INVERSE DYNAMIC SIMULATION OF A HYDRAULIC DRIVE WITH MODELICA. α Cylinder chamber areas ratio... σ Viscous friction coefficient

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1 Proceeding of the ASME 2013 International Mechanical Engineering Congre & Expoition IMECE2013 November 15-21, 2013, San Diego, California, USA IMECE INVERSE DYNAMIC SIMULATION OF A HYDRAULIC DRIVE WITH MODELICA Joeph Saad Department of Mechanical Engineering American Univerity of Beirut Lebanon Jg15@aub.edu.lb Matthia Liermann Department of Mechanical Engineering American Univerity of Beirut Lebanon Matthia.liermann@aub.edu.lb ABSTRACT Invere dynamic imulation of hydraulic drive i helpful in early deign tage of hydraulic machine to anwer the quetion whether the drive can meet dynamic load requirement and at the ame time to predict the energy conumption for required load cycle. While a forward imulation of the hydraulic drive need an implementation of the controller which generate the control input a a function of the control error, the invere dynamic imulation can be implemented without control. Thi i becaue the required motion i imply defined a a contraint and therefore the control error i alway zero. Thi paper urvey example of ucceful ue of invere dynamic imulation in engineering. We ue the example of a hydraulic ervo-drive to explain the procedure how to generate a tate pace decription of the invere problem from the given ytem of differential algebraic equation. Equation baed modeling language uch a Modelica lend themelve naturally for invere imulation becaue the definition of which variable of the model are input and which are output i not made explicit in the model itelf. NOMENCLATURE E Bulk modulu Pa L Inductance H m Q Flow rate S Cylinder troke m V Cylinder volume m 3 p Preure Pa u Input valve voltage V α Cylinder chamber area ratio... σ Vicou friction coefficient N. m γ Approximation factor... Natural undamped valve frequency ω v A p Cylinder piton area m 2 D v Valve damping ratio... F c0 Coulomb friction N F ext External force N F f Friction force N F 0 Static friction N K v Valve voltage gain... Q Le External cylinder Leakage Q Li Internal cylinder leakage m 3 V pl Bottom dead center volume m 3 V 0 Cylinder initial chamber volume m 3 U L Inductor voltage V Hydraulic capacitance m 3 c h c Stribeck velocity rad m 3 Pa m cv Valve flow gain... i L1 Inductor current A m t Total ma kg x v Valve opening... x p Cylinder piton poition m m x p Cylinder piton velocity x p Cylinder piton acceleration m 2 1

2 di L1 Inductor current tate derivative INTRODUCTION & BACKGROUND Mechanical ytem have become complex and include many engineering dicipline. Thi urge companie to pend part of their budget for performing ytem imulation on their deign before they actually manufacture them. Uually, ytem imulation are not introduced in early deign tage. Rather they are implemented in later tage to tet the dynamic of the ytem deigned or to chooe an optimum control that will provide the deired ytem output. The approach ued to perform thee imulation i commonly known a the forward imulation approach, where the tate of the governing differential equation of the ytem are calculated in the direction of cauality from given control and reference input to ytem output. Cloed loop controlled ytem require a forward imulation implementation that include the control deign. Figure 1 illutrate the forward approach where the input to the ytem i the reference and the output i the motion/force obtained from the a control input ignal. Reference Control Control input ignal Sytem Figure 1: Forward Simulation In cae the engineer want to analyze the ytem efficiency, he i intereted in the ytem' input and output variable and not in deigning or tuning a controller. The forward approach can be conidered a a drawback ince the engineer will have to deign a controller in cloed loop ytem to check the efficiency. Therefore, uually deign engineer ue teady tate condition and engineering aumption depending on their expertie to deign their ytem. However, it would be an advantage for engineer to ue the ame imulation model in early deign tage when efficiency of ytem configuration are analyzed a in tage when the control deign take place. Simulation in the deign tage aid the deign engineer to elect the optimal component izing and parameter of their ytem; thu, helping to have a more reliable and efficient ytem once the controller i applied in later tage. Thi i where the backward or invere imulation approach i introduced and can have a major role for helping deign engineer deign their ytem. Invere ytem imulation i alo beneficial in cae where the control deign i difficult. Figure 2 illutrate the invere imulation concept where the deired motion/force output i fed a an input to the ytem thu the output of the imulation correpond to the control input that would otherwie be generated by the controller. A motion/force Output Control Input Sytem Figure 2: Invere Simulation motion/force Output The backward imulation approach from a modeling point of view i not much different from the forward approach. The only difference i the ytem input and output are interchanged. The reult of thi interchange i till a ytem of DAE, Differential Algebraic Equation, that can be olved with the ame technique of any DAE olver [1]. The invere dynamic imulation i baed on two main point which are: inverting the differential equation that decribe the ytem and deigning an output trajectory of the ytem [2]. Thi mean that the output that i deired from the ytem i aumed to be met, and it i fed to the ytem a an input; wherea, the calculated output of uch a imulation are the real required input of the ytem. The benefit of the backward imulation approach i that the imulation aume that the deired output i met; thu, there i no need for a controller. There i low complexity in model where complicated controller deign from partner companie can be avoided [3]. Invere imulation can be introduced into early deign tage to facilitate the election and izing of component. Figure 3 illutrate the difference between the conventional forward imulation (on the left) ued after the deign proce i performed through teady tate analyi from the engineer' expertie and the backward/forward imulation (on the right) ued during and after the deign proce. The conventional imulation ue only the forward approach after the deign parameter have been already choen by the deign engineer and the intention i to elect an optimal controller for the ytem. Wherea, the backward/forward imulation introduce the backward approach in the deign tage to help the deign engineer elect improved deign parameter for hi ytem before the commiioning tage where the commiioning engineer applie the forward imulation approach to elect an optimal controller for the ytem. The advantage of the backward approach i providing a tool that help the deign engineer in electing and izing ytem component more efficiently. The backward/forward approach (on the right) of Figure 3 i deired in order to provide better ytem. It i deired to have tool that allow combined forward/invere imulation to be performed uing the ame model. Thee tool will facilitate the communication among engineer in the firm having different tak and will reduce the poibility of error that might be faced. 2 Copyright 2013 by ASME

3 Modelica Source Code Paring + Preproceing Flattening Caualization Code Generator C Compiler Simulation Figure 4: Tranlation of Model in Modelica Figure 3: Forward imulation only v/ Backward/Forward imulation To illutrate the concept of backward/forward imulation in thi paper, we ue Modelica an equation-baed object oriented modeling language. The benefit of Modelica i that it ue acaual phyical modeling tyle where the model are defined baed on equation rather than aignment tatement [4]. The power of equation baed model i that they do not pecify a priori which variable are declared a input and which are output [4]; however, uing aignment tatement, variable that are aigned on the left-hand ide of an equation are the output of the ytem and variable on the right-hand ide are the aigned input of the ytem. Simulink i one of the tool that i baed on variable aignment. Uing equation-baed object oriented modeling tool uch a Modelica facilitate the ue of backward imulation approach, becaue the uer only build one model and can imulate it either in the forward approach or in the backward approach by changing the ytem input. On the other hand, the ue of traditional variable aignment tool uch a Simulink will require the uer to build two eparate model, one for the traditional forward approach and the other for the backward approach. Thi proce can even be quite tediou, a thi paper will demontrate [5]. The proce of how Modelica handle equation i illutrated in the next ection. EQUATION MANIPULATION IN MODELICA The proce of how Modelica imulate a model make it an important platform to tet and ue the backward modeling approach. Thi proce of tranlating a model i hown in Figure 4 and i illutrated in thi ection with the help of an electric circuit example that i imulated in both forward and invere fahion. Firt, the model i created by the uer uing Modelica ource code with equation a main component. Upon imulation intance, the model i pared to check whether the ue of the Modelica yntax i correct or wrong. After paring, preproceing take place where the model will check whether the clae are ued correctly uch a the extend command that connect different ub-model. After yntax and type checking, flattening take place. In flattening, the hierarchy of the model tructure i detroyed; thu, all the parameter, variable, and equation from all the component model of the ytem are collected in one global et. Thi et contain all the DAE of the ytem that will be olved [6]. A ytem of DAE i repreented implicitly in the following manner: 0 = F dx(t), x(t), u(t), y(t), t The goal i to tranform the implicit DAE to an explicit tate-pace repreentation form that i uited to mot ODE olver. x (t) = f x(t), y(t), u(t) y(t) = g(x(t), u(t)) where, u(t) are input variable x(t) are tate variable y(t) are output variable and alo include algebraic variable x (t) are the derivative of the tate The following example illutrate the way the equation are automatically manipulated in Modelica. Figure 5 below how an electric circuit. The circuit conit of a voltage ource providing an input alternating voltage amplitude of 10V with a frequency of 10Hz, a reitor of reitance R=100Ω, and an inductor of inductance L=1H. The output of the ytem i conidered a the current circuit. 3 Copyright 2013 by ASME

4 of the olution. Thi tep i divided into three part, Figure 6: orting of equation according to the precedence of olution, contructing the cauality graph, and building the tructural incidence matrix. Flattening Caualization Equation Sorting Cauality Graph Incidence Matrix Code Generation Figure 5: Electric Circuit Forward After the ytem model i flattened, the reulting equation are hown below. V S2 = V R1 V R2 = V L1 V L2 = V G i S2 + i R1 = 0 i R2 + i L1 = 0 i L2 + i S1 + i G = 0 i R1 + i R2 = 0 U R = R i R1 V R1 + U R = V R2 V S = V S2 i L1 + i L2 = 0 U L = L di L1 V S1 = V G V G = 0 i S2 + i S1 = 0 V L1 + U L = V L2 Figure 6: Caualization Step The cauality graph i an acyclic directed graph that repreent the cauality of the ytem [6]. Figure 7 i an example of cauality graph. The firt thing that i done i to eliminate all redundant equation or trivial one uch a V L2 = V G and leave out all neceary one. The reult of thi procedure i the 9 equation with their repective variable a hown below. The equation are numbered in order to be orted afterward. Eq.1 V G = 0 Eq.2 U R = R i R1 Eq.3 V R1 + U R = V L1 Eq.4 V G + 10 = V R1 Eq.5 U L = L di L1 Eq.6 V L1 + U L = V G Eq.7 i S1 + i R1 = 0 Eq.8 i R1 + i L1 = 0 Eq.9 i L1 + i S1 + i G = 0 After eliminating trivial equation the ytem i tranformed into explicit tate-pace form where the derivative of the tate i explicitly written in term of the tate itelf and other related variable. In thi cae, the only tate i i L1 ; therefore, the explicit tate equation become: di L1 = U L L After explicitly defining the tate equation, the caualization tep take place, Figure 6. Thi tep deal with caualizing the equation; thu, indicating which equation determine which unknown and tep by tep build the hierarchy Figure 7: Cauality Graph Forward The non-caual lit of equation i repreented in a matrix form called tructural incidence matrix. An example of uch a matrix i hown in the Figure 8. di i G i S1 U L i R1 V G V L1 V R1 U L1 R Eq.1 x Eq.2 x x Eq.3 x x x Eq.4 x x Eq.5 x x Eq.6 x x x Eq.7 x x Eq.8 x Eq.9 x x Figure 8: Structural Incidence Matrix A een from Figure 8, the row of the matrix repreent the equation and the column of the matrix repreent the equation variable or unknown and output. 4 Copyright 2013 by ASME

5 The aim of the caualization a dicued earlier i to ort the equation in a hierarchical fahion. Therefore, the bet equence that i obtained after performing the cauality graph in Figure 7, i a lower triangular form of the incidence matrix. Thi can be een in Figure 9. di V G V R1 i R1 U R V L1 i S1 U L1 L Eq.1 x i G Eq.4 x x Eq.8 x Eq.2 x x Eq.3 x x x Eq.7 x x Eq.6 x x x Eq.5 x x Eq.9 x x Figure 9: Lower Triangular Form of Incidence Matrix It i hown from Figure 9 that the caualization of the equation reult in a permutation of the tructure index matrix to tranform into lower triangular form. The olution i traightforward by imply replacing the variable in a forward manner. Thi LT form follow the ame order of cauality obtained from the acyclic directed graph. The olution i directly performed through forward ubtitution of variable through the chronology obtained from the LT matrix and with any ODE olver uch a Forward-Euler to increment the tate for the next time tep. Figure 10: Invere Electric Circuit After olving for the forward ytem, the invere of thi ytem i olved. Figure 10 how the invere circuit which i the ame a the forward circuit model with interchanging the input/output boundary condition. The ame technique of equation manipulation i ued to find the invere. In thi cae, the input i the current i S1 and the output i the voltage ource U S. The ame equation are ued, but by replacing the new input and the new output. Modelica divide equation Eq.8 to become a function of the tate derivative di L1 which i conidered a an unknown and mandatory to proceed in the olution proce. Therefore, Eq.8 become Eq.8i hown. Eq.8i di L1 = di R1 The new cauality graph i hown in the Figure 11. Figure 11: Cauality Graph of Invere Sytem The LT incidence matrix i then formed and hown in Figure 12. di V G i G i L1 R1 Eq.1 x U L U R V L1 V R1 U S Eq.9 x Eq.7 x Eq.8i x Eq.5 x x Eq.2 x x Eq.6 x x x Eq.3 x x x Eq.4 x x x Figure 12: Lower Triangular Form of Invere Sytem Incidence Matrix However, not all ytem can be permuted into LT form. The forward caualization and the LT form can only be ued for imple problem. For the vat majority of the other cae, the BLT which tand for the Block Lower Triangular form i ued. The BLT form i cloe to the LT form where the block of the BLT matrix at the diagonal are a mall a poible. An example of the BLT matrix i hown in the Figure 13. Matrice that cannot be permuted into LT form uch a in Figure 13 are olved by iolating highlighted block on the diagonal of the matrix of Figure 13 and conidering them a coupled ytem. The reult of thee iolation i a imilar 5 Copyright 2013 by ASME

6 diagonal a the LT and can be olved by imple forward ubtitution on an acyclic directed graph. V G V R1 i R1 U R V L1 i S1 U L Eq.1 x Eq.4 x x Eq.8 x x Eq.2 x x Eq.3 x x x Eq.7 x x di L1 Eq.6 x x x x x Eq.5 x x x Eq.9 x x Figure 13: Block Lower Triangular Form of Incidence Matrix Dymola olve coupled ytem either through ymbolic manipulation of equation which will tranform the BLT form to an LT form or it olve for thee iolated block alone in a proce called tearing [7]. The concept of tearing i to aume a et of variable of an iolated block to be known, where thee variable are called tearing variable, and to olve for the unknown in the block in a caualized fahion. Some equation within a block might be overcontrained, they are conidered a reidual equation and thu the olution i not direct but i olved numerically with an iterative olver. An example of how block could be olved: 1. a + b = 4 2. a b = c Block of Coupled Equation a and b are unknown; wherea, c i known. In order to olve the block, a i aumed to be known; thu i called a tearing variable. After thi, the equation are reorted and caualized to olve for b iteratively a follow: i G otherwie the iteration i repeated to find a new value of x n+1. Thi can be written in peudo code a: while g(a n) > tolerence g (a n ) a n+1 = a n g(a n) ; g (a n ) iteration=iteration+1; a n =a n+1 ; g(a n ) ; g (a n ) end After manipulating the equation and bringing the DAE into a flattened and orted caualized ytem of ODE equation, C code i generated and with the help of any numeric equation olver uch a Forward-Euler, the equation of the ytem model are olved and the imulation reult are plotted. Therefore, the lat 3 tep of Figure 4 are found in other imulation oftware even if they are baed on variable aignment; however, the firt 4 tep which are autonomou and important, are found only in Modelica. Thu, Modelica modeling language i a valuable and eay tool to perform invere or backward imulation. INVERSE DYNAMIC SIMULATION OF A HYDRAULIC DRIVE Servo-Hydraulic linear axe are ued to control poition, velocity, or force in application that require high power-toweight ratio and ytem reliability in harh environmental condition. It i conidered to be a good application to implement both the forward and backward approache on and demontrate their advantage. The ytem tudied in thi example conit of a ynchronizing cylinder and a 4/3 directional control valve, ee Figure b = 4 a = f(a) 2. reidual = c a b = g(a, b) For every b, we get a different reidual value where the aim i to minimize the reidual value. Many method can be ued to olve 0 = f(x) and the motly ued method i the Newton' method. The Newton' method deal with olving for the root of x from the equation 0 = f(x) where an initial gue for x 0 i required. The algorithm find a new value of x uing the following equation [8]: x n+1 = x n f(x n) f (x n ) If y = f(x) = reidual i maller than a certain tolerance value, the iteration i topped and x n+1 become the final value Figure 14: Servo-Hydraulic Linear Actuator The equation to decribe the ytem dynamic can be found in many tandard textbook uch a [9]. 6 Copyright 2013 by ASME

7 MODELING OF VALVE The valve modulate the power provided by the preure ource and delivered to the cylinder. A hown in Figure 14, the poitive ene for the pool motion of the valve i aumed to the right. Thi valve i aumed to be critically lapped. Therefore, the flow orifice equation to the cylinder chamber are: QA = cvg(xv)ign(p p A ) p p A cvg( xv)ign(p A p T ) p A p T QB = cvg( xv)ign(p p B ) p p B cvg(xv)ign(p B p T ) p B p T where xv for xv 0 g(x v ) = 0 for xv < The ign of xv dictate the direction of flow in the ytem. The valve pool piton ha dynamic characteritic with repect to the electrical input. Thee characteritic involve many parameter which are hard to find accurately; however, manufacturer catalogue often how frequency repone plot which can be ued to identify the dynamic of the pool poition control ytem. Often a 2 nd order approximation i ufficient: 1 ω2 x v + 2D v x v + x v ω v = K v u v MODELING OF CYLINDER Figure 16 how the cylinder to be modeled. Introducing the poitive ene for the velocity of the cylinder to be to the right and auming that whatever enter the ytem i poitive and whatever leave the ytem i negative, the equation ued for modeling the cylinder are illutrated below. From the continuity equation, the flow in every chamber of the cylinder i a follow: V A Q A + Q Li = V Ȧ + E(p A ) p A V B Q B Q Li Q Le = V B + E(p B ) p B The volume of the chamber are given by: V A = V A0 + x p A p V B = V B0 x p A p The preure dynamic equation are a follow: p A = 1 (Q c A - A p x p) ha 1 p B= (Q c B + A p x p) hb The hydraulic capacitance of each of the 2 chamber are: c ha = V pl,a+( S 2 + x p)a p E A (p A ) c hb = V pl,b+( S 2 - x p)a p E B (p B ) Newton 2 nd law i applied in order obtain the equation of motion for the cylinder m t x p+ F f x p =(p A -αp B )A p - F ext Where m t i the total ma and conit of the piton ma m p and the hydraulic fluid ma in the cylinder chamber and the pipeline which are m A,fl and m B,fl. The Stribeck friction curve i conidered in our cae for the friction. The problem with friction when it come to modeling i at zero velocity where it i dicontinuou. It ha an equal maximum poitive and negative value depending on the direction of travel. The equation i a follow: x p F f x p =σ x p + ign x p [F c0 + F 0 e c ] In order to make it continuou and monotonou, an approximation can be ued [10]. Thi approximation make the function invertible: ign x p 2 arctan (γx p) π thu, x p = 2 x p arctan (γx p) π INVERSE SIMULATION AND EFFICIENCY ANALYSIS OF SYSTEM The explicit tate differential equation in thi cae are a follow: p A = 1 (Q c A (xv, p A )- A p x p+q Li ) ha 1 p B= (Q c B (xv, p B )+ A p x p- Q Li ) hb Copyright 2013 by ASME

8 x p= 1 m t [(p A -αp B )A p -F ext F f x p ] where Q A and Q B are function of x v a illutrated by equation and repectively. The incidence matrix for the forward imulation i created and i hown in Table 1. The piton poition x p i calculated firt from the initial condition of the preure p A and p B of the cylinder chamber. Then, the new preure derivative are calculated a hown by equation and in Table 1. Any numeric olver can be applied to calculate the next tate obtained by the next time increment. The eaiet olver which i applied in our cae i the Forward-Euler olver that i baed on calculating the next time tate from the previou tate and it derivative. Table 1: Forward Simulation Lower Triangular Form x p p A p B x x x Now, in order to olve for the invere, the boundary condition are interchanged where the new input to our ytem i the piton poition x p and it derivative, and the deired output i the valve opening x v. The explicit equation dealt with are the ame equation , , and of the forward imulation. In thi cae, equation i a contraint equation between tate and mut be derived in order to get thi equation a a function of p A and p which are the unknown B variable. The equation become a follow: x p = 1 [(p A-αp B)A m p -F ėxt F f x p ] t i A Dummy-derivative method can be ued then [11] in cae where the ytem i tructurally ingular due to contraint tate equation. One of the tate derivative i conidered a a dummy derivative which i conidered now a algebraic variable. If the chamber B preure derivative i conidered a dummy derivative, then p B become p B and it relative tate i called dummy tate. Therefore, the ytem will conit now of , , , and i knowing that p B i an algebraic dummy derivative. Thi technique i ued to keep track of the information that the tate are related and to compenate for drift condition during imulation pecially with tiff ytem [12]. The incidence matrix for thi ytem of equation become a hown in Table 2. Table 2: Invere Simulation Block Lower Triangular Form p A x v p B x x x x i x x In order to olve thi BLT, one of the way ued by Modelica i through the ue of ymbolic manipulation by making equation i an explicit function of x v through replacing equation and in equation i. After manipulating i explicitly a function of x v, x v become the only unknown that can be olved for a in the cae of the forward imulation with the help of the initial tate condition a hown in equation ( A p x p x v = + A p x p α 2 c ha c hb α K li c hb ( p p A c ha (p B p A ) + m t x p + F f x p + F ėxt ) A p + α p B p T ) cv c hb After calculating x v at time t=0, the preure derivative p A and p B are calculated from x v. The BLT then become an LT a hown in Table 3, and the ytem can be olved with any ODE olver uch a the Forward-Euler olver dicued above. Table 3: Invere Simulation Lower Triangular Form x v p A p B x x x x x In cae we need to know the input voltage to operate the valve, we add an additional equation which i: x v = 2D v ω v x v ω v 2 x v + K v ω v 2 u And the invere incidence matrix become a hown in Table 4. Table 4: Invere Simulation Lower Triangular Form Larger Sytem x v u p A p B x x x x x x x Figure 15 indicate that if eparate model containing the equation of the valve and cylinder were built in Simulink which i baed on variable aignment, it i not poible to interchange the direction of the boundary condition and the data flow in order imulate the ytem inverely, becaue the preure and flow in both the cylinder and valve are interdependent requiring to know their value beforehand to proceed with the invere imulation. Therefore, in order to 8 Copyright 2013 by ASME

9 perform the imulation in Simulink, the whole ytem hould be implemented after the caualization tep with all it equation manipulation a hown earlier. Figure 15: Backward Simulation Approach in Simulink Meanwhile, after writing the equation and building the model in Modelica, it i poible to imulate the hydraulic ervo-drive either in a forward fahion by providing voltage input to the valve and accordingly attaining the poition and velocity of the hydraulic piton actuator or inverely by providing the poition of the of the cylinder a an input and imulating backward to attain the required valve opening xv and accordingly the required valve voltage input u a it wa hown. Thi i poible thank to the feature of flattening where all the equation from all model are found in one global et a dicued earlier, and thu allowing the tate pace repreentation of the equation, their orting, and the olution of the required one. It wa hown that it i feaible to build eparate valve and cylinder model and imulate the ytem inverely; however, currently the two hydraulic librarie available in Modelica do not allow the invere imulation of hydraulic drive for reaon that are till under invetigation by the author of thi paper. acceleration which i minimum twice differentiable. Thi i done either by introducing a filter to the input ignal or the ignal i differentiable by itelf a in our cae where it i a ine wave [3]. Table 5: Simulation Parameter Parameter Value F ext 1000 N p 75E5 Pa p T 0 Pa α 1(ynchronizing cylinder) cv c 0.01 m D v 0.5 K v 0.1 ω v m t rad kg D (piton large diameter) 0.04m / 0.05m d (piton mall diameter) m S 0.2 m E 15000E5 Pa Q Li 0 Q Le 0 V p1,a 10-5 m 3 V p1,b 10-5 m N 100 N γ 800 σ -4 N. 10 F c0 F 0 The ytem model etup in Modelica i een in Figure 16 below. m Table 5 repreent ome of the parameter that were ued in the imulation of our model. Becaue the aim of the invere or backward imulation i to help the deign engineer in hi election of the izing parameter of the ytem, 2 cae for different piton diameter ize are conidered in the imulation and the reult of the 2 imulation are then compared in order to ee which ize i more efficient to deign the ytem knowing that the deign parameter provided in thi paper are not the bet choice but jut to provide an idea about the technique and that the deign engineer hould keep imulating to have an optimal deign. Figure 16 how the Block.Math.InvereBlockContraint that wa ued to directly interchange ytem input and output; thu, providing the real output deired of our ytem which i the cylinder poition xp a an input and calculating inverely the required input which i the voltage u. An important note when applying the invere i that the input to the ytem hould be differentiable at leat to the number of time one need to differentiate it for example in our cae to acquire an Figure 16: Dymola Sytem Model Figure 17 how the output deired for our piton which i fed a input to the invere imulation. 9 Copyright 2013 by ASME

10 Figure 17: Poition Input The output of the imulation, which i the real voltage input to the valve i hown in Figure 18 for two different configuration. It i een that the voltage required for the valve to provide thi output poition peak to V at 0.5 from tart of imulation for a cylinder piton outer diameter of D=0.05m choen by the deign engineer; wherea, it peak at around -6.51V for a maller choen piton outer diameter of D=0.04m. Figure 20: Power Conumption of Smaller Piton Moreover, and mot importantly, the efficiency of the ytem in both cae i compared and hown Figure 21. It i een that for the maller piton diameter, the efficiency i higher than for a larger diameter. However, one hould note that the efficiencie are till low and the deign engineer can adjut hi parameter in order to reach more efficient ytem. Figure 21: Efficiency Comparion Figure 18: Voltage Output In term of power conumption, Figure 19 and Figure 20 illutrate each of the power conumption for the 2 cae. The figure how that for a bigger piton diameter, the power input peak at W at t=0.5 and the power loe in the valve which alo peak at W at t=0.5 are higher than thoe for a maller piton diameter where the input power required peak at W at t=0.5 and the maximum power lo from the valve i W at t=0.5. One more comparion obtained from the imulation and een in Figure 22 i that the valve i underized in the cae of the larger cylinder piton indicating the need of a larger valve with a larger opening; wherea, for the maller cylinder, the valve doe not operate above it maximum opening limit thu it i working in range and it ize allow the ytem to reach it deired output. Figure 22: Valve Opening Figure 19: Power Conumption of Larger Piton Eventhough the valve allow the ytem having the maller cylinder to operate, it doe not mean that it i the bet valve choice, becaue with the help of the invere imulation, the 10 Copyright 2013 by ASME

11 deign engineer i entitled to tet all the parameter combination poible to elect the mot efficient valve and cylinder ize to be implemented in the later tage of deign. CONCLUSION Thi paper illutrate the concept of invere or backward imulation and it importance when applied on the deign tage of ytem. A illutrated, it i the ame a the forward imulation with the interchange of the input/output boundary condition. It i tediou to obtain the invere ytem by hand; however, Modelica with it equation manipulation technique provide a great advantage becaue it allow the inverion of ytem automatically. Thi paper demontrate the importance of invere imulation in Modelica. We believe that Modelica can become a platform on which deign engineer can rely on for ytem izing and control deign at the ame time. The example of hydraulic ervo-drive demontrated the advantage that the backward or invere imulation provide to the deign engineer in electing hi ytem deign parameter before tarting to apply a controller to the ytem. In thi way, better and more efficient ytem conuming le energy can be deigned in the future. The hydraulic ervo-drive example did not make ue yet of already available librarie. Currently, the author of thi paper are invetigating the reaon why the current hydraulic librarie in Modelica do not allow invere imulation and develop an alternative library which will be compatible with both forward and invere imulation. [6] Zimmer, D. (2011, November 22). (Lecture note). Retrieved from [7] Elmqvit, H., & Otter, M. (1994). Method for Tearing Sytem of Equation in Object-Oriented Modeling. Proceeding ESM'94 European Simulation Multiconference, (pp ). Barcelona. [8] Lindfield, G. R., & Penny, J. E. (2012). Numerical Method uing Matlab (Third Edition ed.). Waltham, MA, USA: Elevier. [9] Jelali, M. K. (2003). Hydraulic Servo-ytem (Vol. 1). London. [10] Liermann, M. (2012). Backward imulation - A tool for deigning more efficient mechatronic ytem. Proceeding of the 9th International Modelica Conference, (pp ). Munich. [11] Matton, S. E., & Söderlind, G. (1993). Index Reduction in Differential-Algebraic Equation uing Dummy Derivative. Siam J. Sci. Comput., 14 (3), (pp ). [12] Bachmann, B. (2012). Mathematical Apect of Object- Oriented Modeling and Simulation. 9th International Modelica Conference, Tutorial 2. Munich. ACKNOWLEDGMENTS The author would like to expre their thank to the American Univerity of Beirut for it upport. REFERENCES [1] Otter M., T. M. (2005). Nonlinear Invere Model for Control. Modelica 4th International Conference, (pp ). Hamburg. [2] Fro berg, A. (2006). Invere Dynamic Simulation of Non- Quadratic MIMO Powertrain Model -Application to Hybrid Vehicle. Vehicle Power and Propulion Conference (pp. 1-6). Windor: IEEE. [3] Bal, J., Hofer, G., Pfeiffer, A., & Schallert, C. (2003). Object-Oriented Invere Modelling of Multi-Domain Aircraft Equipment Sytem with Modelica. Proceeding of the 3rd International Modelica Conference, (pp ). Linkoeping. [4] Fritzon, P. (2004). Principle of Object Oriented Modeling and Simulation with Modelica 2.1. Linkoping: IEEE Pre. [5] Åtröm, K. J., Elmqvit, H., & Matton, S. E. (June 16-19, 1998). Evolution of Continuou-Time Modeling and Simulation. The 12th European Simulation Multiconference, (pp. 9-18). Mancheter, UK. 11 Copyright 2013 by ASME