Pneumatc-Pezoelectrc Hybrd Vbraton Suppresson For a Flexble Translatng Beam Usng Adaptve Fuzzy Sldng Mode Control Algorthm Zh-cheng Qu School of Mechancal and Automotve Engneerng, South Chna Unversty of Technology, Guangzhou 510641, PR Chna, State Key Laboratory of Robotcs, Shenyang Insttute of Automaton, Chnese Academy of Scences, Shenyang 110016, PR Chna. Bn Wang and Xan-mn Zhang School of Mechancal and Automotve Engneerng, South Chna Unversty of Technology, Guangzhou 510641, PR Chna. Jan-da Han State Key Laboratory of Robotcs, Shenyang Insttute of Automaton, Chnese Academy of Scences, Shenyang 110016, PR Chna. (Receved 1 November 2014; accepted 2 July 2015) A translatng flexble beam system drven by a rodless cylnder usng a proportonal valve s proposed. Actve vbraton control of the flexble beam s developed by usng a pneumatc cylnder and a surface-bonded pezoelectrc actuator. The expermental test bed consttutes a pneumatc crcut, pezoelectrc control system, and correspondng nterface crcuts. Then, a proportonal dervatve (PD) controller and an adaptve fuzzy sldng mode control (AFSMC) algorthm are desgned and appled to vbraton control of the expermental system. Experments are conducted for set-pont resdual vbraton suppresson and resonant vbraton control under the exctaton of the frst mode. The expermental results ndcate that the desgned controllers can ncrease the dampng of the system to damp out the resdual vbraton and mnmze certan resonant responses. 1. INTRODUCTION Pneumatc drve systems are wdely used n ndustral systems owng to ther ease of mantenance and cleanng. However, pneumatc systems suffer from several drawbacks, such as 1nonlnear frcton force and hghly nonlnear pneumatc system dynamcs. 1 The nonlneartes n the pneumatc system are manly the couplngs between the pressure and the moton, the flow rate, and the frcton force. 2 Moreover, for a longstroke pneumatc rodless cylnder, hgher nonlnear dynamcs due to a large operaton range may occur; thus, t s dffcult to desgn an effectve controller. 3 Besdes, the hghly nonlnear flow through pneumatc components and the pressure droppng and tme-delay along the connectng tube also result n control errors. 4 For complcated nonlnear pneumatc servo systems, the conventonal PD controller shows a poor capacty to cope wth; t may not gve a satsfactory control performance. To overcome the above-mentoned problems, control algorthms should be nvestgated to guarantee the robustness and performance of the pneumatc control system. To deal wth some of the above-mentoned problems, several advanced schemes were utlzed to control the pneumatc servo system. A fuzzy gan scheduler of the local state feedback controller was desgned for servo-pneumatc actuators.2 Robust loop shapng-fuzzy gan schedulng control (RLSFGS) was appled to desgn a nonlnear controller for a long-stroke pneumatc system. 3 A combned dscrete varable structure control wth self-tunng adaptve control was developed for postonng control strateges of a pneumatc cylnder and a pezoelectrc actuator. 5 Further, a knd of multple-surface sldng controller was desgned for pneumatc servo systems. 6 The sldng mode control was employed to guarantee the robustness of the pneumatc servo system; however, chatterng problem occurred n ths type of controller. Wth the dea of the sldng mode control, a drect adaptve fuzzy sldng mode controller was desgned by combnng a drect adaptve fuzzy control and a fuzzy sldng mode control to reduce the trackng error and chatterng. 7, 8 Adaptve fuzzy sldng mode controller was also appled for vbraton control of magnetorheologcal mount. 9 As for actve vbraton control of flexble pezoelectrc manpulators drven by pneumatc actuators, t s a complcated, rgd-flexble couplng dynamc system, whch ncludes the nteracton of a rgd body moton, flexural vbraton of flexble manpulators, and hghly nonlnear pneumatc actuator dynamcs. 10 Compared wth prevous works, the major contrbutons of ths work are follows: (a) The pneumatc drve translatng flexble pezoelectrc beam system s controlled by usng a proportonal valve to meter the exhaust flow. The hybrd pneumatc and pezoelectrc control strategy s utlzed to suppress the vbraton of the flexble beam. (b) An adaptve fuzzy sldng mode control algorthm s appled to enhance the robustness and adaptablty of the complcated control system. (c) Experments are conducted usng ths desgned adaptve fuzzy sldng mode control, and ts results are compared wth those of the PD controller. 296 https://do.org/10.20855/jav.2017.22.3475 (pp. 296 306) Internatonal Journal of Acoustcs and Vbraton, Vol. 22, No. 3, 2017
2. PNEUMATIC DRIVING FLEXIBLE MANIPULATOR 2.1. Confguraton of a Composte Pneumatc and Pezoelectrc Drvng Flexble Beam System Fgure 1(a) depcts the schematc dagram of the pneumatc drve pezoelectrc flexble beam system. The system s composed of a pneumatc rodless cylnder, pezoelectrc flexble beam, pneumatc drve crcut, gratng dsplacement sensor, data acquston and control components realzed by an ARM controller (Mn2440) wth correspondng perpheral extendng crcuts, and computer. The ARM board communcates wth the computer. The flexble beam can translate n the horzontal drecton drven by the pneumatc rodless cylnder. The cylnder s fxed to the base. The gratng dsplacement sensor s also fxed to the base, whch s used to measure the translatng dsplacement of the slder. The slder s dsplacement s measured by usng the gratng dsplacement sensor, whch s calculated usng a counter crcut. The PZT sensor can measure the excted vbraton of the flexble translatng beam, and t s condtoned by a charge amplfer (YE5850). Then, t s converted nto dgtal data through an analogue-to-dgtal (A/D) converter. The control acton of the proportonal valve s realzed by a dgtalto-analogue (D/A) converter connected to the ARM board and the correspondng crcut. The D/A converter can provde two channels: one converts the analogue sgnal that s modulated to -5 V +5 V by the perpheral crcut for the PZT actuator; and the other converts the analogue sgnal that s modulated to 0 V +5 V for the proportonal valve. A hgh-voltage amplfer (APEX PA240CX) amplfes the output sgnal from - 5 V +5 V V to -130 V +130 V for drvng the PZT actuator. Unlke the prevous work, 10 a proportonal valve s used nstead of the PCM valve to control the pneumatc cylnder. A throttle valve S 1 s connected to the proportonal valve by cascade connecton to lmt the rate of flow. A 5-2 way solenod valve s utlzed to domnate the translatonal drecton of the rodless cylnder. As shown n Fg. 1, P 1 and P 2 ndcate the two-chamber pressures of ths cylnder, respectvely; V 1 and V 2 denote the two chambers volumes, respectvely. y(t) s the translatonal dsplacement of the slder. An ar supply source P S can provde compressed ar by an ar pump. Then, t s dvded nto three ar flows, each connectng wth a relef valve. The relef ar flows, P S1 and P S2, connect to the two ports of the 5-2 way solenod swtchng valve drectly, whch s used as an ar source to drve the moton of the slder n the postve and negatve drecton. P S0 s utlzed as the back pressure of the exhaust chamber, connectng to a check valve. The check valve s connected to one port of the 5-port valve and the ntake port of the proportonal valve. The other two ports of the 5-port valve are connected to those of the rodless cylnder, respectvely. One sde of the flexble beam s clamped nto the cylnder s slder. Fg. 1(b) and Fg. 1(c) show an establshed coordnate system to descrbe the flexble beams moton. The X-axs and the Y -axs are n the longtudnal drecton and n the translatng drecton of the slder n the horzontal plane, respectvely. The coordnate frame XOY s attached to the slder. The Z- axs s specfed n the vertcal drecton. The translatonal dsplacement of the slder s denoted by y(t). The total movng components mass except for the pezoelectrc beam s denoted as m b. An accelerometer s fxed at the free end, and ts mass (a) Schematc dagram of the system. (b) Physcal confguraton dagram of the flexble beam n x-y plane. (c) Physcal confguraton dagram of the flexble beam structure n x-z plane. (d) Photograph of the expermental apparatus. Fgure 1. Pneumatc drve pezoelectrc flexble beam system. s m t. It s consdered as a concentrated payload. The dstance between the accelerometer and the clamped sde s L t. The dstance of the PZT sensor s left edge to the clamped sde s l 1. l 2 s the dstance of the PZT sensor s rght edge to the clamped sde. l 3 and l 4 are the dstance of the PZT actuators. L b, b b, and t b are the length, wdth, and thckness of the flexble beam, respectvely. L p, b p, and t p ndcate the Internatonal Journal of Acoustcs and Vbraton, Vol. 22, No. 3, 2017 297
(a) Drecton dentfy and frequency multpler Crcut. Fgure 3. Interface dagram of AD7862 and Mn2440. (b) Interface dagram of 8254 and Mn2440. Fgure 2. Correspondng nterface crcuts of the counter. length, wdth, and thckness of the PZT sensor, respectvely. As shown n Fg. 1(a), the dsplacement of the slder s measured by the gratng dsplacement sensor, and t s calculated by a counter (counter/tmer chp Intel 8254). The sgnal of the gratng dsplacement sensor s tuned by a drecton dentfy and frequency multpler crcut, as shown n Fg. 2(a). Then, the tunng sgnal s transmtted to the ARM board va the counter chp (8254). The nterface dagram of 8254 and Mn2440 s shown n Fg. 2(b). The PZT sensor s used to measure the excted vbraton of the flexble beam. The measured vbraton sgnal s condtoned by a charge amplfer (YE5850). The obtaned voltage sgnal s n the range of -10 V +10 V, and s converted nto dgtal data through an A/D converter (4-channel chp A/D7862, wth 12-bt resoluton). The nterface dagram of AD7862 and Mn2440 s shown n Fg. 3. Then, both the measured sgnals are sent to the ARM board. A composte drvng scheme s employed for smultaneous postonng and vbraton control by combnng both pneumatc cylnder and PZT actuator. The control values are fgured out by the desgned control algorthms and appled to the proportonal valve and the PZT actuator. The control acton of the proportonal valve s realzed by a D/A converter (2- channel chp AD7847, wth 12-bt resoluton) connected wth the ARM board and the correspondng crcut. The D/A converter can provde two channels: one converted analogue sgnal s modulated to -5 V +5 V by the perpheral crcut for the PZT actuator; and the other converted analogue sgnal s modulated to 0 V +5 V for the proportonal valve. The nterface dagram of AD7847 and Mn2440 s shown n Fg. 4. 2.2. Expermental Apparatus An expermental setup s constructed and ts photograph s dsplayed n Fg. 1(d). In the pneumatc crcut, the ar compressor (SLG50, Botel Machnery) can provde an exhaust pressure of 0.8 MPa. The pneumatc trplet s made n SMC Fgure 4. Interface crcut dagram of AD7847 and Mn2440. Corporaton. The pneumatc rodless cylnder (MYIM25-600L, SMC Corporaton) s used to drve the slder. Three relef valves (AR2500) are used to tune the pressure to the sutable values. After reducng valves, P 1 = P 2 = 0.5 MPa, and P 0 =0.3 MPa. A check valve (Ak2000) s used n the pneumatc crcut to make the exhausted ar 0flow through the proportonal valve. A 5-2 way swtchng valve (VK3120, SMC Corporaton) s appled to control the slder s movng drecton. The swtchng solenod value s drven by the valve drvng board wth a 24 V electrc source. A proportonal valve (ITV2050-212L, SMC Corporaton) together wth a throttle valve s utlzed to meter the exhaust flow. The constant throttle dameter of the throttle valve s 0.79 mm. A gratng dsplacement sensor (MKT-82, FAGOR Corporaton) s used to measure the dsplacement of the slder. A charge amplfer (YE5850) amplfes the measured vbraton of the flexble beam to the range of -5 V +5 V. The output relatonshp of the charge amplfer s 0.1 mv/unt, and ts sens- Table 1. Propertes and geometrc sze of the flexble beam and the PZT patch. Symbol Parameter Value Unt Geometrc sze L b b b t b of the host beam 708 125 2 mm Geometrc sze L p b p t p of the PZT Patch 50 15 1 mm ρ b Densty of the beam 1840 kg/m 3 ρ p Densty of the PZT patch 7650 kg/m 3 Elastc modulus E b of the beam 34.64 GPa PZT patch s E pe elastc modulus 63 GPa ν b Posson rato 0.30 ν p Posson rato 0.30 Pezoelectrc d 31 stran coeffcent -166 10 12 m/v 298 Internatonal Journal of Acoustcs and Vbraton, Vol. 22, No. 3, 2017
Table 2. Parameters of the pneumatc drve system. Symbol Parameter Value Unt A Area of pston 4.9087 10 4 m 2 D Dameter of pston 25 mm The throttle dameter of d c the constant throttle valve 0.79 mm Stroke length L s of the rodless cylnder 600 mm Total mass of the pston m b and added nertal loads 2.81 kg Concentrated mass m t of the tp accelerometer 35 g Supply pressure P S1 of left chamber 5 10 5 Pa Supply pressure P S2 of rght chamber 5 10 5 Pa P S0 Back pressure 3 10 5 Pa P e Exhaust pressure 1.013 10 5 Pa R Gas constant 287 N m/(kg K) K q Gan of charge amplfer 1.0811 10 7 V/C tvty s specfed as 333 pc/unt. A sgnal generator (SPF05) generates the swept sne (chrp) sgnal and the sne sgnal for resonant vbraton exctaton n the experments. The flexble beam s materal s fberglass colophony. The propertes and szes of the beam/pzt patch and pneumatc system are lsted n Table 1 and Table 2, respectvely. 2.3. System Model For modellng of the pneumatc drve pezoelectrc flexble beam system, the prevous work 10 has been dscussed n detal. The man dfference s that the proportonal valve s used nstead of the PCM valve. The flexural dsplacement w(x, t) of the flexble beam at pont P (x, t) s: w(x, t) = n =1 φ (x)q (t); (1) where q (t) and φ (x) are the th generalzed modal coordnate and mode-shape functon, respectvely. The PZT sensor measured vbraton sgnal s: V s (t) = K q b p d 31 E pe ( t n b 2 + t p) [φ (l 1 ) φ (l 2 )] q (t); (2) =1 where K q s the charge amplfer coeffcent; the prmes denote the partal dervatve wth respect to the varable x. The pneumatc cylnder s drvng force F (t) s expressed as: { (P1 (t) P F (t) = 2 (t)) A v c ẏ (t), f y(t) > 0 (P 2 (t) P 1 (t)) A v c ẏ (t), f y(t) < 0 ; (3) where v c s the vscous frctonal coeffcent of the movng par of the pneumatc cylnder. The system dynamcs of the pneumatc drve flexble beam s: (m b + ρ b A b L b + m t )ÿ(t) + v c ẏ(t) [ n ] Lb + ρ b A b φ (x)dx + m t φ (L t ) q (t) = F (t). (4) =1 0 The pezoelectrc drvng equaton of the th mode of vbraton s: [ ] Lb ρ b A b φ (x)dx ÿ(t) 0 + ρ b A b q (t) + v s q (t) + ρ b A b ω 2 q (t) = 1 2 E peb p d 31 (t p + t b ) [ φ (l 4 ) φ (l 3 ) ] V a (t); (5) where V a (t) s the appled voltage; v s s the th modal structural dampng rato. 3. CONTROLLER DESIGN The central objectve of set-pont vbraton suppresson s to mantan the slder at ts specfed poston as the tme scale ends, and cause the resdual vbraton to damp out quckly. The purpose of the resonant vbraton control s to sgnfcantly suppress the resonant vbraton, whch s under frst exctaton usng the PZT actuator, from large ampltude to low ampltude. An adaptve fuzzy sldng mode controller s nvestgated, and a composte PD controller s utlzed for comparson. 3.1. PD Control The measured dsplacement and vbraton sgnals are treated as feedback nformaton. Both pneumatc rodless cylnder and PZT actuator can be used as actuators to suppress the vbraton of the flexble beam. Smlar to the prevous work, 10 three PD controllers are desgned and expressed as: { upn (t)=k p1 [y d y(t)] K d1 ẏ(t) K p2 V s (t) K d2 Vs (t) ; u pe (t)= K p3 V s (t) K d3 Vs (t) (6) where u pn (t) denotes the control voltage appled to the pneumatc proportonal valve; u pe (t) s the control voltage appled to the PZT actuator; K p1 < 0 and K d1 > 0 are PD control gans for postonng control of the slder; K p2 > 0 and K d2 > 0 are PD control gans for controllng the flexble beam s vbraton by employng the rodless cylnder; K p3 > 0 and K d3 > 0 are PD control gans for attenuatng the flexble beam s vbraton by utlzng the PZT actuator; and y d s the slder s desred poston. The control bandwdth of ths pneumatc system s lower than the modal frequency of the second vbraton mode. Therefore, only the frst modal large ampltude vbraton can be effectvely controlled by the pneumatc cylnder. The small ampltude vbraton of the frst mode and the vbraton of the second mode should be controlled only by usng the PZT actuator. 3.2. Adaptve Fuzzy Sldng Mode Control AFSMC s adopted and ncorporated wth the pneumatc drve pezoelectrc flexble beam system. The AFSMC s drectly cted from correspondng adaptve fuzzy sldng-mode 7, 8, 11 13 control references. Consder a nonlnear system s wrtten as x (n) = f (x) + u; (7) where x = ( x, ẋ,, x (n 1)) T R n s the state vector; u R s the nput; f(x) s a nonlnear functon. The control objectve s to make the state x for trackng a desred Internatonal Journal of Acoustcs and Vbraton, Vol. 22, No. 3, 2017 299
where F j ( = 1, 2,, m; j = 1, 2,, N, where N s the number of rules) and B j are fuzzy sets defned on R, and N = m =1 N s the total number of rules. The equvalent controller s: u eq = θ T u ξ(x); (12) (a) Block dagram of the control system of the adaptve fuzzy sldng mode control. (b) Adaptve fuzzy sldng mode control. Fgure 5. Block dagram of the controllers structure. ( ) T state x d = x d, ẋ d,, x (n 1) d. The trackng error s e = x x d = ( e, ė,, e (n 1)) T. Then a sldng surface s defned as: ( s (x, t) = k 1 e + k 2 ė + + k n 1 e (n 2) + e (n 1)) ; (8) where k = (k 1, k 2,, k n 1, 1) are the coeffcents of the Hurwtzan polynomal λ n 1 + k n 1 λ n 2 + k 2 λ + k 1, whch guarantee all the solutons of the equaton λ n 1 + k n 1 λ n 2 + k 2 λ + k 1 = 0 to be n the left half of the plane. A stable condton s s (x, t) ṡ (x, t) η s, where η > 0. The control law becomes: u = u eq u sw ; (9) where u eq s the equvalent controller; u sw s a swtchng-type control term. The swtchng-type control s expressed as: u sw = η sgn(s); (10) where η = u sw, η η > 0. The fuzzy system s characterzed by a set of f-then rules n the followng form: R j : If x 1 s F j 1, and x 2 s F j 2,, and x m s F j m, then u s B j ; (11) where θ u = (θ 1,, θ m ) T s a set of adjustable consequent parameters; ξ(x) = (ξ 1 (x),, ξ m (x)) T s a set of fuzzy bass functons, and ξ (x) s defned as: 13 ξ (x) = n µ F j (x j ) j=1 m n =1 j=1, = 1, 2,, m; (13) µ F j (x j ) where µ F j (x j ) s the membershp functon of the fuzzy controller; n s the number of the nput fuzzy controllers; and m s the number of the fuzzy rules. The swtchng-type control term u sw s approxmately expressed as: u sw = ĥ (s/θ h) = θ T hξ(s). (14) The parameter update law s: θ u = r 1 s ξ(x); (15) θ h = r 2 s ξ(s); (16) where r 1 and r 2 are the postve constants. The block dagram for the control structure of the adaptve fuzzy sldng mode control s shown n Fg. 5(a). The control voltage u pn (t) appled to the proportonal valve s composed of two AFSMC components: AFSMC 1 for postonng the slder and AFSMC 2 for suppressng the large ampltude vbraton of the frst mode. On account of the lmted bandwdth of the pneumatc rodless cylnder, the PZT patch actuator s also utlzed to damp out the resdual vbraton of the frst two modes. The control acton of u pe (t) s appled to the PZT actuator, comprsng one adaptve fuzzy sldng mode control of AFSMC 3. The block dagram of the mentoned three AFSMCs s shown n Fg. 5(b). The parameters used for desgnng the controllers should be provded beforehand, such as the ntal value of θ u (0) and θ h (0). The learnng rates used n the adaptve fuzzy sldng mode control are chosen as: r 1 = 0.4 and r 2 = 0.1 The ntal parameter s chosen as θ (0) = 0. The membershp functons for x are gven as: µ F 1 (x ) = 1/[1+exp(10.0(x +2.5))], µ F 2 (x ) = exp[ 5.0(x + 2.4) 2 ], µ F 3 (x ) = exp[ 5.0(x + 1.8) 2 ], µ F 4 (x ) = exp[ 5.0(x + 1.2) 2 ], µ F 5 (x ) = exp[ 5.0(x + 0.6) 2 ], µ F 6 (x ) = exp[ 5.0 x 2 ], µ F 7 (x ) = exp[ 5.0(x 0.6) 2 ], µ F 8 (x ) = exp[ 5.0(x 1.2) 2 ], µ F 9 (x ) = exp[ 5.0(x 1.8) 2 ], µ F 10(x ) = exp[ 5.0(x 2.4) 2 ], µ F 11(x ) = 1/[1 + exp(10.0(x 2.5))]. Snce the long stroke of the rodless cylnder and large compressblty of ar, the hysteress problem wll emerge. If the control hysteress s not compensated effectvely, the control performance and the stablty of the closed-loop system wll be affected. Therefore, a tme delay compensator s employed to deal wth the problem of hysteress, by applyng phase shftng methods. In addton, the hgh frequency noses wll cause control spll over. To flter out the hgh frequency noses of the measured sgnal, the Butterworth notch flter and low-pass flter are desgned and used n the experments. 300 Internatonal Journal of Acoustcs and Vbraton, Vol. 22, No. 3, 2017
(a) Tme-doman vbraton response. (b) Frequency response. (c) Tme-doman vbraton response. (d) Frequency response. Fgure 6. Free vbraton responses of the frst and the frst two bendng modes. 4. EXPERIMENTAL RESULTS The presented expermental results nclude: modal dentfcaton of the natural frequences, set-pont vbraton suppresson only employng the pneumatc cylnder, set-pont vbraton control usng both pneumatc cylnder and PZT actuator, and resonant vbraton suppresson under persstent exctaton. The samplng nterval s specfed as 5 ms. Two controllers are compared, namely, the composte PD controller and the adaptve fuzzy slde sldng controller. The control acton s appled at the moment of t = 1.0 s. The correspondng control parameters are selected after testng experments. The proportonal valve s stopped when the control value enters the specfed bound for lastng the specfed tme. After several tests, the bound s selected as 0.6 V and the specfed tme s chosen as 50 samplng perods. Ths wll not excte the lmt cycle oscllaton because of the nonlnear propertes of the pneumatc system. Remarks: In the subsequent expermental results, the curves of the control voltages appled to the proportonal valve can be postve and negatve, wth ther ranges of -5 V +5 V. The postve or negatve value can be realzed by swtchng on or swtchng off the 5-2 way swtchng solenod valve. The postve value s the case that the 5-2 way swtchng solenod valve s n the left poston, whle the negatve value s the case n the rght poston. The practcal control voltage appled to the proportonal valve s the absolute values of the sgnals plotted on the expermental curves. 4.1. Expermental Identfcaton When the slder of the pneumatc cylnder stops at some pont, the flexble manpulator s approxmately consdered as a clamped beam wth a concentrated tp mass. The frcton force of the cylnder s prsmatc par can keep the slder statonary wthout pneumatc drve. Exctaton analyses are conducted to dentfy the natural frequences. For expermental dentfcaton, mpulse exctaton was appled. To obtan the frst bendng mode of vbraton, the tp of the beam was pushed to a dstance, and then the free vbraton of the frst mode was caused. To obtan the vbraton of the frst two modes, the flexble beam was ht at some certan ponts by usng a hammer. The measured free vbraton of the frst and the frst two bendng modes by the PZT sensor are shown n Fg. 6. Fgures 6(a) and 6(b) plot the tme-doman response and the frequency response obtaned by employng fast Fourer transform (FFT), respectvely. Fgures 6(c) and 6(d) are the tme-doman response and the frequency response obtaned by employng FFT, respectvely. From Fg. 6(a) and Fg. 6(c), t can be ob- Internatonal Journal of Acoustcs and Vbraton, Vol. 22, No. 3, 2017 301
(a) Tme-doman vbraton response under PD control. (b) Control voltage appled to the proportonal valve. (c) Tme-doman vbraton response under AFSMC. (d) Control voltage appled to the proportonal valve. Fgure 7. Set-pont vbraton suppresson usng the pneumatc cylnder under PD control and AFSMC. served that the vbratons wll take a long tme to decay wthout ntroducng actve control. Next, from Fg. 6(b) and Fg. 6(d), t can be observed that the dentfed natural frequences of the frst two modes are ω 1 =2.08 Hz and ω 2 =13.16 Hz. 4.2. Set-pont Vbraton Control Usng Only the Rodless Cylnder Here, the pneumatc cylnder s employed as the actuator to suppress the vbraton. When the vbraton of the frst mode of the flexble beam s excted, expermental results for vbraton control are shown n Fg. 7, utlzng the PD controller and the AFSMC. The control parameters of the PD control are K p1 = 0.3, K d1 = 0.008, K p2 = 0.25, and K d2 = 0.05. Fgures 7(a) and 7(c) depct the controlled tme-doman vbraton response, respectvely. Fgures 7(b) and 7(d) plot the control voltage appled to the proportonal valve, respectvely. From Fg. 7(a) and Fg. 7(c), t can be observed that the larger ampltude vbraton of the frst mode can be suppressed effectvely. However, the vbraton of the second bendng mode gets unavodably excted n the process of applyng control acton. Ths s the phenomenon where the control splls over nto the hgh order mode of vbraton when controllng the vbraton of the frst mode. Comparng Fg. 7(c) wth Fg. 7(a), t can be observed that AFSMC s better than PD control, although the vbraton of the flexble beam be cannot be elmnated completely. Ths results nto nonlnear frcton of the par, ntrnsc nonlnearty, low stffness, parametrc varatons, hysteress characterstcs of the long-stroke pneumatc cylnder, and the hysteress of the pneumatc servo system caused by the solenod valves. Because of the bandwdth lmtaton of the pneumatc control system, the vbraton of the second mode cannot be suppressed effectvely. Therefore, the PZT actuator should be ntroduced as the actuator to suppress the vbraton of the second mode smultaneously. From Fg. 7(d) and Fg. 7(b), t can be observed that the control value under PD control reaches saturaton abruptly after applyng control effect, whch s undesrable. It s harmful to actuator devces, and t may cause oscllaton of the hgh frequency mode, whle the control voltage of AFSMC ncreases gradually to saturaton. Ths can weaken the control effect at the begnnng, n turn avodng exctaton of the large ampltude oscllatons. Comparson of Fg. 7(a) wth Fg. 7(c) shows that the ampltude of the oscllaton by AFSMC s less than that of the PD controller. 302 Internatonal Journal of Acoustcs and Vbraton, Vol. 22, No. 3, 2017
(a) Tme-doman vbraton response. (b) Control voltage appled to the proportonal valve. (c) Dsplacement curve of the slder. (d) Control voltage appled to the PZT actuator. Fgure 8. Set-pont vbraton suppresson usng both pneumatc cylnder and PZT actuator under PD control. 4.3. Set-pont Vbraton Control Usng Both Pneumatc Cylnder and PZT Actuator The excted vbraton s desred to be attenuated as quckly as possble. Especally, the small ampltude resdual vbraton should be damped out thoroughly. Therefore, the hybrd drvng scheme s used,.e., both pneumatc cylnder and PZT actuator are employed to suppress the set-pont vbraton. The expermental results of the set-pont vbraton control usng PD control are shown n Fg. 8. The control parameters of the PD control are K p1 = 0.3, K d1 = 0.008, K p2 = 0.25, K d2 = 0.05, K p3 = 2.5, and K d3 = 0.02. The control parameters of the AFSMC method are γ 1 = 0.002 and γ 2 = 0.0005. Fgures 8(a), 8(b), 8(c), and 8(d) correspond to the tmedoman vbraton, the control voltage appled to the pneumatc proportonal valve, the dsplacement curve of the slder, and the PZT actuators control voltage, respectvely. Fgure 9 s the result of the set-pont vbraton control usng AFSMC. Fgures 9(a), 9(b), 9(c), and 9(d) show the tme-doman controlled vbraton response curve, the correspondng control voltage appled to the proportonal valve, the dsplacement curve of the slder, and the PZT actuator s control voltage, respectvely. From Fg. 9(a) and Fg. 8(a), t can be observed that the resdual vbraton dsappears as the tme scale ends. From the expermental results, t can be observed that the desgned AFSMC has several advantages over the PD controller: (1) the vbraton s suppressed largely. (2) Snce the control voltage appled to the proportonal valve does not reach ts saturaton nstantly, the control energy s saved and the stablty of the closed-loop system can be guaranteed; furthermore, the hgh frequency chatterng wll not be excted greatly. (3) The resdual vbraton can be effectvely damped out by the PZT actuator, and the performance of the AFSMC s better than the PD controller. The composte strategy can suppress both large and small ampltude vbratons effectvely. 4.4. Vbraton Control Under Persstent Resonant Exctaton The sgnal generator SPF05 generates a sne sgnal wth ts frequency of the frst bendng mode of vbraton,.e., 2.08 Hz. Its ampltude s tuned to an approprate level. Then, the generated sgnal s used to excte the flexble beams resonant vbraton by the PZT actuator after amplfyng wth the hgh voltage amplfer. The control acton by the pneumatc rodless cylnder s appled at the moment of t =1.0 s. The results of tme-doman responses usng the PD controller under the frst resonant exctaton are shown n Fg. 10. Fgure 10(a) plots the tme-doman vbraton response before and after ntroducng control. Fgure 10(b) depcts the control Internatonal Journal of Acoustcs and Vbraton, Vol. 22, No. 3, 2017 303
(a) Tme-doman vbraton response. (b) Control voltage appled to the proportonal valve. (c) Dsplacement of the slder. (d) PZT actuator s control voltage. Fgure 9. Set-pont vbraton suppresson usng both pneumatc cylnder and PZT actuator under AFSMC. voltage appled to the proportonal valve. Fgure 10(c) s the dsplacement of the slder of the cylnder. Fgure 10(d) shows the power spectrum magntude of the vbraton response shown n Fg. 10(a) that s appled control from t = 1.0 s to the moment of t = 25.0 s. The PSD ampltudes of the frst two modes are 31.2 db and 17.6 db. Fgure 11 shows the expermental results on the resonant vbraton suppresson by usng AFSMC. Fgures 11(a), 11(b), and 11(c) depct the tme-doman vbraton response before and after applyng control, the control voltage appled to the pneumatc proportonal valve, and the slder s dsplacement, respectvely. Fgure 11(d) s the power spectrum magntude of the vbraton curve of Fg. 11(a) for control appled from t =1.0 s to the moment of t =25.0 s. The vbraton PSD ampltudes of the frst two modes are 24.3 db and 19.8 db, respectvely. From Fg. 10(a) and Fg. 11(a), t s observed that the vbraton ampltude of the frst mode s reduced sgnfcantly after ntroducng control by the cylnder. However, those of the second mode are excted. From Fg. 10(d) and Fg. 11(d), t s observed that the PSD magntude of the frst mode of vbraton s decreased by usng AFSMC as compared wth the PD controller, whle that of the second mode s ncreased a lttle. Ths s because the AFSMC s a knd of nonlnear control algorthm, whch wll excte the vbraton of the second mode. Further, the goal of suppressng the vbraton of the frst mode can be acheved. From Fg. 10(c) and Fg. 11(c), t can be observed that the slder s oscllatons around the equlbrum pont wll be caused, as the boundary control effect suppresses the resonant vbraton under the frst persstent exctaton, correspondng to the control voltage appled to the proportonal valve shown n Fg. 10(b) and Fg. 11(b). Comparng Fg. 11(d) wth Fg. 10(d), t can be observed that the control value of the AFSMC does not reach saturaton mmedately when the control effect s appled to the large ampltude vbraton; however, that of the PD controller reaches saturaton nstantly for the large ampltude vbraton suppresson acton. Ths s just what we expect not to affect the stablty of the closed-loop control system. The second frequency mode s hgher when compared to PD controller. Ths s the reason why AFSMC s a knd of nonlnear control method. It wll excte the hgh frequency mode vbraton a lttle larger. From the experments, t s observed that the dynamc characterstc of the pneumatc drve flexble beam system s complcated. The desgned controllers can obtan satsfactory control effect, for resdual vbraton suppresson and resonant vbraton control. The control performance of the desgned AF- SMC s better than the PD controller. 304 Internatonal Journal of Acoustcs and Vbraton, Vol. 22, No. 3, 2017
(a) Tme-doman vbraton response. (b) Control voltage appled to the proportonal valve. (c) Dsplacement of the slder. (d) Power spectrum magntude of the vbraton response. Fgure 10. Tme-doman vbraton responses usng the PD controller under the frst resonant exctaton. 5. CONCLUSION A pneumatc rodless cylnder that drves a translatng flexble pezoelectrc beam system s constructed. A pneumatc proportonal valve s used. The classcal PD control and the adaptve fuzzy sldng mode controller are desgned and appled to control the proposed system. Experments are presented ncludng modal frequency dentfcaton, resdual vbraton suppresson and resonant vbraton control under exctaton of the frst mode. Comparsons of the two controllers are dscussed. The expermental results demonstrate that the proposed pneumatc drve flexble beam system can effectvely attenuate the vbraton, by meterng the exhaust ar flow wth the proportonal valve. Moreover, the desgned AFSMC enhances the performance. 6. ACKNOWLEDGEMENTS Ths work was n par supported by the Natonal Natural Scence Foundaton of Chna under Grant 51175181, n part supported by the Fundamental Research Funds for the Central Unverstes, SCUT (2014ZG0019), partally supported by The Natural Scence Foundaton of Guangdong Provnce (S2013030013355) and partally supported by the State Key Laboratory of Robotcs Foundaton. The frst author gratefully acknowledges those agences. REFERENCES 1 Hodgson S., Le M. Q., Tavakol M., Pham M. T. Improved trackng and swtchng performance of an electro-pneumatc postonng system. Mechatroncs, 22 (1), 1 12, (2012). https://dx.do.org/10.1016/j.mechatroncs.2011.10.007 2 Schulte H. and Hahn H. Fuzzy state feedback gan schedulng control of servo-pneumatc actuators. Control Engneerng Practce 12 (5), 639 650, (2004). https://dx.do.org/10.1016/s0967-0661(03)00148-5 3 Katwandvla S. and Olranthchachat P. Robust loop shapng fuzzy gan schedulng control of a servopneumatc system usng partcle swarm optmzaton approach. Mechatroncs 21 (1), 11 21, (2011). https://dx.do.org/10.1016/j.mechatroncs.2010.07.010 4 Chen Y., Zhang J. F., Yang C. J., and Nu B. Desgn and hybrd control of the pneumatc forcefeedback systems for Arm-Exoskeleton by usng on/off valve. Mechatroncs 17 (6), 325 335, (2007). https://dx.do.org/10.1016/j.mechatroncs.2007.04.001 Internatonal Journal of Acoustcs and Vbraton, Vol. 22, No. 3, 2017 305
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