Research ArticleAdaptive Robust Sliding Mode Vibration Controlof a Flexible Beam Using Piezoceramic Sensor and ActuatorAn Experimental Study
Ruo Lin Wang1 H Gu2 and G Song23
1 School of Civil Engineering Wuhan University Wuhan Hubei 430072 China2Department of Mechanical Engineering University of Houston Houston TX 77204 USA3 School of Civil Engineering Dalian University of Technology Dalian Liaoning 116024 China
Correspondence should be addressed to G Song gsonguhedu
Received 28 September 2013 Revised 23 December 2013 Accepted 8 January 2014 Published 26 May 2014
Academic Editor ShengJun Wen
Copyright copy 2014 Ruo Lin Wang et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited
This paper presents an experimental study of an adaptive robust sliding mode control scheme based on the Lyapunovrsquos directmethod for active vibration control of a flexible beam using PZT (lead zirconate titanate) sensor and actuator PZT a type ofpiezoceramic material has the advantages of high reliability high bandwidth and solid state actuation and is adopted here informs of surface-bond patches for vibration control Two adaptive robust sliding mode controllers for vibration suppression aredesigned one uses a discontinuous bang-bang robust compensator and the other uses a smooth compensator with a hyperbolictangent function Both controllers guarantee asymptotic stability as proved by the Lyapunovrsquos direct method Experimental resultsverified the effectiveness and the robustness of both adaptive sliding mode controllers However from the experimental resultsthe bang-bang robust compensator causes small-magnitude chattering because of the discontinuous switching actions With thesmooth compensator vibration is quickly suppressed and no chattering is induced Furthermore the robustness of the controllersis successfully demonstrated with ensured effectiveness in vibration control when masses are added to the flexible beam
1 Introduction
Model inaccuracies or uncertainty can have adverse effectson control systems The adaptive control and the slidingmode control are two major nonlinear control methods todeal with model uncertainties The typical structure of aslidingmode controller is composed of estimated terms of thesystem to the best of knowledge and a robust compensatorwhich deals with model uncertainty and disturbance toensure stability The robust compensator usually consists ofan upper bound of the system uncertainty with a nonsmoothswitching function such as a sign function or a saturationfunction This robust compensator is very important in thesliding mode control design since the robust compensatorwill make the system trajectory converge toward the slidingsurface in a finite time and remain on the sliding surfaceunder system uncertainties and external disturbances Theasymptotic stability is achieved by the proper design of
the robust compensator in the sliding mode control TheLyapunovrsquos direct method is often used to assist the designand to provide a stability proof
Song and Mukherjee [1] presented a comparative studyof two nonsmooth time-invariant robust compensators (thebang-bang compensator and the saturation compensator)with a smooth time-varying compensator The comparativestudy reveals the superiority of the smooth time-varyingcompensator over the nonsmooth robust compensatorManyscholars have studied and developed different kinds of slidingmode control schemes Park et al [2] proposed a two-stage sliding mode controller for vibration suppression of aflexible pointing system Yu et al [3] proposed a fuzzy slidingmode control strategy which consists of fuzzily amalgamatedsliding mode controls of the system linearized around a setof operating points There are also many other sliding modecontrol schemes An adaptive sliding mode control takesadvantage of aspects from both the adaptive control and the
Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2014 Article ID 606817 9 pageshttpdxdoiorg1011552014606817
2 Mathematical Problems in Engineering
slidingmode control schemesThe adaptive control approachis suitable for control of the dynamic systems with constantor slowly varying uncertain parameters The parameters ofthe adaptive controller are updated during operation basedon the measured performance The adaptive control law isdesigned to guarantee the asymptotic stability for the controlsystem under disturbances
With merits of both the adaptive control and the slidingmode control schemes adaptive sliding mode control canachieve satisfactory control performancewith asymptotic sta-bility when the system is subject to model uncertainties anddisturbances Due to these advantages the adaptive slidingmode control has been studied extensively for the control ofdynamic systems with uncertainties and disturbances Yaoand Tomizuka [4] proposed a systematic way to combinethe adaptive control design technique and the sliding modecontrol methodology for trajectory tracking control of robotmanipulators in the presence of parametric uncertainties andexternal disturbance Wang et al [5] proposed an indirectadaptive fuzzy sliding mode control scheme for a class ofnonlinear systems by using the concept of sliding modecontrol design and Lyapunov synthesis approach Simula-tion studies have shown that the adaptive design of fuzzysliding mode controller performs very well in the presenceof an unknown disturbance Song et al [6] proposed anintegrated adaptive-robust control methodology for controlof nonlinear uncertain systems The new approach can takeadvantage of both adaptive and robust control methods andit offers to handle various uncertainties reject external distur-bance and guarantee global asymptotic stability with desiredtransient response Song et al [7] proposed an integrandadaptive-robust approach along with a smooth adaptiverobust friction compensation strategy for tracking the controlof uncertain robot manipulators with joint stick-slip frictionWai [8] proposed an adaptive sliding mode control systemto control the position of an induction servomotor driveIn the adaptive sliding mode control system an adaptivealgorithm is utilized to estimate the bound of uncertaintiesSimulated and experimental results show that the dynamicbehaviors of the proposed control systems are robust withregard to uncertainties Lin and Chen [9] developed anew strategy called adaptive fuzzy sliding mode control(AFSMC) The parameters of the membership functionsin the AFSMC are changed according to some adaptivealgorithm for the purpose of controlling the system statesto converge to a user-defined sliding surface and then slidealong it The sliding mode control can achieve satisfactoryvibration control performance since the vibration controlsystem is usually a nonlinear system with model uncertaintyand external disturbance Fei and Ding [10] proposed aradial basis function neural network adaptive sliding modecontroller to track a microelectromechanical system triaxialgyroscope by integrating the adaptive control neural networkcontrol and sliding mode control Switching function andsliding mode controller were used as the input and outputof the radial basis function of neutral network to deal withnonlinearity with an online learning ability Zhang and Han[11] applied robust sliding mode H
infincontrol using time-
varying delayed states for an offshore steel jacket platform
with self-exciting nonlinear wave force external disturbanceand parametric uncertainties Zhang et al [12] proposed anew variable structure sliding mode control strategy witha combined approaching law for fast tool servo to controlcutting force in diamond-cutting microstructured surfacesLi et al [13] proposed an adaptive sliding mode control viaTakagi-Sugeno (T-S) fuzzy approach to control suspensionsystem The T-S fuzzy control was used to describe theoriginal nonlinear system for the control design via a sectornonlinearity approach A sufficient condition which could beconverted to convex optimization problem is proposed forasymptotical stability of the designing sliding motion
Piezoelectric materials are commonly used in the vibra-tion control as actuators and sensors Piezoelectric materialis an innovative smart material with the advantages of quickresponse low-power cost and easy implementation Duringthe past years piezoelectric material has been successfullyapplied in different kinds of active or passive vibration controlmethods such as PPF control [14 15] fuzzy model referencecontrol [16] vibration control for robotic links [17 18] andpassive vibration control with shunted piezoelectric material[19] Fei et al [20] compares the uses of PZT-based optimalPID strain rate feedback (SRF) and PPF control schemeson the vibration control of a cantilever beam Piezoelectricmaterial-based sliding mode control provides an innovativeand effective strategy for the vibration control of systemswith model inaccuracy and uncertainties Choi et al [21]developed an active vibration control of a hybrid smartstructure consisting of a piezoelectric film actuator andan electrorheological fluid actuator A neuro-sliding modecontroller incorporating neural networks with the conceptof sliding mode control is formulated for a piezoelectric filmactuator Actuators and sensors made of piezoelectric mate-rials have the merits of low weight and distributed natureJha and Inman [22] utilized piezoelectric actuators andsensors in the vibration suppression of gossamer structuresIn order to achieve good vibration reduction performancesunder model uncertainty and external disturbance a slidingmode controller and observer are designed The simulationstudies show that the piezoelectric actuators and sensorsare suitable for vibration suppression of an inflatable torusChoi and Kim [23] proposed a new discrete-time fuzzysliding mode controller and applied it to vibration controlof a smart structure with a piezofilm actuator Experimentalresults of transient forced and random vibration of smartstructures demonstrate the effectiveness of the proposedmethod Recent studies by other researchers have refinedslidingmode controlmethods for use inmicronanoactuators[24ndash26] Hu and Zhu [27] applied a sliding mode controlwith observer to reduce the vibration of a flexible structurewith a piezoelectrical actuator and a strain gauge sensor Astate space dynamic system was derived by using a finiteelement method and the experimental modal test The con-trollerrsquos sliding surface was determined by using optimizationmethod that minimized the cost function of the states Thereare also many other schemes of sliding mode control appliedto vibration control by using piezoelectric material
Many simulation studies have been performed to makecontributions to the theoretical study of the adaptive sliding
Mathematical Problems in Engineering 3
mode control Some recent studies for example have inves-tigated self-tuning laws especially for uncertain nonlinearsystems [28] and advanced observer-based robust control ofpiezoelectric structures [29] However not enough experi-mental study of the adaptive sliding mode control has beenconducted The adaptive sliding mode controller is designedto handle the system uncertainties and disturbances and it ismore important to conduct the experimental study to verifythe robustness of the proposed controller under real systemswith uncertainties and disturbances than under the simulatedsystems Experimental verification of adaptive sliding modecontroller is an important issue for the study of the adaptivesliding mode controller The real-time experimental resultsare more convincing in verifying and proving the robustnessof an adaptive sliding mode control than the simulationresults In this paper experimental comparative study oftwo different adaptive robust sliding mode controllers isconducted An adaptive robust sliding mode control schemebased on the Lyapunovrsquos directmethod is proposed for activevibration control of a flexible beam using PZT (lead zirconatetitanate) sensors and actuators PZT a commonly usedpiezoelectric ceramic material is adopted in this researchand surface-bond PZT patches were used for vibrationcontrol Two adaptive robust sliding mode controllers forvibration suppression were designed one adaptive slidingmode controller uses a discontinuous bang-bang robust com-pensator and the other uses a compensator with a smoothhyperbolic tangent function The Lyapunovrsquos direct methodis used to prove the asymptotic stability of both controllersThe proposed adaptive sliding mode controller reduces theconservatism of the sliding mode controller The asymptoticstability of the proposed control system was proved by usingthe Lyapunovrsquos direct method Experiments have been imple-mented on a flexible beam with PZT actuators and sensorsExperimental results show that both controllers successfullysuppress vibrations of the flexible beam in an efficient wayHowever the bang-bang robust compensator causes small-magnitude chattering because of the discontinuous switchingactions With the smooth compensator vibration is quicklysuppressed and no chattering is induced Furthermore therobustness of the controllers is successfully tested by addingmass to the flexible beam Multimode vibration control ofthe beam with mass added is also conducted to verify theeffectiveness and robustness of the proposed adaptive slidingmode control in vibration control
2 Experimental Setup
A flexible aluminum beam in a cantilevered configuration(Figure 1) is used as the experimental object to test theeffectiveness of the proposed vibration suppression methodThe property of the beam is shown in Table 1 PZT typepiezoelectric patches are used as the smart sensor andactuator One relatively large PZT patch is surface-bonded oneach side of the flexible aluminum beam near its cantileveredroot These two patches are used as actuators to excite and toenable active vibration control of the beam One smaller PZTpatch is surface-bonded on one side of the beam also near its
Table 1 Beam properties
Symbol Quantity Unit Value119871 Beam length mm 7365119908119887
Beam width mm 531119905119887
Beam thickness mm 1120588119887
Beam density kgm3 2690119864 Modulus of elasticity Nm2
703 times 1010
Beam
Sensor
BaseActuator
Figure 1 The experimental setup
cantilevered root This smaller PTZ patch acts as sensor forthe feedback of the signals in the active control algorithmsPhysical position of PZT actuators and sensors is shown inFigure 2 Both the PZT actuators and sensor are installednear the cantilevered root of the beam since this location hasthe highest strain energy along the length of the beam Thisconfiguration of actuator and sensor placement results in themost effective actuation and sensing for vibration controlTheproperties of the PZT patches are shown in Table 2
A PZT actuator has some nonlinear characteristics suchas hysteresis Hysteresis should be compensated for accuratepositioning for position regulation or tracking tasks Thehysteresis width of PZT is about 10 of its stroke andnormally PZT actuator is not classified as a ldquohighrdquo nonlinearactuator In addition the hysteresis width reduces with theincreasing frequency Therefore for active vibration controlit is a common practice that the hysteresis is not consideredin such an application The nonlinear characteristics of PZTtransducers are not considered in this paper
Experiments have shown that the dominant mode of theflexible beam is its first mode and is the major concern forvibration suppression (details can be found in Section 5)Since the first mode is dominant in the experiment highermodes are truncated for simplification Bu et al [30] derivedthe model for a flexible smart beam with piezoceramicactuators and strain gaugesWhen the first mode is dominantin the analysis (ie only the first mode is considered) modelbetween PZT actuator voltage and strain sensor can berepresented as a simplified linear second order system whichvalidates the practice of not considering the nonlinear char-acteristics of PZT transducers in vibration control problemsThe PZT sensor output represents the strain of the sensorlocation Therefore the PZT sensor can be applied as straingauge for control purposes In the control design a simplifiedsecond order model is used as the structure of the modelHowever the model parameters are assumed unknown Thedesign of the controller is not dependent on the parameters
4 Mathematical Problems in Engineering
Table 2 Properties of PZT patches used on the beam
Symbol Quantity Unit PZT actuator PZT sensor119871 times 119908 times 119905 Dimensions mm 46 times 3327 times 025 14 times 7 times 025
11988933
Strain coefficient CN 741 times 10minus10
741 times 10minus10
11988931
Strain coefficient CN minus274 times 10minus10
minus274 times 10minus10
120588119875
PZT density kgm3 7500 7500119864119875
Youngrsquos modulus Nm263 times 10
10
63 times 1010
Beam
Sensor
BaseActuator23mm 992mm
61mm
79mm
735mm
7365mm
Figure 2 The physical position of the piezoelectric actuators andsensor
of the model The plant is represented in the form of thefollowing general simplified dynamic equation
119898 + 119888 + 119896119909 = 119906 (1)
where119898 119888 and 119896 are the coefficients relative with the systemmass damping and stiffness respectively x is the PZT sensorsignal and 119906 is PZT actuator signal
3 Design of Adaptive Robust Sliding ModeController and Stability Proof
The design principle of the proposed adaptive robust slidingmode controller for vibration control is similar to that of theadaptive sliding mode controller for robot manipulator in [16 31]
To assist control design define 119890 = 119909119903minus 119909 where the
subscript ldquo119903rdquo stands for the reference command Define thesliding surface variable as
where 119888 and are the initial estimations of the real valuesof parameters of 119898 119888 and 119896 and these estimates will beadjusted online by the slidingmode based adaptive controllerThe term 119886 sgn(119904) is the bang-bang robust compensator to
ensure stability where 119886 is a positive constant and the sgnfunction is defined as
sgn (119904) = +1 if 119904 gt 0
sgn (119904) = minus1 if 119904 lt 0(4)
With this feedback controller (3) the dynamic equation of thesystem (1) becomes
= minus119886 sdot 119904 sdot sgn (119904) minus 120593119879 (119884119904 minus Γminus1 120593)
(11)
Notice that minus119886 sdot 119904 sdot sgn(119904) le 0The adaptation law will be designed as 119884119904 minus Γminus1 120593 = 0
Therefore V = minus119886 sdot 119904 sdot sgn(119904) le 0 while V = 0 onlywhen 119904 = 0 According to LaSallersquos theorem the control sys-tem is asymptotically stable in the sense of Lyapunov if theadaptation law satisfies 119884119904 minus Γminus1 120593 = 0 In the sliding modecontroller the positive constant 119886 which is the amplitude ofthe robust compensator in control law (3) needs to be theupper bounding of the uncertainty of the system model andthe external disturbance to ensure the stability Due to thelack of the prior knowledge of the system the upper boundingof the uncertainty and the disturbance may be difficult tobe properly or precisely evaluated In order to ensure theasymptotic stability of the system under all conditions theamplitude of the robust compensator usually needs to bechosen large enough to compensate for the worst case of allpossible uncertainty and external disturbances Therefore inthe sliding mode controller the transient performance andthe steady state performance of the controller are usuallysacrificed for the asymptotic stability of the control systemHowever in the proposed adaptive controller the adaptationlaw is designed to guarantee the asymptotic stability underuncertainty and disturbance Therefore there will be lessconstraint on the choice of the amplitude 119886 of the robustcompensator in the proposed method Conservatism of therobust controller would be dramatically reduced by introduc-ing an adaptive scheme into the controller The asymptoticstability of the proposed adaptive sliding mode controller isguaranteed with larger values of allowable uncertainty anddisturbances Both adaptive control part and sliding modecontrol part could compensate for the system uncertaintiesand external disturbancesThe adaptation law can be derivedas
The shortcoming of the adaptive robust sliding mode con-troller (3) is that the bang-bang robust compensatorwill causechattering in implementationThe chattering phenomenon iscaused by the sign function term The sign function switchesthe control action abruptly especially near the equilibriumpoint Chattering is not desirable since it involves extremelyhigh control activity andmay excite high frequency dynamicsof a structure
4 Design of a Smooth Adaptive Robust SlidingMode Controller and Stability Proof
In order to attenuate the chattering associated with the bang-bang compensator a hyperbolic tangent function is used toswitch the control action instead of the sign function Thecontroller becomes
where 119886 and120572 are positive numbers and tanh is the hyperbolictangent function
tanh (119909) = 119890119909
minus 119890minus119909
119890119909 + 119890minus119909 (18)
The 119886 tanh(120572119904) is a robust compensator which provides asmooth control action The hyperbolic function will switchthe control action smoothly and avoid causing chatteringThis robust compensator is continuously differentiable withrespect to the control variable 119904 It generates a smoothcontrol action Compared with the commonly used bang-bang or saturation robust controllers the smooth robustcontroller has advantages in ensuring smooth control inputand the stability of the closed-loop system [32] A detailedcomparative study is provided in Song and Mukherjee [1]Normally the larger the value of the parameter 120572 is the closer
6 Mathematical Problems in Engineering
the tanh(120572119904) function will mimic the sign(119904) function Toavoid chattering a relative smaller value of 120572 can be chosenWhen implementing the controller the value of 120572 is oftenexperimentally determined
To prove the stability of the smooth adaptive robustcontroller (17) define the Lyapunov function candidate as
V =1
21198981199042
+1
2120593119879
Γminus1
120593 (19)
Using the similar process shown in Section 3 we have
V = minus119886 sdot 119904 sdot tanh (120572119904) minus 120593119879 (119884119904 minus Γminus1 120593) (20)
Please note that 119888 and are updated according to (14) tomake 119884119904 minus Γminus1 120593 = 0 Then V = minus119886 sdot 119904 sdot tanh(120572119904) le 0 whileV = 0 only when 119904 = 0 Therefore according to LaSallersquostheorem the closed-loop control system with the smoothadaptive robust controllers (17) is asymptotically stable in thesense of Lyapunov
5 Experiment Results
51 Open Loop Testing of the Aluminum Beam First openloop testing is performed to find the dominant mode of thebeam for active vibration control The beam was subjectedto a mechanical impact at its midlength for a multimodeexcitation The power spectrum density (PSD) plots of the30-second time response of the multimode excitation wereobtainedThe corresponding frequencies of peaks in the PSDplots represent themodal frequencies of the flexible beam Tostudy the attenuation of vibration at each modal frequencythe PSD plots of the first 5 seconds and the last 5 secondsof the beam vibration are shown in Figure 3 The peaks inFigure 3 clearly reveal the modal frequencies of the flexiblebeam the 1st mode at 164Hz the 2nd mode at 977Hzand the 3rd mode at 2685Hz A comparison of PSD plotsof the vibration data for the first 5 seconds and the last 5seconds clearly shows the vibration attenuation at eachmodalfrequency The dB value and dB drop of the first three modesare shown in Table 3 From Figure 3 and Table 3 it can beseen that the dB levels of the second and third modes at thefirst five-second and last five-second periods are significantlylower than that of the 1st mode The dB drop value of thefirst three modes shows that the second and third modesattenuated much faster than the first mode Therefore it canbe concluded that the first modal frequency 164Hz is thedominant one in the vibration and the first mode should bethe target mode in the active vibration control To target atthis mode the vibration control is designed based on thesimplified plant (1) which includes only the dynamics of thefirst mode
52 Experimental Results for the Adaptive Robust SlidingMode Controller To verify the effectiveness of the adaptiverobust sliding mode control (3) and the smooth adaptivesliding mode control (17) experiments were conducted onthe aluminum flexible beam In the experiment 119904 = 119890 +
120582119890 = 119890 + 005119890 and 1199031= 1199032= 1199033= 01 for the diagonal
Table 3 PSD drop comparison of first 5 seconds and last 5 seconds
Mode 1 Mode 2 Mode 3dB level of the first 5 seconds 1673 598 minus1436dB level of the last 5 seconds 291 minus1723 minus3519dB drop 1382 2321 2083
20
10
minus10
minus20
minus30
minus40
minus50
minus60
minus70
minus800
0
10 20 30 40 50 60
Frequency (Hz)
Pow
er sp
ectr
um m
agni
tude
(dB)
First 5 secondsLast 5 seconds
Figure 3 Comparison of PSD plots of first 5 seconds and last 5seconds for time response of multimode excitation of the flexiblebeam
matrix Γ In each test the beam was excited by a sinusoidalsignal at its first modal frequency combined with a lowpower white noise for the initial 5 seconds The amplitudeof the sinusoidal signal was 075V The variance of the whitenoise was 00497 The sinusoidal signal and the white noisewere summed in the Simulink software and downloaded intodSPACE DS1102 Floating-Point Controller Board for real-time implementation The output of the dSPACE controllerboard was amplified by an amplifier for capacitive load beforeexciting the PZT actuators After the initial 5 seconds firstmode excitation was stopped and the active vibration controlstarted to suppress the induced vibration A free vibration testwas conducted for the purpose of comparison In the freevibration test no control signal was implemented after theinitial 5-second excitation
The upper diagram of Figure 4 is the time responsecomparison of free vibration with that of the active control(with the sign function) From the experimental results thevibration is suppressed in about 8 oscillation cycles (about5 seconds) However there is small-magnitude chattering inthe steady state which may cause instability when a certainlevel of disturbance exists The lower diagram of Figure 4 isthe time response comparison of free vibration with that ofthe active control (with the tanh function) It can be seen thatthe robust sliding mode control can suppress the vibration inabout 8 oscillation cycles without the chattering in the steadystate Experimental results show that the chattering can be
Mathematical Problems in Engineering 7
06
04
02
0
minus02
minus04
5 10 15 20
Sens
or v
olta
ge (V
)
Sliding-mode control with sign functionFree vibration
Time (s)
(a)
06
04
02
0
minus02
minus04
5 10 15 20
Sens
or v
olta
ge (V
)
Sliding-mode control with tanh functionFree vibration
Time (s)
(b)
Figure 4 The time response comparisons of the adaptive slidingmode controller (with the sign function) with the adaptive robustsliding mode controller (with tanh function)
eliminated by implementing the hyperbolic tangent function(tanh function) as the switching function instead of the signfunction
The upper diagram of Figure 5 is the control actionwith the bang-bang robust compensator Compared with thetime response in Figure 4 it can be seen that the chatteringhappens when the control action switches abruptly betweenits positive limit and negative limit when the system is alreadyin the steady state (after 10 seconds) The lower diagramof Figure 5 is the control action with the smooth robustcompensator From the diagram it can be seen that thecontrol signal switches smoothly Therefore the chatteringin the output signal is eliminated Figure 6 shows the timehistory of the parametersrsquo updating process Furthermore itis also worthwhile to note that no higher modes have beenexcited even though the plant is actually amultimode systemThis justifies that the primary vibration control target is thefirst mode
53 Experimental Results for the Adaptive Robust SlidingModeControl with Mass Uncertainty To verify the robustness ofthe proposed adaptive robust controller the plant is changedby adding mass on the beam Four 6-gram adhesive masseswere attached to both sides of the beam with two at theremote tip and the other two at the midlength of the beamThe mass of the aluminum beam is increased from 10520gram to 12920 gramThefirstmodal frequency of the originalbeam is 167Hz whereas the first modal frequency of the
Time (s)
200
100
0
minus100
minus2005 10 15 20
Actu
ator
sign
al (V
)
Sliding-mode control with sign function
(a)
Time (s)
200
100
0
minus100
minus2005 10 15 20
Actu
ator
sign
al (V
)Sliding-mode control with tanh function
(b)
Figure 5 The control signal comparisons of the adaptive slidingmode controller (with the sign function) with the adaptive robustsliding mode controller (with tanh function)
00835
0083
008255 10 15 20 25
Time (s)
m
Time (s)
082
081
085 10 15 20 25
c
1100002
1100001
110
Sliding-mode control with sign functionSliding-mode control with tanh function
Time (s)5 10 15 20 25
k
Figure 6 The parametersrsquo updating time history
beam with added mass has been decreased to 1457Hz Theincreasing of the mass has caused the decreasing of thenatural frequency of the beamTherefore in this experimentthe frequency of the sinusoidal excitation signal has beenchanged to 1457Hz to ensure excitation of the beam Thesame controller is used for the vibration control of the beamwith added mass From the experimental results (Figure 7)
8 Mathematical Problems in Engineering
05
04
02
03
01
0
minus01
minus02
minus03
minus04
minus055 10 15 20
Time (s)
Free vibrationSliding-mode control with tanh function
Sens
or v
olta
ge (V
)
Figure 7 The time response comparisons of the adaptive slidingmode controller with tanh function for the flexible beam with 24-gram mass added
1
08
06
04
02
0
minus02
minus04
minus06
minus08
minus10 1 2 3 4 5 6 7
Time (s)
Sliding-mode control with tanh function
Sens
or v
olta
ge (V
)
Free vibration
Figure 8 The time response of comparison of the adaptive slidingmode controller with tanh function for the flexible beam withmultimode excitation
it can be seen that the adaptive robust sliding mode caneffectively suppress the vibration of the beam after its physicalproperty has been changedThis demonstrates the robustnessof the proposed controllers
54 Experimental Results for the Adaptive Robust SlidingModeControl with Multimode Excitation To test its robustness tohigher order dynamics and disturbances multimodal excita-tion of the flexible beam with mass added is conducted Fig-ure 8 shows the comparison of time response of multimodalexcitationwith that of the free vibration Experimental results
clearly show that the proposed method is still effective inmultimodal vibration control and this further demonstratesthe robustness of the proposed controller
6 Conclusion
In this paper an experimental study of the robust adaptivesliding mode control was conducted on a flexible beamwith piezoceramic actuators and sensor for vibration controlpurpose Two different schemes of adaptive sliding modecontrol have been comparatively studied during the experi-ment The adaptive robust sliding mode control with bang-bang action could suppress the induced vibration of a flexiblebeam in a quick fashion However it causes chattering duringsteady state To eliminate the chattering a hyperbolic tangentfunction is used to replace the sign function in the robustcontrol action Both adaptive robust sliding mode controllersare designed based on the Lyapunov direct method In theproposed adaptive sliding mode controller there will beless constraint on the choice of the amplitude of the robustcompensator The conservatism of the control system hasbeen reduced by introducing the adaptive scheme into thesliding mode controlThe asymptotic stability of the adaptiverobust sliding mode control is proved by using Lyapunovdirect method in the paper The experimental results showthat the proposed adaptive robust sliding mode control withthe smooth robust compensator eliminates the chatteringwhile keeping the advantages of adaptive robust slidingmodecontrol Also the proposedmethods are robust to the varianceof the plant parameters as demonstrated experimentally
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
This research was partially supported by Grant no 51278387Grant no 51278084 and Grant no 51121005 (Science Fundfor Creative Research Groups) from the National ScienceFoundation of China (NSFC)
References
[1] G Song and R Mukherjee ldquoA comparative study of conven-tional nonsmooth time-invariant and smooth time-varyingrobust compensatorsrdquo IEEE Transactions on Control SystemsTechnology vol 6 no 4 pp 571ndash576 1998
[2] J-H Park K-W Kim and H-H Yoo ldquoTwo-stage sliding modecontroller for vibration suppression of a flexible pointing sys-temrdquo Proceedings of the Institution ofMechanical Engineers PartC Journal of Mechanical Engineering Science vol 215 no 2 pp155ndash166 2001
[3] X Yu Z Man and B Wu ldquoDesign of fuzzy sliding-mode con-trol systemsrdquo Fuzzy Sets and Systems vol 95 no 3 pp 295ndash3061998
Mathematical Problems in Engineering 9
[4] B Yao and M Tomizuka ldquoSmooth robust adaptive slidingmode control of manipulators with guaranteed transient per-formancerdquo in Proceedings of the American Control Conferencevol 1 pp 1176ndash1180 July 1994
[5] J Wang A B Rad and P T Chan ldquoIndirect adaptive fuzzysliding mode control part I fuzzy switchingrdquo Fuzzy Sets andSystems vol 122 no 1 pp 21ndash30 2001
[6] G Song R W Longman and R Mukherjee ldquoIntegratedsliding-mode adaptive-robust controlrdquo IEE Proceedings ControlTheory and Applications vol 146 no 4 pp 341ndash347 1999
[7] G Song R W Longman and L Cai ldquoIntegrated adaptive-robust control of robot manipulatorsrdquo Journal of Robotic Sys-tems vol 15 no 12 pp 699ndash712 1998
[8] R-J Wai ldquoAdaptive sliding-mode control for induction servo-motor driverdquo IEE Proceedings Electric Power Applications vol147 no 6 pp 553ndash562 2000
[9] S-C Lin and Y-Y Chen ldquoDesign of adaptive fuzzy slidingmode for nonlinear system controlrdquo in Proceedings of the 3rdIEEE Conference on Fuzzy Systems pp 35ndash39 June 1994
[10] J Fei and H Ding ldquoAdaptive vibration control of microelec-tromechanical systems triaxial gyroscope using radial basisfunction sliding mode techniquerdquo Proceedings of the Institutionof Mechanical Engineers Part I Journal of Systems and ControlEngineering vol 227 no 2 pp 264ndash269 2013
[11] B L Zhang and Q L Han ldquoRobust sliding mode 119867infin
controlusing time-varying delayed states for offshore steel jacketplatformsrdquo in Proceedings of the IEEE International Symposiumon Industrial Electronics (ISIE rsquo13) pp 1ndash6 IEEE 2013
[12] H Zhang G Dong M Zhou C Song Y Huang and K Du ldquoAnew variable structure sliding mode control strategy for FTS indiamond-cutting microstructured surfacesrdquo The InternationalJournal of AdvancedManufacturing Technology vol 65 no 5ndash8pp 1177ndash1184 2013
[13] H Li J Yu C Hilton and H Liu ldquoAdaptive sliding modecontrol for nonlinear active suspension vehicle systems usingTS fuzzy approachrdquo IEEE Transactions on Industrial Electronicsvol 60 no 8 2013
[14] J L Fanson and T K Caughey ldquoPositive position feedbackcontrol for large space structuresrdquo AIAA Journal vol 28 no 4pp 717ndash724 1990
[15] G Song P Qiao V Sethi and A Prasad ldquoActive vibrationcontrol of a smart pultruded fiber-reinforced polymer I-beamrdquoin International Symposium on Smart Structures and Materialsvol 4696 of Proceedings of SPIE pp 197ndash208 March 2002
[16] PMayhan andGWashington ldquoFuzzymodel reference learningcontrol a new control paradigm for smart structuresrdquo SmartMaterials and Structures vol 7 no 6 pp 874ndash884 1998
[17] V Bottega AMolter J SO Fonseca andR Pergher ldquoVibrationcontrol of manipulators with flexible nonprismatic links usingpiezoelectric actuators and sensorsrdquo Mathematical Problems inEngineering vol 2009 Article ID 727385 16 pages 2009
[18] A Molter V Bottega O A A Da Silveira and J S O FonsecaldquoSimultaneous piezoelectric actuator and sensor placementoptimization and control design of manipulators with flexiblelinks using SDREmethodrdquoMathematical Problems in Engineer-ing vol 2010 Article ID 362437 23 pages 2010
[19] M Ahmadian and K M Jeric ldquoOn the application of shuntedpiezoceramics for increasing acoustic transmission loss instructuresrdquo Journal of Sound and Vibration vol 243 no 2 pp347ndash359 2001
[20] J Fei Y Fang and C Yan ldquoThe comparative study of vibrationcontrol of flexible structure using smart materialsrdquo Mathemat-ical Problems in Engineering vol 2010 Article ID 768256 13pages 2010
[21] S-B Choi Y-K Park and C-C Cheong ldquoActive vibrationcontrol of hybrid smart structures featuring piezoelectric filmsand electrorheological fluidsrdquo in The International Society forOptical Engineering vol 2717 of Proceedings of the SPIE pp544ndash552 February 1996
[22] A K Jha and D J Inman ldquoSliding mode control of a Gossamerstructure using smart materialsrdquo Journal of Vibration andControl vol 10 no 8 pp 1199ndash1220 2004
[23] S-B Choi and M-S Kim ldquoNew discrete-time fuzzy-sliding-mode control with application to smart structuresrdquo Journal ofGuidance Control and Dynamics vol 20 no 5 pp 857ndash8641997
[24] S Bashash and N Jalili ldquoRobust multiple frequency trajec-tory tracking control of piezoelectrically driven micronano-positioning systemsrdquo IEEE Transactions on Control SystemsTechnology vol 15 no 5 pp 867ndash878 2007
[25] H C Liaw B Shirinzadeh and J Smith ldquoSliding-modeenhanced adaptive motion tracking control of piezoelectricactuation systems formicronanomanipulationrdquo IEEETransac-tions on Control Systems Technology vol 16 no 4 pp 826ndash8332008
[26] Y Li and Q Xu ldquoAdaptive sliding mode control with perturba-tion estimation and PID sliding surface for motion tracking of apiezo-driven micromanipulatorrdquo IEEE Transactions on ControlSystems Technology vol 18 no 4 pp 798ndash810 2010
[27] J Hu and D Zhu ldquoVibration control of smart structure usingsliding mode control with observerrdquo Journal of Computers vol7 no 2 pp 411ndash418 2012
[28] L Zhang Z Zhang Z Long and A Hao ldquoSlidingmode controlwith auto-tuning law forMaglev SystemrdquoEngineering vol 2 no2 pp 107ndash112 2010
[29] T Rittenschober andK Schlacher ldquoObserver-based self sensingactuation of piezoelastic structures for robust vibration controlrdquoAutomatica vol 48 no 6 pp 1123ndash1131 2012
[30] X Bu L Ye Z Su and C Wang ldquoActive control of a flexiblesmart beam using a system identification technique based onARMAXrdquo SmartMaterials and Structures vol 12 no 5 pp 845ndash850 2003
[31] R D Robinett C R Dohrmann G R Eisler et al FlexibleRobot Dynamics and Controls Kluwer AcademicPlenum Pub-lishers New York NY USA 2002
[32] L Cai and G Song ldquoJoint stick-slip friction compensation ofrobotmanipulators by using smooth robust controllersrdquo Journalof Robotic Systems vol 11 no 6 pp 451ndash470 1994
slidingmode control schemesThe adaptive control approachis suitable for control of the dynamic systems with constantor slowly varying uncertain parameters The parameters ofthe adaptive controller are updated during operation basedon the measured performance The adaptive control law isdesigned to guarantee the asymptotic stability for the controlsystem under disturbances
With merits of both the adaptive control and the slidingmode control schemes adaptive sliding mode control canachieve satisfactory control performancewith asymptotic sta-bility when the system is subject to model uncertainties anddisturbances Due to these advantages the adaptive slidingmode control has been studied extensively for the control ofdynamic systems with uncertainties and disturbances Yaoand Tomizuka [4] proposed a systematic way to combinethe adaptive control design technique and the sliding modecontrol methodology for trajectory tracking control of robotmanipulators in the presence of parametric uncertainties andexternal disturbance Wang et al [5] proposed an indirectadaptive fuzzy sliding mode control scheme for a class ofnonlinear systems by using the concept of sliding modecontrol design and Lyapunov synthesis approach Simula-tion studies have shown that the adaptive design of fuzzysliding mode controller performs very well in the presenceof an unknown disturbance Song et al [6] proposed anintegrated adaptive-robust control methodology for controlof nonlinear uncertain systems The new approach can takeadvantage of both adaptive and robust control methods andit offers to handle various uncertainties reject external distur-bance and guarantee global asymptotic stability with desiredtransient response Song et al [7] proposed an integrandadaptive-robust approach along with a smooth adaptiverobust friction compensation strategy for tracking the controlof uncertain robot manipulators with joint stick-slip frictionWai [8] proposed an adaptive sliding mode control systemto control the position of an induction servomotor driveIn the adaptive sliding mode control system an adaptivealgorithm is utilized to estimate the bound of uncertaintiesSimulated and experimental results show that the dynamicbehaviors of the proposed control systems are robust withregard to uncertainties Lin and Chen [9] developed anew strategy called adaptive fuzzy sliding mode control(AFSMC) The parameters of the membership functionsin the AFSMC are changed according to some adaptivealgorithm for the purpose of controlling the system statesto converge to a user-defined sliding surface and then slidealong it The sliding mode control can achieve satisfactoryvibration control performance since the vibration controlsystem is usually a nonlinear system with model uncertaintyand external disturbance Fei and Ding [10] proposed aradial basis function neural network adaptive sliding modecontroller to track a microelectromechanical system triaxialgyroscope by integrating the adaptive control neural networkcontrol and sliding mode control Switching function andsliding mode controller were used as the input and outputof the radial basis function of neutral network to deal withnonlinearity with an online learning ability Zhang and Han[11] applied robust sliding mode H
infincontrol using time-
varying delayed states for an offshore steel jacket platform
with self-exciting nonlinear wave force external disturbanceand parametric uncertainties Zhang et al [12] proposed anew variable structure sliding mode control strategy witha combined approaching law for fast tool servo to controlcutting force in diamond-cutting microstructured surfacesLi et al [13] proposed an adaptive sliding mode control viaTakagi-Sugeno (T-S) fuzzy approach to control suspensionsystem The T-S fuzzy control was used to describe theoriginal nonlinear system for the control design via a sectornonlinearity approach A sufficient condition which could beconverted to convex optimization problem is proposed forasymptotical stability of the designing sliding motion
Piezoelectric materials are commonly used in the vibra-tion control as actuators and sensors Piezoelectric materialis an innovative smart material with the advantages of quickresponse low-power cost and easy implementation Duringthe past years piezoelectric material has been successfullyapplied in different kinds of active or passive vibration controlmethods such as PPF control [14 15] fuzzy model referencecontrol [16] vibration control for robotic links [17 18] andpassive vibration control with shunted piezoelectric material[19] Fei et al [20] compares the uses of PZT-based optimalPID strain rate feedback (SRF) and PPF control schemeson the vibration control of a cantilever beam Piezoelectricmaterial-based sliding mode control provides an innovativeand effective strategy for the vibration control of systemswith model inaccuracy and uncertainties Choi et al [21]developed an active vibration control of a hybrid smartstructure consisting of a piezoelectric film actuator andan electrorheological fluid actuator A neuro-sliding modecontroller incorporating neural networks with the conceptof sliding mode control is formulated for a piezoelectric filmactuator Actuators and sensors made of piezoelectric mate-rials have the merits of low weight and distributed natureJha and Inman [22] utilized piezoelectric actuators andsensors in the vibration suppression of gossamer structuresIn order to achieve good vibration reduction performancesunder model uncertainty and external disturbance a slidingmode controller and observer are designed The simulationstudies show that the piezoelectric actuators and sensorsare suitable for vibration suppression of an inflatable torusChoi and Kim [23] proposed a new discrete-time fuzzysliding mode controller and applied it to vibration controlof a smart structure with a piezofilm actuator Experimentalresults of transient forced and random vibration of smartstructures demonstrate the effectiveness of the proposedmethod Recent studies by other researchers have refinedslidingmode controlmethods for use inmicronanoactuators[24ndash26] Hu and Zhu [27] applied a sliding mode controlwith observer to reduce the vibration of a flexible structurewith a piezoelectrical actuator and a strain gauge sensor Astate space dynamic system was derived by using a finiteelement method and the experimental modal test The con-trollerrsquos sliding surface was determined by using optimizationmethod that minimized the cost function of the states Thereare also many other schemes of sliding mode control appliedto vibration control by using piezoelectric material
Many simulation studies have been performed to makecontributions to the theoretical study of the adaptive sliding
Mathematical Problems in Engineering 3
mode control Some recent studies for example have inves-tigated self-tuning laws especially for uncertain nonlinearsystems [28] and advanced observer-based robust control ofpiezoelectric structures [29] However not enough experi-mental study of the adaptive sliding mode control has beenconducted The adaptive sliding mode controller is designedto handle the system uncertainties and disturbances and it ismore important to conduct the experimental study to verifythe robustness of the proposed controller under real systemswith uncertainties and disturbances than under the simulatedsystems Experimental verification of adaptive sliding modecontroller is an important issue for the study of the adaptivesliding mode controller The real-time experimental resultsare more convincing in verifying and proving the robustnessof an adaptive sliding mode control than the simulationresults In this paper experimental comparative study oftwo different adaptive robust sliding mode controllers isconducted An adaptive robust sliding mode control schemebased on the Lyapunovrsquos directmethod is proposed for activevibration control of a flexible beam using PZT (lead zirconatetitanate) sensors and actuators PZT a commonly usedpiezoelectric ceramic material is adopted in this researchand surface-bond PZT patches were used for vibrationcontrol Two adaptive robust sliding mode controllers forvibration suppression were designed one adaptive slidingmode controller uses a discontinuous bang-bang robust com-pensator and the other uses a compensator with a smoothhyperbolic tangent function The Lyapunovrsquos direct methodis used to prove the asymptotic stability of both controllersThe proposed adaptive sliding mode controller reduces theconservatism of the sliding mode controller The asymptoticstability of the proposed control system was proved by usingthe Lyapunovrsquos direct method Experiments have been imple-mented on a flexible beam with PZT actuators and sensorsExperimental results show that both controllers successfullysuppress vibrations of the flexible beam in an efficient wayHowever the bang-bang robust compensator causes small-magnitude chattering because of the discontinuous switchingactions With the smooth compensator vibration is quicklysuppressed and no chattering is induced Furthermore therobustness of the controllers is successfully tested by addingmass to the flexible beam Multimode vibration control ofthe beam with mass added is also conducted to verify theeffectiveness and robustness of the proposed adaptive slidingmode control in vibration control
2 Experimental Setup
A flexible aluminum beam in a cantilevered configuration(Figure 1) is used as the experimental object to test theeffectiveness of the proposed vibration suppression methodThe property of the beam is shown in Table 1 PZT typepiezoelectric patches are used as the smart sensor andactuator One relatively large PZT patch is surface-bonded oneach side of the flexible aluminum beam near its cantileveredroot These two patches are used as actuators to excite and toenable active vibration control of the beam One smaller PZTpatch is surface-bonded on one side of the beam also near its
Table 1 Beam properties
Symbol Quantity Unit Value119871 Beam length mm 7365119908119887
Beam width mm 531119905119887
Beam thickness mm 1120588119887
Beam density kgm3 2690119864 Modulus of elasticity Nm2
703 times 1010
Beam
Sensor
BaseActuator
Figure 1 The experimental setup
cantilevered root This smaller PTZ patch acts as sensor forthe feedback of the signals in the active control algorithmsPhysical position of PZT actuators and sensors is shown inFigure 2 Both the PZT actuators and sensor are installednear the cantilevered root of the beam since this location hasthe highest strain energy along the length of the beam Thisconfiguration of actuator and sensor placement results in themost effective actuation and sensing for vibration controlTheproperties of the PZT patches are shown in Table 2
A PZT actuator has some nonlinear characteristics suchas hysteresis Hysteresis should be compensated for accuratepositioning for position regulation or tracking tasks Thehysteresis width of PZT is about 10 of its stroke andnormally PZT actuator is not classified as a ldquohighrdquo nonlinearactuator In addition the hysteresis width reduces with theincreasing frequency Therefore for active vibration controlit is a common practice that the hysteresis is not consideredin such an application The nonlinear characteristics of PZTtransducers are not considered in this paper
Experiments have shown that the dominant mode of theflexible beam is its first mode and is the major concern forvibration suppression (details can be found in Section 5)Since the first mode is dominant in the experiment highermodes are truncated for simplification Bu et al [30] derivedthe model for a flexible smart beam with piezoceramicactuators and strain gaugesWhen the first mode is dominantin the analysis (ie only the first mode is considered) modelbetween PZT actuator voltage and strain sensor can berepresented as a simplified linear second order system whichvalidates the practice of not considering the nonlinear char-acteristics of PZT transducers in vibration control problemsThe PZT sensor output represents the strain of the sensorlocation Therefore the PZT sensor can be applied as straingauge for control purposes In the control design a simplifiedsecond order model is used as the structure of the modelHowever the model parameters are assumed unknown Thedesign of the controller is not dependent on the parameters
4 Mathematical Problems in Engineering
Table 2 Properties of PZT patches used on the beam
Symbol Quantity Unit PZT actuator PZT sensor119871 times 119908 times 119905 Dimensions mm 46 times 3327 times 025 14 times 7 times 025
11988933
Strain coefficient CN 741 times 10minus10
741 times 10minus10
11988931
Strain coefficient CN minus274 times 10minus10
minus274 times 10minus10
120588119875
PZT density kgm3 7500 7500119864119875
Youngrsquos modulus Nm263 times 10
10
63 times 1010
Beam
Sensor
BaseActuator23mm 992mm
61mm
79mm
735mm
7365mm
Figure 2 The physical position of the piezoelectric actuators andsensor
of the model The plant is represented in the form of thefollowing general simplified dynamic equation
119898 + 119888 + 119896119909 = 119906 (1)
where119898 119888 and 119896 are the coefficients relative with the systemmass damping and stiffness respectively x is the PZT sensorsignal and 119906 is PZT actuator signal
3 Design of Adaptive Robust Sliding ModeController and Stability Proof
The design principle of the proposed adaptive robust slidingmode controller for vibration control is similar to that of theadaptive sliding mode controller for robot manipulator in [16 31]
To assist control design define 119890 = 119909119903minus 119909 where the
subscript ldquo119903rdquo stands for the reference command Define thesliding surface variable as
where 119888 and are the initial estimations of the real valuesof parameters of 119898 119888 and 119896 and these estimates will beadjusted online by the slidingmode based adaptive controllerThe term 119886 sgn(119904) is the bang-bang robust compensator to
ensure stability where 119886 is a positive constant and the sgnfunction is defined as
sgn (119904) = +1 if 119904 gt 0
sgn (119904) = minus1 if 119904 lt 0(4)
With this feedback controller (3) the dynamic equation of thesystem (1) becomes
= minus119886 sdot 119904 sdot sgn (119904) minus 120593119879 (119884119904 minus Γminus1 120593)
(11)
Notice that minus119886 sdot 119904 sdot sgn(119904) le 0The adaptation law will be designed as 119884119904 minus Γminus1 120593 = 0
Therefore V = minus119886 sdot 119904 sdot sgn(119904) le 0 while V = 0 onlywhen 119904 = 0 According to LaSallersquos theorem the control sys-tem is asymptotically stable in the sense of Lyapunov if theadaptation law satisfies 119884119904 minus Γminus1 120593 = 0 In the sliding modecontroller the positive constant 119886 which is the amplitude ofthe robust compensator in control law (3) needs to be theupper bounding of the uncertainty of the system model andthe external disturbance to ensure the stability Due to thelack of the prior knowledge of the system the upper boundingof the uncertainty and the disturbance may be difficult tobe properly or precisely evaluated In order to ensure theasymptotic stability of the system under all conditions theamplitude of the robust compensator usually needs to bechosen large enough to compensate for the worst case of allpossible uncertainty and external disturbances Therefore inthe sliding mode controller the transient performance andthe steady state performance of the controller are usuallysacrificed for the asymptotic stability of the control systemHowever in the proposed adaptive controller the adaptationlaw is designed to guarantee the asymptotic stability underuncertainty and disturbance Therefore there will be lessconstraint on the choice of the amplitude 119886 of the robustcompensator in the proposed method Conservatism of therobust controller would be dramatically reduced by introduc-ing an adaptive scheme into the controller The asymptoticstability of the proposed adaptive sliding mode controller isguaranteed with larger values of allowable uncertainty anddisturbances Both adaptive control part and sliding modecontrol part could compensate for the system uncertaintiesand external disturbancesThe adaptation law can be derivedas
The shortcoming of the adaptive robust sliding mode con-troller (3) is that the bang-bang robust compensatorwill causechattering in implementationThe chattering phenomenon iscaused by the sign function term The sign function switchesthe control action abruptly especially near the equilibriumpoint Chattering is not desirable since it involves extremelyhigh control activity andmay excite high frequency dynamicsof a structure
4 Design of a Smooth Adaptive Robust SlidingMode Controller and Stability Proof
In order to attenuate the chattering associated with the bang-bang compensator a hyperbolic tangent function is used toswitch the control action instead of the sign function Thecontroller becomes
where 119886 and120572 are positive numbers and tanh is the hyperbolictangent function
tanh (119909) = 119890119909
minus 119890minus119909
119890119909 + 119890minus119909 (18)
The 119886 tanh(120572119904) is a robust compensator which provides asmooth control action The hyperbolic function will switchthe control action smoothly and avoid causing chatteringThis robust compensator is continuously differentiable withrespect to the control variable 119904 It generates a smoothcontrol action Compared with the commonly used bang-bang or saturation robust controllers the smooth robustcontroller has advantages in ensuring smooth control inputand the stability of the closed-loop system [32] A detailedcomparative study is provided in Song and Mukherjee [1]Normally the larger the value of the parameter 120572 is the closer
6 Mathematical Problems in Engineering
the tanh(120572119904) function will mimic the sign(119904) function Toavoid chattering a relative smaller value of 120572 can be chosenWhen implementing the controller the value of 120572 is oftenexperimentally determined
To prove the stability of the smooth adaptive robustcontroller (17) define the Lyapunov function candidate as
V =1
21198981199042
+1
2120593119879
Γminus1
120593 (19)
Using the similar process shown in Section 3 we have
V = minus119886 sdot 119904 sdot tanh (120572119904) minus 120593119879 (119884119904 minus Γminus1 120593) (20)
Please note that 119888 and are updated according to (14) tomake 119884119904 minus Γminus1 120593 = 0 Then V = minus119886 sdot 119904 sdot tanh(120572119904) le 0 whileV = 0 only when 119904 = 0 Therefore according to LaSallersquostheorem the closed-loop control system with the smoothadaptive robust controllers (17) is asymptotically stable in thesense of Lyapunov
5 Experiment Results
51 Open Loop Testing of the Aluminum Beam First openloop testing is performed to find the dominant mode of thebeam for active vibration control The beam was subjectedto a mechanical impact at its midlength for a multimodeexcitation The power spectrum density (PSD) plots of the30-second time response of the multimode excitation wereobtainedThe corresponding frequencies of peaks in the PSDplots represent themodal frequencies of the flexible beam Tostudy the attenuation of vibration at each modal frequencythe PSD plots of the first 5 seconds and the last 5 secondsof the beam vibration are shown in Figure 3 The peaks inFigure 3 clearly reveal the modal frequencies of the flexiblebeam the 1st mode at 164Hz the 2nd mode at 977Hzand the 3rd mode at 2685Hz A comparison of PSD plotsof the vibration data for the first 5 seconds and the last 5seconds clearly shows the vibration attenuation at eachmodalfrequency The dB value and dB drop of the first three modesare shown in Table 3 From Figure 3 and Table 3 it can beseen that the dB levels of the second and third modes at thefirst five-second and last five-second periods are significantlylower than that of the 1st mode The dB drop value of thefirst three modes shows that the second and third modesattenuated much faster than the first mode Therefore it canbe concluded that the first modal frequency 164Hz is thedominant one in the vibration and the first mode should bethe target mode in the active vibration control To target atthis mode the vibration control is designed based on thesimplified plant (1) which includes only the dynamics of thefirst mode
52 Experimental Results for the Adaptive Robust SlidingMode Controller To verify the effectiveness of the adaptiverobust sliding mode control (3) and the smooth adaptivesliding mode control (17) experiments were conducted onthe aluminum flexible beam In the experiment 119904 = 119890 +
120582119890 = 119890 + 005119890 and 1199031= 1199032= 1199033= 01 for the diagonal
Table 3 PSD drop comparison of first 5 seconds and last 5 seconds
Mode 1 Mode 2 Mode 3dB level of the first 5 seconds 1673 598 minus1436dB level of the last 5 seconds 291 minus1723 minus3519dB drop 1382 2321 2083
20
10
minus10
minus20
minus30
minus40
minus50
minus60
minus70
minus800
0
10 20 30 40 50 60
Frequency (Hz)
Pow
er sp
ectr
um m
agni
tude
(dB)
First 5 secondsLast 5 seconds
Figure 3 Comparison of PSD plots of first 5 seconds and last 5seconds for time response of multimode excitation of the flexiblebeam
matrix Γ In each test the beam was excited by a sinusoidalsignal at its first modal frequency combined with a lowpower white noise for the initial 5 seconds The amplitudeof the sinusoidal signal was 075V The variance of the whitenoise was 00497 The sinusoidal signal and the white noisewere summed in the Simulink software and downloaded intodSPACE DS1102 Floating-Point Controller Board for real-time implementation The output of the dSPACE controllerboard was amplified by an amplifier for capacitive load beforeexciting the PZT actuators After the initial 5 seconds firstmode excitation was stopped and the active vibration controlstarted to suppress the induced vibration A free vibration testwas conducted for the purpose of comparison In the freevibration test no control signal was implemented after theinitial 5-second excitation
The upper diagram of Figure 4 is the time responsecomparison of free vibration with that of the active control(with the sign function) From the experimental results thevibration is suppressed in about 8 oscillation cycles (about5 seconds) However there is small-magnitude chattering inthe steady state which may cause instability when a certainlevel of disturbance exists The lower diagram of Figure 4 isthe time response comparison of free vibration with that ofthe active control (with the tanh function) It can be seen thatthe robust sliding mode control can suppress the vibration inabout 8 oscillation cycles without the chattering in the steadystate Experimental results show that the chattering can be
Mathematical Problems in Engineering 7
06
04
02
0
minus02
minus04
5 10 15 20
Sens
or v
olta
ge (V
)
Sliding-mode control with sign functionFree vibration
Time (s)
(a)
06
04
02
0
minus02
minus04
5 10 15 20
Sens
or v
olta
ge (V
)
Sliding-mode control with tanh functionFree vibration
Time (s)
(b)
Figure 4 The time response comparisons of the adaptive slidingmode controller (with the sign function) with the adaptive robustsliding mode controller (with tanh function)
eliminated by implementing the hyperbolic tangent function(tanh function) as the switching function instead of the signfunction
The upper diagram of Figure 5 is the control actionwith the bang-bang robust compensator Compared with thetime response in Figure 4 it can be seen that the chatteringhappens when the control action switches abruptly betweenits positive limit and negative limit when the system is alreadyin the steady state (after 10 seconds) The lower diagramof Figure 5 is the control action with the smooth robustcompensator From the diagram it can be seen that thecontrol signal switches smoothly Therefore the chatteringin the output signal is eliminated Figure 6 shows the timehistory of the parametersrsquo updating process Furthermore itis also worthwhile to note that no higher modes have beenexcited even though the plant is actually amultimode systemThis justifies that the primary vibration control target is thefirst mode
53 Experimental Results for the Adaptive Robust SlidingModeControl with Mass Uncertainty To verify the robustness ofthe proposed adaptive robust controller the plant is changedby adding mass on the beam Four 6-gram adhesive masseswere attached to both sides of the beam with two at theremote tip and the other two at the midlength of the beamThe mass of the aluminum beam is increased from 10520gram to 12920 gramThefirstmodal frequency of the originalbeam is 167Hz whereas the first modal frequency of the
Time (s)
200
100
0
minus100
minus2005 10 15 20
Actu
ator
sign
al (V
)
Sliding-mode control with sign function
(a)
Time (s)
200
100
0
minus100
minus2005 10 15 20
Actu
ator
sign
al (V
)Sliding-mode control with tanh function
(b)
Figure 5 The control signal comparisons of the adaptive slidingmode controller (with the sign function) with the adaptive robustsliding mode controller (with tanh function)
00835
0083
008255 10 15 20 25
Time (s)
m
Time (s)
082
081
085 10 15 20 25
c
1100002
1100001
110
Sliding-mode control with sign functionSliding-mode control with tanh function
Time (s)5 10 15 20 25
k
Figure 6 The parametersrsquo updating time history
beam with added mass has been decreased to 1457Hz Theincreasing of the mass has caused the decreasing of thenatural frequency of the beamTherefore in this experimentthe frequency of the sinusoidal excitation signal has beenchanged to 1457Hz to ensure excitation of the beam Thesame controller is used for the vibration control of the beamwith added mass From the experimental results (Figure 7)
8 Mathematical Problems in Engineering
05
04
02
03
01
0
minus01
minus02
minus03
minus04
minus055 10 15 20
Time (s)
Free vibrationSliding-mode control with tanh function
Sens
or v
olta
ge (V
)
Figure 7 The time response comparisons of the adaptive slidingmode controller with tanh function for the flexible beam with 24-gram mass added
1
08
06
04
02
0
minus02
minus04
minus06
minus08
minus10 1 2 3 4 5 6 7
Time (s)
Sliding-mode control with tanh function
Sens
or v
olta
ge (V
)
Free vibration
Figure 8 The time response of comparison of the adaptive slidingmode controller with tanh function for the flexible beam withmultimode excitation
it can be seen that the adaptive robust sliding mode caneffectively suppress the vibration of the beam after its physicalproperty has been changedThis demonstrates the robustnessof the proposed controllers
54 Experimental Results for the Adaptive Robust SlidingModeControl with Multimode Excitation To test its robustness tohigher order dynamics and disturbances multimodal excita-tion of the flexible beam with mass added is conducted Fig-ure 8 shows the comparison of time response of multimodalexcitationwith that of the free vibration Experimental results
clearly show that the proposed method is still effective inmultimodal vibration control and this further demonstratesthe robustness of the proposed controller
6 Conclusion
In this paper an experimental study of the robust adaptivesliding mode control was conducted on a flexible beamwith piezoceramic actuators and sensor for vibration controlpurpose Two different schemes of adaptive sliding modecontrol have been comparatively studied during the experi-ment The adaptive robust sliding mode control with bang-bang action could suppress the induced vibration of a flexiblebeam in a quick fashion However it causes chattering duringsteady state To eliminate the chattering a hyperbolic tangentfunction is used to replace the sign function in the robustcontrol action Both adaptive robust sliding mode controllersare designed based on the Lyapunov direct method In theproposed adaptive sliding mode controller there will beless constraint on the choice of the amplitude of the robustcompensator The conservatism of the control system hasbeen reduced by introducing the adaptive scheme into thesliding mode controlThe asymptotic stability of the adaptiverobust sliding mode control is proved by using Lyapunovdirect method in the paper The experimental results showthat the proposed adaptive robust sliding mode control withthe smooth robust compensator eliminates the chatteringwhile keeping the advantages of adaptive robust slidingmodecontrol Also the proposedmethods are robust to the varianceof the plant parameters as demonstrated experimentally
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
This research was partially supported by Grant no 51278387Grant no 51278084 and Grant no 51121005 (Science Fundfor Creative Research Groups) from the National ScienceFoundation of China (NSFC)
References
[1] G Song and R Mukherjee ldquoA comparative study of conven-tional nonsmooth time-invariant and smooth time-varyingrobust compensatorsrdquo IEEE Transactions on Control SystemsTechnology vol 6 no 4 pp 571ndash576 1998
[2] J-H Park K-W Kim and H-H Yoo ldquoTwo-stage sliding modecontroller for vibration suppression of a flexible pointing sys-temrdquo Proceedings of the Institution ofMechanical Engineers PartC Journal of Mechanical Engineering Science vol 215 no 2 pp155ndash166 2001
[3] X Yu Z Man and B Wu ldquoDesign of fuzzy sliding-mode con-trol systemsrdquo Fuzzy Sets and Systems vol 95 no 3 pp 295ndash3061998
Mathematical Problems in Engineering 9
[4] B Yao and M Tomizuka ldquoSmooth robust adaptive slidingmode control of manipulators with guaranteed transient per-formancerdquo in Proceedings of the American Control Conferencevol 1 pp 1176ndash1180 July 1994
[5] J Wang A B Rad and P T Chan ldquoIndirect adaptive fuzzysliding mode control part I fuzzy switchingrdquo Fuzzy Sets andSystems vol 122 no 1 pp 21ndash30 2001
[6] G Song R W Longman and R Mukherjee ldquoIntegratedsliding-mode adaptive-robust controlrdquo IEE Proceedings ControlTheory and Applications vol 146 no 4 pp 341ndash347 1999
[7] G Song R W Longman and L Cai ldquoIntegrated adaptive-robust control of robot manipulatorsrdquo Journal of Robotic Sys-tems vol 15 no 12 pp 699ndash712 1998
[8] R-J Wai ldquoAdaptive sliding-mode control for induction servo-motor driverdquo IEE Proceedings Electric Power Applications vol147 no 6 pp 553ndash562 2000
[9] S-C Lin and Y-Y Chen ldquoDesign of adaptive fuzzy slidingmode for nonlinear system controlrdquo in Proceedings of the 3rdIEEE Conference on Fuzzy Systems pp 35ndash39 June 1994
[10] J Fei and H Ding ldquoAdaptive vibration control of microelec-tromechanical systems triaxial gyroscope using radial basisfunction sliding mode techniquerdquo Proceedings of the Institutionof Mechanical Engineers Part I Journal of Systems and ControlEngineering vol 227 no 2 pp 264ndash269 2013
[11] B L Zhang and Q L Han ldquoRobust sliding mode 119867infin
controlusing time-varying delayed states for offshore steel jacketplatformsrdquo in Proceedings of the IEEE International Symposiumon Industrial Electronics (ISIE rsquo13) pp 1ndash6 IEEE 2013
[12] H Zhang G Dong M Zhou C Song Y Huang and K Du ldquoAnew variable structure sliding mode control strategy for FTS indiamond-cutting microstructured surfacesrdquo The InternationalJournal of AdvancedManufacturing Technology vol 65 no 5ndash8pp 1177ndash1184 2013
[13] H Li J Yu C Hilton and H Liu ldquoAdaptive sliding modecontrol for nonlinear active suspension vehicle systems usingTS fuzzy approachrdquo IEEE Transactions on Industrial Electronicsvol 60 no 8 2013
[14] J L Fanson and T K Caughey ldquoPositive position feedbackcontrol for large space structuresrdquo AIAA Journal vol 28 no 4pp 717ndash724 1990
[15] G Song P Qiao V Sethi and A Prasad ldquoActive vibrationcontrol of a smart pultruded fiber-reinforced polymer I-beamrdquoin International Symposium on Smart Structures and Materialsvol 4696 of Proceedings of SPIE pp 197ndash208 March 2002
[16] PMayhan andGWashington ldquoFuzzymodel reference learningcontrol a new control paradigm for smart structuresrdquo SmartMaterials and Structures vol 7 no 6 pp 874ndash884 1998
[17] V Bottega AMolter J SO Fonseca andR Pergher ldquoVibrationcontrol of manipulators with flexible nonprismatic links usingpiezoelectric actuators and sensorsrdquo Mathematical Problems inEngineering vol 2009 Article ID 727385 16 pages 2009
[18] A Molter V Bottega O A A Da Silveira and J S O FonsecaldquoSimultaneous piezoelectric actuator and sensor placementoptimization and control design of manipulators with flexiblelinks using SDREmethodrdquoMathematical Problems in Engineer-ing vol 2010 Article ID 362437 23 pages 2010
[19] M Ahmadian and K M Jeric ldquoOn the application of shuntedpiezoceramics for increasing acoustic transmission loss instructuresrdquo Journal of Sound and Vibration vol 243 no 2 pp347ndash359 2001
[20] J Fei Y Fang and C Yan ldquoThe comparative study of vibrationcontrol of flexible structure using smart materialsrdquo Mathemat-ical Problems in Engineering vol 2010 Article ID 768256 13pages 2010
[21] S-B Choi Y-K Park and C-C Cheong ldquoActive vibrationcontrol of hybrid smart structures featuring piezoelectric filmsand electrorheological fluidsrdquo in The International Society forOptical Engineering vol 2717 of Proceedings of the SPIE pp544ndash552 February 1996
[22] A K Jha and D J Inman ldquoSliding mode control of a Gossamerstructure using smart materialsrdquo Journal of Vibration andControl vol 10 no 8 pp 1199ndash1220 2004
[23] S-B Choi and M-S Kim ldquoNew discrete-time fuzzy-sliding-mode control with application to smart structuresrdquo Journal ofGuidance Control and Dynamics vol 20 no 5 pp 857ndash8641997
[24] S Bashash and N Jalili ldquoRobust multiple frequency trajec-tory tracking control of piezoelectrically driven micronano-positioning systemsrdquo IEEE Transactions on Control SystemsTechnology vol 15 no 5 pp 867ndash878 2007
[25] H C Liaw B Shirinzadeh and J Smith ldquoSliding-modeenhanced adaptive motion tracking control of piezoelectricactuation systems formicronanomanipulationrdquo IEEETransac-tions on Control Systems Technology vol 16 no 4 pp 826ndash8332008
[26] Y Li and Q Xu ldquoAdaptive sliding mode control with perturba-tion estimation and PID sliding surface for motion tracking of apiezo-driven micromanipulatorrdquo IEEE Transactions on ControlSystems Technology vol 18 no 4 pp 798ndash810 2010
[27] J Hu and D Zhu ldquoVibration control of smart structure usingsliding mode control with observerrdquo Journal of Computers vol7 no 2 pp 411ndash418 2012
[28] L Zhang Z Zhang Z Long and A Hao ldquoSlidingmode controlwith auto-tuning law forMaglev SystemrdquoEngineering vol 2 no2 pp 107ndash112 2010
[29] T Rittenschober andK Schlacher ldquoObserver-based self sensingactuation of piezoelastic structures for robust vibration controlrdquoAutomatica vol 48 no 6 pp 1123ndash1131 2012
[30] X Bu L Ye Z Su and C Wang ldquoActive control of a flexiblesmart beam using a system identification technique based onARMAXrdquo SmartMaterials and Structures vol 12 no 5 pp 845ndash850 2003
[31] R D Robinett C R Dohrmann G R Eisler et al FlexibleRobot Dynamics and Controls Kluwer AcademicPlenum Pub-lishers New York NY USA 2002
[32] L Cai and G Song ldquoJoint stick-slip friction compensation ofrobotmanipulators by using smooth robust controllersrdquo Journalof Robotic Systems vol 11 no 6 pp 451ndash470 1994
mode control Some recent studies for example have inves-tigated self-tuning laws especially for uncertain nonlinearsystems [28] and advanced observer-based robust control ofpiezoelectric structures [29] However not enough experi-mental study of the adaptive sliding mode control has beenconducted The adaptive sliding mode controller is designedto handle the system uncertainties and disturbances and it ismore important to conduct the experimental study to verifythe robustness of the proposed controller under real systemswith uncertainties and disturbances than under the simulatedsystems Experimental verification of adaptive sliding modecontroller is an important issue for the study of the adaptivesliding mode controller The real-time experimental resultsare more convincing in verifying and proving the robustnessof an adaptive sliding mode control than the simulationresults In this paper experimental comparative study oftwo different adaptive robust sliding mode controllers isconducted An adaptive robust sliding mode control schemebased on the Lyapunovrsquos directmethod is proposed for activevibration control of a flexible beam using PZT (lead zirconatetitanate) sensors and actuators PZT a commonly usedpiezoelectric ceramic material is adopted in this researchand surface-bond PZT patches were used for vibrationcontrol Two adaptive robust sliding mode controllers forvibration suppression were designed one adaptive slidingmode controller uses a discontinuous bang-bang robust com-pensator and the other uses a compensator with a smoothhyperbolic tangent function The Lyapunovrsquos direct methodis used to prove the asymptotic stability of both controllersThe proposed adaptive sliding mode controller reduces theconservatism of the sliding mode controller The asymptoticstability of the proposed control system was proved by usingthe Lyapunovrsquos direct method Experiments have been imple-mented on a flexible beam with PZT actuators and sensorsExperimental results show that both controllers successfullysuppress vibrations of the flexible beam in an efficient wayHowever the bang-bang robust compensator causes small-magnitude chattering because of the discontinuous switchingactions With the smooth compensator vibration is quicklysuppressed and no chattering is induced Furthermore therobustness of the controllers is successfully tested by addingmass to the flexible beam Multimode vibration control ofthe beam with mass added is also conducted to verify theeffectiveness and robustness of the proposed adaptive slidingmode control in vibration control
2 Experimental Setup
A flexible aluminum beam in a cantilevered configuration(Figure 1) is used as the experimental object to test theeffectiveness of the proposed vibration suppression methodThe property of the beam is shown in Table 1 PZT typepiezoelectric patches are used as the smart sensor andactuator One relatively large PZT patch is surface-bonded oneach side of the flexible aluminum beam near its cantileveredroot These two patches are used as actuators to excite and toenable active vibration control of the beam One smaller PZTpatch is surface-bonded on one side of the beam also near its
Table 1 Beam properties
Symbol Quantity Unit Value119871 Beam length mm 7365119908119887
Beam width mm 531119905119887
Beam thickness mm 1120588119887
Beam density kgm3 2690119864 Modulus of elasticity Nm2
703 times 1010
Beam
Sensor
BaseActuator
Figure 1 The experimental setup
cantilevered root This smaller PTZ patch acts as sensor forthe feedback of the signals in the active control algorithmsPhysical position of PZT actuators and sensors is shown inFigure 2 Both the PZT actuators and sensor are installednear the cantilevered root of the beam since this location hasthe highest strain energy along the length of the beam Thisconfiguration of actuator and sensor placement results in themost effective actuation and sensing for vibration controlTheproperties of the PZT patches are shown in Table 2
A PZT actuator has some nonlinear characteristics suchas hysteresis Hysteresis should be compensated for accuratepositioning for position regulation or tracking tasks Thehysteresis width of PZT is about 10 of its stroke andnormally PZT actuator is not classified as a ldquohighrdquo nonlinearactuator In addition the hysteresis width reduces with theincreasing frequency Therefore for active vibration controlit is a common practice that the hysteresis is not consideredin such an application The nonlinear characteristics of PZTtransducers are not considered in this paper
Experiments have shown that the dominant mode of theflexible beam is its first mode and is the major concern forvibration suppression (details can be found in Section 5)Since the first mode is dominant in the experiment highermodes are truncated for simplification Bu et al [30] derivedthe model for a flexible smart beam with piezoceramicactuators and strain gaugesWhen the first mode is dominantin the analysis (ie only the first mode is considered) modelbetween PZT actuator voltage and strain sensor can berepresented as a simplified linear second order system whichvalidates the practice of not considering the nonlinear char-acteristics of PZT transducers in vibration control problemsThe PZT sensor output represents the strain of the sensorlocation Therefore the PZT sensor can be applied as straingauge for control purposes In the control design a simplifiedsecond order model is used as the structure of the modelHowever the model parameters are assumed unknown Thedesign of the controller is not dependent on the parameters
4 Mathematical Problems in Engineering
Table 2 Properties of PZT patches used on the beam
Symbol Quantity Unit PZT actuator PZT sensor119871 times 119908 times 119905 Dimensions mm 46 times 3327 times 025 14 times 7 times 025
11988933
Strain coefficient CN 741 times 10minus10
741 times 10minus10
11988931
Strain coefficient CN minus274 times 10minus10
minus274 times 10minus10
120588119875
PZT density kgm3 7500 7500119864119875
Youngrsquos modulus Nm263 times 10
10
63 times 1010
Beam
Sensor
BaseActuator23mm 992mm
61mm
79mm
735mm
7365mm
Figure 2 The physical position of the piezoelectric actuators andsensor
of the model The plant is represented in the form of thefollowing general simplified dynamic equation
119898 + 119888 + 119896119909 = 119906 (1)
where119898 119888 and 119896 are the coefficients relative with the systemmass damping and stiffness respectively x is the PZT sensorsignal and 119906 is PZT actuator signal
3 Design of Adaptive Robust Sliding ModeController and Stability Proof
The design principle of the proposed adaptive robust slidingmode controller for vibration control is similar to that of theadaptive sliding mode controller for robot manipulator in [16 31]
To assist control design define 119890 = 119909119903minus 119909 where the
subscript ldquo119903rdquo stands for the reference command Define thesliding surface variable as
where 119888 and are the initial estimations of the real valuesof parameters of 119898 119888 and 119896 and these estimates will beadjusted online by the slidingmode based adaptive controllerThe term 119886 sgn(119904) is the bang-bang robust compensator to
ensure stability where 119886 is a positive constant and the sgnfunction is defined as
sgn (119904) = +1 if 119904 gt 0
sgn (119904) = minus1 if 119904 lt 0(4)
With this feedback controller (3) the dynamic equation of thesystem (1) becomes
= minus119886 sdot 119904 sdot sgn (119904) minus 120593119879 (119884119904 minus Γminus1 120593)
(11)
Notice that minus119886 sdot 119904 sdot sgn(119904) le 0The adaptation law will be designed as 119884119904 minus Γminus1 120593 = 0
Therefore V = minus119886 sdot 119904 sdot sgn(119904) le 0 while V = 0 onlywhen 119904 = 0 According to LaSallersquos theorem the control sys-tem is asymptotically stable in the sense of Lyapunov if theadaptation law satisfies 119884119904 minus Γminus1 120593 = 0 In the sliding modecontroller the positive constant 119886 which is the amplitude ofthe robust compensator in control law (3) needs to be theupper bounding of the uncertainty of the system model andthe external disturbance to ensure the stability Due to thelack of the prior knowledge of the system the upper boundingof the uncertainty and the disturbance may be difficult tobe properly or precisely evaluated In order to ensure theasymptotic stability of the system under all conditions theamplitude of the robust compensator usually needs to bechosen large enough to compensate for the worst case of allpossible uncertainty and external disturbances Therefore inthe sliding mode controller the transient performance andthe steady state performance of the controller are usuallysacrificed for the asymptotic stability of the control systemHowever in the proposed adaptive controller the adaptationlaw is designed to guarantee the asymptotic stability underuncertainty and disturbance Therefore there will be lessconstraint on the choice of the amplitude 119886 of the robustcompensator in the proposed method Conservatism of therobust controller would be dramatically reduced by introduc-ing an adaptive scheme into the controller The asymptoticstability of the proposed adaptive sliding mode controller isguaranteed with larger values of allowable uncertainty anddisturbances Both adaptive control part and sliding modecontrol part could compensate for the system uncertaintiesand external disturbancesThe adaptation law can be derivedas
The shortcoming of the adaptive robust sliding mode con-troller (3) is that the bang-bang robust compensatorwill causechattering in implementationThe chattering phenomenon iscaused by the sign function term The sign function switchesthe control action abruptly especially near the equilibriumpoint Chattering is not desirable since it involves extremelyhigh control activity andmay excite high frequency dynamicsof a structure
4 Design of a Smooth Adaptive Robust SlidingMode Controller and Stability Proof
In order to attenuate the chattering associated with the bang-bang compensator a hyperbolic tangent function is used toswitch the control action instead of the sign function Thecontroller becomes
where 119886 and120572 are positive numbers and tanh is the hyperbolictangent function
tanh (119909) = 119890119909
minus 119890minus119909
119890119909 + 119890minus119909 (18)
The 119886 tanh(120572119904) is a robust compensator which provides asmooth control action The hyperbolic function will switchthe control action smoothly and avoid causing chatteringThis robust compensator is continuously differentiable withrespect to the control variable 119904 It generates a smoothcontrol action Compared with the commonly used bang-bang or saturation robust controllers the smooth robustcontroller has advantages in ensuring smooth control inputand the stability of the closed-loop system [32] A detailedcomparative study is provided in Song and Mukherjee [1]Normally the larger the value of the parameter 120572 is the closer
6 Mathematical Problems in Engineering
the tanh(120572119904) function will mimic the sign(119904) function Toavoid chattering a relative smaller value of 120572 can be chosenWhen implementing the controller the value of 120572 is oftenexperimentally determined
To prove the stability of the smooth adaptive robustcontroller (17) define the Lyapunov function candidate as
V =1
21198981199042
+1
2120593119879
Γminus1
120593 (19)
Using the similar process shown in Section 3 we have
V = minus119886 sdot 119904 sdot tanh (120572119904) minus 120593119879 (119884119904 minus Γminus1 120593) (20)
Please note that 119888 and are updated according to (14) tomake 119884119904 minus Γminus1 120593 = 0 Then V = minus119886 sdot 119904 sdot tanh(120572119904) le 0 whileV = 0 only when 119904 = 0 Therefore according to LaSallersquostheorem the closed-loop control system with the smoothadaptive robust controllers (17) is asymptotically stable in thesense of Lyapunov
5 Experiment Results
51 Open Loop Testing of the Aluminum Beam First openloop testing is performed to find the dominant mode of thebeam for active vibration control The beam was subjectedto a mechanical impact at its midlength for a multimodeexcitation The power spectrum density (PSD) plots of the30-second time response of the multimode excitation wereobtainedThe corresponding frequencies of peaks in the PSDplots represent themodal frequencies of the flexible beam Tostudy the attenuation of vibration at each modal frequencythe PSD plots of the first 5 seconds and the last 5 secondsof the beam vibration are shown in Figure 3 The peaks inFigure 3 clearly reveal the modal frequencies of the flexiblebeam the 1st mode at 164Hz the 2nd mode at 977Hzand the 3rd mode at 2685Hz A comparison of PSD plotsof the vibration data for the first 5 seconds and the last 5seconds clearly shows the vibration attenuation at eachmodalfrequency The dB value and dB drop of the first three modesare shown in Table 3 From Figure 3 and Table 3 it can beseen that the dB levels of the second and third modes at thefirst five-second and last five-second periods are significantlylower than that of the 1st mode The dB drop value of thefirst three modes shows that the second and third modesattenuated much faster than the first mode Therefore it canbe concluded that the first modal frequency 164Hz is thedominant one in the vibration and the first mode should bethe target mode in the active vibration control To target atthis mode the vibration control is designed based on thesimplified plant (1) which includes only the dynamics of thefirst mode
52 Experimental Results for the Adaptive Robust SlidingMode Controller To verify the effectiveness of the adaptiverobust sliding mode control (3) and the smooth adaptivesliding mode control (17) experiments were conducted onthe aluminum flexible beam In the experiment 119904 = 119890 +
120582119890 = 119890 + 005119890 and 1199031= 1199032= 1199033= 01 for the diagonal
Table 3 PSD drop comparison of first 5 seconds and last 5 seconds
Mode 1 Mode 2 Mode 3dB level of the first 5 seconds 1673 598 minus1436dB level of the last 5 seconds 291 minus1723 minus3519dB drop 1382 2321 2083
20
10
minus10
minus20
minus30
minus40
minus50
minus60
minus70
minus800
0
10 20 30 40 50 60
Frequency (Hz)
Pow
er sp
ectr
um m
agni
tude
(dB)
First 5 secondsLast 5 seconds
Figure 3 Comparison of PSD plots of first 5 seconds and last 5seconds for time response of multimode excitation of the flexiblebeam
matrix Γ In each test the beam was excited by a sinusoidalsignal at its first modal frequency combined with a lowpower white noise for the initial 5 seconds The amplitudeof the sinusoidal signal was 075V The variance of the whitenoise was 00497 The sinusoidal signal and the white noisewere summed in the Simulink software and downloaded intodSPACE DS1102 Floating-Point Controller Board for real-time implementation The output of the dSPACE controllerboard was amplified by an amplifier for capacitive load beforeexciting the PZT actuators After the initial 5 seconds firstmode excitation was stopped and the active vibration controlstarted to suppress the induced vibration A free vibration testwas conducted for the purpose of comparison In the freevibration test no control signal was implemented after theinitial 5-second excitation
The upper diagram of Figure 4 is the time responsecomparison of free vibration with that of the active control(with the sign function) From the experimental results thevibration is suppressed in about 8 oscillation cycles (about5 seconds) However there is small-magnitude chattering inthe steady state which may cause instability when a certainlevel of disturbance exists The lower diagram of Figure 4 isthe time response comparison of free vibration with that ofthe active control (with the tanh function) It can be seen thatthe robust sliding mode control can suppress the vibration inabout 8 oscillation cycles without the chattering in the steadystate Experimental results show that the chattering can be
Mathematical Problems in Engineering 7
06
04
02
0
minus02
minus04
5 10 15 20
Sens
or v
olta
ge (V
)
Sliding-mode control with sign functionFree vibration
Time (s)
(a)
06
04
02
0
minus02
minus04
5 10 15 20
Sens
or v
olta
ge (V
)
Sliding-mode control with tanh functionFree vibration
Time (s)
(b)
Figure 4 The time response comparisons of the adaptive slidingmode controller (with the sign function) with the adaptive robustsliding mode controller (with tanh function)
eliminated by implementing the hyperbolic tangent function(tanh function) as the switching function instead of the signfunction
The upper diagram of Figure 5 is the control actionwith the bang-bang robust compensator Compared with thetime response in Figure 4 it can be seen that the chatteringhappens when the control action switches abruptly betweenits positive limit and negative limit when the system is alreadyin the steady state (after 10 seconds) The lower diagramof Figure 5 is the control action with the smooth robustcompensator From the diagram it can be seen that thecontrol signal switches smoothly Therefore the chatteringin the output signal is eliminated Figure 6 shows the timehistory of the parametersrsquo updating process Furthermore itis also worthwhile to note that no higher modes have beenexcited even though the plant is actually amultimode systemThis justifies that the primary vibration control target is thefirst mode
53 Experimental Results for the Adaptive Robust SlidingModeControl with Mass Uncertainty To verify the robustness ofthe proposed adaptive robust controller the plant is changedby adding mass on the beam Four 6-gram adhesive masseswere attached to both sides of the beam with two at theremote tip and the other two at the midlength of the beamThe mass of the aluminum beam is increased from 10520gram to 12920 gramThefirstmodal frequency of the originalbeam is 167Hz whereas the first modal frequency of the
Time (s)
200
100
0
minus100
minus2005 10 15 20
Actu
ator
sign
al (V
)
Sliding-mode control with sign function
(a)
Time (s)
200
100
0
minus100
minus2005 10 15 20
Actu
ator
sign
al (V
)Sliding-mode control with tanh function
(b)
Figure 5 The control signal comparisons of the adaptive slidingmode controller (with the sign function) with the adaptive robustsliding mode controller (with tanh function)
00835
0083
008255 10 15 20 25
Time (s)
m
Time (s)
082
081
085 10 15 20 25
c
1100002
1100001
110
Sliding-mode control with sign functionSliding-mode control with tanh function
Time (s)5 10 15 20 25
k
Figure 6 The parametersrsquo updating time history
beam with added mass has been decreased to 1457Hz Theincreasing of the mass has caused the decreasing of thenatural frequency of the beamTherefore in this experimentthe frequency of the sinusoidal excitation signal has beenchanged to 1457Hz to ensure excitation of the beam Thesame controller is used for the vibration control of the beamwith added mass From the experimental results (Figure 7)
8 Mathematical Problems in Engineering
05
04
02
03
01
0
minus01
minus02
minus03
minus04
minus055 10 15 20
Time (s)
Free vibrationSliding-mode control with tanh function
Sens
or v
olta
ge (V
)
Figure 7 The time response comparisons of the adaptive slidingmode controller with tanh function for the flexible beam with 24-gram mass added
1
08
06
04
02
0
minus02
minus04
minus06
minus08
minus10 1 2 3 4 5 6 7
Time (s)
Sliding-mode control with tanh function
Sens
or v
olta
ge (V
)
Free vibration
Figure 8 The time response of comparison of the adaptive slidingmode controller with tanh function for the flexible beam withmultimode excitation
it can be seen that the adaptive robust sliding mode caneffectively suppress the vibration of the beam after its physicalproperty has been changedThis demonstrates the robustnessof the proposed controllers
54 Experimental Results for the Adaptive Robust SlidingModeControl with Multimode Excitation To test its robustness tohigher order dynamics and disturbances multimodal excita-tion of the flexible beam with mass added is conducted Fig-ure 8 shows the comparison of time response of multimodalexcitationwith that of the free vibration Experimental results
clearly show that the proposed method is still effective inmultimodal vibration control and this further demonstratesthe robustness of the proposed controller
6 Conclusion
In this paper an experimental study of the robust adaptivesliding mode control was conducted on a flexible beamwith piezoceramic actuators and sensor for vibration controlpurpose Two different schemes of adaptive sliding modecontrol have been comparatively studied during the experi-ment The adaptive robust sliding mode control with bang-bang action could suppress the induced vibration of a flexiblebeam in a quick fashion However it causes chattering duringsteady state To eliminate the chattering a hyperbolic tangentfunction is used to replace the sign function in the robustcontrol action Both adaptive robust sliding mode controllersare designed based on the Lyapunov direct method In theproposed adaptive sliding mode controller there will beless constraint on the choice of the amplitude of the robustcompensator The conservatism of the control system hasbeen reduced by introducing the adaptive scheme into thesliding mode controlThe asymptotic stability of the adaptiverobust sliding mode control is proved by using Lyapunovdirect method in the paper The experimental results showthat the proposed adaptive robust sliding mode control withthe smooth robust compensator eliminates the chatteringwhile keeping the advantages of adaptive robust slidingmodecontrol Also the proposedmethods are robust to the varianceof the plant parameters as demonstrated experimentally
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
This research was partially supported by Grant no 51278387Grant no 51278084 and Grant no 51121005 (Science Fundfor Creative Research Groups) from the National ScienceFoundation of China (NSFC)
References
[1] G Song and R Mukherjee ldquoA comparative study of conven-tional nonsmooth time-invariant and smooth time-varyingrobust compensatorsrdquo IEEE Transactions on Control SystemsTechnology vol 6 no 4 pp 571ndash576 1998
[2] J-H Park K-W Kim and H-H Yoo ldquoTwo-stage sliding modecontroller for vibration suppression of a flexible pointing sys-temrdquo Proceedings of the Institution ofMechanical Engineers PartC Journal of Mechanical Engineering Science vol 215 no 2 pp155ndash166 2001
[3] X Yu Z Man and B Wu ldquoDesign of fuzzy sliding-mode con-trol systemsrdquo Fuzzy Sets and Systems vol 95 no 3 pp 295ndash3061998
Mathematical Problems in Engineering 9
[4] B Yao and M Tomizuka ldquoSmooth robust adaptive slidingmode control of manipulators with guaranteed transient per-formancerdquo in Proceedings of the American Control Conferencevol 1 pp 1176ndash1180 July 1994
[5] J Wang A B Rad and P T Chan ldquoIndirect adaptive fuzzysliding mode control part I fuzzy switchingrdquo Fuzzy Sets andSystems vol 122 no 1 pp 21ndash30 2001
[6] G Song R W Longman and R Mukherjee ldquoIntegratedsliding-mode adaptive-robust controlrdquo IEE Proceedings ControlTheory and Applications vol 146 no 4 pp 341ndash347 1999
[7] G Song R W Longman and L Cai ldquoIntegrated adaptive-robust control of robot manipulatorsrdquo Journal of Robotic Sys-tems vol 15 no 12 pp 699ndash712 1998
[8] R-J Wai ldquoAdaptive sliding-mode control for induction servo-motor driverdquo IEE Proceedings Electric Power Applications vol147 no 6 pp 553ndash562 2000
[9] S-C Lin and Y-Y Chen ldquoDesign of adaptive fuzzy slidingmode for nonlinear system controlrdquo in Proceedings of the 3rdIEEE Conference on Fuzzy Systems pp 35ndash39 June 1994
[10] J Fei and H Ding ldquoAdaptive vibration control of microelec-tromechanical systems triaxial gyroscope using radial basisfunction sliding mode techniquerdquo Proceedings of the Institutionof Mechanical Engineers Part I Journal of Systems and ControlEngineering vol 227 no 2 pp 264ndash269 2013
[11] B L Zhang and Q L Han ldquoRobust sliding mode 119867infin
controlusing time-varying delayed states for offshore steel jacketplatformsrdquo in Proceedings of the IEEE International Symposiumon Industrial Electronics (ISIE rsquo13) pp 1ndash6 IEEE 2013
[12] H Zhang G Dong M Zhou C Song Y Huang and K Du ldquoAnew variable structure sliding mode control strategy for FTS indiamond-cutting microstructured surfacesrdquo The InternationalJournal of AdvancedManufacturing Technology vol 65 no 5ndash8pp 1177ndash1184 2013
[13] H Li J Yu C Hilton and H Liu ldquoAdaptive sliding modecontrol for nonlinear active suspension vehicle systems usingTS fuzzy approachrdquo IEEE Transactions on Industrial Electronicsvol 60 no 8 2013
[14] J L Fanson and T K Caughey ldquoPositive position feedbackcontrol for large space structuresrdquo AIAA Journal vol 28 no 4pp 717ndash724 1990
[15] G Song P Qiao V Sethi and A Prasad ldquoActive vibrationcontrol of a smart pultruded fiber-reinforced polymer I-beamrdquoin International Symposium on Smart Structures and Materialsvol 4696 of Proceedings of SPIE pp 197ndash208 March 2002
[16] PMayhan andGWashington ldquoFuzzymodel reference learningcontrol a new control paradigm for smart structuresrdquo SmartMaterials and Structures vol 7 no 6 pp 874ndash884 1998
[17] V Bottega AMolter J SO Fonseca andR Pergher ldquoVibrationcontrol of manipulators with flexible nonprismatic links usingpiezoelectric actuators and sensorsrdquo Mathematical Problems inEngineering vol 2009 Article ID 727385 16 pages 2009
[18] A Molter V Bottega O A A Da Silveira and J S O FonsecaldquoSimultaneous piezoelectric actuator and sensor placementoptimization and control design of manipulators with flexiblelinks using SDREmethodrdquoMathematical Problems in Engineer-ing vol 2010 Article ID 362437 23 pages 2010
[19] M Ahmadian and K M Jeric ldquoOn the application of shuntedpiezoceramics for increasing acoustic transmission loss instructuresrdquo Journal of Sound and Vibration vol 243 no 2 pp347ndash359 2001
[20] J Fei Y Fang and C Yan ldquoThe comparative study of vibrationcontrol of flexible structure using smart materialsrdquo Mathemat-ical Problems in Engineering vol 2010 Article ID 768256 13pages 2010
[21] S-B Choi Y-K Park and C-C Cheong ldquoActive vibrationcontrol of hybrid smart structures featuring piezoelectric filmsand electrorheological fluidsrdquo in The International Society forOptical Engineering vol 2717 of Proceedings of the SPIE pp544ndash552 February 1996
[22] A K Jha and D J Inman ldquoSliding mode control of a Gossamerstructure using smart materialsrdquo Journal of Vibration andControl vol 10 no 8 pp 1199ndash1220 2004
[23] S-B Choi and M-S Kim ldquoNew discrete-time fuzzy-sliding-mode control with application to smart structuresrdquo Journal ofGuidance Control and Dynamics vol 20 no 5 pp 857ndash8641997
[24] S Bashash and N Jalili ldquoRobust multiple frequency trajec-tory tracking control of piezoelectrically driven micronano-positioning systemsrdquo IEEE Transactions on Control SystemsTechnology vol 15 no 5 pp 867ndash878 2007
[25] H C Liaw B Shirinzadeh and J Smith ldquoSliding-modeenhanced adaptive motion tracking control of piezoelectricactuation systems formicronanomanipulationrdquo IEEETransac-tions on Control Systems Technology vol 16 no 4 pp 826ndash8332008
[26] Y Li and Q Xu ldquoAdaptive sliding mode control with perturba-tion estimation and PID sliding surface for motion tracking of apiezo-driven micromanipulatorrdquo IEEE Transactions on ControlSystems Technology vol 18 no 4 pp 798ndash810 2010
[27] J Hu and D Zhu ldquoVibration control of smart structure usingsliding mode control with observerrdquo Journal of Computers vol7 no 2 pp 411ndash418 2012
[28] L Zhang Z Zhang Z Long and A Hao ldquoSlidingmode controlwith auto-tuning law forMaglev SystemrdquoEngineering vol 2 no2 pp 107ndash112 2010
[29] T Rittenschober andK Schlacher ldquoObserver-based self sensingactuation of piezoelastic structures for robust vibration controlrdquoAutomatica vol 48 no 6 pp 1123ndash1131 2012
[30] X Bu L Ye Z Su and C Wang ldquoActive control of a flexiblesmart beam using a system identification technique based onARMAXrdquo SmartMaterials and Structures vol 12 no 5 pp 845ndash850 2003
[31] R D Robinett C R Dohrmann G R Eisler et al FlexibleRobot Dynamics and Controls Kluwer AcademicPlenum Pub-lishers New York NY USA 2002
[32] L Cai and G Song ldquoJoint stick-slip friction compensation ofrobotmanipulators by using smooth robust controllersrdquo Journalof Robotic Systems vol 11 no 6 pp 451ndash470 1994
Table 2 Properties of PZT patches used on the beam
Symbol Quantity Unit PZT actuator PZT sensor119871 times 119908 times 119905 Dimensions mm 46 times 3327 times 025 14 times 7 times 025
11988933
Strain coefficient CN 741 times 10minus10
741 times 10minus10
11988931
Strain coefficient CN minus274 times 10minus10
minus274 times 10minus10
120588119875
PZT density kgm3 7500 7500119864119875
Youngrsquos modulus Nm263 times 10
10
63 times 1010
Beam
Sensor
BaseActuator23mm 992mm
61mm
79mm
735mm
7365mm
Figure 2 The physical position of the piezoelectric actuators andsensor
of the model The plant is represented in the form of thefollowing general simplified dynamic equation
119898 + 119888 + 119896119909 = 119906 (1)
where119898 119888 and 119896 are the coefficients relative with the systemmass damping and stiffness respectively x is the PZT sensorsignal and 119906 is PZT actuator signal
3 Design of Adaptive Robust Sliding ModeController and Stability Proof
The design principle of the proposed adaptive robust slidingmode controller for vibration control is similar to that of theadaptive sliding mode controller for robot manipulator in [16 31]
To assist control design define 119890 = 119909119903minus 119909 where the
subscript ldquo119903rdquo stands for the reference command Define thesliding surface variable as
where 119888 and are the initial estimations of the real valuesof parameters of 119898 119888 and 119896 and these estimates will beadjusted online by the slidingmode based adaptive controllerThe term 119886 sgn(119904) is the bang-bang robust compensator to
ensure stability where 119886 is a positive constant and the sgnfunction is defined as
sgn (119904) = +1 if 119904 gt 0
sgn (119904) = minus1 if 119904 lt 0(4)
With this feedback controller (3) the dynamic equation of thesystem (1) becomes
= minus119886 sdot 119904 sdot sgn (119904) minus 120593119879 (119884119904 minus Γminus1 120593)
(11)
Notice that minus119886 sdot 119904 sdot sgn(119904) le 0The adaptation law will be designed as 119884119904 minus Γminus1 120593 = 0
Therefore V = minus119886 sdot 119904 sdot sgn(119904) le 0 while V = 0 onlywhen 119904 = 0 According to LaSallersquos theorem the control sys-tem is asymptotically stable in the sense of Lyapunov if theadaptation law satisfies 119884119904 minus Γminus1 120593 = 0 In the sliding modecontroller the positive constant 119886 which is the amplitude ofthe robust compensator in control law (3) needs to be theupper bounding of the uncertainty of the system model andthe external disturbance to ensure the stability Due to thelack of the prior knowledge of the system the upper boundingof the uncertainty and the disturbance may be difficult tobe properly or precisely evaluated In order to ensure theasymptotic stability of the system under all conditions theamplitude of the robust compensator usually needs to bechosen large enough to compensate for the worst case of allpossible uncertainty and external disturbances Therefore inthe sliding mode controller the transient performance andthe steady state performance of the controller are usuallysacrificed for the asymptotic stability of the control systemHowever in the proposed adaptive controller the adaptationlaw is designed to guarantee the asymptotic stability underuncertainty and disturbance Therefore there will be lessconstraint on the choice of the amplitude 119886 of the robustcompensator in the proposed method Conservatism of therobust controller would be dramatically reduced by introduc-ing an adaptive scheme into the controller The asymptoticstability of the proposed adaptive sliding mode controller isguaranteed with larger values of allowable uncertainty anddisturbances Both adaptive control part and sliding modecontrol part could compensate for the system uncertaintiesand external disturbancesThe adaptation law can be derivedas
The shortcoming of the adaptive robust sliding mode con-troller (3) is that the bang-bang robust compensatorwill causechattering in implementationThe chattering phenomenon iscaused by the sign function term The sign function switchesthe control action abruptly especially near the equilibriumpoint Chattering is not desirable since it involves extremelyhigh control activity andmay excite high frequency dynamicsof a structure
4 Design of a Smooth Adaptive Robust SlidingMode Controller and Stability Proof
In order to attenuate the chattering associated with the bang-bang compensator a hyperbolic tangent function is used toswitch the control action instead of the sign function Thecontroller becomes
where 119886 and120572 are positive numbers and tanh is the hyperbolictangent function
tanh (119909) = 119890119909
minus 119890minus119909
119890119909 + 119890minus119909 (18)
The 119886 tanh(120572119904) is a robust compensator which provides asmooth control action The hyperbolic function will switchthe control action smoothly and avoid causing chatteringThis robust compensator is continuously differentiable withrespect to the control variable 119904 It generates a smoothcontrol action Compared with the commonly used bang-bang or saturation robust controllers the smooth robustcontroller has advantages in ensuring smooth control inputand the stability of the closed-loop system [32] A detailedcomparative study is provided in Song and Mukherjee [1]Normally the larger the value of the parameter 120572 is the closer
6 Mathematical Problems in Engineering
the tanh(120572119904) function will mimic the sign(119904) function Toavoid chattering a relative smaller value of 120572 can be chosenWhen implementing the controller the value of 120572 is oftenexperimentally determined
To prove the stability of the smooth adaptive robustcontroller (17) define the Lyapunov function candidate as
V =1
21198981199042
+1
2120593119879
Γminus1
120593 (19)
Using the similar process shown in Section 3 we have
V = minus119886 sdot 119904 sdot tanh (120572119904) minus 120593119879 (119884119904 minus Γminus1 120593) (20)
Please note that 119888 and are updated according to (14) tomake 119884119904 minus Γminus1 120593 = 0 Then V = minus119886 sdot 119904 sdot tanh(120572119904) le 0 whileV = 0 only when 119904 = 0 Therefore according to LaSallersquostheorem the closed-loop control system with the smoothadaptive robust controllers (17) is asymptotically stable in thesense of Lyapunov
5 Experiment Results
51 Open Loop Testing of the Aluminum Beam First openloop testing is performed to find the dominant mode of thebeam for active vibration control The beam was subjectedto a mechanical impact at its midlength for a multimodeexcitation The power spectrum density (PSD) plots of the30-second time response of the multimode excitation wereobtainedThe corresponding frequencies of peaks in the PSDplots represent themodal frequencies of the flexible beam Tostudy the attenuation of vibration at each modal frequencythe PSD plots of the first 5 seconds and the last 5 secondsof the beam vibration are shown in Figure 3 The peaks inFigure 3 clearly reveal the modal frequencies of the flexiblebeam the 1st mode at 164Hz the 2nd mode at 977Hzand the 3rd mode at 2685Hz A comparison of PSD plotsof the vibration data for the first 5 seconds and the last 5seconds clearly shows the vibration attenuation at eachmodalfrequency The dB value and dB drop of the first three modesare shown in Table 3 From Figure 3 and Table 3 it can beseen that the dB levels of the second and third modes at thefirst five-second and last five-second periods are significantlylower than that of the 1st mode The dB drop value of thefirst three modes shows that the second and third modesattenuated much faster than the first mode Therefore it canbe concluded that the first modal frequency 164Hz is thedominant one in the vibration and the first mode should bethe target mode in the active vibration control To target atthis mode the vibration control is designed based on thesimplified plant (1) which includes only the dynamics of thefirst mode
52 Experimental Results for the Adaptive Robust SlidingMode Controller To verify the effectiveness of the adaptiverobust sliding mode control (3) and the smooth adaptivesliding mode control (17) experiments were conducted onthe aluminum flexible beam In the experiment 119904 = 119890 +
120582119890 = 119890 + 005119890 and 1199031= 1199032= 1199033= 01 for the diagonal
Table 3 PSD drop comparison of first 5 seconds and last 5 seconds
Mode 1 Mode 2 Mode 3dB level of the first 5 seconds 1673 598 minus1436dB level of the last 5 seconds 291 minus1723 minus3519dB drop 1382 2321 2083
20
10
minus10
minus20
minus30
minus40
minus50
minus60
minus70
minus800
0
10 20 30 40 50 60
Frequency (Hz)
Pow
er sp
ectr
um m
agni
tude
(dB)
First 5 secondsLast 5 seconds
Figure 3 Comparison of PSD plots of first 5 seconds and last 5seconds for time response of multimode excitation of the flexiblebeam
matrix Γ In each test the beam was excited by a sinusoidalsignal at its first modal frequency combined with a lowpower white noise for the initial 5 seconds The amplitudeof the sinusoidal signal was 075V The variance of the whitenoise was 00497 The sinusoidal signal and the white noisewere summed in the Simulink software and downloaded intodSPACE DS1102 Floating-Point Controller Board for real-time implementation The output of the dSPACE controllerboard was amplified by an amplifier for capacitive load beforeexciting the PZT actuators After the initial 5 seconds firstmode excitation was stopped and the active vibration controlstarted to suppress the induced vibration A free vibration testwas conducted for the purpose of comparison In the freevibration test no control signal was implemented after theinitial 5-second excitation
The upper diagram of Figure 4 is the time responsecomparison of free vibration with that of the active control(with the sign function) From the experimental results thevibration is suppressed in about 8 oscillation cycles (about5 seconds) However there is small-magnitude chattering inthe steady state which may cause instability when a certainlevel of disturbance exists The lower diagram of Figure 4 isthe time response comparison of free vibration with that ofthe active control (with the tanh function) It can be seen thatthe robust sliding mode control can suppress the vibration inabout 8 oscillation cycles without the chattering in the steadystate Experimental results show that the chattering can be
Mathematical Problems in Engineering 7
06
04
02
0
minus02
minus04
5 10 15 20
Sens
or v
olta
ge (V
)
Sliding-mode control with sign functionFree vibration
Time (s)
(a)
06
04
02
0
minus02
minus04
5 10 15 20
Sens
or v
olta
ge (V
)
Sliding-mode control with tanh functionFree vibration
Time (s)
(b)
Figure 4 The time response comparisons of the adaptive slidingmode controller (with the sign function) with the adaptive robustsliding mode controller (with tanh function)
eliminated by implementing the hyperbolic tangent function(tanh function) as the switching function instead of the signfunction
The upper diagram of Figure 5 is the control actionwith the bang-bang robust compensator Compared with thetime response in Figure 4 it can be seen that the chatteringhappens when the control action switches abruptly betweenits positive limit and negative limit when the system is alreadyin the steady state (after 10 seconds) The lower diagramof Figure 5 is the control action with the smooth robustcompensator From the diagram it can be seen that thecontrol signal switches smoothly Therefore the chatteringin the output signal is eliminated Figure 6 shows the timehistory of the parametersrsquo updating process Furthermore itis also worthwhile to note that no higher modes have beenexcited even though the plant is actually amultimode systemThis justifies that the primary vibration control target is thefirst mode
53 Experimental Results for the Adaptive Robust SlidingModeControl with Mass Uncertainty To verify the robustness ofthe proposed adaptive robust controller the plant is changedby adding mass on the beam Four 6-gram adhesive masseswere attached to both sides of the beam with two at theremote tip and the other two at the midlength of the beamThe mass of the aluminum beam is increased from 10520gram to 12920 gramThefirstmodal frequency of the originalbeam is 167Hz whereas the first modal frequency of the
Time (s)
200
100
0
minus100
minus2005 10 15 20
Actu
ator
sign
al (V
)
Sliding-mode control with sign function
(a)
Time (s)
200
100
0
minus100
minus2005 10 15 20
Actu
ator
sign
al (V
)Sliding-mode control with tanh function
(b)
Figure 5 The control signal comparisons of the adaptive slidingmode controller (with the sign function) with the adaptive robustsliding mode controller (with tanh function)
00835
0083
008255 10 15 20 25
Time (s)
m
Time (s)
082
081
085 10 15 20 25
c
1100002
1100001
110
Sliding-mode control with sign functionSliding-mode control with tanh function
Time (s)5 10 15 20 25
k
Figure 6 The parametersrsquo updating time history
beam with added mass has been decreased to 1457Hz Theincreasing of the mass has caused the decreasing of thenatural frequency of the beamTherefore in this experimentthe frequency of the sinusoidal excitation signal has beenchanged to 1457Hz to ensure excitation of the beam Thesame controller is used for the vibration control of the beamwith added mass From the experimental results (Figure 7)
8 Mathematical Problems in Engineering
05
04
02
03
01
0
minus01
minus02
minus03
minus04
minus055 10 15 20
Time (s)
Free vibrationSliding-mode control with tanh function
Sens
or v
olta
ge (V
)
Figure 7 The time response comparisons of the adaptive slidingmode controller with tanh function for the flexible beam with 24-gram mass added
1
08
06
04
02
0
minus02
minus04
minus06
minus08
minus10 1 2 3 4 5 6 7
Time (s)
Sliding-mode control with tanh function
Sens
or v
olta
ge (V
)
Free vibration
Figure 8 The time response of comparison of the adaptive slidingmode controller with tanh function for the flexible beam withmultimode excitation
it can be seen that the adaptive robust sliding mode caneffectively suppress the vibration of the beam after its physicalproperty has been changedThis demonstrates the robustnessof the proposed controllers
54 Experimental Results for the Adaptive Robust SlidingModeControl with Multimode Excitation To test its robustness tohigher order dynamics and disturbances multimodal excita-tion of the flexible beam with mass added is conducted Fig-ure 8 shows the comparison of time response of multimodalexcitationwith that of the free vibration Experimental results
clearly show that the proposed method is still effective inmultimodal vibration control and this further demonstratesthe robustness of the proposed controller
6 Conclusion
In this paper an experimental study of the robust adaptivesliding mode control was conducted on a flexible beamwith piezoceramic actuators and sensor for vibration controlpurpose Two different schemes of adaptive sliding modecontrol have been comparatively studied during the experi-ment The adaptive robust sliding mode control with bang-bang action could suppress the induced vibration of a flexiblebeam in a quick fashion However it causes chattering duringsteady state To eliminate the chattering a hyperbolic tangentfunction is used to replace the sign function in the robustcontrol action Both adaptive robust sliding mode controllersare designed based on the Lyapunov direct method In theproposed adaptive sliding mode controller there will beless constraint on the choice of the amplitude of the robustcompensator The conservatism of the control system hasbeen reduced by introducing the adaptive scheme into thesliding mode controlThe asymptotic stability of the adaptiverobust sliding mode control is proved by using Lyapunovdirect method in the paper The experimental results showthat the proposed adaptive robust sliding mode control withthe smooth robust compensator eliminates the chatteringwhile keeping the advantages of adaptive robust slidingmodecontrol Also the proposedmethods are robust to the varianceof the plant parameters as demonstrated experimentally
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
This research was partially supported by Grant no 51278387Grant no 51278084 and Grant no 51121005 (Science Fundfor Creative Research Groups) from the National ScienceFoundation of China (NSFC)
References
[1] G Song and R Mukherjee ldquoA comparative study of conven-tional nonsmooth time-invariant and smooth time-varyingrobust compensatorsrdquo IEEE Transactions on Control SystemsTechnology vol 6 no 4 pp 571ndash576 1998
[2] J-H Park K-W Kim and H-H Yoo ldquoTwo-stage sliding modecontroller for vibration suppression of a flexible pointing sys-temrdquo Proceedings of the Institution ofMechanical Engineers PartC Journal of Mechanical Engineering Science vol 215 no 2 pp155ndash166 2001
[3] X Yu Z Man and B Wu ldquoDesign of fuzzy sliding-mode con-trol systemsrdquo Fuzzy Sets and Systems vol 95 no 3 pp 295ndash3061998
Mathematical Problems in Engineering 9
[4] B Yao and M Tomizuka ldquoSmooth robust adaptive slidingmode control of manipulators with guaranteed transient per-formancerdquo in Proceedings of the American Control Conferencevol 1 pp 1176ndash1180 July 1994
[5] J Wang A B Rad and P T Chan ldquoIndirect adaptive fuzzysliding mode control part I fuzzy switchingrdquo Fuzzy Sets andSystems vol 122 no 1 pp 21ndash30 2001
[6] G Song R W Longman and R Mukherjee ldquoIntegratedsliding-mode adaptive-robust controlrdquo IEE Proceedings ControlTheory and Applications vol 146 no 4 pp 341ndash347 1999
[7] G Song R W Longman and L Cai ldquoIntegrated adaptive-robust control of robot manipulatorsrdquo Journal of Robotic Sys-tems vol 15 no 12 pp 699ndash712 1998
[8] R-J Wai ldquoAdaptive sliding-mode control for induction servo-motor driverdquo IEE Proceedings Electric Power Applications vol147 no 6 pp 553ndash562 2000
[9] S-C Lin and Y-Y Chen ldquoDesign of adaptive fuzzy slidingmode for nonlinear system controlrdquo in Proceedings of the 3rdIEEE Conference on Fuzzy Systems pp 35ndash39 June 1994
[10] J Fei and H Ding ldquoAdaptive vibration control of microelec-tromechanical systems triaxial gyroscope using radial basisfunction sliding mode techniquerdquo Proceedings of the Institutionof Mechanical Engineers Part I Journal of Systems and ControlEngineering vol 227 no 2 pp 264ndash269 2013
[11] B L Zhang and Q L Han ldquoRobust sliding mode 119867infin
controlusing time-varying delayed states for offshore steel jacketplatformsrdquo in Proceedings of the IEEE International Symposiumon Industrial Electronics (ISIE rsquo13) pp 1ndash6 IEEE 2013
[12] H Zhang G Dong M Zhou C Song Y Huang and K Du ldquoAnew variable structure sliding mode control strategy for FTS indiamond-cutting microstructured surfacesrdquo The InternationalJournal of AdvancedManufacturing Technology vol 65 no 5ndash8pp 1177ndash1184 2013
[13] H Li J Yu C Hilton and H Liu ldquoAdaptive sliding modecontrol for nonlinear active suspension vehicle systems usingTS fuzzy approachrdquo IEEE Transactions on Industrial Electronicsvol 60 no 8 2013
[14] J L Fanson and T K Caughey ldquoPositive position feedbackcontrol for large space structuresrdquo AIAA Journal vol 28 no 4pp 717ndash724 1990
[15] G Song P Qiao V Sethi and A Prasad ldquoActive vibrationcontrol of a smart pultruded fiber-reinforced polymer I-beamrdquoin International Symposium on Smart Structures and Materialsvol 4696 of Proceedings of SPIE pp 197ndash208 March 2002
[16] PMayhan andGWashington ldquoFuzzymodel reference learningcontrol a new control paradigm for smart structuresrdquo SmartMaterials and Structures vol 7 no 6 pp 874ndash884 1998
[17] V Bottega AMolter J SO Fonseca andR Pergher ldquoVibrationcontrol of manipulators with flexible nonprismatic links usingpiezoelectric actuators and sensorsrdquo Mathematical Problems inEngineering vol 2009 Article ID 727385 16 pages 2009
[18] A Molter V Bottega O A A Da Silveira and J S O FonsecaldquoSimultaneous piezoelectric actuator and sensor placementoptimization and control design of manipulators with flexiblelinks using SDREmethodrdquoMathematical Problems in Engineer-ing vol 2010 Article ID 362437 23 pages 2010
[19] M Ahmadian and K M Jeric ldquoOn the application of shuntedpiezoceramics for increasing acoustic transmission loss instructuresrdquo Journal of Sound and Vibration vol 243 no 2 pp347ndash359 2001
[20] J Fei Y Fang and C Yan ldquoThe comparative study of vibrationcontrol of flexible structure using smart materialsrdquo Mathemat-ical Problems in Engineering vol 2010 Article ID 768256 13pages 2010
[21] S-B Choi Y-K Park and C-C Cheong ldquoActive vibrationcontrol of hybrid smart structures featuring piezoelectric filmsand electrorheological fluidsrdquo in The International Society forOptical Engineering vol 2717 of Proceedings of the SPIE pp544ndash552 February 1996
[22] A K Jha and D J Inman ldquoSliding mode control of a Gossamerstructure using smart materialsrdquo Journal of Vibration andControl vol 10 no 8 pp 1199ndash1220 2004
[23] S-B Choi and M-S Kim ldquoNew discrete-time fuzzy-sliding-mode control with application to smart structuresrdquo Journal ofGuidance Control and Dynamics vol 20 no 5 pp 857ndash8641997
[24] S Bashash and N Jalili ldquoRobust multiple frequency trajec-tory tracking control of piezoelectrically driven micronano-positioning systemsrdquo IEEE Transactions on Control SystemsTechnology vol 15 no 5 pp 867ndash878 2007
[25] H C Liaw B Shirinzadeh and J Smith ldquoSliding-modeenhanced adaptive motion tracking control of piezoelectricactuation systems formicronanomanipulationrdquo IEEETransac-tions on Control Systems Technology vol 16 no 4 pp 826ndash8332008
[26] Y Li and Q Xu ldquoAdaptive sliding mode control with perturba-tion estimation and PID sliding surface for motion tracking of apiezo-driven micromanipulatorrdquo IEEE Transactions on ControlSystems Technology vol 18 no 4 pp 798ndash810 2010
[27] J Hu and D Zhu ldquoVibration control of smart structure usingsliding mode control with observerrdquo Journal of Computers vol7 no 2 pp 411ndash418 2012
[28] L Zhang Z Zhang Z Long and A Hao ldquoSlidingmode controlwith auto-tuning law forMaglev SystemrdquoEngineering vol 2 no2 pp 107ndash112 2010
[29] T Rittenschober andK Schlacher ldquoObserver-based self sensingactuation of piezoelastic structures for robust vibration controlrdquoAutomatica vol 48 no 6 pp 1123ndash1131 2012
[30] X Bu L Ye Z Su and C Wang ldquoActive control of a flexiblesmart beam using a system identification technique based onARMAXrdquo SmartMaterials and Structures vol 12 no 5 pp 845ndash850 2003
[31] R D Robinett C R Dohrmann G R Eisler et al FlexibleRobot Dynamics and Controls Kluwer AcademicPlenum Pub-lishers New York NY USA 2002
[32] L Cai and G Song ldquoJoint stick-slip friction compensation ofrobotmanipulators by using smooth robust controllersrdquo Journalof Robotic Systems vol 11 no 6 pp 451ndash470 1994
= minus119886 sdot 119904 sdot sgn (119904) minus 120593119879 (119884119904 minus Γminus1 120593)
(11)
Notice that minus119886 sdot 119904 sdot sgn(119904) le 0The adaptation law will be designed as 119884119904 minus Γminus1 120593 = 0
Therefore V = minus119886 sdot 119904 sdot sgn(119904) le 0 while V = 0 onlywhen 119904 = 0 According to LaSallersquos theorem the control sys-tem is asymptotically stable in the sense of Lyapunov if theadaptation law satisfies 119884119904 minus Γminus1 120593 = 0 In the sliding modecontroller the positive constant 119886 which is the amplitude ofthe robust compensator in control law (3) needs to be theupper bounding of the uncertainty of the system model andthe external disturbance to ensure the stability Due to thelack of the prior knowledge of the system the upper boundingof the uncertainty and the disturbance may be difficult tobe properly or precisely evaluated In order to ensure theasymptotic stability of the system under all conditions theamplitude of the robust compensator usually needs to bechosen large enough to compensate for the worst case of allpossible uncertainty and external disturbances Therefore inthe sliding mode controller the transient performance andthe steady state performance of the controller are usuallysacrificed for the asymptotic stability of the control systemHowever in the proposed adaptive controller the adaptationlaw is designed to guarantee the asymptotic stability underuncertainty and disturbance Therefore there will be lessconstraint on the choice of the amplitude 119886 of the robustcompensator in the proposed method Conservatism of therobust controller would be dramatically reduced by introduc-ing an adaptive scheme into the controller The asymptoticstability of the proposed adaptive sliding mode controller isguaranteed with larger values of allowable uncertainty anddisturbances Both adaptive control part and sliding modecontrol part could compensate for the system uncertaintiesand external disturbancesThe adaptation law can be derivedas
The shortcoming of the adaptive robust sliding mode con-troller (3) is that the bang-bang robust compensatorwill causechattering in implementationThe chattering phenomenon iscaused by the sign function term The sign function switchesthe control action abruptly especially near the equilibriumpoint Chattering is not desirable since it involves extremelyhigh control activity andmay excite high frequency dynamicsof a structure
4 Design of a Smooth Adaptive Robust SlidingMode Controller and Stability Proof
In order to attenuate the chattering associated with the bang-bang compensator a hyperbolic tangent function is used toswitch the control action instead of the sign function Thecontroller becomes
where 119886 and120572 are positive numbers and tanh is the hyperbolictangent function
tanh (119909) = 119890119909
minus 119890minus119909
119890119909 + 119890minus119909 (18)
The 119886 tanh(120572119904) is a robust compensator which provides asmooth control action The hyperbolic function will switchthe control action smoothly and avoid causing chatteringThis robust compensator is continuously differentiable withrespect to the control variable 119904 It generates a smoothcontrol action Compared with the commonly used bang-bang or saturation robust controllers the smooth robustcontroller has advantages in ensuring smooth control inputand the stability of the closed-loop system [32] A detailedcomparative study is provided in Song and Mukherjee [1]Normally the larger the value of the parameter 120572 is the closer
6 Mathematical Problems in Engineering
the tanh(120572119904) function will mimic the sign(119904) function Toavoid chattering a relative smaller value of 120572 can be chosenWhen implementing the controller the value of 120572 is oftenexperimentally determined
To prove the stability of the smooth adaptive robustcontroller (17) define the Lyapunov function candidate as
V =1
21198981199042
+1
2120593119879
Γminus1
120593 (19)
Using the similar process shown in Section 3 we have
V = minus119886 sdot 119904 sdot tanh (120572119904) minus 120593119879 (119884119904 minus Γminus1 120593) (20)
Please note that 119888 and are updated according to (14) tomake 119884119904 minus Γminus1 120593 = 0 Then V = minus119886 sdot 119904 sdot tanh(120572119904) le 0 whileV = 0 only when 119904 = 0 Therefore according to LaSallersquostheorem the closed-loop control system with the smoothadaptive robust controllers (17) is asymptotically stable in thesense of Lyapunov
5 Experiment Results
51 Open Loop Testing of the Aluminum Beam First openloop testing is performed to find the dominant mode of thebeam for active vibration control The beam was subjectedto a mechanical impact at its midlength for a multimodeexcitation The power spectrum density (PSD) plots of the30-second time response of the multimode excitation wereobtainedThe corresponding frequencies of peaks in the PSDplots represent themodal frequencies of the flexible beam Tostudy the attenuation of vibration at each modal frequencythe PSD plots of the first 5 seconds and the last 5 secondsof the beam vibration are shown in Figure 3 The peaks inFigure 3 clearly reveal the modal frequencies of the flexiblebeam the 1st mode at 164Hz the 2nd mode at 977Hzand the 3rd mode at 2685Hz A comparison of PSD plotsof the vibration data for the first 5 seconds and the last 5seconds clearly shows the vibration attenuation at eachmodalfrequency The dB value and dB drop of the first three modesare shown in Table 3 From Figure 3 and Table 3 it can beseen that the dB levels of the second and third modes at thefirst five-second and last five-second periods are significantlylower than that of the 1st mode The dB drop value of thefirst three modes shows that the second and third modesattenuated much faster than the first mode Therefore it canbe concluded that the first modal frequency 164Hz is thedominant one in the vibration and the first mode should bethe target mode in the active vibration control To target atthis mode the vibration control is designed based on thesimplified plant (1) which includes only the dynamics of thefirst mode
52 Experimental Results for the Adaptive Robust SlidingMode Controller To verify the effectiveness of the adaptiverobust sliding mode control (3) and the smooth adaptivesliding mode control (17) experiments were conducted onthe aluminum flexible beam In the experiment 119904 = 119890 +
120582119890 = 119890 + 005119890 and 1199031= 1199032= 1199033= 01 for the diagonal
Table 3 PSD drop comparison of first 5 seconds and last 5 seconds
Mode 1 Mode 2 Mode 3dB level of the first 5 seconds 1673 598 minus1436dB level of the last 5 seconds 291 minus1723 minus3519dB drop 1382 2321 2083
20
10
minus10
minus20
minus30
minus40
minus50
minus60
minus70
minus800
0
10 20 30 40 50 60
Frequency (Hz)
Pow
er sp
ectr
um m
agni
tude
(dB)
First 5 secondsLast 5 seconds
Figure 3 Comparison of PSD plots of first 5 seconds and last 5seconds for time response of multimode excitation of the flexiblebeam
matrix Γ In each test the beam was excited by a sinusoidalsignal at its first modal frequency combined with a lowpower white noise for the initial 5 seconds The amplitudeof the sinusoidal signal was 075V The variance of the whitenoise was 00497 The sinusoidal signal and the white noisewere summed in the Simulink software and downloaded intodSPACE DS1102 Floating-Point Controller Board for real-time implementation The output of the dSPACE controllerboard was amplified by an amplifier for capacitive load beforeexciting the PZT actuators After the initial 5 seconds firstmode excitation was stopped and the active vibration controlstarted to suppress the induced vibration A free vibration testwas conducted for the purpose of comparison In the freevibration test no control signal was implemented after theinitial 5-second excitation
The upper diagram of Figure 4 is the time responsecomparison of free vibration with that of the active control(with the sign function) From the experimental results thevibration is suppressed in about 8 oscillation cycles (about5 seconds) However there is small-magnitude chattering inthe steady state which may cause instability when a certainlevel of disturbance exists The lower diagram of Figure 4 isthe time response comparison of free vibration with that ofthe active control (with the tanh function) It can be seen thatthe robust sliding mode control can suppress the vibration inabout 8 oscillation cycles without the chattering in the steadystate Experimental results show that the chattering can be
Mathematical Problems in Engineering 7
06
04
02
0
minus02
minus04
5 10 15 20
Sens
or v
olta
ge (V
)
Sliding-mode control with sign functionFree vibration
Time (s)
(a)
06
04
02
0
minus02
minus04
5 10 15 20
Sens
or v
olta
ge (V
)
Sliding-mode control with tanh functionFree vibration
Time (s)
(b)
Figure 4 The time response comparisons of the adaptive slidingmode controller (with the sign function) with the adaptive robustsliding mode controller (with tanh function)
eliminated by implementing the hyperbolic tangent function(tanh function) as the switching function instead of the signfunction
The upper diagram of Figure 5 is the control actionwith the bang-bang robust compensator Compared with thetime response in Figure 4 it can be seen that the chatteringhappens when the control action switches abruptly betweenits positive limit and negative limit when the system is alreadyin the steady state (after 10 seconds) The lower diagramof Figure 5 is the control action with the smooth robustcompensator From the diagram it can be seen that thecontrol signal switches smoothly Therefore the chatteringin the output signal is eliminated Figure 6 shows the timehistory of the parametersrsquo updating process Furthermore itis also worthwhile to note that no higher modes have beenexcited even though the plant is actually amultimode systemThis justifies that the primary vibration control target is thefirst mode
53 Experimental Results for the Adaptive Robust SlidingModeControl with Mass Uncertainty To verify the robustness ofthe proposed adaptive robust controller the plant is changedby adding mass on the beam Four 6-gram adhesive masseswere attached to both sides of the beam with two at theremote tip and the other two at the midlength of the beamThe mass of the aluminum beam is increased from 10520gram to 12920 gramThefirstmodal frequency of the originalbeam is 167Hz whereas the first modal frequency of the
Time (s)
200
100
0
minus100
minus2005 10 15 20
Actu
ator
sign
al (V
)
Sliding-mode control with sign function
(a)
Time (s)
200
100
0
minus100
minus2005 10 15 20
Actu
ator
sign
al (V
)Sliding-mode control with tanh function
(b)
Figure 5 The control signal comparisons of the adaptive slidingmode controller (with the sign function) with the adaptive robustsliding mode controller (with tanh function)
00835
0083
008255 10 15 20 25
Time (s)
m
Time (s)
082
081
085 10 15 20 25
c
1100002
1100001
110
Sliding-mode control with sign functionSliding-mode control with tanh function
Time (s)5 10 15 20 25
k
Figure 6 The parametersrsquo updating time history
beam with added mass has been decreased to 1457Hz Theincreasing of the mass has caused the decreasing of thenatural frequency of the beamTherefore in this experimentthe frequency of the sinusoidal excitation signal has beenchanged to 1457Hz to ensure excitation of the beam Thesame controller is used for the vibration control of the beamwith added mass From the experimental results (Figure 7)
8 Mathematical Problems in Engineering
05
04
02
03
01
0
minus01
minus02
minus03
minus04
minus055 10 15 20
Time (s)
Free vibrationSliding-mode control with tanh function
Sens
or v
olta
ge (V
)
Figure 7 The time response comparisons of the adaptive slidingmode controller with tanh function for the flexible beam with 24-gram mass added
1
08
06
04
02
0
minus02
minus04
minus06
minus08
minus10 1 2 3 4 5 6 7
Time (s)
Sliding-mode control with tanh function
Sens
or v
olta
ge (V
)
Free vibration
Figure 8 The time response of comparison of the adaptive slidingmode controller with tanh function for the flexible beam withmultimode excitation
it can be seen that the adaptive robust sliding mode caneffectively suppress the vibration of the beam after its physicalproperty has been changedThis demonstrates the robustnessof the proposed controllers
54 Experimental Results for the Adaptive Robust SlidingModeControl with Multimode Excitation To test its robustness tohigher order dynamics and disturbances multimodal excita-tion of the flexible beam with mass added is conducted Fig-ure 8 shows the comparison of time response of multimodalexcitationwith that of the free vibration Experimental results
clearly show that the proposed method is still effective inmultimodal vibration control and this further demonstratesthe robustness of the proposed controller
6 Conclusion
In this paper an experimental study of the robust adaptivesliding mode control was conducted on a flexible beamwith piezoceramic actuators and sensor for vibration controlpurpose Two different schemes of adaptive sliding modecontrol have been comparatively studied during the experi-ment The adaptive robust sliding mode control with bang-bang action could suppress the induced vibration of a flexiblebeam in a quick fashion However it causes chattering duringsteady state To eliminate the chattering a hyperbolic tangentfunction is used to replace the sign function in the robustcontrol action Both adaptive robust sliding mode controllersare designed based on the Lyapunov direct method In theproposed adaptive sliding mode controller there will beless constraint on the choice of the amplitude of the robustcompensator The conservatism of the control system hasbeen reduced by introducing the adaptive scheme into thesliding mode controlThe asymptotic stability of the adaptiverobust sliding mode control is proved by using Lyapunovdirect method in the paper The experimental results showthat the proposed adaptive robust sliding mode control withthe smooth robust compensator eliminates the chatteringwhile keeping the advantages of adaptive robust slidingmodecontrol Also the proposedmethods are robust to the varianceof the plant parameters as demonstrated experimentally
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
This research was partially supported by Grant no 51278387Grant no 51278084 and Grant no 51121005 (Science Fundfor Creative Research Groups) from the National ScienceFoundation of China (NSFC)
References
[1] G Song and R Mukherjee ldquoA comparative study of conven-tional nonsmooth time-invariant and smooth time-varyingrobust compensatorsrdquo IEEE Transactions on Control SystemsTechnology vol 6 no 4 pp 571ndash576 1998
[2] J-H Park K-W Kim and H-H Yoo ldquoTwo-stage sliding modecontroller for vibration suppression of a flexible pointing sys-temrdquo Proceedings of the Institution ofMechanical Engineers PartC Journal of Mechanical Engineering Science vol 215 no 2 pp155ndash166 2001
[3] X Yu Z Man and B Wu ldquoDesign of fuzzy sliding-mode con-trol systemsrdquo Fuzzy Sets and Systems vol 95 no 3 pp 295ndash3061998
Mathematical Problems in Engineering 9
[4] B Yao and M Tomizuka ldquoSmooth robust adaptive slidingmode control of manipulators with guaranteed transient per-formancerdquo in Proceedings of the American Control Conferencevol 1 pp 1176ndash1180 July 1994
[5] J Wang A B Rad and P T Chan ldquoIndirect adaptive fuzzysliding mode control part I fuzzy switchingrdquo Fuzzy Sets andSystems vol 122 no 1 pp 21ndash30 2001
[6] G Song R W Longman and R Mukherjee ldquoIntegratedsliding-mode adaptive-robust controlrdquo IEE Proceedings ControlTheory and Applications vol 146 no 4 pp 341ndash347 1999
[7] G Song R W Longman and L Cai ldquoIntegrated adaptive-robust control of robot manipulatorsrdquo Journal of Robotic Sys-tems vol 15 no 12 pp 699ndash712 1998
[8] R-J Wai ldquoAdaptive sliding-mode control for induction servo-motor driverdquo IEE Proceedings Electric Power Applications vol147 no 6 pp 553ndash562 2000
[9] S-C Lin and Y-Y Chen ldquoDesign of adaptive fuzzy slidingmode for nonlinear system controlrdquo in Proceedings of the 3rdIEEE Conference on Fuzzy Systems pp 35ndash39 June 1994
[10] J Fei and H Ding ldquoAdaptive vibration control of microelec-tromechanical systems triaxial gyroscope using radial basisfunction sliding mode techniquerdquo Proceedings of the Institutionof Mechanical Engineers Part I Journal of Systems and ControlEngineering vol 227 no 2 pp 264ndash269 2013
[11] B L Zhang and Q L Han ldquoRobust sliding mode 119867infin
controlusing time-varying delayed states for offshore steel jacketplatformsrdquo in Proceedings of the IEEE International Symposiumon Industrial Electronics (ISIE rsquo13) pp 1ndash6 IEEE 2013
[12] H Zhang G Dong M Zhou C Song Y Huang and K Du ldquoAnew variable structure sliding mode control strategy for FTS indiamond-cutting microstructured surfacesrdquo The InternationalJournal of AdvancedManufacturing Technology vol 65 no 5ndash8pp 1177ndash1184 2013
[13] H Li J Yu C Hilton and H Liu ldquoAdaptive sliding modecontrol for nonlinear active suspension vehicle systems usingTS fuzzy approachrdquo IEEE Transactions on Industrial Electronicsvol 60 no 8 2013
[14] J L Fanson and T K Caughey ldquoPositive position feedbackcontrol for large space structuresrdquo AIAA Journal vol 28 no 4pp 717ndash724 1990
[15] G Song P Qiao V Sethi and A Prasad ldquoActive vibrationcontrol of a smart pultruded fiber-reinforced polymer I-beamrdquoin International Symposium on Smart Structures and Materialsvol 4696 of Proceedings of SPIE pp 197ndash208 March 2002
[16] PMayhan andGWashington ldquoFuzzymodel reference learningcontrol a new control paradigm for smart structuresrdquo SmartMaterials and Structures vol 7 no 6 pp 874ndash884 1998
[17] V Bottega AMolter J SO Fonseca andR Pergher ldquoVibrationcontrol of manipulators with flexible nonprismatic links usingpiezoelectric actuators and sensorsrdquo Mathematical Problems inEngineering vol 2009 Article ID 727385 16 pages 2009
[18] A Molter V Bottega O A A Da Silveira and J S O FonsecaldquoSimultaneous piezoelectric actuator and sensor placementoptimization and control design of manipulators with flexiblelinks using SDREmethodrdquoMathematical Problems in Engineer-ing vol 2010 Article ID 362437 23 pages 2010
[19] M Ahmadian and K M Jeric ldquoOn the application of shuntedpiezoceramics for increasing acoustic transmission loss instructuresrdquo Journal of Sound and Vibration vol 243 no 2 pp347ndash359 2001
[20] J Fei Y Fang and C Yan ldquoThe comparative study of vibrationcontrol of flexible structure using smart materialsrdquo Mathemat-ical Problems in Engineering vol 2010 Article ID 768256 13pages 2010
[21] S-B Choi Y-K Park and C-C Cheong ldquoActive vibrationcontrol of hybrid smart structures featuring piezoelectric filmsand electrorheological fluidsrdquo in The International Society forOptical Engineering vol 2717 of Proceedings of the SPIE pp544ndash552 February 1996
[22] A K Jha and D J Inman ldquoSliding mode control of a Gossamerstructure using smart materialsrdquo Journal of Vibration andControl vol 10 no 8 pp 1199ndash1220 2004
[23] S-B Choi and M-S Kim ldquoNew discrete-time fuzzy-sliding-mode control with application to smart structuresrdquo Journal ofGuidance Control and Dynamics vol 20 no 5 pp 857ndash8641997
[24] S Bashash and N Jalili ldquoRobust multiple frequency trajec-tory tracking control of piezoelectrically driven micronano-positioning systemsrdquo IEEE Transactions on Control SystemsTechnology vol 15 no 5 pp 867ndash878 2007
[25] H C Liaw B Shirinzadeh and J Smith ldquoSliding-modeenhanced adaptive motion tracking control of piezoelectricactuation systems formicronanomanipulationrdquo IEEETransac-tions on Control Systems Technology vol 16 no 4 pp 826ndash8332008
[26] Y Li and Q Xu ldquoAdaptive sliding mode control with perturba-tion estimation and PID sliding surface for motion tracking of apiezo-driven micromanipulatorrdquo IEEE Transactions on ControlSystems Technology vol 18 no 4 pp 798ndash810 2010
[27] J Hu and D Zhu ldquoVibration control of smart structure usingsliding mode control with observerrdquo Journal of Computers vol7 no 2 pp 411ndash418 2012
[28] L Zhang Z Zhang Z Long and A Hao ldquoSlidingmode controlwith auto-tuning law forMaglev SystemrdquoEngineering vol 2 no2 pp 107ndash112 2010
[29] T Rittenschober andK Schlacher ldquoObserver-based self sensingactuation of piezoelastic structures for robust vibration controlrdquoAutomatica vol 48 no 6 pp 1123ndash1131 2012
[30] X Bu L Ye Z Su and C Wang ldquoActive control of a flexiblesmart beam using a system identification technique based onARMAXrdquo SmartMaterials and Structures vol 12 no 5 pp 845ndash850 2003
[31] R D Robinett C R Dohrmann G R Eisler et al FlexibleRobot Dynamics and Controls Kluwer AcademicPlenum Pub-lishers New York NY USA 2002
[32] L Cai and G Song ldquoJoint stick-slip friction compensation ofrobotmanipulators by using smooth robust controllersrdquo Journalof Robotic Systems vol 11 no 6 pp 451ndash470 1994
the tanh(120572119904) function will mimic the sign(119904) function Toavoid chattering a relative smaller value of 120572 can be chosenWhen implementing the controller the value of 120572 is oftenexperimentally determined
To prove the stability of the smooth adaptive robustcontroller (17) define the Lyapunov function candidate as
V =1
21198981199042
+1
2120593119879
Γminus1
120593 (19)
Using the similar process shown in Section 3 we have
V = minus119886 sdot 119904 sdot tanh (120572119904) minus 120593119879 (119884119904 minus Γminus1 120593) (20)
Please note that 119888 and are updated according to (14) tomake 119884119904 minus Γminus1 120593 = 0 Then V = minus119886 sdot 119904 sdot tanh(120572119904) le 0 whileV = 0 only when 119904 = 0 Therefore according to LaSallersquostheorem the closed-loop control system with the smoothadaptive robust controllers (17) is asymptotically stable in thesense of Lyapunov
5 Experiment Results
51 Open Loop Testing of the Aluminum Beam First openloop testing is performed to find the dominant mode of thebeam for active vibration control The beam was subjectedto a mechanical impact at its midlength for a multimodeexcitation The power spectrum density (PSD) plots of the30-second time response of the multimode excitation wereobtainedThe corresponding frequencies of peaks in the PSDplots represent themodal frequencies of the flexible beam Tostudy the attenuation of vibration at each modal frequencythe PSD plots of the first 5 seconds and the last 5 secondsof the beam vibration are shown in Figure 3 The peaks inFigure 3 clearly reveal the modal frequencies of the flexiblebeam the 1st mode at 164Hz the 2nd mode at 977Hzand the 3rd mode at 2685Hz A comparison of PSD plotsof the vibration data for the first 5 seconds and the last 5seconds clearly shows the vibration attenuation at eachmodalfrequency The dB value and dB drop of the first three modesare shown in Table 3 From Figure 3 and Table 3 it can beseen that the dB levels of the second and third modes at thefirst five-second and last five-second periods are significantlylower than that of the 1st mode The dB drop value of thefirst three modes shows that the second and third modesattenuated much faster than the first mode Therefore it canbe concluded that the first modal frequency 164Hz is thedominant one in the vibration and the first mode should bethe target mode in the active vibration control To target atthis mode the vibration control is designed based on thesimplified plant (1) which includes only the dynamics of thefirst mode
52 Experimental Results for the Adaptive Robust SlidingMode Controller To verify the effectiveness of the adaptiverobust sliding mode control (3) and the smooth adaptivesliding mode control (17) experiments were conducted onthe aluminum flexible beam In the experiment 119904 = 119890 +
120582119890 = 119890 + 005119890 and 1199031= 1199032= 1199033= 01 for the diagonal
Table 3 PSD drop comparison of first 5 seconds and last 5 seconds
Mode 1 Mode 2 Mode 3dB level of the first 5 seconds 1673 598 minus1436dB level of the last 5 seconds 291 minus1723 minus3519dB drop 1382 2321 2083
20
10
minus10
minus20
minus30
minus40
minus50
minus60
minus70
minus800
0
10 20 30 40 50 60
Frequency (Hz)
Pow
er sp
ectr
um m
agni
tude
(dB)
First 5 secondsLast 5 seconds
Figure 3 Comparison of PSD plots of first 5 seconds and last 5seconds for time response of multimode excitation of the flexiblebeam
matrix Γ In each test the beam was excited by a sinusoidalsignal at its first modal frequency combined with a lowpower white noise for the initial 5 seconds The amplitudeof the sinusoidal signal was 075V The variance of the whitenoise was 00497 The sinusoidal signal and the white noisewere summed in the Simulink software and downloaded intodSPACE DS1102 Floating-Point Controller Board for real-time implementation The output of the dSPACE controllerboard was amplified by an amplifier for capacitive load beforeexciting the PZT actuators After the initial 5 seconds firstmode excitation was stopped and the active vibration controlstarted to suppress the induced vibration A free vibration testwas conducted for the purpose of comparison In the freevibration test no control signal was implemented after theinitial 5-second excitation
The upper diagram of Figure 4 is the time responsecomparison of free vibration with that of the active control(with the sign function) From the experimental results thevibration is suppressed in about 8 oscillation cycles (about5 seconds) However there is small-magnitude chattering inthe steady state which may cause instability when a certainlevel of disturbance exists The lower diagram of Figure 4 isthe time response comparison of free vibration with that ofthe active control (with the tanh function) It can be seen thatthe robust sliding mode control can suppress the vibration inabout 8 oscillation cycles without the chattering in the steadystate Experimental results show that the chattering can be
Mathematical Problems in Engineering 7
06
04
02
0
minus02
minus04
5 10 15 20
Sens
or v
olta
ge (V
)
Sliding-mode control with sign functionFree vibration
Time (s)
(a)
06
04
02
0
minus02
minus04
5 10 15 20
Sens
or v
olta
ge (V
)
Sliding-mode control with tanh functionFree vibration
Time (s)
(b)
Figure 4 The time response comparisons of the adaptive slidingmode controller (with the sign function) with the adaptive robustsliding mode controller (with tanh function)
eliminated by implementing the hyperbolic tangent function(tanh function) as the switching function instead of the signfunction
The upper diagram of Figure 5 is the control actionwith the bang-bang robust compensator Compared with thetime response in Figure 4 it can be seen that the chatteringhappens when the control action switches abruptly betweenits positive limit and negative limit when the system is alreadyin the steady state (after 10 seconds) The lower diagramof Figure 5 is the control action with the smooth robustcompensator From the diagram it can be seen that thecontrol signal switches smoothly Therefore the chatteringin the output signal is eliminated Figure 6 shows the timehistory of the parametersrsquo updating process Furthermore itis also worthwhile to note that no higher modes have beenexcited even though the plant is actually amultimode systemThis justifies that the primary vibration control target is thefirst mode
53 Experimental Results for the Adaptive Robust SlidingModeControl with Mass Uncertainty To verify the robustness ofthe proposed adaptive robust controller the plant is changedby adding mass on the beam Four 6-gram adhesive masseswere attached to both sides of the beam with two at theremote tip and the other two at the midlength of the beamThe mass of the aluminum beam is increased from 10520gram to 12920 gramThefirstmodal frequency of the originalbeam is 167Hz whereas the first modal frequency of the
Time (s)
200
100
0
minus100
minus2005 10 15 20
Actu
ator
sign
al (V
)
Sliding-mode control with sign function
(a)
Time (s)
200
100
0
minus100
minus2005 10 15 20
Actu
ator
sign
al (V
)Sliding-mode control with tanh function
(b)
Figure 5 The control signal comparisons of the adaptive slidingmode controller (with the sign function) with the adaptive robustsliding mode controller (with tanh function)
00835
0083
008255 10 15 20 25
Time (s)
m
Time (s)
082
081
085 10 15 20 25
c
1100002
1100001
110
Sliding-mode control with sign functionSliding-mode control with tanh function
Time (s)5 10 15 20 25
k
Figure 6 The parametersrsquo updating time history
beam with added mass has been decreased to 1457Hz Theincreasing of the mass has caused the decreasing of thenatural frequency of the beamTherefore in this experimentthe frequency of the sinusoidal excitation signal has beenchanged to 1457Hz to ensure excitation of the beam Thesame controller is used for the vibration control of the beamwith added mass From the experimental results (Figure 7)
8 Mathematical Problems in Engineering
05
04
02
03
01
0
minus01
minus02
minus03
minus04
minus055 10 15 20
Time (s)
Free vibrationSliding-mode control with tanh function
Sens
or v
olta
ge (V
)
Figure 7 The time response comparisons of the adaptive slidingmode controller with tanh function for the flexible beam with 24-gram mass added
1
08
06
04
02
0
minus02
minus04
minus06
minus08
minus10 1 2 3 4 5 6 7
Time (s)
Sliding-mode control with tanh function
Sens
or v
olta
ge (V
)
Free vibration
Figure 8 The time response of comparison of the adaptive slidingmode controller with tanh function for the flexible beam withmultimode excitation
it can be seen that the adaptive robust sliding mode caneffectively suppress the vibration of the beam after its physicalproperty has been changedThis demonstrates the robustnessof the proposed controllers
54 Experimental Results for the Adaptive Robust SlidingModeControl with Multimode Excitation To test its robustness tohigher order dynamics and disturbances multimodal excita-tion of the flexible beam with mass added is conducted Fig-ure 8 shows the comparison of time response of multimodalexcitationwith that of the free vibration Experimental results
clearly show that the proposed method is still effective inmultimodal vibration control and this further demonstratesthe robustness of the proposed controller
6 Conclusion
In this paper an experimental study of the robust adaptivesliding mode control was conducted on a flexible beamwith piezoceramic actuators and sensor for vibration controlpurpose Two different schemes of adaptive sliding modecontrol have been comparatively studied during the experi-ment The adaptive robust sliding mode control with bang-bang action could suppress the induced vibration of a flexiblebeam in a quick fashion However it causes chattering duringsteady state To eliminate the chattering a hyperbolic tangentfunction is used to replace the sign function in the robustcontrol action Both adaptive robust sliding mode controllersare designed based on the Lyapunov direct method In theproposed adaptive sliding mode controller there will beless constraint on the choice of the amplitude of the robustcompensator The conservatism of the control system hasbeen reduced by introducing the adaptive scheme into thesliding mode controlThe asymptotic stability of the adaptiverobust sliding mode control is proved by using Lyapunovdirect method in the paper The experimental results showthat the proposed adaptive robust sliding mode control withthe smooth robust compensator eliminates the chatteringwhile keeping the advantages of adaptive robust slidingmodecontrol Also the proposedmethods are robust to the varianceof the plant parameters as demonstrated experimentally
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
This research was partially supported by Grant no 51278387Grant no 51278084 and Grant no 51121005 (Science Fundfor Creative Research Groups) from the National ScienceFoundation of China (NSFC)
References
[1] G Song and R Mukherjee ldquoA comparative study of conven-tional nonsmooth time-invariant and smooth time-varyingrobust compensatorsrdquo IEEE Transactions on Control SystemsTechnology vol 6 no 4 pp 571ndash576 1998
[2] J-H Park K-W Kim and H-H Yoo ldquoTwo-stage sliding modecontroller for vibration suppression of a flexible pointing sys-temrdquo Proceedings of the Institution ofMechanical Engineers PartC Journal of Mechanical Engineering Science vol 215 no 2 pp155ndash166 2001
[3] X Yu Z Man and B Wu ldquoDesign of fuzzy sliding-mode con-trol systemsrdquo Fuzzy Sets and Systems vol 95 no 3 pp 295ndash3061998
Mathematical Problems in Engineering 9
[4] B Yao and M Tomizuka ldquoSmooth robust adaptive slidingmode control of manipulators with guaranteed transient per-formancerdquo in Proceedings of the American Control Conferencevol 1 pp 1176ndash1180 July 1994
[5] J Wang A B Rad and P T Chan ldquoIndirect adaptive fuzzysliding mode control part I fuzzy switchingrdquo Fuzzy Sets andSystems vol 122 no 1 pp 21ndash30 2001
[6] G Song R W Longman and R Mukherjee ldquoIntegratedsliding-mode adaptive-robust controlrdquo IEE Proceedings ControlTheory and Applications vol 146 no 4 pp 341ndash347 1999
[7] G Song R W Longman and L Cai ldquoIntegrated adaptive-robust control of robot manipulatorsrdquo Journal of Robotic Sys-tems vol 15 no 12 pp 699ndash712 1998
[8] R-J Wai ldquoAdaptive sliding-mode control for induction servo-motor driverdquo IEE Proceedings Electric Power Applications vol147 no 6 pp 553ndash562 2000
[9] S-C Lin and Y-Y Chen ldquoDesign of adaptive fuzzy slidingmode for nonlinear system controlrdquo in Proceedings of the 3rdIEEE Conference on Fuzzy Systems pp 35ndash39 June 1994
[10] J Fei and H Ding ldquoAdaptive vibration control of microelec-tromechanical systems triaxial gyroscope using radial basisfunction sliding mode techniquerdquo Proceedings of the Institutionof Mechanical Engineers Part I Journal of Systems and ControlEngineering vol 227 no 2 pp 264ndash269 2013
[11] B L Zhang and Q L Han ldquoRobust sliding mode 119867infin
controlusing time-varying delayed states for offshore steel jacketplatformsrdquo in Proceedings of the IEEE International Symposiumon Industrial Electronics (ISIE rsquo13) pp 1ndash6 IEEE 2013
[12] H Zhang G Dong M Zhou C Song Y Huang and K Du ldquoAnew variable structure sliding mode control strategy for FTS indiamond-cutting microstructured surfacesrdquo The InternationalJournal of AdvancedManufacturing Technology vol 65 no 5ndash8pp 1177ndash1184 2013
[13] H Li J Yu C Hilton and H Liu ldquoAdaptive sliding modecontrol for nonlinear active suspension vehicle systems usingTS fuzzy approachrdquo IEEE Transactions on Industrial Electronicsvol 60 no 8 2013
[14] J L Fanson and T K Caughey ldquoPositive position feedbackcontrol for large space structuresrdquo AIAA Journal vol 28 no 4pp 717ndash724 1990
[15] G Song P Qiao V Sethi and A Prasad ldquoActive vibrationcontrol of a smart pultruded fiber-reinforced polymer I-beamrdquoin International Symposium on Smart Structures and Materialsvol 4696 of Proceedings of SPIE pp 197ndash208 March 2002
[16] PMayhan andGWashington ldquoFuzzymodel reference learningcontrol a new control paradigm for smart structuresrdquo SmartMaterials and Structures vol 7 no 6 pp 874ndash884 1998
[17] V Bottega AMolter J SO Fonseca andR Pergher ldquoVibrationcontrol of manipulators with flexible nonprismatic links usingpiezoelectric actuators and sensorsrdquo Mathematical Problems inEngineering vol 2009 Article ID 727385 16 pages 2009
[18] A Molter V Bottega O A A Da Silveira and J S O FonsecaldquoSimultaneous piezoelectric actuator and sensor placementoptimization and control design of manipulators with flexiblelinks using SDREmethodrdquoMathematical Problems in Engineer-ing vol 2010 Article ID 362437 23 pages 2010
[19] M Ahmadian and K M Jeric ldquoOn the application of shuntedpiezoceramics for increasing acoustic transmission loss instructuresrdquo Journal of Sound and Vibration vol 243 no 2 pp347ndash359 2001
[20] J Fei Y Fang and C Yan ldquoThe comparative study of vibrationcontrol of flexible structure using smart materialsrdquo Mathemat-ical Problems in Engineering vol 2010 Article ID 768256 13pages 2010
[21] S-B Choi Y-K Park and C-C Cheong ldquoActive vibrationcontrol of hybrid smart structures featuring piezoelectric filmsand electrorheological fluidsrdquo in The International Society forOptical Engineering vol 2717 of Proceedings of the SPIE pp544ndash552 February 1996
[22] A K Jha and D J Inman ldquoSliding mode control of a Gossamerstructure using smart materialsrdquo Journal of Vibration andControl vol 10 no 8 pp 1199ndash1220 2004
[23] S-B Choi and M-S Kim ldquoNew discrete-time fuzzy-sliding-mode control with application to smart structuresrdquo Journal ofGuidance Control and Dynamics vol 20 no 5 pp 857ndash8641997
[24] S Bashash and N Jalili ldquoRobust multiple frequency trajec-tory tracking control of piezoelectrically driven micronano-positioning systemsrdquo IEEE Transactions on Control SystemsTechnology vol 15 no 5 pp 867ndash878 2007
[25] H C Liaw B Shirinzadeh and J Smith ldquoSliding-modeenhanced adaptive motion tracking control of piezoelectricactuation systems formicronanomanipulationrdquo IEEETransac-tions on Control Systems Technology vol 16 no 4 pp 826ndash8332008
[26] Y Li and Q Xu ldquoAdaptive sliding mode control with perturba-tion estimation and PID sliding surface for motion tracking of apiezo-driven micromanipulatorrdquo IEEE Transactions on ControlSystems Technology vol 18 no 4 pp 798ndash810 2010
[27] J Hu and D Zhu ldquoVibration control of smart structure usingsliding mode control with observerrdquo Journal of Computers vol7 no 2 pp 411ndash418 2012
[28] L Zhang Z Zhang Z Long and A Hao ldquoSlidingmode controlwith auto-tuning law forMaglev SystemrdquoEngineering vol 2 no2 pp 107ndash112 2010
[29] T Rittenschober andK Schlacher ldquoObserver-based self sensingactuation of piezoelastic structures for robust vibration controlrdquoAutomatica vol 48 no 6 pp 1123ndash1131 2012
[30] X Bu L Ye Z Su and C Wang ldquoActive control of a flexiblesmart beam using a system identification technique based onARMAXrdquo SmartMaterials and Structures vol 12 no 5 pp 845ndash850 2003
[31] R D Robinett C R Dohrmann G R Eisler et al FlexibleRobot Dynamics and Controls Kluwer AcademicPlenum Pub-lishers New York NY USA 2002
[32] L Cai and G Song ldquoJoint stick-slip friction compensation ofrobotmanipulators by using smooth robust controllersrdquo Journalof Robotic Systems vol 11 no 6 pp 451ndash470 1994
Sliding-mode control with sign functionFree vibration
Time (s)
(a)
06
04
02
0
minus02
minus04
5 10 15 20
Sens
or v
olta
ge (V
)
Sliding-mode control with tanh functionFree vibration
Time (s)
(b)
Figure 4 The time response comparisons of the adaptive slidingmode controller (with the sign function) with the adaptive robustsliding mode controller (with tanh function)
eliminated by implementing the hyperbolic tangent function(tanh function) as the switching function instead of the signfunction
The upper diagram of Figure 5 is the control actionwith the bang-bang robust compensator Compared with thetime response in Figure 4 it can be seen that the chatteringhappens when the control action switches abruptly betweenits positive limit and negative limit when the system is alreadyin the steady state (after 10 seconds) The lower diagramof Figure 5 is the control action with the smooth robustcompensator From the diagram it can be seen that thecontrol signal switches smoothly Therefore the chatteringin the output signal is eliminated Figure 6 shows the timehistory of the parametersrsquo updating process Furthermore itis also worthwhile to note that no higher modes have beenexcited even though the plant is actually amultimode systemThis justifies that the primary vibration control target is thefirst mode
53 Experimental Results for the Adaptive Robust SlidingModeControl with Mass Uncertainty To verify the robustness ofthe proposed adaptive robust controller the plant is changedby adding mass on the beam Four 6-gram adhesive masseswere attached to both sides of the beam with two at theremote tip and the other two at the midlength of the beamThe mass of the aluminum beam is increased from 10520gram to 12920 gramThefirstmodal frequency of the originalbeam is 167Hz whereas the first modal frequency of the
Time (s)
200
100
0
minus100
minus2005 10 15 20
Actu
ator
sign
al (V
)
Sliding-mode control with sign function
(a)
Time (s)
200
100
0
minus100
minus2005 10 15 20
Actu
ator
sign
al (V
)Sliding-mode control with tanh function
(b)
Figure 5 The control signal comparisons of the adaptive slidingmode controller (with the sign function) with the adaptive robustsliding mode controller (with tanh function)
00835
0083
008255 10 15 20 25
Time (s)
m
Time (s)
082
081
085 10 15 20 25
c
1100002
1100001
110
Sliding-mode control with sign functionSliding-mode control with tanh function
Time (s)5 10 15 20 25
k
Figure 6 The parametersrsquo updating time history
beam with added mass has been decreased to 1457Hz Theincreasing of the mass has caused the decreasing of thenatural frequency of the beamTherefore in this experimentthe frequency of the sinusoidal excitation signal has beenchanged to 1457Hz to ensure excitation of the beam Thesame controller is used for the vibration control of the beamwith added mass From the experimental results (Figure 7)
8 Mathematical Problems in Engineering
05
04
02
03
01
0
minus01
minus02
minus03
minus04
minus055 10 15 20
Time (s)
Free vibrationSliding-mode control with tanh function
Sens
or v
olta
ge (V
)
Figure 7 The time response comparisons of the adaptive slidingmode controller with tanh function for the flexible beam with 24-gram mass added
1
08
06
04
02
0
minus02
minus04
minus06
minus08
minus10 1 2 3 4 5 6 7
Time (s)
Sliding-mode control with tanh function
Sens
or v
olta
ge (V
)
Free vibration
Figure 8 The time response of comparison of the adaptive slidingmode controller with tanh function for the flexible beam withmultimode excitation
it can be seen that the adaptive robust sliding mode caneffectively suppress the vibration of the beam after its physicalproperty has been changedThis demonstrates the robustnessof the proposed controllers
54 Experimental Results for the Adaptive Robust SlidingModeControl with Multimode Excitation To test its robustness tohigher order dynamics and disturbances multimodal excita-tion of the flexible beam with mass added is conducted Fig-ure 8 shows the comparison of time response of multimodalexcitationwith that of the free vibration Experimental results
clearly show that the proposed method is still effective inmultimodal vibration control and this further demonstratesthe robustness of the proposed controller
6 Conclusion
In this paper an experimental study of the robust adaptivesliding mode control was conducted on a flexible beamwith piezoceramic actuators and sensor for vibration controlpurpose Two different schemes of adaptive sliding modecontrol have been comparatively studied during the experi-ment The adaptive robust sliding mode control with bang-bang action could suppress the induced vibration of a flexiblebeam in a quick fashion However it causes chattering duringsteady state To eliminate the chattering a hyperbolic tangentfunction is used to replace the sign function in the robustcontrol action Both adaptive robust sliding mode controllersare designed based on the Lyapunov direct method In theproposed adaptive sliding mode controller there will beless constraint on the choice of the amplitude of the robustcompensator The conservatism of the control system hasbeen reduced by introducing the adaptive scheme into thesliding mode controlThe asymptotic stability of the adaptiverobust sliding mode control is proved by using Lyapunovdirect method in the paper The experimental results showthat the proposed adaptive robust sliding mode control withthe smooth robust compensator eliminates the chatteringwhile keeping the advantages of adaptive robust slidingmodecontrol Also the proposedmethods are robust to the varianceof the plant parameters as demonstrated experimentally
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
This research was partially supported by Grant no 51278387Grant no 51278084 and Grant no 51121005 (Science Fundfor Creative Research Groups) from the National ScienceFoundation of China (NSFC)
References
[1] G Song and R Mukherjee ldquoA comparative study of conven-tional nonsmooth time-invariant and smooth time-varyingrobust compensatorsrdquo IEEE Transactions on Control SystemsTechnology vol 6 no 4 pp 571ndash576 1998
[2] J-H Park K-W Kim and H-H Yoo ldquoTwo-stage sliding modecontroller for vibration suppression of a flexible pointing sys-temrdquo Proceedings of the Institution ofMechanical Engineers PartC Journal of Mechanical Engineering Science vol 215 no 2 pp155ndash166 2001
[3] X Yu Z Man and B Wu ldquoDesign of fuzzy sliding-mode con-trol systemsrdquo Fuzzy Sets and Systems vol 95 no 3 pp 295ndash3061998
Mathematical Problems in Engineering 9
[4] B Yao and M Tomizuka ldquoSmooth robust adaptive slidingmode control of manipulators with guaranteed transient per-formancerdquo in Proceedings of the American Control Conferencevol 1 pp 1176ndash1180 July 1994
[5] J Wang A B Rad and P T Chan ldquoIndirect adaptive fuzzysliding mode control part I fuzzy switchingrdquo Fuzzy Sets andSystems vol 122 no 1 pp 21ndash30 2001
[6] G Song R W Longman and R Mukherjee ldquoIntegratedsliding-mode adaptive-robust controlrdquo IEE Proceedings ControlTheory and Applications vol 146 no 4 pp 341ndash347 1999
[7] G Song R W Longman and L Cai ldquoIntegrated adaptive-robust control of robot manipulatorsrdquo Journal of Robotic Sys-tems vol 15 no 12 pp 699ndash712 1998
[8] R-J Wai ldquoAdaptive sliding-mode control for induction servo-motor driverdquo IEE Proceedings Electric Power Applications vol147 no 6 pp 553ndash562 2000
[9] S-C Lin and Y-Y Chen ldquoDesign of adaptive fuzzy slidingmode for nonlinear system controlrdquo in Proceedings of the 3rdIEEE Conference on Fuzzy Systems pp 35ndash39 June 1994
[10] J Fei and H Ding ldquoAdaptive vibration control of microelec-tromechanical systems triaxial gyroscope using radial basisfunction sliding mode techniquerdquo Proceedings of the Institutionof Mechanical Engineers Part I Journal of Systems and ControlEngineering vol 227 no 2 pp 264ndash269 2013
[11] B L Zhang and Q L Han ldquoRobust sliding mode 119867infin
controlusing time-varying delayed states for offshore steel jacketplatformsrdquo in Proceedings of the IEEE International Symposiumon Industrial Electronics (ISIE rsquo13) pp 1ndash6 IEEE 2013
[12] H Zhang G Dong M Zhou C Song Y Huang and K Du ldquoAnew variable structure sliding mode control strategy for FTS indiamond-cutting microstructured surfacesrdquo The InternationalJournal of AdvancedManufacturing Technology vol 65 no 5ndash8pp 1177ndash1184 2013
[13] H Li J Yu C Hilton and H Liu ldquoAdaptive sliding modecontrol for nonlinear active suspension vehicle systems usingTS fuzzy approachrdquo IEEE Transactions on Industrial Electronicsvol 60 no 8 2013
[14] J L Fanson and T K Caughey ldquoPositive position feedbackcontrol for large space structuresrdquo AIAA Journal vol 28 no 4pp 717ndash724 1990
[15] G Song P Qiao V Sethi and A Prasad ldquoActive vibrationcontrol of a smart pultruded fiber-reinforced polymer I-beamrdquoin International Symposium on Smart Structures and Materialsvol 4696 of Proceedings of SPIE pp 197ndash208 March 2002
[16] PMayhan andGWashington ldquoFuzzymodel reference learningcontrol a new control paradigm for smart structuresrdquo SmartMaterials and Structures vol 7 no 6 pp 874ndash884 1998
[17] V Bottega AMolter J SO Fonseca andR Pergher ldquoVibrationcontrol of manipulators with flexible nonprismatic links usingpiezoelectric actuators and sensorsrdquo Mathematical Problems inEngineering vol 2009 Article ID 727385 16 pages 2009
[18] A Molter V Bottega O A A Da Silveira and J S O FonsecaldquoSimultaneous piezoelectric actuator and sensor placementoptimization and control design of manipulators with flexiblelinks using SDREmethodrdquoMathematical Problems in Engineer-ing vol 2010 Article ID 362437 23 pages 2010
[19] M Ahmadian and K M Jeric ldquoOn the application of shuntedpiezoceramics for increasing acoustic transmission loss instructuresrdquo Journal of Sound and Vibration vol 243 no 2 pp347ndash359 2001
[20] J Fei Y Fang and C Yan ldquoThe comparative study of vibrationcontrol of flexible structure using smart materialsrdquo Mathemat-ical Problems in Engineering vol 2010 Article ID 768256 13pages 2010
[21] S-B Choi Y-K Park and C-C Cheong ldquoActive vibrationcontrol of hybrid smart structures featuring piezoelectric filmsand electrorheological fluidsrdquo in The International Society forOptical Engineering vol 2717 of Proceedings of the SPIE pp544ndash552 February 1996
[22] A K Jha and D J Inman ldquoSliding mode control of a Gossamerstructure using smart materialsrdquo Journal of Vibration andControl vol 10 no 8 pp 1199ndash1220 2004
[23] S-B Choi and M-S Kim ldquoNew discrete-time fuzzy-sliding-mode control with application to smart structuresrdquo Journal ofGuidance Control and Dynamics vol 20 no 5 pp 857ndash8641997
[24] S Bashash and N Jalili ldquoRobust multiple frequency trajec-tory tracking control of piezoelectrically driven micronano-positioning systemsrdquo IEEE Transactions on Control SystemsTechnology vol 15 no 5 pp 867ndash878 2007
[25] H C Liaw B Shirinzadeh and J Smith ldquoSliding-modeenhanced adaptive motion tracking control of piezoelectricactuation systems formicronanomanipulationrdquo IEEETransac-tions on Control Systems Technology vol 16 no 4 pp 826ndash8332008
[26] Y Li and Q Xu ldquoAdaptive sliding mode control with perturba-tion estimation and PID sliding surface for motion tracking of apiezo-driven micromanipulatorrdquo IEEE Transactions on ControlSystems Technology vol 18 no 4 pp 798ndash810 2010
[27] J Hu and D Zhu ldquoVibration control of smart structure usingsliding mode control with observerrdquo Journal of Computers vol7 no 2 pp 411ndash418 2012
[28] L Zhang Z Zhang Z Long and A Hao ldquoSlidingmode controlwith auto-tuning law forMaglev SystemrdquoEngineering vol 2 no2 pp 107ndash112 2010
[29] T Rittenschober andK Schlacher ldquoObserver-based self sensingactuation of piezoelastic structures for robust vibration controlrdquoAutomatica vol 48 no 6 pp 1123ndash1131 2012
[30] X Bu L Ye Z Su and C Wang ldquoActive control of a flexiblesmart beam using a system identification technique based onARMAXrdquo SmartMaterials and Structures vol 12 no 5 pp 845ndash850 2003
[31] R D Robinett C R Dohrmann G R Eisler et al FlexibleRobot Dynamics and Controls Kluwer AcademicPlenum Pub-lishers New York NY USA 2002
[32] L Cai and G Song ldquoJoint stick-slip friction compensation ofrobotmanipulators by using smooth robust controllersrdquo Journalof Robotic Systems vol 11 no 6 pp 451ndash470 1994
Free vibrationSliding-mode control with tanh function
Sens
or v
olta
ge (V
)
Figure 7 The time response comparisons of the adaptive slidingmode controller with tanh function for the flexible beam with 24-gram mass added
1
08
06
04
02
0
minus02
minus04
minus06
minus08
minus10 1 2 3 4 5 6 7
Time (s)
Sliding-mode control with tanh function
Sens
or v
olta
ge (V
)
Free vibration
Figure 8 The time response of comparison of the adaptive slidingmode controller with tanh function for the flexible beam withmultimode excitation
it can be seen that the adaptive robust sliding mode caneffectively suppress the vibration of the beam after its physicalproperty has been changedThis demonstrates the robustnessof the proposed controllers
54 Experimental Results for the Adaptive Robust SlidingModeControl with Multimode Excitation To test its robustness tohigher order dynamics and disturbances multimodal excita-tion of the flexible beam with mass added is conducted Fig-ure 8 shows the comparison of time response of multimodalexcitationwith that of the free vibration Experimental results
clearly show that the proposed method is still effective inmultimodal vibration control and this further demonstratesthe robustness of the proposed controller
6 Conclusion
In this paper an experimental study of the robust adaptivesliding mode control was conducted on a flexible beamwith piezoceramic actuators and sensor for vibration controlpurpose Two different schemes of adaptive sliding modecontrol have been comparatively studied during the experi-ment The adaptive robust sliding mode control with bang-bang action could suppress the induced vibration of a flexiblebeam in a quick fashion However it causes chattering duringsteady state To eliminate the chattering a hyperbolic tangentfunction is used to replace the sign function in the robustcontrol action Both adaptive robust sliding mode controllersare designed based on the Lyapunov direct method In theproposed adaptive sliding mode controller there will beless constraint on the choice of the amplitude of the robustcompensator The conservatism of the control system hasbeen reduced by introducing the adaptive scheme into thesliding mode controlThe asymptotic stability of the adaptiverobust sliding mode control is proved by using Lyapunovdirect method in the paper The experimental results showthat the proposed adaptive robust sliding mode control withthe smooth robust compensator eliminates the chatteringwhile keeping the advantages of adaptive robust slidingmodecontrol Also the proposedmethods are robust to the varianceof the plant parameters as demonstrated experimentally
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
This research was partially supported by Grant no 51278387Grant no 51278084 and Grant no 51121005 (Science Fundfor Creative Research Groups) from the National ScienceFoundation of China (NSFC)
References
[1] G Song and R Mukherjee ldquoA comparative study of conven-tional nonsmooth time-invariant and smooth time-varyingrobust compensatorsrdquo IEEE Transactions on Control SystemsTechnology vol 6 no 4 pp 571ndash576 1998
[2] J-H Park K-W Kim and H-H Yoo ldquoTwo-stage sliding modecontroller for vibration suppression of a flexible pointing sys-temrdquo Proceedings of the Institution ofMechanical Engineers PartC Journal of Mechanical Engineering Science vol 215 no 2 pp155ndash166 2001
[3] X Yu Z Man and B Wu ldquoDesign of fuzzy sliding-mode con-trol systemsrdquo Fuzzy Sets and Systems vol 95 no 3 pp 295ndash3061998
Mathematical Problems in Engineering 9
[4] B Yao and M Tomizuka ldquoSmooth robust adaptive slidingmode control of manipulators with guaranteed transient per-formancerdquo in Proceedings of the American Control Conferencevol 1 pp 1176ndash1180 July 1994
[5] J Wang A B Rad and P T Chan ldquoIndirect adaptive fuzzysliding mode control part I fuzzy switchingrdquo Fuzzy Sets andSystems vol 122 no 1 pp 21ndash30 2001
[6] G Song R W Longman and R Mukherjee ldquoIntegratedsliding-mode adaptive-robust controlrdquo IEE Proceedings ControlTheory and Applications vol 146 no 4 pp 341ndash347 1999
[7] G Song R W Longman and L Cai ldquoIntegrated adaptive-robust control of robot manipulatorsrdquo Journal of Robotic Sys-tems vol 15 no 12 pp 699ndash712 1998
[8] R-J Wai ldquoAdaptive sliding-mode control for induction servo-motor driverdquo IEE Proceedings Electric Power Applications vol147 no 6 pp 553ndash562 2000
[9] S-C Lin and Y-Y Chen ldquoDesign of adaptive fuzzy slidingmode for nonlinear system controlrdquo in Proceedings of the 3rdIEEE Conference on Fuzzy Systems pp 35ndash39 June 1994
[10] J Fei and H Ding ldquoAdaptive vibration control of microelec-tromechanical systems triaxial gyroscope using radial basisfunction sliding mode techniquerdquo Proceedings of the Institutionof Mechanical Engineers Part I Journal of Systems and ControlEngineering vol 227 no 2 pp 264ndash269 2013
[11] B L Zhang and Q L Han ldquoRobust sliding mode 119867infin
controlusing time-varying delayed states for offshore steel jacketplatformsrdquo in Proceedings of the IEEE International Symposiumon Industrial Electronics (ISIE rsquo13) pp 1ndash6 IEEE 2013
[12] H Zhang G Dong M Zhou C Song Y Huang and K Du ldquoAnew variable structure sliding mode control strategy for FTS indiamond-cutting microstructured surfacesrdquo The InternationalJournal of AdvancedManufacturing Technology vol 65 no 5ndash8pp 1177ndash1184 2013
[13] H Li J Yu C Hilton and H Liu ldquoAdaptive sliding modecontrol for nonlinear active suspension vehicle systems usingTS fuzzy approachrdquo IEEE Transactions on Industrial Electronicsvol 60 no 8 2013
[14] J L Fanson and T K Caughey ldquoPositive position feedbackcontrol for large space structuresrdquo AIAA Journal vol 28 no 4pp 717ndash724 1990
[15] G Song P Qiao V Sethi and A Prasad ldquoActive vibrationcontrol of a smart pultruded fiber-reinforced polymer I-beamrdquoin International Symposium on Smart Structures and Materialsvol 4696 of Proceedings of SPIE pp 197ndash208 March 2002
[16] PMayhan andGWashington ldquoFuzzymodel reference learningcontrol a new control paradigm for smart structuresrdquo SmartMaterials and Structures vol 7 no 6 pp 874ndash884 1998
[17] V Bottega AMolter J SO Fonseca andR Pergher ldquoVibrationcontrol of manipulators with flexible nonprismatic links usingpiezoelectric actuators and sensorsrdquo Mathematical Problems inEngineering vol 2009 Article ID 727385 16 pages 2009
[18] A Molter V Bottega O A A Da Silveira and J S O FonsecaldquoSimultaneous piezoelectric actuator and sensor placementoptimization and control design of manipulators with flexiblelinks using SDREmethodrdquoMathematical Problems in Engineer-ing vol 2010 Article ID 362437 23 pages 2010
[19] M Ahmadian and K M Jeric ldquoOn the application of shuntedpiezoceramics for increasing acoustic transmission loss instructuresrdquo Journal of Sound and Vibration vol 243 no 2 pp347ndash359 2001
[20] J Fei Y Fang and C Yan ldquoThe comparative study of vibrationcontrol of flexible structure using smart materialsrdquo Mathemat-ical Problems in Engineering vol 2010 Article ID 768256 13pages 2010
[21] S-B Choi Y-K Park and C-C Cheong ldquoActive vibrationcontrol of hybrid smart structures featuring piezoelectric filmsand electrorheological fluidsrdquo in The International Society forOptical Engineering vol 2717 of Proceedings of the SPIE pp544ndash552 February 1996
[22] A K Jha and D J Inman ldquoSliding mode control of a Gossamerstructure using smart materialsrdquo Journal of Vibration andControl vol 10 no 8 pp 1199ndash1220 2004
[23] S-B Choi and M-S Kim ldquoNew discrete-time fuzzy-sliding-mode control with application to smart structuresrdquo Journal ofGuidance Control and Dynamics vol 20 no 5 pp 857ndash8641997
[24] S Bashash and N Jalili ldquoRobust multiple frequency trajec-tory tracking control of piezoelectrically driven micronano-positioning systemsrdquo IEEE Transactions on Control SystemsTechnology vol 15 no 5 pp 867ndash878 2007
[25] H C Liaw B Shirinzadeh and J Smith ldquoSliding-modeenhanced adaptive motion tracking control of piezoelectricactuation systems formicronanomanipulationrdquo IEEETransac-tions on Control Systems Technology vol 16 no 4 pp 826ndash8332008
[26] Y Li and Q Xu ldquoAdaptive sliding mode control with perturba-tion estimation and PID sliding surface for motion tracking of apiezo-driven micromanipulatorrdquo IEEE Transactions on ControlSystems Technology vol 18 no 4 pp 798ndash810 2010
[27] J Hu and D Zhu ldquoVibration control of smart structure usingsliding mode control with observerrdquo Journal of Computers vol7 no 2 pp 411ndash418 2012
[28] L Zhang Z Zhang Z Long and A Hao ldquoSlidingmode controlwith auto-tuning law forMaglev SystemrdquoEngineering vol 2 no2 pp 107ndash112 2010
[29] T Rittenschober andK Schlacher ldquoObserver-based self sensingactuation of piezoelastic structures for robust vibration controlrdquoAutomatica vol 48 no 6 pp 1123ndash1131 2012
[30] X Bu L Ye Z Su and C Wang ldquoActive control of a flexiblesmart beam using a system identification technique based onARMAXrdquo SmartMaterials and Structures vol 12 no 5 pp 845ndash850 2003
[31] R D Robinett C R Dohrmann G R Eisler et al FlexibleRobot Dynamics and Controls Kluwer AcademicPlenum Pub-lishers New York NY USA 2002
[32] L Cai and G Song ldquoJoint stick-slip friction compensation ofrobotmanipulators by using smooth robust controllersrdquo Journalof Robotic Systems vol 11 no 6 pp 451ndash470 1994
[4] B Yao and M Tomizuka ldquoSmooth robust adaptive slidingmode control of manipulators with guaranteed transient per-formancerdquo in Proceedings of the American Control Conferencevol 1 pp 1176ndash1180 July 1994
[5] J Wang A B Rad and P T Chan ldquoIndirect adaptive fuzzysliding mode control part I fuzzy switchingrdquo Fuzzy Sets andSystems vol 122 no 1 pp 21ndash30 2001
[6] G Song R W Longman and R Mukherjee ldquoIntegratedsliding-mode adaptive-robust controlrdquo IEE Proceedings ControlTheory and Applications vol 146 no 4 pp 341ndash347 1999
[7] G Song R W Longman and L Cai ldquoIntegrated adaptive-robust control of robot manipulatorsrdquo Journal of Robotic Sys-tems vol 15 no 12 pp 699ndash712 1998
[8] R-J Wai ldquoAdaptive sliding-mode control for induction servo-motor driverdquo IEE Proceedings Electric Power Applications vol147 no 6 pp 553ndash562 2000
[9] S-C Lin and Y-Y Chen ldquoDesign of adaptive fuzzy slidingmode for nonlinear system controlrdquo in Proceedings of the 3rdIEEE Conference on Fuzzy Systems pp 35ndash39 June 1994
[10] J Fei and H Ding ldquoAdaptive vibration control of microelec-tromechanical systems triaxial gyroscope using radial basisfunction sliding mode techniquerdquo Proceedings of the Institutionof Mechanical Engineers Part I Journal of Systems and ControlEngineering vol 227 no 2 pp 264ndash269 2013
[11] B L Zhang and Q L Han ldquoRobust sliding mode 119867infin
controlusing time-varying delayed states for offshore steel jacketplatformsrdquo in Proceedings of the IEEE International Symposiumon Industrial Electronics (ISIE rsquo13) pp 1ndash6 IEEE 2013
[12] H Zhang G Dong M Zhou C Song Y Huang and K Du ldquoAnew variable structure sliding mode control strategy for FTS indiamond-cutting microstructured surfacesrdquo The InternationalJournal of AdvancedManufacturing Technology vol 65 no 5ndash8pp 1177ndash1184 2013
[13] H Li J Yu C Hilton and H Liu ldquoAdaptive sliding modecontrol for nonlinear active suspension vehicle systems usingTS fuzzy approachrdquo IEEE Transactions on Industrial Electronicsvol 60 no 8 2013
[14] J L Fanson and T K Caughey ldquoPositive position feedbackcontrol for large space structuresrdquo AIAA Journal vol 28 no 4pp 717ndash724 1990
[15] G Song P Qiao V Sethi and A Prasad ldquoActive vibrationcontrol of a smart pultruded fiber-reinforced polymer I-beamrdquoin International Symposium on Smart Structures and Materialsvol 4696 of Proceedings of SPIE pp 197ndash208 March 2002
[16] PMayhan andGWashington ldquoFuzzymodel reference learningcontrol a new control paradigm for smart structuresrdquo SmartMaterials and Structures vol 7 no 6 pp 874ndash884 1998
[17] V Bottega AMolter J SO Fonseca andR Pergher ldquoVibrationcontrol of manipulators with flexible nonprismatic links usingpiezoelectric actuators and sensorsrdquo Mathematical Problems inEngineering vol 2009 Article ID 727385 16 pages 2009
[18] A Molter V Bottega O A A Da Silveira and J S O FonsecaldquoSimultaneous piezoelectric actuator and sensor placementoptimization and control design of manipulators with flexiblelinks using SDREmethodrdquoMathematical Problems in Engineer-ing vol 2010 Article ID 362437 23 pages 2010
[19] M Ahmadian and K M Jeric ldquoOn the application of shuntedpiezoceramics for increasing acoustic transmission loss instructuresrdquo Journal of Sound and Vibration vol 243 no 2 pp347ndash359 2001
[20] J Fei Y Fang and C Yan ldquoThe comparative study of vibrationcontrol of flexible structure using smart materialsrdquo Mathemat-ical Problems in Engineering vol 2010 Article ID 768256 13pages 2010
[21] S-B Choi Y-K Park and C-C Cheong ldquoActive vibrationcontrol of hybrid smart structures featuring piezoelectric filmsand electrorheological fluidsrdquo in The International Society forOptical Engineering vol 2717 of Proceedings of the SPIE pp544ndash552 February 1996
[22] A K Jha and D J Inman ldquoSliding mode control of a Gossamerstructure using smart materialsrdquo Journal of Vibration andControl vol 10 no 8 pp 1199ndash1220 2004
[23] S-B Choi and M-S Kim ldquoNew discrete-time fuzzy-sliding-mode control with application to smart structuresrdquo Journal ofGuidance Control and Dynamics vol 20 no 5 pp 857ndash8641997
[24] S Bashash and N Jalili ldquoRobust multiple frequency trajec-tory tracking control of piezoelectrically driven micronano-positioning systemsrdquo IEEE Transactions on Control SystemsTechnology vol 15 no 5 pp 867ndash878 2007
[25] H C Liaw B Shirinzadeh and J Smith ldquoSliding-modeenhanced adaptive motion tracking control of piezoelectricactuation systems formicronanomanipulationrdquo IEEETransac-tions on Control Systems Technology vol 16 no 4 pp 826ndash8332008
[26] Y Li and Q Xu ldquoAdaptive sliding mode control with perturba-tion estimation and PID sliding surface for motion tracking of apiezo-driven micromanipulatorrdquo IEEE Transactions on ControlSystems Technology vol 18 no 4 pp 798ndash810 2010
[27] J Hu and D Zhu ldquoVibration control of smart structure usingsliding mode control with observerrdquo Journal of Computers vol7 no 2 pp 411ndash418 2012
[28] L Zhang Z Zhang Z Long and A Hao ldquoSlidingmode controlwith auto-tuning law forMaglev SystemrdquoEngineering vol 2 no2 pp 107ndash112 2010
[29] T Rittenschober andK Schlacher ldquoObserver-based self sensingactuation of piezoelastic structures for robust vibration controlrdquoAutomatica vol 48 no 6 pp 1123ndash1131 2012
[30] X Bu L Ye Z Su and C Wang ldquoActive control of a flexiblesmart beam using a system identification technique based onARMAXrdquo SmartMaterials and Structures vol 12 no 5 pp 845ndash850 2003
[31] R D Robinett C R Dohrmann G R Eisler et al FlexibleRobot Dynamics and Controls Kluwer AcademicPlenum Pub-lishers New York NY USA 2002
[32] L Cai and G Song ldquoJoint stick-slip friction compensation ofrobotmanipulators by using smooth robust controllersrdquo Journalof Robotic Systems vol 11 no 6 pp 451ndash470 1994
