Active Damping Control using Optimal Integral Force Feedback

Transcription

1 214 America Cotrol Coferece (ACC) Jue Portlad Orego USA Active Dampig Cotrol usig Optimal Itegral Force Feedback Yik R. Teo Adrew J. Flemig Abstract This article shows a improvemet to Itegral Force Feedback (IFF) for active dampig cotrol of precisio mechaisms. The beefits of IFF iclude robustess guarateed stability ad simplicity. However the dampig performace depeds o the stiffess of the system; hece some systems caot be adequately cotrolled. I this article a extesio to the classical itegral force feedback cotrol scheme is proposed. The ew method achieves arbitrary dampig for ay mechaical system by itroducig a feed-through term i the system. I. INTRODUCTION The speed ad resolutio of may scietific ad idustrial machies are limited by the presece of lightly damped mechaical resoaces. Examples iclude scaig probe microscopy [1] [5] aofabricatio [6] precisio optics [7] ad aerospace systems [8]. Traditioal passive dampig methods have icludes viscoelastic dampig ad tued-mass absorbers; however these methods ca be bulky ad may ot perform well at low frequecies. I cotrast active dampig is a alterative method to icrease the performace. Active dampig cotrol utilizes a sesor ad feedback loop to artificially icrease the dampig ratio of a system. A umber of successful dampig cotrol techiques iclude Positive Positio Feedback (PPF) [9] polyomial based cotrol [1] shut cotrol [11] [13] resoat cotrol [14] Force Feedback [15] [18] ad Itegral Resoace Cotrol (IRC) [19] [21]. Amog these techiques PPF cotrollers velocity feedback cotrollers force feedback cotrollers ad IRC cotrollers have bee show to guaratee stability whe the plat is strictly egative imagiary [22]. Optimal cotrollers with automatic sythesis have also bee successfully applied to dampig cotrol applicatios for example robust H cotrollers [23] [24]. I refereces [15] [18] itegral force feedback (IFF) is described for vibratio cotrol ad positioig applicatios. This techique utilizes a force sesor ad itegral cotroller to directly augmet the dampig of a mechaical system. The major advatages of IFF are the simplicity of the cotroller guarateed stability ad excellet performace robustess. However the maximum dampig achievable with IFF is a fuctio of the system properties i particular the system stiffess relative to the actuator stiffess. Hece some systems ca be critically damped usig IFF while other exhibit egligible dampig improvemet. Yik R. Teo ad Adrew J. Flemig are with Precisio Mechatroics Lab at the School of Electrical Egieerig ad Computer Sciece The Uiversity of Newcastle 238 Callagha NSW Australia. (yik.teo adrew.flemig)@ewcastle.edu.au Fig. 2. Fig. 1. Structure G with a piezoelectric trasducer Block Diagram of the Classical Itegral Force Feedback (CIFF). I this work a extesio to IFF is described for a arbitrary dampig ratio to be achieved for ay mechaical system. The modificatio amouts to replacig the itegral cotroller with a first-order equivalet low-pass filter. Although the additioal complexity is egligible the dampig performace is sigificatly improved. This is a exceptioal result that allows itegral force feedback to be exteded to systems that were ot previously suited. II. CLASSICAL INTEGRAL FORCE FEEDBACK (CIFF) Itegral Force Feedback (IFF) cotrol has bee widely applied for augmetig the dampig of flexible structures [16]. The feedback law is simple to implemet ad uder commo circumstaces provides excellet dampig performace with guarateed stability. Fig. 1 illustrates a structure G equipped with a piezoelectric actuator that produces a force F a with iteral stiffess K a. A force sesor is collocated with the piezoelectric actuator ad measures the axial force F s actig o the system G. The variable d represets the mechaical displacemet. The classical itegral force feedback cotroller (CIFF) has a block diagram represetatio illustrated i Fig. 2. The trasfer fuctio betwee the ucostraied piezo expasio δ to the sesor force F s is [25] { } G Fs δ (s) = F s δ = K a 1 v i 1 + s 2 /ω 2 i (1) /$ AACC 1637

2 (a) Classical method. Fig. 3. (b) Optimal method. Typical root locus plots. the system as show i Fig. 4(a). The ew system ca be coverted ito the classical form by equatig the systems i Fig. 4(a) ad Fig. 4(b) that is } { } v i v i K a {1 2 + β = K a 1 2 K a v i K a v i K a + β 2 = K a 2. (6) From (6) we obtai the ew expressios ad K a = K a + β (7) (a) Block Diagram of the OIFF with ew feed-through term β. v i = K av i K a = K av i K a + β. (8) The trasfer fuctio from δ to F s is Ĝ Fs δ (s) = K a {1 v i 1 + s 2 /ω 2 i }. (9) (b) Block Diagram of the OIFF i the classical form with K a ad v i. Fig. 4. Block Diagrams of Optimal Itegral Force Feedback (OIFF). where ω i is the atural frequecy of the system ad v i is the fractio of modal strai eergy for the i th mode. The modal zeros are [16] The itegral force feedback cotroller is z 2 i = ω 2 i (1 v i ). (2) C d1 (s) = K d1 K a s (3) where K d is the gai of cotroller. The maximum modal dampig is ad is achieved for ζ max i = ω i z i 2ω i (4) K d1 = ω i ωi z i. (5) The root locus plot correspodig to CIFF is show i Fig. 3(a). A key limitatio of the classical method is that the maximum modal dampig (4) depeds o the distace betwee the system poles ω i ad modal zeros z i. If the distace betwee the pole ad zero is small the maximum dampig achievable with CIFF is ot sigificat. III. OPTIMAL INTEGRAL FORCE FEEDBACK (OIFF) I this work we exted the classical itegral force feedback methodology by itroducig a feed-through term β i The modal zeros are ow a fuctio of β ( ẑ i (β) = ωi 2 1 K ) a K a + β v i. (1) This results i a extra degree of freedom that allows the positio of the zeros to be modified. As β decreases the zeros move closer to the real axis uder the coditio that K a (v i 1) < β < is satisfied. The itegral force feedback cotroller is C d2 (s) = K d2 K a s (11) The root locus plot for a typical OIFF system is show i Fig. 3(b). Notice that the locatio of the zero chages with respect to β. The ew maximum modal dampig is ζ max i = ω i ẑ i (β) (12) 2ẑ i (β) which correspods to the ew optimal gai K d2 = ω i ωi ẑ i (β). (13) O the other had give a desired modal dampig ζ d the value of β required is β = K a + K av i (2ζ d + 1) 2 4ζ d (1 + ζ d ). (14) IV. CASE STUDY I this sectio the performace of CIFF ad OIFF is evaluated o a simple mechaical system. 1638

3 Magitude (db) 5 Fig. 5. Mechaical diagram of a secod-order mechaical system where F s is the measured force actig betwee the actuator ad the mass i the vertical directio. A. Mechaical Dyamics ad System Properties Fig. 5 shows a secod-order mechaical system with mass M p = 25 g flexure stiffess k f = 3 N/µm actuator stiffess K a = 1 N/µm ad flexure dampig c f = 1 N/ms 1. The equatio of motio for this system is M p d + c f d + (K a + k f )d = F a (15) where M p is the mass of the platform ad the stiffess ad dampig coefficiet of the flexures are deoted by k f ad c f respectively. The force of the actuator is F a ad the stiffess is K a. A force sesor is collocated with the actuator ad measures the load force F s. The cofiguratio of the system is such that the actuator ad flexure appear mechaically i parallel hece the stiffess coefficiets ca be grouped together k = K a + k f. This simplifies the equatio of motio (15) to M p d + c f d + k = F a. (16) The trasfer fuctio from actuator force F a to the displacemet of the mass d is G dfa (s) = d F a = The sesor force F s ca be writte as F s = F a dk a 1 M p s 2 + c f s + k. (17) = F a K a F a G dfa (s) = F a (1 K a G dfa (s)). (18) The trasfer fuctio betwee the applied force F a ad measured force F s is foud by rearragig (18). G Fs F a (s) = F s F a The force developed by the actuator F a is = 1 K a G dfa (s). (19) F a = K a δ (2) recall that δ is the ucostraied piezo expasio. Substitutig (2) ito (19) we obtai the trasfer fuctio from the Frequecy (Hz) 1 5 Fig. 6. Case Study: Frequecy Respose of G FsF a (s). ucostraied piezo expasio δ to the force of the sesor F s G Fs δ = F s δ = K a F s F a = K a (1 K a G dfa (s)). (21) A valid assumptio is that the effect of the dampig i the flexure c f is small ad thus egligible. The imagiary parts of the ope-loop poles ad zeros are k ω 1 = M p K a + k f = M p z 1 = k f M p. (22) For this system the ope-loop poles ad zeros of the system are ω 1 = 6.37 khz ad z 1 = 5.5 khz. B. Dampig Cotrol Desig 1) Classical Itegral Force Feedback (CIFF): The opeloop frequecy respose of G Fs F a (s) is show i Fig. 6. Oe key observatio is that its phase respose of the system lies betwee ad 18. This is a geeral property of flexible structures with iputs ad outputs proportioal to the applied ad measured forces. Recall that the classical itegral force feedback cotroller is C d1 (s) = K d1 K a s. (23) The itegral cotroller has a costat phase lag of 9 so the loop-gai of the system lies betwee -9 ad 9. Hece the closed-loop system has a ifiite gai margi ad phase margi of 9. The solutio for the optimal feedback gai 1639

4 Imagiary Axis (secods 1 ) 5 x Real Axis (secods 1 ) x 1 4 Fig. 7. Case Study: Root locus of the system usig CIFF ad OIFF at differet values of ζ max. Fig. 8. Detail Block Diagram of the CIFF system for aalysis K d has already bee derived i [16] ad further geeralised for aopositioig systems i [17]. The optimal feedback gai K d ad correspodig maximum closed-loop dampig ratio ζ max 1 are ad K d1 = ω 1 ω1 z 1 = (24) ζ max 1 = ω 1 z 1 2z 1 =.77. (25) The umerical root-locus plot i Fig. 7 validate these values. The umerically optimal gai is ad the correspodig dampig ratio is.77. This correlates closely with the predicted values which supports the accuracy of the assumptios made i derivig the optimal gai. Fig. 8 shows a detailed block diagram of the system with sesor force F s ad platform displacemet d as outputs. A disturbace w is also cosidered. The trasfer fuctio from the disturbace w to the sesor force F s is G Fs w(s) = F s w G Fs δ =. (26) 1 +C d G Fs δ The simulated ope-loop ad closed-loop frequecy resposes of (26) are plotted i Fig. 9. Magitude (db) Ope Loop CIFF OIFF Frequecy (Hz) Fig. 9. Case Study: Frequecy respose from the iput disturbace w to the sesor force F s. Magitude (db) Ope Loop CIFF OIFF Frequecy (Hz) Fig. 1. Case Study: Frequecy respose from the iput disturbace w to the displacemet of the platform d. The trasfer fuctio from the disturbace w to the displacemet of the platform d is G dw (s) = d w = K ag dfa. (27) 1 +C d G Fs δ The simulated ope-loop ad closed-loop frequecy resposes of (27) are plotted i Fig. 1. 2) Optimal Itegral Force Feedback (OIFF): The CIFF method ca be exteded to iclude the feed-through term β as illustrated i Fig. 4(a). The equivalet cotroller Ĉ d (s) is foud by equatig the systems i Fig. 4(b) ad Fig. 11 Ĉ d (s) = C d2(s) 1 +C d2 (s)β. (28) For the case study the relatioship betwee β ad ζ described i (14) is plotted i Fig. 12. The maximum 164

5 Fig x Block Diagram of the example system usig OIFF. TABLE I COMPARISON BETWEEN ANALYTIC AND NUMERICALLY OBTAINED DAMPING RATIO ζ max AND FEEDBACK GAIN K d2. Aalytic Numerical β ζ max K d2 ζ max K d Fig. 12. Case Study: The relatioship betwee β ad ζ max for OIFF. Magitude (db) Fig Detail Block Diagram of the OIFF system for aalysis modal dampig with CIFF is.77; however with OIFF the maximum modal dampig ca be varied from.77 to 1 at differet values of β. The root locus of the system is show i Fig. 7. The optimal feedback gai maximum dampig ratio ad correspodig value of β is give i Table I. These values ca be validated by the umerical root-locus plot i Fig. 7 ad is summarize i Table I. These values correlates closely with the predicted values which supports the accuracy of the assumptios made i derivig the optimal gai. Fig. 13 shows a detailed block diagram of the system with sesor force F s ad platform displacemet d as outputs. A disturbace w is also cosidered. The trasfer fuctio from the disturbaces w to the sesor force F s is G Fs w(s) = F s w G Fs δ =. (29) 1 +Ĉ d G Fs δ The simulated ope-loop ad closed-loop frequecy resposes of (29) is show i Fig. 9. The closed-loop trasfer fuctio measured from the referece r to the sesor force F s is G Fs r(s) = F s r = ĈdG Fs δ. (3) 1 +Ĉ d G Fs δ Frequecy (Hz) Fig. 14. Case Study: Frequecy respose from the iput disturbace w to the cotrol actio u as β is decreased. whe s = G Fs r() = ĈdG Fs δ () 1 +Ĉ d G Fs δ () = G F s δ () G Fs δ () + β. (31) This shows that the DC gai of the closed-loop icreases as β decreases such that K a (v i 1) < β < holds. Recall that the maximum dampig ratio of the closed-loop system icreases as β is decrease. The trasfer fuctio from the disturbace w to the displacemet of the platform d is G dw (s) = d w = K ag dfa. (32) 1 +Ĉ d G Fs δ The simulated ope-loop ad closed-loop frequecy resposes of (32) are plotted i Fig. 14. The frequecy respose 1641

6 of this trasfer fuctio is show i Fig. 14. The sesitivity of the cotrol actio toward iput disturbace icreases as the desired dampig ratio is icreased. V. CONCLUSION This article describes a extesio to itegral force feedback cotrol that allows arbitrary mechaical dampig to be achieved for ay mechaical system. A additioal feedthrough term is added to the system to provide a extra degree of freedom that ca be used to arbitrarily maipulate the modal zeros ad maximum dampig. Simulatio results o a simple mechaical system demostrate a icrease i the maximum achievable dampig from.77 to 1 usig itegral force feedback. This result will allow high-performace mechaical systems to be critically damped with a first-order cotrol law. Future work will iclude experimetal applicatio extesio to systems with multiple actuators ad modelig usig a egative imagiary framework. REFERENCES [1] N. Jalili ad K. Laxmiarayaa A review of atomic force microscopy imagig systems: applicatio to molecular metrology ad biological scieces Mechatroics vol. 14 o. 8 pp October 24. [2] A. J. Flemig B. J. Keto ad K. K. Leag Bridgig the gap betwee covetioal ad video-speed scaig probe microscopes Ultramicroscopy vol. 11 o. 9 pp August 21. [3] K. K. Leag ad A. J. Flemig High-speed serial-kiematic AFM scaer: desig ad drive cosideratios Asia Joural of Cotrol vol. 11 o. 2 pp March 29. [4] A. J. Flemig ad Y. K. Yog Piezoelectric actuators with itegrated high voltage power electroics IEEE/ASME Trasactios o Mechatroics vol. (I Press) 214. [5] A. J. Flemig ad K. K. Leag Desig Modelig ad Cotrol of Naopositioig Systems. Lodo UK: Spriger 214. [6] A. Ferreira ad C. Mavroidis Virtual reality ad haptics for aorobotics IEEE Robotics ad Automatio Magazie vol. 13 o. 3 pp September 26. [7] L. Vaillo ad C. Philippe Passive ad active microvibratio cotrol for very high poitig accuracy space systems Smart Materials ad Structures vol. 8 o. 6 December [8] V. Giurgiutiu Review of smart-materials actuatio solutios for aeroelastic ad vibratio cotrol Joural of Itelliget Material Systems ad Structures vol. 11 pp July 2. [9] J. L. Faso ad T. K. Caughey Positive positio feedback cotrol for large space structures AIAA Joural vol. 28 o. 4 pp [1] S. S. Aphale B. Bhikkaji ad S. O. R. Moheimai Miimizig scaig errors i piezoelectric stack-actuated aopositioig platforms IEEE Trasactios o Naotechology vol. 7 o. 1 pp Jauary 28. [11] A. J. Flemig ad S. O. R. Moheimai Sesorless vibratio suppressio ad sca compesatio for piezoelectric tube aopositioers IEEE Trasactios o Cotrol Systems Techology vol. 14 o. 1 pp Jauary 26. [12] A. J. Flemig S. Behres ad S. O. R. Moheimai Optimizatio ad implemetatio of multi-mode piezoelectric shut dampig systems IEEE/ASME Trasactios o Mechatroics vol. 7 o. 1 pp March 22. [13] A. A. Eielse ad A. J. Flemig Passive shut dampig of a piezoelectric stack aopositioer i Proc. America Cotrol Coferece Baltimore MD Jue 21 pp [14] A. Sebastia A. Patazi S. O. R. Moheimai H. Pozidis ad E. Eleftheriou A self servo writig scheme for a MEMS storage device with sub-aometer precisio i Proc. IFAC World Cogress Seoul Korea July 28 pp [15] A. Preumot J. P. Dufour ad C. Malekia Active dampig by a local force feedback with piezoelectric actuators. AIAA Joural of Guidace ad Cotrol vol. 15 o [16] A. Preumot Mechatroics Dyamics of electromechaical ad piezoelectric systems. Dordrecht The Netherlads: Spriger 26. [17] A. J. Flemig Naopositioig system with force feedback for highperformace trackig ad vibratio cotrol IEEE Trasactios o Mechatroics vol. 15 o. 3 pp Jue 21. [18] A. J. Flemig ad K. K. Leag Itegrated strai ad force feedback for high performace cotrol of piezoelectric actuators Sesors ad Actuators A vol. 161 o. 1-2 pp Jue 21. [19] S. S. Aphale A. J. Flemig ad S. O. R. Moheimai Itegral resoat cotrol of collocated smart structures IOP Smart materials ad Structures vol. 16 pp April 27. [2] B. Bhikkaji ad S. O. R. Moheimai Itegral resoat cotrol of a piezoelectric tube actuator for fast aoscale positioig IEEE/ASME Trasactios o Mechatroics vol. 13 o. 5 pp October 28. [21] A. J. Flemig S. S. Aphale ad S. O. R. Moheimai A ew method for robust dampig ad trackig cotrol of scaig probe microscope positioig stages IEEE Trasactios o Naotechology vol. 9 o. 4 pp September 21. [22] I. R. Peterse ad A. Lazo Feedback cotrol of egative-imagiary systems Cotrol Systems IEEE vol. 3 o. 5 pp oct. 21. [23] S. Salapaka A. Sebastia J. P. Clevelad ad M. V. Salapaka High badwidth ao-positioer: A robust cotrol approach Review of Scietific Istrumets vol. 75 o. 9 pp September 22. [24] A. Sebastia ad S. M. Salapaka Desig methodologies for robust ao-positioig IEEE Trasactios o Cotrol Systems Techology vol. 13 o. 6 pp November 25. [25] A. Preumot B. de Mareffe A. Deraemaeker ad F. Bosses The dampig of a truss structure with a piezoelectric trasducer Computers ad Structures vol