A General Approach for Analysis of Actuator Delay Compensation Methods for Real-Time Testing

Transcription

1 The th World Conferene on Earthquake Engineering Otober -7, 8, Beijing, China A General Aroah for Analysis of Atuator Delay Comensation Methods for Real-Time Testing Cheng Chen and James M. Riles Post-dotoral Researh Assoiate, ATLSS Researh Center, Lehigh University, Bethlehem, PA, USA Brue G. Johnston Professor, ATLSS Researh Center, Lehigh University, Bethlehem, PA, USA h@lehigh.edu, jmr5@lehigh.edu ABSTRACT : Real-time testing is a useful tehnique to evaluate the erformane of strutural systems subjeted to seismi loading. Servo-hydrauli atuators an introdue an inevitable delay when alying ommand dislaements to a struture during a real-time test due to their inherent dynamis. Various omensation methods have been develoed and alied in real-time testing to minimize the effet of atuator delay. This aer rooses a general aroah to analyze atuator delay omensation methods using an equivalent disrete transfer funtion. Disrete ontrol theory is introdued and used to formulate the disrete transfer funtion in the frequeny domain and the differene equation in the time domain for an atuator delay omensation method. The disrete transfer funtion is used to ondut frequeny resonse analysis to rovide insight into the exeted erformane of the omensation method by grahially illustrating the harateristis of the omensation method as a funtion of frequeny of ommand dislaement and the arameters for the omensator. The differene equation gives the form of the mathematial extraolation in the time domain for the omensation method. Three oular omensation methods are seleted to illustrate the roosed aroah. Exeriments with redefined sine swee dislaement are onduted to exerimentally validate the analysis of the seleted omensation methods using the roosed aroah. KEYWORDS: real-time testing, atuator delay, omensation, disrete transfer funtion, frequeny resonse, differene equation, hase, magnitude. Introdution Strutural exeriments are imortant for earthquake engineering researh when the strutural system or its omonents are diffiult to model. Various methods have been develoed for strutural exeriments, these inlude the dislaement-based seudodynami and hybrid testing methods whih rovide viable tehniques to evaluate the erformane of strutural system subjet to dynami loading (Mahin et al. 985, Shing et al. 996). Reently, several devies that are sensitive to loading-rate have been develoed for seismi hazard mitigation, inluding visous, visoelasti and elastomeri damers (Soong and Sener ). An evaluation of the erformane of strutures with these rate-deendent devies requires the strutural exeriments be erformed in real-time. Real-time seudodynami and hybrid testing therefore resent themselves as eonomi and viable tehniques to shake table testing and have been investigated by numerous researhers (Jung and Shing 6, Chen et al. 8). In real-time testing, the equations of motion are numerially solved by an integrated algorithm. Servo hydrauli atuators are often utilized in real-time tests to aly the ommand dislaement(s) to the exerimental struture(s). Due to the inherent atuator servo-hydrauli dynamis, the atuator has an inevitable delay in resonse to the ommand dislaement. This delay is usually referred to as atuator delay. The effet of atuator delay on real-time testing has attrated the attention of many researhers (Horiuhi et al. 999, Wallae el. 5, Chen and Riles 7). Their studies show that atuator delay is equivalent to a negative daming that an destabilize a real-time test if not omensated roerly. Various omensation methods have been roosed to minimize the effet of atuator delay for real-time testing. Horiuhi et al. (999, ) roosed two omensation shemes based on olynomial extraolation and a linear aeleration assumtion, resetively. Other omensation methods originating from ontrol engineering ratie have also been investigated suh as

2 The th World Conferene on Earthquake Engineering Otober -7, 8, Beijing, China derivative feedforward (Jung and Shing 6). Chen (7) roosed a simlified disrete transfer funtion model for the servo hydrauli atuator and alied the inverse of the model for atuator delay omensation for real-time testing. The analysis of omensation methods is neessary in order to gain an understanding and insight into the behavior of the methods. This will enable an aroriate atuator delay omensation method to be seleted and the values for the arameters of the omensation method to be assigned in order to ahieve a suessful real-time test. A general aroah is roosed in this aer to analyze different atuator delay omensation methods. Disrete ontrol theory is introdued to formulate the equivalent disrete transfer funtion for a seleted omensation method. Frequeny resonse analysis is onduted using the equivalent disrete transfer funtion to rovide insight into the erformane of the omensation method by grahially illustrating the harateristis of the omensation method as a funtion of frequeny of the ommand dislaement and the arameter values for the omensator. It is also shown that the differene equation orresonding to the equivalent disrete transfer funtion for the omensation method gives the exression for extraolation in the time domain. Three oular atuator delay omensation methods are seleted to illustrate the roosed aroah to demonstrate the effetiveness of the roosed aroah.. DISCRETE TRANSFER FUNCTION AND FREQUENCY RESPONSE The disrete transfer funtion of a disrete system is defined as the ratio of the disrete z-transforms of the inut and outut (Ogata 995). For a linear disrete-time system with inut F and outut x, the disrete transfer funtion G(z) an be written in the following general form: G X F n nn z L n z n m dm z L d z d (.) In Eqn.., n n, L n and d m, L d are oeffiients of the linear differene equation; z is the omlex variable in the disrete z-domain; and X(z) and F(z) are the disrete z-transforms of system inut x and system outut F, resetively; and m and n are integer numbers. The frequeny resonse of a dynami system is usually reresented by the magnitude and hase roerties, where the magnitude is defined as the amlitude hange of the outut omared with the inut and the hase is the hase shift between the two signals. For a disrete transfer funtion G(z) in Eqn.., the frequeny resonse jωt funtion an be obtained by substituting z e, whereby the magnitude M and hase θ an be written as M jωt M ( ω ) G( e ) (.a) j ω T θ θ ( ω) G( e ) (.b) In Eqns..a and.b, j is the imaginary unit and defined as ; ω is the irular frequeny of the inut signal; and T is the samling time of G(z). M and θ an be readily alulated using Matlab (8). In real-time testing when the atuator is subjet to a sinusoidal ommand dislaement ( t) Asin( ωt), where A and ω are the amlitude and irular frequeny of the harmoni inut, resetively, the redited dislaement outut x (t) from a omensator method an be written as x ( t) A sin( ω t φ ), where A and φ are the amlitude and hase shift of x (t). The atuator measured dislaement (t) an be written as x m ( t) A sin( ω t φ ), where A and φ are the amlitude and hase shift of x m (t). By definition, A / A and φ are frequeny resonse roerties of the omensation method, while A / A and x x m

3 The th World Conferene on Earthquake Engineering Otober -7, 8, Beijing, China φ φ are frequeny resonse roerties of the atuator dynamis. To ahieve reise atuator ontrol in real-time testing, the measured dislaement should be the same as the ommand dislaement, whih leads to A A A (.a) A A A φ φ ( φ φ ) (.b). EQUIVALENT DISCRETE TRANSFER FUNCTIONS FOR SELECTED COMPENSATION METHODS Three different omensation methods are seleted to illustrate the roosed aroah, inluding the linear aeleration extraolation method (Horiuhi et al. ), the derivative feedforward omensation method (Jung and Shing 7) and the inverse omensation method (Chen 7).. Linear Aeleration Extraolation Horiuhi et al. () roosed an atuator delay omensation method based on a linear aeleration assumtion. Assuming that strutural aeleration & x i and & x& i at the (i-) th and i th time ste are known, the linear aeleration assumtion gives && τ τ τ && && && && & xi xi ( xi xi ) ( ) xi xi (.) where & x i is the redited aeleration for the (i) th time ste; and and τ are the integration time ste and the atuator delay, resetively. The redited dislaement xi an be alulated using the redited aeleration with the Newmark family of integration algorithms (Newmark 959) as (. β ) ( τ ) && x β ( τ ) & x τ & & (.) xi xi ( ) xi 5 i i where γ and β is integration arameter (where γ aears below in Table.); and x i, x& and & x& i are the ommand dislaement, veloity and aeleration resonse, resetively, for the i th time ste alulated by the integration algorithm. Alying the disrete z-transforms to Eqns.. and. leads to an equivalent disrete transfer funtion G X d / X d for the linear aeleration extraolation written in the general form of Eqn.., where X d (z) and X d (z) are disrete z-transforms of xi and x i, resetively. The oeffiients of the equivalent disrete transfer funtion G (z) are tabulated in Table.. Table. Coeffiients of equivalent disrete transfer funtion G (z) for the linear aeleration extraolation method using the Newmark family of integration algorithms Denominator Numerator β n d d d ( γ β ) n ( γ β ) n d n d n ( γ β ) ( 6β ) τ ( γ β ) τ βτ ( γ β ) ( 9β ) τ ( γ 9β ) τ 6βτ 8β ( γ 9β ) τ ( 8β ) τ 6βτ β 6βτ 6βτ βτ i

4 The th World Conferene on Earthquake Engineering Otober -7, 8, Beijing, China Alying the inverse disrete-z transform to the equivalent disrete transfer funtion for the linear aeleration extraolation method leads to n n n n d d d xi xi xi xi xi xi xi xi (.) d d d d d d d Eqn.. is the differene equation for the linear aeleration extraolation omensation method, and indiates that the method an be interreted as an extraolation in the time domain using revious ommand dislaements and revious redited dislaements.. Derivative Feedforward Comensation Jung and Shing (6) used derivative feedforward omensation in real-time testing. The blok diagram for real-time testing using the derivative feedforward omensation method is given in Figure, where the disrete transfer funtion G ff (z) alulates the derivative of the ommand dislaements from the ram generator and adds them to the dislaement signal for the servo-hydrauli atuator. A ram generator is used in Figure to smoothly aly the ommand dislaement to the exerimental struture. If the disrete transfer funtion G PID (z) reresents the digital PID servo-ontroller and G a (z) reresents the servo-hydrauli dynamis, the disrete transfer funtion G h (z) that relates the measured dislaement to ommand dislaement an be written as G h [ GPID G ff ] Ga GPID G ff GPID Ga (.) G G G G G PID a Similarly the transfer funtion for the servo-hydrauli ontrol loo without derivative feedforward omensation an be written as Gh GPID Ga /( GPID Ga ). The equivalent disrete transfer funtion for derivative feedforward omensation an therefore be written as PID PID GPID G ff G (.5) G If G PID (z) for the servo ontroller is written as GPID K ( δ t / Ti /( z )), where K and T i are a roortional gain and integrative time onstant, and G ff (z) is written as G ff K ff ( z ) / δt / z, where K ff is the derivative feedforward gain, from Eqn..5 the equivalent disrete transfer funtion G (z) for derivative feedforward omensation an be written as PID ( K δt K δt δt / Ti K ff ) z (K ff K δt) z K ff G (.6) K δt ( δt / T ) z K δt z i G (z) ff y ( i a Exitation Fore ( Integration x Ram S i d i Algorithm Generator S - - feedbak m( dislaement d i restoring fore r i Servo Controller S Exerimental Substruture Servo-hydrauli Atuator Figure Blok diagram reresentation of real-time testing with derivative feedforward omensation Alying the inverse disrete z-transform to Eqn..6 leads to

5 The th World Conferene on Earthquake Engineering Otober -7, 8, Beijing, China ( ) ( K / ) ( ) ( ) δt K δtδt Ti K ff di K j ff K δt di K ff di K δtd i di (.7) K δt ( δt / Ti ) Eqn..7 again indiates that the derivative feedforward omensation an be interreted as an extraolation in the time domain using the ommand dislaements and the redited dislaement.. Inverse Comensation Method ( ) Under the interolated ommand dislaement d i, the atual resonse of the atuator an be idealized as a linear resonse shown in Figure. The duration for the atuator to ahieve the ommand dislaement is m( ) assumed to be t d and designated as α δt. The delayed atuator dislaement of d an be exressed as ( i ( j ) ( m( ) m( ( ) m( d i di ( di di α ) (.8) Alying the disrete z-transform to Eqn..8 leads to m X d z Gd (.9) X α z ( α ) d where X m d (z) and X d (z) are disrete z-transforms of d i and d i, resetively. Chen (7) roosed to use the inverse of Eqn..9 for atuator delay omensation in real-time testing, whereby the equivalent disrete transfer funtion for the resulting inverse omensation method an be written as m( ) ( ) G X X d d α z ( α ) z (.) Alying inverse disrete z-transform to Eqn.., the extraolation in the time domain an be exressed as ( ) ( ) ( d i di ( α ) di α (.) Eqn.. indiates that the inverse omensation an be interreted as an extraolation in the time domain using the revious ommand dislaements. Dislaement t d adt ( ) d i dt m ( ) d i mj () d i Command from ram generator Atuator resonse South A-Frame (Wx9) Loading RTMD Stub Atuator Damers North A-Frame (Wx9) o () j ( ) t i t i Time Atuator Suort Roller Bearings N Figure Conetual atuator resonse Figure Frequeny resonse of servo-hydrauli system

6 The th World Conferene on Earthquake Engineering Otober -7, 8, Beijing, China. FREQUENCY RESPONSE ANALYSIS OF SELECTED COMPENSATION METHODS A real-time hybrid testing bed at the NEES Lehigh RTMD Equiment Site (Chen et al. 8) is used in this aer to exerimentally validate the roosed aroah. Figure shows the exerimental setu. Exeriments with redefined sinusoidal dislaements were onduted to identify the dynamis of the atuator attahed to the elastomeri damer. The exerimental results are onverted to the frequeny resonse roerties in magnitude (db) and hase (degrees), and are resented in Figure. Phase error an be observed in Figure (b), where for the frequeny range of to 5 Hz the hase lag varies almost linearly with reset to frequeny. The amlitude error is also observed in Figure (a) to be frequeny deendent, whih is observed to be db for small frequenies and -.9 db for 5 Hz. For the urose of illustration, a onstant atuator delay of τ6/ se. is seleted to determine the arameters for the three omensation methods. When the integration arameters of γ/ and β are used and the time ste t is seleted to be / se. for the linear aeleration extraolation method, the resulting equivalent disrete transfer funtions an be written as G (5.76z.z 6.56) / z. Considering the PID ontroller used in the exeriments, the equivalent disrete transfer funtion for derivative feedforward omensation an be determined using where the ontroller arameters of K, T i 5 and K ff.7. The equivalent disrete transfer funtion for the inverse omensators an be written as G (6z 5) / z. Figures 5(a) and 5() resent the frequeny resonse of G (z) for the three seleted omensation methods between and 5 Hz. For the linear aeleration method, the magnitude inreases and reahes a magnitude of. db (%) at 5 Hz. The hase varies almost linearly with reset to frequeny where the maximum hase angle reahes 75 degree at 5 Hz. The derivative feedforward omensation introdues both amlifiation and hase lead. The maximum magnitude and hase our at 5 Hz and are.5 db (5%) and 8 degrees, resetively. The frequeny resonse for the inverse omensator is shown to have the maximum amlifiation and hase lead of.5 db (5%) and 5 degree at 5 Hz. The exeted frequeny resonse of the servo-hydrauli atuator under the three seleted omensation methods an be derived by ombining the frequeny resonse of the atuator dynamis with eah of the omensation methods, and is resented in Figures 5(b) and 5(d). For the linear aeleration method, the magnitude has a minimum value of around - db (9%) at 5 Hz. An over-omensation for hase is observed to our, where a maximum hase lead value of degrees ours at 5 Hz. For derivative feedforward omensation, the hase will remain around degree and have a minimum value of -5.8 degree at 5 Hz. The magnitude has a maximum value of.8 db (8%). Thus the derivative feedforward omensation with a feedforward gain of.7 is observed to redue the hase error due to the atuator dynamis but have a slight amlitude error when alied to real-time testing. The exeted frequeny resonse for the servo hydrauli atuator with the inverse omensator has a maximum magnitude of.7 db at Hz and a slight under-omensation for hase an be observed to be -6. degrees around 5 Hz. 5. EXPERIMENTAL VALIDATION Exeriments with redefined sine swee dislaement are onduted to validate the above frequeny resonse analysis of the seleted omensation methods using the equivalent disrete transfer funtion. The sine swee signal has a magnitude of 5 mm and a frequeny ontent of to 5 Hz. The three seleted omensation methods are imlemented by rogramming their orresonding equivalent disrete transfer funtions onto the Lehigh RTMD real-time integrated ontrol system. Figure 6 shows the exerimental results when different methods are used for atuator delay omensation. The ourrene of a derease in amlitude and a hase lead an be observed between the ommand and redited dislaements. For the linear aeleration extraolation, Figure 6(a) ~ 6() show a slight derease in the amlitude ours at 5 se. (orresonding to.5 Hz) and 5 se. (orresonding to.5 Hz), whih is exeted in aordane with the frequeny resonse analysis results, while a slight hase lead between the measured and

7 The th World Conferene on Earthquake Engineering Otober -7, 8, Beijing, China ommand dislaement an be observed. This indiates a slight hase over-omensation. This hase lead is also exeted in aordane with the frequeny resonse analysis results. The exerimental results for derivative feedforward omensation (Figure 6(d) ~ 6(f) and inverse omensation (Figure 6(g) ~ 6(i)) are almost the same as they have also almost the same antiiated atuator resonse. Consistent with the frequeny resonse analysis resented, the amlitude of the measured dislaement is a little larger than that of the ommand dislaement for frequenies around.5 Hz and.5 Hz. A slight hase lag also ours around.5 Hz. magnitude (db) hase (degree) 5 frequeny (Hz) (b) frequeny (Hz) Figure Frequeny resonse of atuator dynamis (a) magnitude(db) hase(degree) (a) 5 frequeny(hz) () frequeny(hz) magnitude(db) hase(degree) (b) 5 frequeny(hz) (d) lin. ael. deriv. feedforward inverse 5 5 frequeny(hz) Figure 5 Frequeny resonse of different omensation methods dislaement (mm) dislaement (mm) dislaement (mm) (a) (d) (g) time (se) ommand measured redited (b).5.5 (e).5.5 (h).5.5 time (se) () (f) (i) time (se) Figure 6 Exerimental results using different omensation methods Comaring the exeriment results using the three omensation methods, the derivative feedforward omensation and the inverse omensation methods have a small hase under-omensation and slight amlitude inrease, while the linear aeleration extraolation method has a small hase over-omensation and slight amlitude derease. In ratie, a researher must tune the omensator over the frequeny range of interest, and attemt to minimize hase and magnitude error in the measured dislaement resonse.

8 The th World Conferene on Earthquake Engineering Otober -7, 8, Beijing, China 6. SUMMARY AND CONCLUSION A general aroah that is based on using a disrete transfer funtion and frequeny resonse analysis of the funtion is roosed for analyzing atuator delay omensation methods for real-time testing. Three omensation methods are seleted to illustrate the roosed aroah. The seleted omensation methods are shown to have different harateristis of amlitude and hase in their frequeny resonse. Using the same value of atuator delay, the three omensation methods are exerimentally alied in redefined sine swee tests. The frequeny analysis using the equivalent disrete transfer funtions is validated by the exerimental results, whih illustrates the effetiveness of the roosed general aroah. The roosed aroah is shown to resent a useful tool to analyze omensation methods for real-time testing. It is also worth noting that the roosed aroah an also be extended to other omensation methods used for real-time testing. ACKNOWLEDGEMENTS The researh resented in this aer is suorted by a grant from the Pennsylvania Deartment of Community and Eonomi Develoment through the Pennsylvania Infrastruture Tehnial Alliane, and by the National Siene Foundation (NSF) under Grant No. CMS-9 within the George E. Brown, Jr. Network for Earthquake Engineering Simulation Consortium Oeration. Their suort is gratefully areiated. REFERENCES Mahin, S.A., Shing, P.B. (985). Pseudodynami method for seismi testing. Journal of Strutural Engineering :7, 8-5. Shing, P.B., Nakashima, M., and Bursi, O.S. (996). Aliation of seudodynami test method to strutural researh. Earthquake Setra :, -. Soong, T.T. and Sener, Jr. B.F. (). Sulemental energy dissiation: state-of-the-art and state-of-the-ratie. Engineering Strutures :, -59. Jung, R.Y. and Shing, P.B. (6). Performane evaluation of a real-time seudodynami test system. Earthquake Engineering and Strutural Dynamis 5:7, Horiuhi, T., Inoue, M., Konno, T. and Namita, Y. (999). Real-time hybrid exerimental system with atuator delay omensation and its aliation to a iing system with energy absorber. Earthquake Engineering and Strutural Dynamis 8:, -. Chen, C., Riles, J.M., Marullo T.M. and Meran O. (8). Real-time hybrid testing using an unonditionally stable exliit integration algorithm. Earthquake Engineering and Strutural Dynamis Published online, early view. Wallae, M.I., Sieber, J., Neild, S.A., Wagg, D.J. and Krauskof, B. (5). Stability analysis of real-time dynami substruturing using delay differential equation models. Earthquake Engineering and Strutural Dynamis :5, Chen, C. and Riles, J.M. (8). Stability analysis of exliit integration algorithms with atuator delay for real-time hybrid testing. Earthquake Engineering and Strutural Dynamis 7:, Horiuhi, T. and Konno, T. (). A new method for omensating atuator delay in real-time hybrid exeriment. Philosohial Transations of the Royal Soiety of London A 59, Chen, C. (7). Develoment and numerial simulation of hybrid effetive testing method. PhD Dissertation, Lehigh University, Bethlehem, PA. Ogata, K. (995) Disrete-time ontrol systems, nd Edition, Prentie-Hall: New Jersey. MATLAB (7). Version 7., Mathworks, In., Natik, MA. Newmark, N.M. (959). A method of omutation for strutural dynamis. Journal of Engineering Mehanis 85:, 67-9.