October 12-17, 28, Beijing, China USING NONLINEAR CONTR ALGORITHMS TO IMPROVE THE QUALITY OF SHAKING TABLE TESTS T.Y. Yang 1 and A. Schellenberg 2 1 Pot Doctoral Scholar, Dept. of Civil and Env. Eng., Univ. of California, Berkeley, CA, USA 2 Ph.D. Candidate, Dept. of Civil and Env. Eng., Univ. of California, Berkeley, CA, USA Email: yangtony2@gmail.com, andrea.chellenberg@gmx.net ABSTRACT: Among experimental teting method the haking table teting method i one of the mot realitic technique to tudy the eimic behavior of tructural ytem or component thereof. Shaking table tet are able to produce nearly realitic prototype condition, giving important inight into critical iue uch a collape mechanim, component failure, acceleration amplification, reidual diplacement and pot-earthquake capacitie. Ideally, the pecimen teted on a haking table i excited a it would be during an actual earthquake event. However, due to the nonlinear behavior of the teted pecimen and the limitation of the hydraulic control ytem, the motion of the haking table i uually different from the commanded motion recorded during an earthquake event. Thi difference may produce an unintended repone of the teted tructure; ultimately, it may reult in a pre-mature failure of the pecimen. Traditional tuning of haking table control ytem ha focued on linear control algorithm, where the tructural ytem i aumed to remain linear-elatic and where there are no model uncertaintie and diturbance in the ytem. In thi paper, a nonlinear control algorithm, baed on the Lyapunov tability theorem, i utilized to control a haking table tet of a nonlinear ingle degree-of-freedom ytem with the goal to account for nonlinear repone, model uncertaintie and other diturbance affecting the tet a the tet i being performed. Simulation reult indicate that uch nonlinear control algorithm can be ued to achieve excellent tracking, even when the teted tructure behave nonlinearly. The example alo demontrate the ability of the nonlinear control algorithm to compenate for diturbance, where the actuator force input to the haking table plant are altered with repect to the force command that the control algorithm put out. Thu, the propoed nonlinear haking table control algorithm i not only a viable alternative, but alo a way to ignificantly improve the quality of haking table tet. KEYWORDS: Sliding Mode Control, Nonlinear Control Algorithm, Shaking Table Tet. 1. INTRODUCTION Traditional control of haking table tet i baed on the ue of linear control algorithm to minimizing tracking error between the command and feedback ignal. However due to the nonlinear nature of mot teted pecimen, it i very difficult, or impoible, to achieve excellent tracking uing uch linear control algorithm. Thu, a nonlinear control algorithm, the liding mode control () trategy, i choen to regulate the movement of the haking table. The liding mode controller i deigned uing a Lyapunov-like tability function, which permit the algorithm to control nonlinear ytem. In addition, the liding mode controller contain an inherent robutne term, which i capable of compenating for any model uncertaintie. Thi i epecially ueful in the environment of experimental teting, where mot of the model parameter are not eaily aeable. In thi paper, a liding mode control algorithm i derived to regulate a unidirectional haking table, where a nonlinear ingle-degree-of-freedom pecimen i teted. Similar approache could be utilized for multidegree-of-freedom ytem teted on multidirectional haking table. To evaluate the performance and accuracy of the developed liding mode controller, the repone of the haking table and the ingle-degree-of-freedom bridge column pecimen are compared againt the open-loop () and the traditional control algorithm.
October 12-17, 28, Beijing, China 2. MODELLING OF THE SYSTEM Figure 1 how the model of a cantilever bridge column that i being teted on a unidirectional haking table. For modeling purpoe the column i idealized a a ingle-degree-of-freedom pecimen. Hence, the only two free degree-of-freedom of the ytem are d t and d, which repreent the abolute diplacement of the haking table and the relative diplacement of the pecimen, repectively. dt d t = t = ti m F mt P,C Figure 1: Model of the idealized ytem. To imulate the eimic repone of the implified model hown above, parameter from the haking table tet conducted by (Mahin et. al., 26) (Figure 2) are adopted for thi tudy. The teted pecimen had an unbounded pot-tenioning tendon in the center of the column, which provided a retoring force to re-center the pecimen after ever inelatic damage. To implify the modeling aumption, the flag-haped hyterei-loop of the pecimen wa replaced by a hyterei-loop without re-centering capability. The adopted hyterei-loop repreented a column with imilar initial tiffne and trength but without the tiffne degradation after cycle of inelatic deformation. Figure 3 how the diplacement loading hitory and the correponding hytereiloop of the idealized cantilever column pecimen. 1 1 2 6 8 1 12 2 1 u y 2u y 3u y u y u y 1 Reiting Force [kip] 1 1 Figure 2: Bridge column tet pecimen on EERC haking table. 2 8 6 2 2 6 8 Figure 3: SAC loading hitory and hytereiloop of the idealized model The dynamic repone of the haking table/pecimen configuration (Figure 1) can be modeled uing the following ytem of two econd-order nonlinear differential equation. m dt + ( d )+ c d + P ( d, d )= m dt t c d P ( d, d ) F = (2.1)
October 12-17, 28, Beijing, China where d t, d t, d and d are the abolute velocity and acceleration of the table and the relative velocity and acceleration of the pecimen, repectively. m t i the ma of the haking table, m i the ma of the pecimen, c i the damping of the pecimen, P i the nonlinear reiting force of the pecimen and F i the control force of the actuator. To model the nonlinear reiting force of the pecimen, a Bouc-Wen model (Bouc, 1971; Wen, 1976) a hown in Equation (2.2) i ued. P ( d, d )= k e, d + ( 1 )k e, d y, z = d y, z + d zz n1 + d z n d (2.2) where i the ratio of the pot-yield to elatic tiffne, k e, i the elatic tiffne, d y, i the yield diplacement, z i a time varying hyteretic parameter which can be olved for uing Equation (2.2b) and, and n are dimenionle parameter that control the hape of the hyterei-loop. To olve the ytem of equation, the two econd-order differential equation hown in Equation (2.1) are tranformed into a tate pace repreentation a hown in Equation (2.3). The dynamic repone of the ytem i olved uing Matlab ode23 olver (MathWork, 28). Value of 1e-6 were elected for both the relative and abolute tolerance. x 1 x 2 x 3 x x = d t d t d d z x 2 x 1 1 P m ( x 3, x )+ c x + F x t ( ) 2 x x 3 = x 1 P x ( m 3, x ) + c x + F x t ( ) 1 P m ( x 3, x )+ c x ( ) 1 n1 n x d x x x x x y, ( ) (2.3) 3. SELECTION OF CONTR ALGORITHMS To compare the effectivene of the different control algorithm, the eimic repone of the idealized model wa analyzed uing the SACNF1 near-fault ground motion of the SAC databae (Somerville, 1997) (Figure ). 2 3 Velocity [in./ec] 1 1 1 1 Acceleration [in./ec 2 ] 2 1 1 2 1 1 2 2 2 1 1 2 2 Figure : SACNF1 near-fault ground motion record. 3 1 1 2 2
October 12-17, 28, Beijing, China The ground motion wa originally recorded during the 1978 Taba earthquake. The record wa caled and baeline corrected to match the capacitie of the EERC haking table located at the Univerity of California, Berkeley. The dynamic repone of the pecimen excited by the elected ground motion wa firt imulated uing the exact ground acceleration hitory. Thee reult, a hown in Figure, erved a the benchmark repone to evaluate the effectivene of the different control algorithm. Reiting Force [kip] 1 1 1 1 2 2 Velocity [in./ec] Acceleration [in./ec 2 ] 2 2 1 1 2 2 2 2 2 2 1 1 2 2 1 1 2 2 Figure : Repone of the pecimen ubjected to the SACNF1 near-fault ground motion. 3.1 Open-Loop Control () The implet control algorithm to regulate the haking table i to ue open-loop control. The controller calculate the actuator force by multiplying the um of the table and pecimen mae with the input acceleration. Thi control algorithm ignore the nonlinear interaction force between the pecimen and the table and doe not adjut it input uing any feedback meaurement. 3.2 Control () To improve the performance of the control ytem, the open-loop controller i replaced by a cloed-loop control technique, where the input ignal to the table i adjuted according to the feedback error meaured during teting. The controller i one of the mot common linear cloed-loop control algorithm, where the driving ignal i adjuted baed on a linear combination of the feedback error, the integral of the feedback error and the derivative of the feedback error. By electing the appropriate gain that multiply thee three term, the controller modifie the cloed-loop dynamic, which in term force the error dynamic of the ytem aymptotically to zero. Once the error dynamic reache zero perfect tracking i obtained. The implementation of the controller (in the Matlab/Simulink environment (MathWork, 28)) for the haking table tet i illutrated in Figure 6. The plant repreent the ytem dynamic of both, the haking table and the cantilever column pecimen. The input to the plant i the control force and the output of the plant are the different tate of the table and pecimen. The error dynamic of the ytem wa elected baed on the error of the table diplacement. To minimize uch error dynamic, three contant gain that multiply the feedback error, the integration of the feedback error and the derivative of the feedback error were elected. For the purpoe of tuning the controller, the pecimen wa replaced by a rigid block of equal ma. Thi mean that the interaction force between the table and the pecimen were ignored. Thi i a common approach for tuning haking table without damaging pecimen prior to teting. However, the tuning of the controller till required ome engineering judgment and experience. Once the controller had been tuned to atifaction, the rigid block wa replaced by the pecimen.
October 12-17, 28, Beijing, China 3 Out 3 du /dt at Out SACNF 1.mat dtm m Controller cout D Diturbance pin Shaking Table & Specimen x m Plant Scope XY Graph 1 Out 1 2 Out 2 Out du /dt a 6 Out 6 Figure 6: Simulink model of ytem with controller. 3.3 Sliding Mode Control () The controller provide ignificant performance advantage over the open-loop controller. However, the performance of the controller i ignificantly affected by ignoring the nonlinear interaction force of the pecimen and the influence of external diturbance. To overcome thi diadvantage, it i propoed to intead utilize a liding mode control algorithm to regulate the haking table. Sliding mode control i one of the robut control technique that aim at controlling nonlinear ytem including model uncertaintie and diturbance. The control algorithm employ a Lyapunov-like function to drive the tracking error aymptotically toward zero, which again provide the deired tracking. Even if there are model uncertaintie and/or diturbance preent, the liding mode control algorithm i able to overcome thee nonlinearitie uing it robutne term and achieve excellent tracking. 3.3.1 Deign of the Sliding Mode Controller The liding mode controller i deigned uing a Lyapunov-like function uch a the one hown in Equation (3.1). If Equation (3.1a) i a poitive define function, and Equation (3.1b) i elected to be a negative definite function, the Lyapunov tability theorem guarantee that the calar function, S, will go to zero aymptotically. V = 1 2 S 2 V = S S (3.1) If S i now defined a a combination of the diplacement and velocity tracking error a hown in Equation (3.2) and if in addition Equation (3.1b) i elected to be equal to KS 2 (a negative definite function), the Lyapunov tability theorem guarantee that S aymptotically converge to zero. Thi mean that the olution x 1 = e t of the tracking error equation will alo aymptotically converge to zero and therefore achieve perfect tracking. S = x 1 x 1 (3.2) where x 1 = x 1 x 1d, x 1 = x 1 x 1d, i a trictly poitive contant (that define the exponential convergence rate on the liding urface), x 1, x 1 are the actual diplacement and velocity of the table and x 1d, x 1d are the deired diplacement and deired velocity of the table, repectively.
October 12-17, 28, Beijing, China Taking the derivative of Equation (3.2) and ubtituting the tate from Equation (2.3), the derivative of S can be expreed in the following manner. S = x1 x1 = 1 m t ( P + c x + F ) x1d + ( x 2 x1d )= CE( x )+ f + bf (3.3) where CE( x )= ( x 2 x 1d ), f = 1 ( P m + c x ) x1d and b = 1. t m t Finally, ubtituting Equation (3.3) and V = KS 2 into Equation (3.1b), the following liding mode control law i obtained. S = KS F = 1 b ( CE( x )+ f + KS) (3.). EVALUATION OF CONTR ALGORITHMS To achieve a more realitic imulation of the ytem dynamic, an external diturbance and command aturation were added to the control force. For the purpoe of thi tudy, the control force wa limited to 8% of the actuator capacity, which correponded to 2 kip. The diturbance wa modeled by adjuting the actuator force utilizing a normal ditributed random variable with mean of 1. and variance of.1. Figure 7 how the tracking indicator for the table diplacement, velocity and acceleration. The tracking indicator were calculated according to the formula preented in (Mercan and Ricle, 27). The indicator i baed on the encloed area of the hyterei loop when actual output i plotted veru deired input. If the value of the indicator i increaing the repone i leading. On the other hand, if the indicator i decreaing the repone i lagging. The reult how that the controller did not produce good tracking. The tracking indicator increae rapidly, leaving the graph after the firt 2. econd. On the other hand, the and controller were able to achieved good tracking. The controller achieved the bet tracking performance and in addition wa not affected by the diturbance in any way. The controller performed lightly wore than the controller. In addition, the tracking performance wa clearly affected by the preence of the diturbance. Thee trend are even more pronounced for velocitie and acceleration than for diplacement. Table.1 how the maximum control force in the actuator. The reult indicate that the controller require the larget control force and reache the aturation limited of the actuator. The controller require the leat amount of control force. With the diturbance added to the ytem, larger actuator force are required. Table.1: Maximum control force in the actuator. Control algorithm Without diturbance With diturbance 133.7 kip 173.6 kip 2. kip 2. kip 11.9 kip 1.7 kip Figure 8 illutrate the tracking indicator for the pecimen repone compared againt the benchmark repone preented in Figure. The reult indicate that the controller damage the pecimen pre-maturely. The diplacement tracking indicator increae out of bound. The and controller both are able to achieve accurate diplacement tracking. Similar trend are oberved for the velocity tracking indicator. However, the acceleration tracking indicator demontrate that the controller produce large acceleration error, which i mainly caued by the aturation of the actuator force. On the other hand, the controller produce excellent acceleration tracking. Similar tructural repone are oberved with and without the added diturbance.
October 12-17, 28, Beijing, China Diplacement TI [in. 2 ].3.2.1 & Noie 1 1 2 2 Velocity TI [in. 2 /ec 2 ] 1 & Noie 1 1 1 2 2 Acceleration TI [in. 2 /ec ].. 1 x 16 & Noie 1 1 1 2 2 Figure 7: Tracking indicator of the table repone for different control law. Diplacement TI [in. 2 ] 1 8 6 2 & Noie 1 1 2 2 Velocity TI [in. 2 /ec 2 ] 1 & Noie 1 1 1 2 2 Acceleration TI [in. 2 /ec ].. 1 x 16 & Noie 1 1 1 2 2 Figure 8: Tracking indicator of the pecimen repone for different control law.
October 12-17, 28, Beijing, China. CONCLUSIONS The haking table teting method i one of the mot realitic experimental teting technique to evaluate the dynamic repone of the tructure or component thereof. However, the accuracy of a haking table tet depend trongly on the control algorithm ued to regulate the haking table. In thi paper, an idealized nonlinear ingledegree-of-freedom ytem teted on a unidirectional haking table ha been tudied. To properly model the ytem dynamic, the nonlinearity of the pecimen, the aturation of the actuator control force and an additional external diturbance have been conidered. Three different controller were deigned and implemented. The repone of the table and the pecimen were then imulated for the different controller with and without diturbance. A expected, the reult indicated that the controller produced poor tracking in all ituation. On the other hand, the controller wa able to produce good diplacement tracking, but performed poorly in tracking velocitie and acceleration. Contrary, the controller achieved excellent tracking for all repone quantitie. Furthermore, performance wa not affected by external diturbance, confirming the robutne of the controller. The controller wa able to control the table with lower actuator force and thu it wa le likely that the actuator capacity wa exhauted. Thi provide a ignificant advantage over the traditional controller and allow for a larger range of pecimen and configuration to be teted. Further reearch hould be conducted on the liding mode control algorithm. Specifically, the performance of the controller hould be verified in a real application in the laboratory environment. Alo, the theory hould be extended to multi-degree-of-freedom ytem teted on multidirectional haking table. Once the implementation of the liding mode controller ha been validated experimentally, thee reearch reult could ignificantly contribute to the improvement of the haking table teting method. REFERENCES Bouc, R. (1971). Mathematical model for hyterei. Report to the Centre de Recherche Phyique, pp16-2, Mareille, France. Hedrick, J. K. (2). Control of Nonlinear Dynamic Sytem, ME237 cla note, Spring 2. Univerity of California, Berkeley, CA, United State. Khalil, H. (22). Nonlinear Sytem, 3 rd edition. Prentice-Hall, NJ, United State. Mahin S. A., Sakai J. and Jeong H. (26). Ue of Partially Pretreed Reinforced Concrete Column to Reduce Pot-Earthquake Reidual Diplacement of Bridge. Proceeding of Fifth National Seimic Conference on Bridge & Highway, September 18-2, San Francico, CA, United State. MathWork (28). MATLAB and Simulink for Technical Computing. The MathWork Inc., Natick, MA, United State. Mercan, O. and Ricle, J. M. (27). Stability and accuracy analyi of outer loop dynamic in real-time peudodynamic teting of SDOF ytem. Earthquake Engineering and Structural Dynamic, 36(11), 123-13. Slotine, J. E. and Li W. (1991). Applied Nonlinear Control. Prentice-Hall, NJ, United State. Somerville, P. et al. (1997). Development of Ground Motion Time Hitorie for Phae 2 of the FEMA/SAC Steel Project. Report SAC/BD-97/, SAC Steel Project, Sacramento, CA, United State. Wen, Y. K. (1976). Method for random vibration of hyteretic ytem. Journal of Engineering Mechanic Diviion, 12(EM2), 29 263.