FLOW RATE CONTROL OF VARIABLE DISPLACEMENT PISTON PUMP USING GENETIC ALGORITHM TECHNIQUE

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1 FLOW RATE CONTROL OF VARIABLE DISPLACEMENT PISTON PUMP USING GENETIC ALGORITHM TECHNIQUE Ayman A. Aly 1,2 and A. Al-Marakeby 1,3 1) Mechatronics Section, Mech. Eng. Det., Taif University, Taif, 888, Saudi Arabia 2) Mechatronics Section, Faculty of Engineering, Assiut University, Assiut, 71516, Egyt 3) Systems & Comuters Det., Faculty of Engineering, Al-Azhar University. Abstract:- One of the most imortant comonents in hydraulic circuits is the um that generates hydraulic ower suly. The erformance and the versatility of a variable dislacement iston um are, to a large extent, determined by its controller. Ugrading of existing controllers is considered one of the major means of imroving the characteristics of the um. The dynamic erformance of hydraulic systems with a demand flow suly not only deends on the erformance of the flow modulation valve, but also on the erformance of the um. In order to increase the reliability, controllability and utilizing the suerior seed of resonse achievable from electrohydraulic systems, further research is required to develo a control software that has the ability of overcoming the roblems of system nonlinearities. In the resent work, a Proortional Integral Derivative (PID) controller is designed and attached to electrohydraulic servo system to control its swash-late angular osition. The PID arameters are otimized by the Genetic Algorithm (GA). The controller is verified on the state sace model of servo valve attached to a rotary actuator by SIMULINK rogram. The aroriate secifications of the GA for the rotary osition control of an actuator system are resented. It is seen that the use of the roosed methodology results in better erformance. Key words: Genetic Algorithm, Flow Rate Control, Variable Dislacement Axial Piston Pum. 1. Introduction Variable dislacement iston ums have found widesread alication in the field of fluid ower industry. The most common way to change the flow rate of a um is to change its dislacement or iston stroke when it is oerated under a constant rotational seed. A variable dislacement um is designed such that the dislacement can be varied from zero to some maximum value while the um is oerating. The

2 variable dislacement axial iston um which is shown in Fig.1 has many alications in fluid ower systems. Changing the angle of the swashlate can change the iston stroke. Since the dislacement of the um is roortional to the iston stroke, the dislacement can be changed by varying the angle of the swashlate, [10]. A DC motor is directly couled to the intle of the swashlate as shown in Fig. 2. It is anticiated that a DC motor should rovide a raid dynamic resonse to the um swashlate. The reason for this anticiation is that the maximum torque rovided by the DC motor is about 60 Nm [5], which is much higher than the torque generated by its hydraulic counterart (13 Nm to fully stroke the um). Then, the dynamic resonse of the um flow rate should be increased. Further, it is much easier to integrate a DC motor to an electronic feedback circuit. This design strategy enables sohisticated electronic control algorithms to be alied for the DC motor controller. Since a DC motor can initially locate the swashlate at any angular osition, even at zero osition, it is much easier to control the initial flow rate of the um and to build the system ressure using this design. Because there is no return sring in the um, the torque generated by the DC motor is mainly used to overcome the friction torque and the back torque [13] roduced by the um ressure. Fig.1 Schematic of variable dislacement Piston um Fig.2 Direct swashlate control with a DC motor Several investigators [7, 17] have alied research about the dynamic roerties of a variable dislacement iston um. Most of these investigations are based on a linearized model of the um dynamics. In industrial alication, the dynamic characteristics of the variable delivery um are always comlex and highly nonlinear, [14]. Moreover, there are too many uncertainties in it; as the viscosity of the oil, the bulk modulus, leakage coefficient, equivalent torque coefficient, volumetric dislacement and others. So, the design of such ums control flow at different um ressure levels needs various controllers that cause the um outut to match different load characteristics more efficiently and effectively. The design of these controllers,

3 however, is often based on comromise and thus their erformances are very oerating condition deendent. Over the ast a few years, many different techniques have been develoed to acquire the otimum control arameters for PID controllers. The academic control community has develoed many new techniques for tuning PID controllers. They have not been slow in seeking to exloit the emerging methods based on the rinciles of evolution. A GA is one such direct search otimization technique which is based on the mechanics of natural genetics. An advantage of the GA for auto-tuning is that it does not need gradient information and therefore can oerate to minimize naturally defined cost functions without comlex mathematical oerations. This article describes the alication of GA Technique based on a fitness function to otimally tune the three terms of the classical PID controller to regulate the osition of the um swash-late as a nonlinear rocess. 2. Mathematical Model of the DC Motor Controlled Pum One aroach to model a dynamic system is to use linear or small signal analysis. The linear analysis method is based on the assumtion that a linear transfer function can be used to describe the behavior of the lant over the comlete oerating range. On the other hand, the small signal analysis method assumes that the lant behavior is nonlinear but the model can be linearized over a small range near an oerating oint. Both methods are very owerful analytical tools but have limitations, esecially for a highly nonlinear dynamic system such as the DC motor controlled um. In this study, the um is modeled using nonlinear large signal techniques which are reresented by a series of differential equations. Although it is difficult to analyze the dynamic erformance of a nonlinear model using conventional control theories (transfer function aroaches), it is feasible to do this using a simulation rogram. The flow rate is determined by the angle of the swashlate which, is controlled using a ermanent magnet servo DC motor, [5]. From the viewoint of the um control, the DC motor can be considered as a art of the um. Hence, the model of the DC motor is also a art of the um model. The mathematic model of a DC motor can be derived using a schematic diagram of the motor circuit shown in Fig.3. The DC motor is assumed to consist of inertia, J d, with daming, B d. The torque develoed by the current in motor windings not only overcomes the friction in the DC motor and load torque, T dl, on the motor shaft but also accelerates the rotor, [4].

4 Fig.3 Schematic diagram of a DC motor Fig.4 Pum-controlled system with relief valve The electrical circuit of the motor can be simly described by: di V Vemf Ri L dt. Vemf K b (2) The torque develoed at the shaft of the motor is roortional to the armature current and given by: K i J.. B. sgn(. )( T T ) T (3) t d d ds dc The friction torque consists of three terms: static friction, coulomb friction, and viscous daming. Normally, the static friction and coulomb friction of the DC motor are negligible comared to that of the um swashlate. This is evident by the effortless torque that is required to manually turn only the shaft of the DC motor. Neglecting the static and coulomb friction and taking Lalace transforms of equations 1 to 3 yields the model of the DC motor as described by the following transfer function. KtV ( s) ( Ls R) Tdl( s) ( s) (4) s(( Ls R)( J dms Bdm) Kt Kb ) The numerator of equation 4 includes two terms. One term is the inut signal and the other one is the load, which can be considered as a disturbance inut signal. In 1987, Kavanagh [7] develoed a comrehensive model for a variable dislacement axial iston um which is used as the basis for modeling the um in this study. The um model consisted of the torque model and fluid flow model. The motion of the swashlate is described by the torque model; and the flow rate of the um is described by the flow model. Some general assumtions are made regarding the um model. They are: Constant rime drive seed on the um, Zero suction and drain ressure, Constant chamber volume Constant fluid density and temerature. dl (1)

5 In Kavanagh s study, the swashlate is controlled by a control iston and balanced by a return sring. However, in this study, the swashlate is actuated by a DC motor. Under these conditions, Kavanagh s model can be simlified to yield J.... θ Td S1 S2θ sgn(θ )Tfc B θ K 1P K 2Pθ (5) The dislacement of the um is defined as follows: D NA R P tan / (6) Assuming that the rotational seed of the rime mover is ω, the ideal flow rate of the um is as follows: Q idea D NA R tan / (7) The actual flow rate of the um is less than the ideal flow rate due to the fluid leakage and fluid comression. There are two tyes of leakage flows in the um, one is the internal leakage flow between the suction ort and the discharge ort of the um and the other is the external leakage from the high-ressure chamber to the case drain through the um casing. From the continuity equation, the flow equation of the um can be written as V dp Qidea Qi Qe Q (8) e dt Since the suction ressure is assumed to be zero, the leakage flow of the um (including the internal leakage and the external leakage flow) can be aroximated by Q Q Q C P (9) l i e t Substituting equations.7 and 9 into equation.8, yields V dp NA R tan / CtP Q (10) e dt As shown in Fig.4, there is a two-stage relief valve (RV), worked as a constant resistive load. It is used to adjust the backressure on the system. 3. Closed Loo Control System The feedback signal is the angular osition of the um swashlate, which is also the controlled variable. The closed loo system is including a controller, a ower amlifier, a DC motor and a variable dislacement um. The urose for controlling the swashlate angle is to control the flow rate of the um. Before designing the controller, it is imortant to determine the dynamic erformance of the DC motor and um swashlate assembly. As a result, a model of the DC motor and um is attemted. Based on this model, a motor controller is designed based on Ziegler- Nichols turning PID rules, [17]. A tyical PID controller has the following transfer function form,

6 K I Gc ( s) K K d s (11) s As the system gains changed with ressure changes, the critical gain and oscillation frequency are not the same under different loading. It is interesting to note that at the same ressure level, the um oeration tended to be stabilized by decreasing the gain and destabilized by increasing the gain. On the other hand, at the same gain, the um tends to be stable with increasing the ressure and unstable with decreasing the ressure. Thus, the um demonstrates a highly nonlinear characteristic which is strongly deendent on the oerating ressure and controller gains, [15]. So at every oerating oint a new PID gains setting is needed. However, the linear PID is difficult to aly to this highly nonlinear lant. Even with a erfect feedforward controller, a feedback controller is also required to correct for noise and unmeasured disturbances. The requirement for the controller design at this stage is to design a DC motor controller which could drive the DC motor and um swashlate at any ressure levels with a fast dynamic resonse but without exhibiting any limit cycle oscillations. Many methods can be used to design the controller for a dynamic system; however, most of them are limited to linear systems. As a roosed solution, a GA based PID controller is used. 3.1 Genetic Algorithm Genetic algorithms (GA) are abstract imlementations of natural evolutionary rocesses used to solve search and otimization roblems [1]. GAs include the oerators: initialization, selection, crossover, and mutation, in addition to four control arameters: oulation size, selection ressure, crossover and mutation rate[zhang]. Initialization Initially many individual solutions are randomly generated to form an initial oulation. The oulation size deends on the nature of the roblem, but tyically contains several hundreds or thousands of ossible solutions. Traditionally, the oulation is generated randomly, covering the entire range of ossible solutions (the search sace). Occasionally, the solutions may be seeded in areas where otimal solutions are likely to be found,[16]. Selection During each successive generation, a roortion of the existing oulation is selected to breed a new generation. Individual solutions are selected through a fitness-based rocess, where fitter solutions (as measured by a fitness function) are tyically more likely to be selected. Certain selection methods rate the fitness of each solution and referentially select the best solutions. Other methods rate only a random samle of the oulation, as this rocess may be very time-consuming [8].

7 Mutation Mutation is a genetic oerator that alters one or more gene values in a chromosome from its initial state. This can result in entirely new individuals being added to the oulation. With these new individuals, GAs may be able to arrive at a better solution, [16]. Fig 5 shows the mutation rocess. Fig.5 mutation rocess in GA Genetic algorithms are used in closed loo systems to imrove the erformance of the system and to find best design arameters. Mehrdad has used GAs to design an adative PID controller, [9]. He used a Genetic Algorithm imlemented in hardware for setting the K values of the PID Controller and minimizing the integral error in the system. Zhang designed a self-organizing genetic algorithm (SOGA) based tuning of PID controller with good global search roerties and a high convergence seed, [16]. Renato designed an otimal disturbance rejection PID Controller using genetic algorithms, [12 ]. He formulated the design as a constrained otimization roblem. It consists of minimizing a erformance index, i.e., the integral of the time weighted squared error subject to the disturbance rejection constraint. In this system we used the GA to find the best values of K, K i, and K d of the PID controller to imrove the system erformance. Fitness Function Fitness function is very imortant in system design using GA. The good formulation of fitness function hels in obtaining better results. Fitness function has chosen in our system to increase the rise time and decrease overshoot and oscillation. The rise time and overshoot is measured numerically in simulation using numerical analysis method. 4. Results and discussion By using Ziegler-Nichols method the PID gains K, K I, and K d will be [ x ], and it will roduce an overshoot resonse of the swashlate and consequently of the um flow rate as dislayed in Fig.6 and 7. As overshoot of

8 the um flow rate increases, the overshoot of the hydraulic motor outut and the dynamic load also will increase. So the PID control need to imrove its gains setting to give minimum overshoots as ossible. The trail and error technique is considered to adjust manually the PID gains to get the acceted resonse of the um flow rate. Fig. 6 the swashlate angle with reference 10 Deg. based on PID control action Fig. 7 The um flow rate Based on PID control action To imrove the erformance of the system we used Genetic Algorithm to find the best values for K, Ki, and Kd. Fig.8 shows the best and mean values of fitness function. We used oulation size 10 with gausian mutation and scatterd crossover. The fitness function has been chosen to minmize the rise time and overshoot. The result values of Kd, Ki, and K is 731, 0.53, and 0.74 resectively after running the genetic Algorithm. Secification of GA Poulation size 10 Crossover rate 0.8 Mutation Gaussian Fig. 8 Genetic Algorithm best and mean values of fitness functions In Fig. 9, the system resonse based on the roosal G-PID controller has a robustness behavior because of at every ressure level the controller model can kee an adequate swashlate angle to regulate the flow rate regardless to the ressure level.

9 Fig. 9 the swashlate angle with reference 10 Deg. based on G-PID control action Fig.10 system resonse To imrove the dynamic resonse of the um, a DC motor is imlemented to control the um swashlate (and hence flow rate) directly. The um and DC motor are mathematically modeled and their arameters are simulated via MATLAB 6.3 and SIMULINK 5.0 software. By means of the DC motor s quick dynamic resonse, the DC motor controlled um demonstrated a fast dynamic resonse indeendent on the um ressure. For recise control, a GA is introduced. The strategy of design is also given. It is imortant to note that the GA have the distinct advantage of not relying on the system arameters and it deal with system as a black box. The simulation results indicate the accuracy of tracking. Also the effectively of the roosed controller with the system nonlinearities is cleared. It gave a robustness resonse, as the ressure changes, the um does not affect to a great extent the outut flow rate accuracy. 5. Conclusion Genetic Algorithm is a owerful tool to solve many comlex roblems. In this work, a Proortional Integral Derivative (PID) controller is designed and attached to electrohydraulic servo system to control its swash-late angular osition. The PID arameters are otimized by the Genetic Algorithm (GA). The aroriate secifications of the GA for the rotary osition control of an actuator system are resented. It is seen that the use of the roosed methodology results in better erformance.

10 5. Reference [1] A. AL-Marakeby,FPGA on FPGA: Imlementation of Fine-grained Parallel Genetic Algorithm on Field Programmable Gate Array, International Journal of Comuter Alications, Volume 80 No.6, October 2013 [2] D. Nauck and R. Kruse, 1999, Neuro-Fuzzy Systems for Function Aroximation, Fuzzy Sets and Systems (101) [3] G.S. Virk and A. Al-Dmour, Jan.1994 System Simulation Using Neural Networks, Deartmental Research Reort No. 537, University of Bradford. [4] Habibi, S., 2001, "Lecture Notes: Control System I", Deartment of Mechanical Engineering, University of Saskatchewan, Canada. [5] HT-High Torque, Direct Drive Series, Emoteq Inc., Tulsa, Oklahoma [6] J. Jantzen, 30 Oct 1998, Neurofuzzy Modelling, Bldg 326, DK-2800 Lyngby, DENMARK Tech. reort no 98-H-874 (nfmod). [7] Kavanagh, G. P., 1987, "The Dynamic Modelling of an Axial Piston Hydraulic Pum", MSc Thesis, Deartment of Mechanical Engineering, University of Saskatchewan, Canada. [8] Manoj Kumar1, Mohammad Husian2, Naveen Ureti3 & Deeti Guta4,GENETIC ALGORITHM: REVIEW AND APPLICATION, International Journal of Information Technology and Knowledge Management, Volume 2, No. 2, 2010 [9] Mehrdad Salami and Greg Cain,An Adative PID Controller Based on Genetic Algorithm Processor, Genetic Algorithms in Engineering Systems: Innovations and Alications,se., 1995 [10] Merritt, H., 1967, "Hydraulic Control Systems", John Wiley & Sons, Inc., New York, [11] R.J. Schalkoff, 1997, Artificial Neural Networks, McGraw-Hill, Inc. [12] Renato A. Krohling and Joost P. Rey, Design of Otimal Disturbance Rejection PID Controllers Using Genetic Algorithms, IEEE TRANSACTIONS ON EVOLUTIONARY COMPUTATION, VOL. 5, NO. 1, FEBRUARY 2001 [13] Tonglin Shang, 2004, Imroving Performance of an Energy Efficient Hydraulic Circuit, MSc. University of Saskatchewan, Saskatoon, Saskatchewan, Canada. [14] Wahid, A. El-Sayed, and Yehia. El-Mashad, May 2002, Fuzzy Control of Highly Sensitive Dynamics Behaviours of Variable Dislacement Axial Piston Pum, roceeding of the 10 th Int. AMME conference, Egyt. [15] You, Z., 1993, Sliding Mode Control of A Variable Dislacement Hydraulic Pum, PhD. Thesis, University of Saskatchewan, Saskatoon, Canada.

11 [16] Zhang Jinhua, Zhuang Jian, Du Haifeng, Wang Sun an,self-organizing genetic algorithm based tuning of PID controllers, Information Sciences 179 (2009) [17] Zeiger, g and Akers, A., 1986, Dynamics Analysis of an Axial Piston Pum Swashlate Control, roceeding of Instaan. Mech. Engrs art C, [18] Ziegler, J. G. and N. B. Nichols, 1942, "Otimum Settings for Automatic Controllers", ASME Trans. 64 (1942), Nomenclature A Area of the iston, (8305x10-6 m 2 ) B d Viscous daming coefficient, (1.43 x 10-3 Nm.rad -1 s) B Daming coefficient of the swashlate yoke assembly, (0.28 Nm.rad -1 s) C t Total leakage flow coefficient, (4.3 x10-13 m 3 s -1.Pa -1 ) D Dislacement of the um, (1.95 x10-6 m 3.rad -1 ) i Armature current, (A) J d Moment of inertia of the motor rotator, (1.4x10-3 Nm.rad -1 s 2 ) J Average moment of inertia of swashlate yoke assembly, (1.06 x10-3 Nm.rad -1 s 2 ) K b Back EMF constant of the DC motor, (2.27 V.rad -1 s) K d Derivative gain Integral gain K i K Proortional gain K 1 Pressure torque constant, (7.46 x10-7 Nm.Pa -1 ) K 2 Pressure torque constant, (8.3x10-7 Nm.Pa -1.rad -1 ) K t Motor torque sensitivity, (2.27 NmA -1 ) L Terminal inductance of the DC motor windings, ( H) N Number of istons, (9 istons) P Pum ressure, (Pa) Q e External leakage flow of the um, (m 3 s -1 ) Q i Internal leakage flow of the um, (m 3 s -1 ) Q Outut flow of the um, (m 3 s -1 ) R Terminal resistance of the DC motor windings, (4.83 Ω) R Radius of the iston itch, ( m) S 1 Simlified um model constant, (0.096 Nm) S 2 Simlified um model constant, (2.36 Nm.rad -1 ), T d Torque alied to the yoke by the DC motor, (Nm) T dc Coulomb friction torque, (Nm) T dl Load torque acting on the DC motor shaft, (Nm) T ds Static friction torque, (Nm) T e Motor electrical time constant, (6.87 x10-3 sec) Torque roduced by the coulomb friction force, (0.36 Nm) T fc

12 T m Motor mechanical time constant, (1.3 x10-3 sec) V Inut voltage, (V) V emf back EMF voltage, (V) V Volume of the um forward chamber, (3x10-5 m 3 ) β e Bulk modulus of the fluid, (1.45x10 9 Pa) θ Angular osition of the DC motor shaft and um swashlate, (rad) ω Pum rotational seed, (183.3 rad s -1 ) γ Daming factor.