l) (k) (e) (f) (d) Figure. Experimental setup (a) signal generator (b) antilever beam () aelerometer (d) dynami signal aquisition (e) hammer (f) dynami strain gauge (g) strain gauge (h) piezoeletri path (i) DS3 onnetor panel (j) eletri bridge box (k) voltage amplifier (l) industrial omputer he length of the antilever beam is 3 mm, its width and thikness are mm,.5 mm, respetively. Its material is steel with elasti module, density and Poisson s ratio GPa, 78 kg/m3,.3, respetively. he beam is divided into 7 beam elements and 8 generalized oordinates shown as in the Fig. 3. he lengths of the elements are 5 mm, mm, mm, mm, mm, mm, 5 mm, respetively. he symbols N, E, A, S denote node, atuator, sensor, respetively, in the Fig. 3. he atuators are made of PZ-5H piezoeletri erami, whih its thikness is.5 mm, length is mm and width is mm, piezoeletri onstant d 3, elasti module and density is m/v, GPa, 765 kg/m 3, respetively. he sensors are resistane strain gage. he onfiguration of the atuators and sensors are shown in Fig.3. he three pairs atuator bonded on the link are loated at elements E, E and E 6. wo sensors S, S are loated at the midpoint of (a) () (b) (g) elements E3 and E 5. hree atuators A, A, A 3 are loated at elements E, E and E 6, respetively. Fig. illustrates the experimental setup that onsists of a flexible beam bonded on PZ and strain gauge. By the way, the experimental devie is also used to study the vibration ontrol of high-speed 5-bar mehanism. As the bolt is tighten, the rotational pair beomes a fixed pair, that is, the flexible link an be onsidered as a antilever beam. he high-speed analog input and output ports are provided with dspace DS3. he eletri amplifier made in Harbin ore tomorrow siene and tehnology Co., Ltd is used to drive piezoeletri path. Resistane strain gauges are made in ZEMIC Co., Ltd, its type is BE-3AA (), the resistane is Ω, and the sensitivity oeffiient is.7. he strain gauge is onneted to dynami strain gauge through / eletri bridge whih is used to transform the strain signal to voltage. B. Experimental modal test Experimental modal analysis is a ase of system identifiation where a priori model form onsisting of modal parameters is assumed. Beause it is hard to obtain damping ratio of struture by finite element method (FEM), the experimental modal test is a good method to get aurate natural frequeny and damping ratio, whih provide a basis for adjustment of the ontrol model of the flexible struture. he setup of the experimental modal test is shown as Fig.. Kistler 869C5-type piezoeletri aelerometer is used as aeleration sensor. he ZoniBook/68E is the dynami signal aquisition system. he ez-analyst software is real-time vibration analysis software equipped with a dynami signal aquisition system whih provides a real-time analysis apability in the frequeny domain and time domain. ME sopeves software is used to be a post-proessing test whih is apable of analyzing mehanial and strutural stati and dynami harateristis to obtain modal parameters. Impulse hammer method is applied to perform experimental modal test. o exite the bending vibration, the antilever beam was hit with a hammer at the speified exitation points. Using a hammer to produe a wide band of exitation, it an exite eah mode in a wider frequeny range. he loations of hammer and aelerometers are loated at midpoints of element E 3 and elements E ~ E 7, respetively. A miniaturized aelerometer PCB is sequentially plaed at different loations. he tap position of the hammer is fixed, and the measurement points are 7 different positions. In order to eliminate measurement noise, the multiple measurements value of measurement point is averaged, the times of measurement is set to 5. he ez-analyst software is used to ollet data of exitation point and all measurement points, to obtain the frequeny response funtion of the measuring point, and then export the data format whih ME sopeves an support. he ME sopeves software is used to identify the modal parameters of the flexible beam. he synthesis of the ACADEMY PUBLISHER

6 6 JOURNAL OF COMPUERS, VOL. 7, NO., FEBRUARY ontroller is based on a nominal model onstruted by low-frequeny modes. In the present ase, the first fourorder mode of the link is identified. he first four-order nature frequenies by using experimental identifiation and FEM are shown in able, respetively, and the orresponding damping ratios are identified and tabulated in able. As an be seen from the table, relative error of alulated and experimental values of natural frequeny is lose to -6 %, whih indiates that the finite element model is not fully onsistent in the atual system. hat is to say, the model used to design ontroller is unertain. ontroller. As the ontrol voltage from D/A port is relatively low, the voltage exerted on the piezoeletri path must be amplified by the voltage amplifier to implement the vibration ontrol. he ontrol signal alulated by the dspace is onverted into an analog signal by a D/A onverter, and then is magnified 5 times by a voltage amplifier. Signals are then amplified and fed to a digital ontrol system. he ontrol algorithms are implemented using dspace DS3 system with neessary Matlab/Simulink software installed in an industrial omputer. he ontrol algorithm is implemented using Simulink software and Real ime Workshop (RW) is used to generate C ode from the developed Simulink model. he C ode is then onverted to target speifi ode by real time interfae (RI) and target language ompiler (LC) supported by DS3 ontroller board. hen we an design a vibration ontrol experiment in real time by using ControlDestk software provided by dspace. he ontrol objetive is to minimize the output strain of two sensor outputs within the ontrol bandwidth under the exitation of the disturbane fore indued by external fore. DS3 platform Exitation Voltage amplifier Dynami strain gauge Exitation ABLE I. Mode order 3 Figure 5. Fitting urve of frequeny response data HE FIRS FOUR-ORDER NAURAL FREQUENCY AND DAMPING RAIO Natural frequeny (Hz) Calulation Experimental value value Relative error Damping ratio (%) C. Experimental results Shemati diagram of vibration ontrol experiment is shown as Fig. 6. he prinipal of the vibration ontrol is desribed as follows. When the exogenous disturbanes are exerted on the beam, the vibration response will be generated. he output of the strain gage is given as input to the dynami strain gauge whih filters out the noise ontents. he onditioned sensor signal is given as analog input ard through the eletri bridge onnetor box. he vibration signal measured by a sensor is transformed into a voltage signal by a dynami strain transduer, and through a low-pass filter and an A/D onverter, the analog voltage signal is onverted into a digital signal to the dspace ontroller board. And the ontrol voltage applied to atuator an be obtained through the designed S A A A 3 Figure 6. Shemati diagram of vibration ontrol experiment he objetive in the experimental studies is to ontrol the first three vibration modes. In order to generate the disturbane fore exerted on the beam, we use the atuator A 3 as exitation soure. he atuators A, A are applied to vibration ontrol. he gain of the dynami strain gauge is set to, the bridge voltage 6 V, utoff frequeny of filter is set to Hz. he sampling frequeny of analog input and output port is set to Hz. he ovariane of measurement noise is estimated as. he sampling period of the ontroller is ms. he real-time ontrol time is set to s. he diagonal matrix δ in Eq. () is δ = diag(,, ). he shemati diagram of the feedbak ontrol system is depited in Fig. 6. In order to experimentally investigate the ontrol performane, the two exitation soures are exerted to the antilever beam, respetively. he first exitation soure alled exitation is the free-vibration one by taping instantly the free end of the beam. he seond exitation soure named as exitation is the fored one by applying a voltage to the piezoeletri erami path A 3. he exitation voltage is generated by the signal generator whih generates a white S ACADEMY PUBLISHER

7 JOURNAL OF COMPUERS, VOL. 7, NO., FEBRUARY 7 noise signal, amplified by voltage amplifier. And the ovariane of the exitation signal is set to V. As we design the Kalman state estimator, the parameters are set as Q d =, R e = 3. In order to validate that the proposed estimator an effetively estimate the state values, we will arry out the following test. As the atuator A 3 is applied to a sinusoidal signal, we ompared the measured values and estimated values of output strain from sensor S and sensor S. he sinusoidal signal is generated by the signal generator, and is amplified by voltage amplifier, its amplitude V, frequeny 75 Hz, whih the frequeny is lose to the seond order natural one of the system to make the beam vibrate signifiantly. Figure 7 and Fig. 8 denote the measured values and estimated values of output strain from sensor S and sensor S. From the two figures, we an observe the measurement values are oinide with estimated values of output strain to illustrate the effetiveness of the estimator. Strain Strain.. -. measurement value estimated value ime (s) Figure 7. Measuremetn and estimated values of output strain from sensor S measurement value estimated value ime (s) Strain 3 x w ith ontroller w ithout ontroller ime (s) Figure 9. Strain from sensor S in two ases of without ontroller and with ontroller under the exitation Strain 6 x w ith ontroller w ithout ontroller ime (s) Figure. Strain from sensor S in two ases of without ontroller and with ontroller under the exitation Strain w ith ontroller w ithout ontroller ime (s) Figure. Strain from sensor S in two ases of without ontroller and with ontroller under the exitation Figure 8. Measuremetn and estimated values of output strain from sensor S ACADEMY PUBLISHER

8 8 JOURNAL OF COMPUERS, VOL. 7, NO., FEBRUARY Strain.5 w ith ontroller w ithout ontroller ime (s) Figure. Strain from sensor S in two ases of without ontroller and with ontroller under the exitation Figure 9 and Fig. show the strains of the flexible link from sensor S and S in two ases of without ontroller and with ontroller under the exitation, respetively. Figure and Fig. show the strains of the flexible link from sensor S and S in two ases of without ontroller and with ontroller under the exitation, respetively. In these figures, solid line represents the strains in ase of with ontroller, and dotted line denotes the strains in ase of without ontroller. By the ontrast to the results of Fig. 9 and, it an be observed that the strains generated by the first exitation soure from sensor S and S are remarkably redued under the ase where sliding mode ontroller is adopted. Comparing the experimental results of Fig. and Fig., it an be seen that the strains aused by the seond exitation soure from sensor S and S an also be inhibited greatly in the ase of with ontroller. hus, it is onluded that the proposed ontroller is effetive to suppress the vibration response of the flexible beam. V. CONCLUSION he vibration ontrol of a smart flexible beam has been investigated by applying sliding mode ontrol. he sliding surfae is determined by onstruting a linear quadrati optimization problem, and Lyapunov diret method is used to design the sliding ontroller based on the Lyapunov asymptoti stability ondition. he Kalman filter is applied onsidering unertain disturbane and measurement noise. And the effetiveness of the proposed estimator was proved by omparing the measured and estimated values of the output strains. he experimental results have been showed that the proposed ontroller is valid to suppress the vibration response of the flexible beam. And it has been shown that the proposed ontroller guaranties robust performane. ACKNOWLEDGMEN his work was supported by National Natural Siene Foundation of China (Grant No ). REFERENCES [] S.B. Choi, Ative strutural aousti ontrol of a smart plate featuring piezoeletri atuators, Journal of Sound and Vibration, vol. 9(-), pp. -9, 6. [] K. Ma, Adaptive nonlinear ontrol of a lamped retangular plate with PZ pathes, Journal of Sound and Vibration, vol. 6(), pp , 3. [3] K. Ma, M. N. Ghasemi-Nejhad, Frequeny-weighted adaptive ontrol for simultaneous preision positioning and vibration suppression of smart strutures, Smart Materials and Strutures, vol. 3(5), pp. 3 5,. [] G. L. C. M. Abreu, J. F. Ribeiro, and V. Steffen, Experiments on optimal vibration ontrol of a flexible beam ontaining piezoeletri sensors and atuators, Shok and vibration, vol., pp. 83 3, 3. [5] Y. Y. Li, L. H. L. Yam, Roubust vibration ontrol unertain systems using variable parameter feedbak and model-based fuzzy strategies, Computers and Strutures, vol. 79, pp. 9 9,. [6] P. Mayhan, G. Washington, Fuzzy model referene learning ontrol: a new ontrol paradigm for smart strutures, Smart Mater. Strut., vol. 7, pp , 998. [7] X. M. Zhang, C. J. Shao, and A. G. Erdman, Ative vibration ontroller design and omparison study of flexible linkage mehanism systems, Mehanism & Mahine heory, vol. 37, pp ,. [8] X. M. Zhang, C. J. Shao, and S. Li, et al., Robust H vibration ontrol for flexible linkage mehanism systems with piezoeletri sensors and atuators, Journal of Sound and Vibration, vol. 3(), pp. 5-55,. [9] H. Kawabe, N. sukiyama, and K. Yoshida, Ative vibration damping based on neural network theory, Materials Siene and Engineering A, vol., pp , 6. [] C. Y. Liao, C. K. Sung, An elastodynami analysis and ontrol of flexible linkages using piezoerami sensors and atuators, ransations of the ASME, vol. 5, pp , 993. [] L. Iorga, H. Baruh, and I. Ursu, A Review of H robust ontrol of piezoeletri smart strutures, Appl. Meh. Rev., vol. 6(), pp. -6, 8. Junfeng Hu Jiangxi Provine, China. born in 978. He is Engineering Ph.D., graduated from South China University of ehnology. His researh interests inlude mehanial dynamis, smart struture and ative vibration ontrol. He is urrently a leturer in Shool of Mehanial & Eletrial Engineering, Jiangxi University of Siene and ehnology, China. Dahang Zhu, Jiangxi Provine, China, born in 973. He is Engineering Ph.D., graduated from Beijing Jiaotong University. His researh interests inlude parallel mehanism and ompliant parallel mehanism. He is urrently a assoiate professor in Shool of Mehanial & Eletrial Engineering, Jiangxi University of Siene and ehnology, China. ACADEMY PUBLISHER