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Original Article

Proc IMechE Part I:J Systems and Control Engineering1–12� IMechE 2017Reprints and permissions:sagepub.co.uk/journalsPermissions.navDOI: 10.1177/0959651817711859journals.sagepub.com/home/pii

A new self-tuning fuzzy controller forvibration of a flexible structuresubjected to multi-frequencyexcitations

Jie Fu1, Peidong Li1, Seung-Bok Choi2, Guanyao Liao1,Yuan Wang1 and Miao Yu1

AbstractThis work proposes a new self-tuning fuzzy controller for vibration control of a flexible structure subjected to bothsingle-frequency and multi-frequency excitations. A piezoelectric stack actuator is used to generate control force toattenuate vibrations in the resonant and non-resonant frequency ranges. As a first step, modal frequencies and modalshapes of the flexible beam structure are obtained via the finite element method, and these are verified through compar-ison with the measured values. Second, in order to attenuate multi-frequency vibration of the beam, a new fuzzy control-ler with self-tuning factors is proposed. This controller is independent of the structure model and very adaptive tovariable excitations. Then, vibration control performances of the self-tuning fuzzy controller are experimentally investi-gated under two different excitation conditions: single-frequency and multi-frequency excitations. It is shown that vibra-tion control capability of the proposed controller outperforms conventional fuzzy controller under two differentexcitation conditions. In addition, it is demonstrated by experimentally implementing the proposed self-tuning fuzzy con-troller that high performances of multi-frequency vibration control are successfully achieved in which both amplitudeand frequency of sinusoidal excitations are varied.

KeywordsVibration control, multi-frequency excitation, self-tuning fuzzy controller, flexible structure, piezoelectric stack actuator,modal analysis

Date received: 8 October 2016; accepted: 23 April 2017

Introduction

Flexible structures in engineering fields are widely used:civil engineering (long-span bridge),1 aerospace engi-neering (cabin of a space station or solar panels)2 andmechanical engineering (machine structure).3,4 Despiteseveral advantages of the flexible structures such aslight weight, unwanted vibrations are inevitablyoccurred and this may cause instability of the systemresulting in degraded performances and catastrophicfailure. In order to suppress vibrations and improvesafety, numerous types of passive mounts5 such as rub-ber mount and hydraulic mount have been developed.The rubber mount with low damping shows efficientvibration isolation performance in high-frequency andnon-resonant excitations. However, it cannot provide afavorable performance in low-frequency and resonantexcitations. The hydraulic mount with large damping is

effectively utilized to isolate low-frequency vibrations,but not effective in high-frequency band.6 To overcomethe disadvantages of passive mounts, active mounts

1Key Laboratory for Optoelectronic Technology and Systems, Ministry of

Education, School of Optoelectronic Engineering, Chongqing University,

Chongqing, China2Smart Structures and Systems Laboratory, Department of Mechanical

Engineering, Inha University, Incheon, Korea

Corresponding authors:

Seung-Bok Choi, Smart Structures and Systems Laboratory, Department

of Mechanical Engineering, Inha University, Yonghyun-Dong, Namgu,

Incheon 402-751, Korea.

Email: [email protected]

Jie Fu, Key Laboratory for Optoelectronic Technology and Systems,

Ministry of Education, School of Optoelectronic Engineering, Chongqing

University, Chongqing 400044, China.

Email: [email protected]

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utilizing smart materials such as shape memory alloys,7

magneto-strictive materials8 and piezoelectric materi-als9–14 have been actively studied in recent years.Among these active mounts, the piezoelectric actuatoris featured by small stroke, fast response time, highforce generation and wide working frequency band-width.15 Using these salient features, an effective vibra-tion control performance of various flexible structuresystems subjected to small-magnitude and high-frequency bandwidth excitations can be accomplished.In general, the performance of the dynamic responsesof flexible structures highly depends on an appropriatecontrol strategy. Choi et al. designed a linear quadraticGaussian (LQG) controller to suppress the vibration ofa flexible structural system with inertial type piezoelec-tric mount. They experimentally verified good control-ler performance by evaluating acceleration response ofthe structural system in frequency and time domains.6

Hanagud et al.16 developed an acceleration feedbackcontroller for the piezoelectric stack actuator (PSA) toreduce the amplitude of aircraft’s tail buffet in aselected frequency range (8–80Hz). Nitsche and Gaulused the PSA to control the normal force in the frictioninterface of a rotational joint connecting two linearelastic beams. They formulated a damping controller todissipate the system energy additionally based onLyapunov function.17 Choi et al. developed a hybridmount consisting of elastic rubber and PSA for theeffective vibration control of a flexible beam structure.In this work, a sliding mode controller was designedand implemented to attenuate the vibration of thestructure subjected to high-frequency and small-magnitude excitations.18 It is noted here that in orderto successfully achieve vibration control performance,accurate system models are absolutely required for theabove-mentioned control methods. However, in prac-tice, it is difficult to formulate an accurate mathemati-cal or analytical model of the flexible structure due toseveral uncertainties such as parameter variation.Therefore, the fuzzy controller (FC) which is indepen-dent of complicated mathematical model of the struc-ture system is frequently adopted.

One of salient features of the FC is that an expertknowledge is incorporated into the controller designprocess using some linguistic rules and this leads to anonlinear controller which can describe any complexrelationships between input and output variables. Inaddition, the FC is very robust against uncertainties,19–21

and hence, it has been extensively researched in variousfields of engineering.22–25 Xu et al.26 designed a FC tocontrol vibrations of the solar panel using the PSA andshowed that the FC could effectively suppress thevibrations in a short time with appropriate choice ofinput and output scaling factors. However, for thesinusoidal excitations in which both the amplitude andfrequency of the excitations are varied, conventionalFC with the fixed quantification factors, scalingfactors, rules and membership functions could notguarantee stable control performance. Therefore, the

self-tuning FC is required to deal with this kind of exci-tation condition.27–31 It is well known that the gain-tuning FC provides much better control responses inreal-time environment than the FC associated with thetuning of rule base and membership functions.27–31

Gao et al.31 proposed a self-tuning FC for a class ofindustrial temperature control system and showed sig-nificant improvement in maintaining stability over awide range of fixed temperature. The self-tuning factorused in Gao et al.31 was adjusted by another fuzzy rulewhich could lead large computation costs, and hence, itwas not suitable for real-time vibration control of flex-ible structures with multi-frequency excitations. Otherresearch works on the gain-tuning FC showed onlysimulation results by focusing on the control strat-egy.11,29,30 As surveyed, the study on the gain-tuningFC associated with experiment verification is consider-ably rare. Moreover, it is still challenge to experimen-tally implement the gain-tuning FC for vibrationcontrol of flexible structures subjected to variable exci-tations in terms of amplitude and frequency.

Consequently, the main technical originality of thiswork is to propose a new FC with self-tuning factorsand experimentally implement it to verify excellentvibration control performance of a flexible structuresubjected to multi-frequency excitations. The controlforce associated with the new FC controller is generatedby adopting the PSA and the flexible beam structure issubjected to both single-frequency and multi-frequencyexcitations. In section ‘‘Modal analysis,’’ the finite ele-ment method is utilized to obtain modal frequencies andmode shapes of the beam structure, and these are vali-dated via experimental identification. In section ‘‘Designof FC with self-tuning factors,’’ considering a strongnonlinearity of the flexible beam structure, a self-tuningFC is designed considering self-tuning laws of the quan-tification factors and scaling factor obtained from thelinear fitting. And in section ‘‘Experimental results anddiscussions,’’ experimental realizations of the proposedcontroller are undertaken and vibration control perfor-mances are presented in time and frequency domains. Inaddition, a comparative work between the proposedcontroller and conventional FC is performed undersame experimental conditions. It is shown from experi-mental results that vibrations at the resonant and non-resonant frequency ranges are well controlled despitemulti-frequency excitations in which both amplitudeand frequency are varied during control action.

Modal analysis

It is necessary to obtain dominant resonance frequen-cies of the flexible beam structure before designing thecontroller. Therefore, both modal analysis based onfinite element analysis and experimental identificationare carried out to obtain modal frequencies and modeshapes of the beam.

Figure 1 shows the schematic configuration of theflexible beam structure incorporated with the PSA. The

2 Proc IMechE Part I: J Systems and Control Engineering 00(0)

dimension of the beam (304 stainless steel material) is600mm (length) 3 50mm (width) 3 2mm (thickness),and the constraints used for the end beam is fixed sup-port. The PSA is mounted just below the center of thebeam denoted as position 2. And positions 1 and 3 areplaced at 150 and 450mm from the left end of thebeam, respectively. The finite element analysis is per-formed using ANSYS software to identify modal fre-quencies and shapes. The results are shown in Figure 2.It is identified from the results that the maximum vibra-tion amplitude of the beam is occurred at positions 1and 3, and the first two-order modal frequencies areidentified by 71 and 90Hz, respectively.

The modal characteristics of the proposed flexiblestructure are also investigated via experiment to vali-date the results obtained from the finite element analy-sis. Figure 3 shows an experimental setup for modaltest. The dSPACE/AutoBox (Model ds1005; dSPACE,Germany) generates a sine sweep signal with the fre-quency range of 20–300Hz, which is amplified by thepower amplifier (Model JF2; Chongqing University,China) to drive the exciter (Model JZ1; Far EastVibration, China). The response of the acceleration atposition 1 is measured using an accelerometer (Model333B52; Piezotronics, USA). It is remarked that inorder to experimentally obtain the modal frequenciesor resonant frequencies of the flexible structure, theaccelerometer cannot be placed on the vibration nodes.From the results of ANSYS modal analysis, it is knownthat position 2 in Figure 1 is the vibration node. So theaccelerometer should be placed at position 1 or 3.However, in this work, the accelerometer is mounted atposition 3 since PSA is put on the node rather thanposition 1 or 3. If the PSA is mounted on position 1 or

3, the mounting point will be a new node instead of theposition with the maximum amplitude. Another reasonis that since the vibration magnitude at position 2 isvery small, the PSA can be protected from the failurewhich may be caused from the high tensile stress. Thebasic performances of the PSA (Model 40VS12; CoreTomorrow, China) used in this work are experimen-tally investigated and shown in Figure 4. It can be seenclearly that the output displacement and force increasewith the increase of the input voltage, and the fre-quency response of the output force and displacementare almost invariable within the frequency range of100Hz. Now, the frequency response function of theflexible beam structure is obtained using a sine sweepmethod and presented in Figure 5. As mentioned ear-lier, this response is measured at position 1 using anaccelerometer. It is identified from this result that thedominant modes are the first mode (72Hz) and secondmode (91Hz), which agree well to the analysis results.Thus, the first two modes will be considered as domi-nant modes of the flexible structure for vibrationcontrol.

Design of FC with self-tuning factors

The main idea of the FC is the use of linguistic instruc-tions based on human skills as a basis for control. Bytransferring human skills into linguistic IF-THENrules, the FC can be embedded into a closed-loop sys-tem, which is similar to conventional controllers. Thesynthesis of a conventional FC involves fuzzifier, fuzzyrules and fuzzy inference and defuzzifier, as shown inFigure 6. The input variables to the developed FC areselected as the error of acceleration e and its rate ofchange ec, while the output is chosen as the appliedvoltage to the PSA. The error signal is obtained by

e ¼ ar � am ð1Þ

where am is the acceleration measured by the acceler-ometer, ar is the reference value, which is set as zero inthis work, and e denotes the error of acceleration.

Fuzzifier

The fuzzifier converts the input variables from numeri-cal form into linguistic form. The structure of FC willbe more complex if many linguistic variables are chosen

Figure 1. Schematic configuration of the flexible beamstructure.

Figure 2. Mode shapes of the beam: (a) the first mode shape and (b) the second mode shape.

Fu et al. 3

to describe inputs or outputs. However, the lesser oneswill cause worse performance of the system. In thiswork, considering the controller’s complexity and

accurate performance of the system, the input and out-put linguistic variables are assigned by seven fuzzy sub-sets. These are denoted by negative big (NB), negativemedium (NM), negative small (NS), zero (ZO), positivesmall (PS), positive medium (PM) and positive big(PB). The discourse domain of the inputs is set as [26,6]. Hence, the input variables need to be scaled in sucha manner that their minima are 26 and maxima are 6

ke ¼ Emax

emax

kec ¼ ECmax

ecmax

(ð2Þ

where ke and kec represent the quantification factor ofthe error and the error’s change, respectively. emax andecmax are the maxima of the input variables, and Emax

and ECmax are the maxima of the discourse domain ofthe inputs. In conventional FC, the quantification fac-tors are normally set based on specific signals and thusconstant.

Fuzzy rules and fuzzy inference

Fuzzy rules play an important role in a fuzzy controlsystem. In this work, for the active vibration controlsystem, E and EC are selected as PB, and U (the outputof the FC) is selected as NB to prevent the change of E.If E is PB and EC is negative, U should be NS or ZOto keep the system stable. So the ith rule can be writtenas follows

If E ¼ Ai and EC ¼ Bi then U ¼ Ci; i ¼ 1; 2; . . . ; 49

ð3Þ

where Ai;Bi;Ci are the linguistic values of the fuzzyvariables. Considering the number of fuzzy subsetsgiven for each input and output in the fuzzifier section,it can be concluded that the number of fuzzy rules is49, and the rules are listed in Table 1.

The fuzzy inference of the controller is based onMamdani’s method, which is associated with the max–min composition. The membership functions of theinput and output are illustrated in Figures 7 and 8. It isnoted that an ‘‘S’’-shaped membership function isselected for NB and PB to enhance the smoothness of

Figure 3. Experimental setup for the modal test.

Figure 4. Dynamic characteristics of the PSA: (a) photograph,(b) generated output displacement and (c) generated outputforce.

Figure 5. Acceleration response at position 1.


the control processing and guarantee the stability of

the controlled system. And different triangle-shapedmembership functions are used for the rest of linguisticvariables to improve the sensitivity of the controlledsystem, especially under small-magnitude excitations.

Defuzzifier

Through a process referred to as defuzzifier, the resultsof the fuzzy inference are transformed back to numeri-cal form to be implemented in practical control. Thecenter-of-gravity method is adopted as the defuzzifica-tion technique to compute the output voltage U, whichis widely utilized in the Mamdani inference method–based fuzzy control system.32 Since the discoursedomain of the output U is set as [27, 7], it must bescaled prior to its control action on the PSA

u ¼ ku sgn Uð ÞU ð4Þ

where

sgnðUÞ ¼ 1 Uø 00 U\ 0

�

where ku represents the output scaling factor and u is thecontrol voltage to be applied to the PSA. A sign function(sgn(�)) is used here to guarantee that the voltage u isalways non-negative to protect the PSA, which cannottolerate negative driving voltages. In other words, thisindicates a semi-active input voltage which always guar-antees the stability of the proposed control system.

Factor adjustment mechanism

When the frequency and amplitude of the excitationsignal are known, the calculated quantification factorsand scaling factor according to equations (2) and (4)are constant. However, there is variation of excitationin many practical applications. In this case, the FCwith constant factors may not be able to achieve satis-fying control effect. To improve control performance,the factor adjustment mechanisms needs to be designedon the basis of the excitation estimate result. A newstructure of the FC with self-tuning factors is shown inFigure 9.

The peak observer first computes and stores localmaxima or minima P of the input signal xðtÞ, which canbe obtained by the following equation

P ¼ x t� Tð Þ; x tð Þ � x t� Tð Þ½ �x t� Tð Þ � x t� 2Tð Þ½ �ł 0 ð5Þ

where T is the sampling period of the controlled sys-tem. Then, the absolute maximum AðtÞ in n cycles canbe expressed by

A tð Þ ¼ max P1j j; P2j j; . . . ; Pnj j½ �ð Þ ð6Þ

where n is determined by the experimental test for vari-able excitation which is 15 in this work. The observerin equation (6) is sensitive to the noise, and hence, theinput error is preprocessed by a band-pass filter. InFigure 9, the peak observer identifies the peak value ofthe error AeðtÞ and the one of the error’s change rateAecðtÞ on-line. And the quantification factor and scal-ing factor adjustment mechanism (adaptation law) aredetermined as follows

Figure 6. Structure of a conventional FC.

Table 1. Fuzzy rules of the designed FC.

UEC

NB NM NS ZO PS PM PB

E

NB PB PB PM PM PM PS PSNM PB PB PM PM PM ZO ZONS PB PB PM PS PS ZO ZOZO PM PM PS ZO ZO NS NSPS PM PM ZO NS NS NM NMPM PS PS NS NM NM NM NBPB ZO ZO NS NM NM NB NB

NB: negative big; NM: negative medium; NS: negative small; ZO: zero;

PS: positive small; PM: positive medium; PB: positive big.

Figure 7. Input membership functions.

Figure 8. Output membership function.

Fu et al. 5

ke tð Þ ¼ Emax

Ae tð Þ

kec tð Þ ¼ ECmax

Aec tð Þ

ku tð Þ ¼ gAe tð Þ

8>><>>: ð7Þ

where AeðtÞ;AecðtÞ are obtained by the peak observer,and g is identified to be 0.016 through the linear fittingbased on the measured values, as shown in Figure 10.

The effectiveness of this method will be verified in thefollowing section.

Experimental results and discussions

In order to evaluate vibration control performance ofthe proposed self-tuning FC, an experimental setup isestablished, as shown in Figure 11. The desired wave-form is generated from the signal generator (Model33120A; Agilent, USA) and amplified by the poweramplifier to drive the exciter. The acceleration of thebeam at position 3 is measured by an accelerometer.After being conditioned and filtered, the accelerationsignal is sent to the designed self-tuning FC, which isrealized in the MATLAB/Simulink environment fol-lowed by the downloading to the dSPACE/AutoBox.The piezoelectric amplifier (Model XE501A; CoreTomorrow) is used to supply high voltage to the PSAto achieve enough actuating force. By considering thelimitation of computing speed of AutoBox and vibra-tion modes to be controlled, the sampling rate in theexperimental implementation is chosen by 600Hz.

Single-frequency excitation

The excitation signals used in this work are denoted byacceleration and frequency as follows: accelerationamplitude—0.5, 1.0 and 1.5m/s2 and frequencychange—60, 70, 80 and 95Hz. The used frequency cov-ers resonant frequencies and non-resonant frequencies.The quantification and scaling factors in the conven-tional FC are regulated with the amplitude of 0.5m/s2

and keep constant. Table 2 lists the factors of both con-ventional FC and self-tuning FC under different excita-tions. Vibration control performances of conventionalFC and the proposed controller are measured at differ-ent excitations and presented in Figures 12–15. It isclearly observed that control performance of the pro-posed self-tuning FC is superior to that of conventionalFC as expected. The root-mean-square (RMS) valuesand percentage decrement of the accelerations at posi-tion 3 are calculated based on the result of the

Figure 9. Schematic diagram of the self-tuning FC of the flexible structure with PSA.

Figure 10. Linear fitting of the factor g.

Figure 11. Experimental schematic for vibration control of theflexible structure.


Figure 12. Experimental results under 0.5 m/s2 and 80 Hz excitation: (a) measured acceleration response and (b) control voltage.



Figure 15. Measured results under 1.5 m/s2 and 95 Hz excitation: (a) measured acceleration response and (b) control voltage.

Fu et al. 7

uncontrolled case and given in Table 3. From theseresults, the followings can be identified: (1) conven-tional FC with constant factors could only achievesatisfying control effect when the excitation signal isconstant or a small variation. For instance, when theacceleration amplitude is two times bigger than 0.5m/s2, the percentage decrement is just over 10%. (2) Theself-tuning FC results in greater percentage decrementof acceleration RMS values showing around 40% inboth resonant and non-resonant frequencies. To sumup, it clearly indicates that the factor adjustmentmechanism proposed in this work performs well and

the self-tuning FC could effectively attenuate the vari-able single-frequency vibrations.

Multi-frequency excitation

To further demonstrate the effectiveness of the pro-posed FC with self-tuning factors, the response accel-erations at position 3 are measured under multi-frequency excitations. The excitations consist of fourdifferent frequency compositions: 70+80Hz,70+95Hz, 80+95Hz and 70+80+95Hz, and thepeak value in time domain of each excitation

Table 2. Quantification and scaling factors of the conventional and self-tuning FCs in each case under single-frequency excitations.

Acceleration amplitude (m/s2) Excitation frequency (Hz) Conventional FC Self-tuning FC

ke kec ku ke kec ku

0.5 60 12 0.0318 0.008 12 0.0318 0.00880 0.0239

1.0 70 6 0.0136 0.01680 0.011995 0.0101

1.5 70 4 0.0091 0.02495 0.0067

FC: fuzzy controller.

Table 3. Comparison of the accelerations at position 3 under different excitations and controllers.

Accelerationamplitude (m/s2)

Excitationfrequency (Hz)

Uncontrolled Conventional FC Self-tuning FC

RMS (m/s2) RMS (m/s2) Decrease (%) RMS (m/s2) Decrease (%)

0.5 60 0.3536 0.2196 38.04 0.2207 37.5980 0.2525 28.59 0.2226 37.05

1.0 70 0.7071 0.5657 20.00 0.3988 43.6080 0.6437 8.97 0.3967 43.9095 0.6069 14.17 0.3733 47.21

1.5 70 1.0607 0.9210 13.17 0.6401 39.6595 0.9137 13.86 0.5984 43.59

FC: fuzzy controller; RMS: root mean square.

Table 4. Quantification and scaling factors of the conventional and self-tuning FCs in each case under multi-frequency excitations.

Frequency composition (Hz) Acceleration peak value (m/s2) Conventional FC Self-tuning FC

ke kec ku ke kec ku

70 + 80 0.5 12 0.0269 0.008 12 0.0269 0.0081.0 6 0.0138 0.0161.5 4 0.0091 0.024

70 + 95 0.5 0.0231 12 0.0231 0.0081.0 6 0.0118 0.0161.5 4 0.008 0.024

80 + 95 0.5 0.0224 12 0.0224 0.0081.0 6 0.0114 0.0161.5 4 0.0075 0.024

70 + 80 + 95 0.5 0.024 12 0.0243 0.0081.0 6 0.0121 0.0161.5 4 0.0081 0.024



composition is set as 0.5, 1.0 and 1.5m/s2, respectively.Each frequency component of multi-frequency excita-tions corresponds to each different fuzzy variable ofconventional FCs. It is noted that constant quantifica-tion factors and scaling factors of each fuzzy variableare optimized for small-amplitude excitation (0.5m/s2).Table 4 lists the factors of both conventional FC andself-tuning one under different excitations.Experimental results are presented in both time and fre-quency domains, as shown in Figures 16–19. Table 5

calculates the RMS values and percentage decrementof the accelerations in each case, while Table 6 presentsthe peak values of acceleration frequency spectrumsunder different excitations and controllers. Now, fromthese results, the followings are identified: (1) For theconventional FC with constant factors, the percentagedecrement of RMS acceleration is about 20%, and thecontrol effect is limited. Especially, the peak value ofthe acceleration is decreased only 2.27% at worst case,while the one using the FC with self-tuning is decreased

Table 5. Comparison of the acceleration at position 3 under different excitations and controllers.

Frequencycomposition (Hz)

Accelerationpeak value (m/s2)

Uncontrolled Conventional FC Self-tuning FC

RMS (m/s2) RMS (m/s2) Decrease (%) RMS (m/s2) Decrease (%)

70 + 80 0.5 0.2528 0.1576 37.65 0.1576 37.651.0 0.5045 0.3823 24.22 0.3016 40.221.5 0.7535 0.6375 15.40 0.4754 36.91

70 + 95 0.5 0.2698 0.1653 38.74 0.1653 38.741.0 0.5179 0.4049 21.84 0.3253 37.201.5 0.7712 0.6742 12.57 0.4554 40.95

80 + 95 0.5 0.2646 0.1543 41.69 0.1543 41.691.0 0.5173 0.3965 23.69 0.2917 43.861.5 0.7886 0.6548 16.97 0.4423 43.91

70 + 80 + 95 0.5 0.2237 0.1310 41.47 0.1310 41.471.0 0.4526 0.3307 27.14 0.2528 44.321.5 0.6757 0.5497 18.65 0.3975 41.17

FC: fuzzy controller; RMS: root mean square.

Table 6. Comparison of the peak values of acceleration spectrums under different excitations.

Frequencycomposition (Hz)

Accelerationpeak value (m/s2)

Frequency (Hz) Uncontrolled Conventional FC Self-tuning FC

Peak Peak Decrease (%) Peak Decrease (%)

70 + 80 0.5 70 0.257 0.1301 49.38 0.1301 49.3880 0.2429 0.1771 27.09 0.1771 27.09

1.0 70 0.5159 0.3243 37.14 0.2831 45.1380 0.4861 0.3716 23.55 0.3095 36.33

1.5 70 0.7625 0.5604 26.50 0.4227 44.5680 0.7205 0.5763 20.01 0.4935 31.51

70 + 95 0.5 70 0.227 0.1223 46.12 0.1223 46.1295 0.288 0.1773 38.44 0.1773 38.44

1.0 70 0.4209 0.3058 27.35 0.2187 48.0495 0.5884 0.3936 33.11 0.3933 33.16

1.5 70 0.6294 0.5182 17.67 0.3474 44.8095 0.8698 0.7797 10.36 0.5277 39.33

80 + 95 0.5 80 0.2272 0.1451 36.14 0.1451 36.1495 0.2837 0.1459 48.57 0.1459 48.57

1.0 80 0.4411 0.3765 14.65 0.292 33.8095 0.5031 0.3997 20.55 0.2604 48.24

1.5 80 0.6619 0.6009 9.22 0.3961 40.1695 0.6193 0.4953 20.02 0.4317 30.29

70 + 80 + 95 0.5 70 0.1843 0.1189 35.49 0.1189 35.4980 0.08288 0.0548 33.92 0.0548 33.9295 0.2186 0.1097 49.82 0.1097 49.82

1.0 70 0.3655 0.2787 23.75 0.2306 36.9180 0.1655 0.1183 28.52 0.1082 34.6295 0.3762 0.3190 15.20 0.2049 45.53

1.5 70 0.5463 0.4618 15.47 0.3849 29.5480 0.2493 0.2053 17.65 0.1717 31.1395 0.5586 0.5459 2.27 0.2996 46.37


Fu et al. 9

up to 46.37%. (2) For the FC with self-tuning factors,the peak observer estimates the excitation changes on-line and the factors are adjusted according to equation(7). The RMS value of the acceleration decreased by44.32%, while the one using the conventional FCreduced by 27.14%. (3) In general, the vibration con-trol effect is more eminent in the resonant frequenciesthan the non-resonant frequencies as expected.

Conclusion

In this work, a new self-tuning FC was designed andexperimentally realized for vibration control of a flex-ible structure in which the PSA was used as an actua-tor. As a first step, the first two modal frequencies of

the flexible structure were calculated from the finite ele-ment analysis and also measured by sine sweepingmethod in frequency domain. Subsequently, a new FCwith self-tuning factors was formulated for effectivevibration control under multi-frequency excitationsand experimentally realized using dSpace and othercontrol devices such as filter, voltage amplifier and sig-nal generator. It has been demonstrated that the self-tuning FC can attenuate unwanted vibrations of theflexible beam structure under both single-frequencyand multi-frequency excitations. More specifically,about 40% decrement of the RMS acceleration hasbeen achieved for both excitations using the proposedcontroller, while it is less than 20% using the conven-tional FC with constant tuning factors. It is noted that

Figure 16. Experimental results under 1.0 m/s2 and 70 Hz + 80 Hz excitation: (a) accelerations in time domain, (b) control voltagesand (c) accelerations in frequency domain.




the most salient benefit of the proposed controller is apotential capability of vibration control of flexiblestructures subjected to multi-frequency excitations inwhich both amplitude and frequency are varied duringcontrol action. This is possible due to the use of theadjustment mechanism in which tuning factors (adapta-tion laws) are determined in real time based on the exci-tation estimation result achieved from the observer.

The results presented in this work are quite self-explanatory justifying that the proposed self-tuning FCcan be effectively applied to vibration control of variousflexible dynamic systems subjected to multi-frequencyexcitations in which both amplitude and frequency arechanged during control action. It is finally remarkedthat the control performance of the proposed controllerheavily depends on the excitation estimation result fromthe observer and the output scaling factor of ku, whichfinally determines the magnitude of the input voltage tothe PSA. The computation time, of course, depends onthe characteristics of the excitations. Therefore, in orderto achieve vibration controllability in high-frequencyexcitations (say above 300Hz in this work), an optimi-zation of the observer design and scaling adjustmentmechanism should be undertaken considering the high-frequency excitations to reduce the computation time.

Declaration of conflicting interests

The author(s) declared no potential conflicts of interestwith respect to the research, authorship and/or publica-tion of this article.

Funding

The author(s) disclosed receipt of the following finan-cial support for the research, authorship, and/or publi-cation of this article: This research was supported byChongqing Research Program of Basic Research andFrontier Technology (grant no. cstc2015jcyBX0069), theNatural Science Foundation of Chongqing (grant no.cstc2015jcyjA0848) and the Fundamental ResearchFunds for the Central Universities (grant no.CDJZR13120090). The authors are grateful for the help-ful comments from the reviewers and editor to improvethis article.

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