# Modeling, simulation and control of high speed nonlinear hydraulic servo system

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2 28 D. Maneetham & N. Afzulpurkar: High speed nonlinear hydraulic servo system the processing speed of micro-controller. At the same time, PD controller is designed to verify the performance of the model, which is developed by simulation and compared to experimental results. The paper is organized as follows, In Section 2, a linearized mathematical model for the system is presented. Section 3 presents the development and stability analysis of PD controller. Section 4 describes hardware system architectures. Simulations and experimental results are presented in Section 5 and finally the paper is concluded. 2 Mathematical model We describe detailed mathematical modeling of HSS for hydraulic mini press machine. The System consists of high-speed, electronic drives, hydraulic actuators, and position transducers. The mathematical model behavior of servo valves can be developed from the relationship between the displacement (x p ) and input voltage (µ) for the proportional valve. A third order model is sufficient for HSS which is described by the equations given in following sections [11]. 2.1 System modeling In pressing machine, the hydraulic actuator is typically a double-acting hydraulic piston. The cylinder ports are connected to a proportional valve, and piston motion is obtained by modulating the oil flow into and out of the cylinder chambers. A servo valve provides this modulation as shown in Fig. 1. The actuator can be precisely controlled by regulating the flow rates Q 1 and Q 2. However, the relationship between the piston position (x p ), and the flow rates depends on the dynamic properties of the load acting on the piston [1]. cylinder piston load x P m P1, Q1 P2, Q2 spool valve C tp x v PR return port P S supply port PR return port Fig. 1. A hydraulic actuator with four-way valve configuration The mathematical model used in simulation is represented by the following system of equations: Where V t 4β e P L = Aẋ C tp P L + Q L. (1) V t : total actuator volume; β e : effective bulk modulus; P L : load pressure; A : actuator ram area; x p : actuator piston position; C tp : total coefficient of leakage; Q L : load flow. WJMS for contribution:

3 World Journal of Modelling and Simulation, Vol. 6 (21) No. 1, pp Hydraulic actuator system dynamics The objective for developing the actuator system dynamics is to construct a strict-feedback control with fixed boundary layer to obtain precise position control of a nonlinear electro-hydraulic servo system. In order to represent servo valve dynamics through a wider frequency range, a first order transfer function is used as approximation of the valve dynamics. The relation between the servo valve spool position x y and the input voltage u can be considered as a second order system. T (s) = x v(s) u(s) = ωe 2 s 2 + 2ζω e s + ωe 2 ; x v s 2 = ωex 2 v 2ζω e x v s + ωeµ; 2 x v s 2 + 2ζω e x v s + ωex 2 v = ωeµ; 2 ẍ v = ωe 2 2ζω e ẋ v + ωeµ. 2 (2) Substituting ẍ v = ẋ v we obtain the valve velocity ẋ v as: ẍ 5 = ωex 2 4 2ζω e x 5 + ωeµ. 2 (3) where, T (s) is the transfer function, ω e is the undamped natural frequency and ζ is the damping ratio of the system. Defining the load pressure P L as P L = P 1 P 2 and the load flow Q L as Q L = (Q 1 + Q 2 )/2, the relationship between the load pressure P L and the load flow Q L for an ideal critical servo valve with a matched and symmetric orifice can be expressed as follows: The piston force equation is given by: Q L = C d wx v P s sgn(x v )P L. (4) ρ P L A = mẍ + kx + F f. (5) The equation of motion of the load mass is mẋ 2 = F hyd, where m is the total mass of the load, and F hyd = AP L is the hydraulic force due to pressure differential across area, we obtain mẋ 2 = F hyd = AP L = (F spring + F vis cos city ) + F load pressure. (6) Defining the load pressure from Eq. (1) and Eq. (4). where P L = ( ) ( 4βe A 4βe C tp ẋ V t P L = αẋ βp L + V t ) P L + ( 4βe V t ) Q t, (7) ( 4βe V t C d w ρ ) x v Ps sgn(x v P L ), (8) α : (4β e A/V t ); β : (4β e C tp /V t ); γ : (4β ecd w/v t 1/ρ ). Combining Eq. (1) Eq. (5) with other system parameter results in the system state equation given below: ẋ 1 = x 2 ; ẋ 2 = 1 m (Ax 3 F f ); ẋ 3 = αx 2 βx 3 + (γ ) P s sgn(x 4 )x 3 x 4 ; ẋ 4 = x 5 ; ẋ 5 = ω e 2x 4 2ζω e x 5 + ω e 2µ. (9) WJMS for subscription:

4 3 D. Maneetham & N. Afzulpurkar: High speed nonlinear hydraulic servo system 2.3 Hydraulic servo system parameter identification The data is separated into estimation data, which is used for identifying unknown system parameters and measurement data. Both factors are used for modeling in the experiments [4]. In order to excite all the relevant frequencies of the systems and to construct a good model, the frequencies are set the sinusoidal inputs with range of 1 to 6 Hz and pressure of 5 Mpa. In conventional design of a hydraulic servo system, third order transfer function is generally used, as given below: Gs = K q ω 2 n A 1 s(s 2 + 2δω n s + ω 2 n). (1) In frequency response analysis, we measure the amplitude of oscillations at the signal frequency. At first, we carried out a set of experiments using the open loop system to determine the amplitudes of oscillations, which occur at the signal frequency. In order to observe the signal frequency component of the response only, experiments are carried out using signal frequencies of 1, 1.5, 2, 2.5, 3, 3.5, 4, 4.5, 5, 5.5 and 6 Hz at 5 Mpa pressure. Two different values of signal frequency are used in the experimental determination of sine wave and step response. Overall, the system identification is done by fitting a third order polynomial. The system transfer function is found to be: 9272 G(s) = s s e4s. (11) The correspondence levels between predicted model and the experimental data for second order and the third order models are about 74.16% and 8.77%, respectively. Fig. 2. shows a comparison between measured frequency responses. In this experiment, the input test signal during 1 to 6 Hz is applied as the estimation data. Magnitude(dB) Estimated1 Estimated2 Measured Frequency (Hz) Phase(degree) Estimated1 Estimated2 Measured Frequency (Hz) Fig. 2. Bode plot of the open loop transfer function: (estimated 1) second order and (estimated 2) third order Then the parameter values are varied by optimization until best fitting is reached. Matlab identification toolbox software is used to create the mathematical model of the system. The process-modeling tool is selected to customize the structure of identified model based on the knowledge of the second order and third order of hydraulic plant. Finally, comparison between the experimental outputs and its predicted model using estimation data is obtained as shown in Fig. 2. Results reveal similarity between the measured and simulated response is fitting curve. WJMS for contribution:

5 World Journal of Modelling and Simulation, Vol. 6 (21) No. 1, pp Simulink model A simulink model for HSS was constructed and verified using an experimental system. The dynamic model includes load flow, pressure, velocity, force, and other dynamics in hydraulic systems. Simulation test included tests of the sensitivity of observer with respect to changes in the model parameters. The parameter variations used in simulations for input, output, velocity, pressure, and flow rate. The system parameters were varied so as to keep the damping coefficient ζ constant. The values of the system and observer parameters are given in Tab. 1. Table 1. Physical parameters for the HSS Name System Nominal value Unit Supply pressure P s 5 16 P a Total actuator volume V t m 3 Effective modulus β e N/m 2 Actuator ram area A m 2 Actuator piston position x v.2 m Total leakage coefficient C tp m 3 /(s.p a) Discharge coefficient C d.61 Spool valve area gradient w m Spool valve position (stroke) x m Load flow Q L m 3 /sec Fluid mass density ρ 87 kg/m 3 Mass of actuator and load m 9.19 kg The main idea of a new structure of HSS is the simulink model of an open loop system. Eq. (1) is obtained in system state equation of the HSS. At first, the system is characterized by position control (x 1 ). The piston stroke is ±.2m, mass of actuator and load m = 9.9kg. The friction relation is approximated as linear F f :.363N. Secondly, the system is characterized by velocity control. Let ẋ 1 = x 2 be a differentiable function, given by ẋ 1 dt = x 2 dt : i. e, we obtain the result in terms of x 1 = 1 s (x 2). The response is a set of the velocity and its top most displacement range is ±.16m/s. Referring the Eq. (9) [ẋ 2 = 1 m (Ax 3 F f )] where 1 m is gain. The line block is required for each channel, where velocity x 2 is calculated. For that, area & force is added in to the block & friction is subtracted from the block. Fig. 3. The simulink model of the open loop system WJMS for subscription:

6 32 D. Maneetham & N. Afzulpurkar: High speed nonlinear hydraulic servo system The third method is load pressure system x 3. This is the most effective method. The pressure of the system is set considering the pressure of the load. Pressure limit is the supply pressure P L : 5Mpa. The value of P L is determined by substituting P L = x 3. To solve this equation, we may assume a solution x 3 of the form: ( ) 1 x 3 = Ax 2 + Q L V t. (12) 4β e + C tm Overall, the main idea of the modified hydraulic subsystems is to use valve position x v, valve velocity (ẋ v ) and input current (µ). The upper and lower limits in servo valve are equal to the stroke of the servo valve x v : ± m and the maximum flow limit through the valve Q L : m 3 /s. There are more models that can be used. At first line, the rate limiter block limits the first derivative of the signal passing to the saturation block. The saturation limit of the input signal is reached, which are carried out for a range of signal frequencies passing through the transfer functions of hydraulic subsystems. The block diagrams are generated Fig. 4. Modified of force, velocity, and friction model using force and velocity from friction line. When friction line is greater than or equal to the threshold of the switch block, the switch block switches that changes to one of its three inputs to pass via the output. According to the absolute block, static friction and kinematics friction are selected for switch function block. To get static friction, we used signum function for velocity line. 2.5 Simulation result The results of an open loop test and corresponding simulation model is presented for 1 Hz sine signal. From the result the correlation between the control signal frequency and error signal can be compared as follows: Open loop response Open loop response control voltage (v) 5-5 control voltage (v) Fig. 5. Experimental of the open loop system for 5 Mpa and 1 Hz sine wave Fig. 6. Simulink model of the open loop system for 5 Mpa and 1 Hz sine wave WJMS for contribution:

7 World Journal of Modelling and Simulation, Vol. 6 (21) No. 1, pp Open loop response Open loop response Fig. 7. Experimantal: velocity of the open loop system for 5 Mpa and 1 Hz sine wave Fig. 8. Simulink model: velocity of the open loop system for 5 Mpa and 1 Hz sine wave Open loop response Open loop response Fig. 9. Experimental: velocity of the open loop system for 5 Mpa and step response Fig. 1. Simulink model: velocity of the open loop system for 5 Mpa and step response 3 Control design The PD regulators for process control is the basis for many hydraulic control systems. Many variations in the basic PD algorithm substantially improve its performance and operability. A PD controller is parameterized by the following transfer function. Y (s) R(s) = This can be compared with the following, general third-order characteristic C(s)G(s) 1 + C(s)G(s). (13) Thus from Eq. (14) we obtain G(s) = 9272 s s s. (14) G(s) = 9272K d (s + Kp K d ) s s 2 + ( K d )s K p. (15) The parameters of the PD controller are obtained as K p = 1.46 and K d =.74. This controller is applied to both the plant and model. WJMS for subscription:

8 34 D. Maneetham & N. Afzulpurkar: High speed nonlinear hydraulic servo system 4 Hardware system architecture The HSS must follow the control theory guidelines, which is the main purpose to improve the velocity of piston in HSS. The hardware system of HSS is divided into two parts, which are explained below. 4.1 Hardware design Firstly, we construct the mechanical model of an electro-hydraulic system. The simulated response of the model provides insight into the behavior of electro-hydraulic system. As shown in Fig. 11. (1) is the 2 F= 1 A B 3 4 P T 5 6 ADC Microcontroller DAC Fig. 11. Schematic diagram of the electro-hydraulic position control system linear potentiometer, (2) is double cylinder, (3) is servo valve, (4) pressure relief valve represent fluid flows in out of the valve, (5) pressure unit is the input and output line pressures and (6) is the micro-controller to control system. The HSS consists of a hydraulic pump, servo valve, actuator, transducer, power supply, and micro-controller. Hydraulic system model is shown in Fig. 12. A micro-controller based control system has been developed and used to control the hydraulic servo system. We used micro-controller PIC 18F458 to control the hydraulic servo system, in conjunction with the data acquisition processor. The schematic of micro-controller system design is shown in Fig. 13. The mass flow rate across the five-port valve is controlled by manipulating the spool offset, by controlling the current supplied to the solenoid. 5 Simulation and experiment results The model is based on electro-hydraulic system control. A simulation of Matlab/Simulink based study is used to evaluate the performance of the PD controllers on the nonlinear model of HSS and compared with the performance of the controllers on linearized approximation of the system for displacement and velocity control. 5.1 Simulink model for closed loop control The core of simulink is the HSS for hydraulic mini press machine. The parameters of a model must be identified for designing control algorithm and the steps of the identifying dynamic models of the hydraulic system involved designing an experiment, selecting a model structure, choosing a criterion to fit, and devising a procedure to validate the chosen model. Figure below shows the simulink model with Matlab program. WJMS for contribution:

9 World Journal of Modelling and Simulation, Vol. 6 (21) No. 1, pp Digital Pressure Gauge Power Unit Power Supply Pressure Gauge Pressure sensor P1 Cylinder Pressure sensor P2 Potentiometer Mass Oscilloscope Encoder Function Generator Linear Guide Filter LCD Counter Keypad Microcontroller Control Box Proportional Valve Amplifier Fig. 12. Electro-hydraulic servo system model Fig. 13. Schematic of microcontroller system design In simulink model, recall that for open loop system we used simulation for input and output with frequency response. Matlab simulink model is built such as, continuous microcontroller controls block, discrete microcontroller controls block, velocity, force and friction block and linear potentiometer block. The proposed model system is designed based on the use of a single-chip microcontroller PIC 18F458 with the EPROM emulator for programming the computer software. The block diagram of velocity is shown in Fig. 15. WJMS for subscription:

10 36 D. Maneetham & N. Afzulpurkar: High speed nonlinear hydraulic servo system Fig. 14. Matlab simulink model of closed loop system Fig. 15. Block diagram of velocity, force, and friction control 5.2 Simulation results The simulation is used to examine the effect of errors in the system parameters. These parameters include the total leakage coefficient, effective bulk modulus system mass, friction terms and so on. The results of simulations are presented in this section. Following are the system parameters used for simulation m = 9.19kg, A = 2.4e 4, P s = 5e6P a, ζ =.7, beta = 29.5, V t = 2.44e 5m 3, F f =.247N, β e = 7.6e8N/m 2, w =.239m, alpha = 7.2e9, C m = m 3 /(s.p a), gamma = 2.917e9, C d =.61, and C o = e Experimental set up A set of experiments are performed to study the effect of variation in supply pressure, hydraulic parameters, environmental stiffness and position control on velocity piston in HSS. In order to test the performance of developed system, we compared the system output with simulink model. A typical closed loop control hydraulic system consists of power supply with 4 lit/min flow rate and 5 Mpa supply pressure. Load piston position is measured by FESTO Model TP 51 with 2 mm length of stroke and four way 3 state proportional directional control valve, which is measured by VICKERS Model KBSDG4V-3. Linear potentiometer from FESTO Model FP112 is used to record the position of the cylinder and DUPOMATIC Model PTH-1/2E pressure sensors are used to measure P1 and P2 (Vickers). WJMS for contribution:

11 World Journal of Modelling and Simulation, Vol. 6 (21) No. 1, pp Validation To verify the model, both static and dynamics tests are performed in simulation and compared with the experimental results. In case of the model validation and PD controller experiments, the controller is modeled using simulink approach in Matlab program. Experimental results for PD controller which are based on the control of position and velocity of position at an operating supply pressure of 5 Mpa are given below. During Fig. 16. Observer performance of step responses with K p = 1 and K d = position (m) position (m) Fig. 17. Unit step responses of experiment with Kp = 1 and Kd =.74 Fig. 18. Unit step responses of simulation with Kp = 1 and Kd = Fig. 19. Unit velocity responses experiment with Kp = 1 and Kd =.74 Fig. 2. Unit velocity responses simulation with Kp = 1 and Kd =.74 WJMS for subscription:

12 38 D. Maneetham & N. Afzulpurkar: High speed nonlinear hydraulic servo system Fig. 21. Unit velocity responses simulation [4] Fig. 22. Unit velocity responses simulation [1] Fig. 23. Unit velocity responses simulation [5] the experimental test, only linear position of the movable cylinder is measured for feedback information. The velocity is obtained via differentiating the position with respect to time at high speed and pressure. To test the effectiveness of the proposed controllers, we first compared it with a high-speed control system. The proposed control velocity is shown in Fig. 2, with the gains K p = 1, K d =.74, rise time =.28s, P s = 5Mpa, mass = 9.19kg, β e = P a, velocity =.125m/s. In [4], the control result of the proposed controller is shown in Fig. 21, with corresponding feedback gains P s = 5Mpa, β e = P a, mass = 2Kg, Q = 28l/ min, velocity =.165m/s. In [1], the control result of the proposed controller is shown in Fig. 22, with corresponding feedback gains P s = 2Mpa, β e = P a, distace = 2mm, velocity =.13m/s, mass = 5kg. In [5], the control result of the proposed controller is shown in Fig. 23. with corresponding feedback result time =.35s, velocity = 1.25m/s, overshooting is less than 5%. The difference in the results shown in Fig. 2 demonstrates that the proposed controller is much better than Fig. 21, Fig. 22 and Fig. 23, in terms of high speed, transient response and settling time. For the hydraulic mini press machine, the smaller the settling time is, the higher the production rate will be. Therefore, it is significant for practical applications. In addition, overshoot is much smaller. This is essential for hydraulic mini press machine, because a large overshoot will generate a large impact force, which would damage the sheet metal forming. 6 Conclusion We have presented deviation, simulation, and implementation of the nonlinear control law for HSS. The proposed controller provides performance of the PD controller for high-speed control. PD controller theory is introduced as the control technique to accomplish this goal in this study, and the controllers designed using WJMS for contribution:

13 World Journal of Modelling and Simulation, Vol. 6 (21) No. 1, pp this method are validated using experimental tests. From these tests, it can be seen that, for hydraulic systems, which have nonlinear characteristics, control theory provides a powerful control strategy that clearly improves on PD control in terms of high speed and transient response. The result shows the time constant of simulation and experiment, proportional - derivative controller with K p = 1, K d =.74. The comparison of the results between simulink and hardware system for optimal PD controller indicates the improvement of the simulation research, where the reference velocity is.125m/s. Thus, the response of the model gives a good control performance prediction of the PD controller. References [1] A. Alleyne, R. Liu. Systematic control of a class of nonlinear systems with application to electro-hydraulic cylinder pressure control. IEEE transactions on control systems technology, 2, 8(4): [2] J. Ferreira, P. Sun, J. Gracio. Close loop control of a hydraulic press for springback analysis. Journal of Materials Processing Technology, 26, [3] R. Fung, Y. Wang, et al. A variable structure control with proportional and integral compensations for electrohydraulic position servo control system. Mechatronics, 1996, 7: [4] H. Khan, S. Abou, N. Sepehri. Nonlinear observer-based fault detection technique for electro-hydraulic servopositioning systems. Mechatronics, 25, [5] C. Lin, L. Ren. The integrated Simulation Analysis on 6-DOFElectrohydraulic Servo-Control Parallel Platform. JSME International Journal, 25, 48(9): [6] H. Merritt. Hydraulic Control System. John Wiley & Son, Inc, [7] Moog. Industrial control division. [8] V. Product. Proportional Directional valve with Feedback. [9] M. Sirouspour, S. Salcudean. On the Nonlinear Control of Hydraulic Servo System. in: International Conference on Robotics & Automation, San Francisco CA, 2. [1] P. Sun, J. Gracio, J. Ferreira. Control system of a mini-hydraulic press for evaluating springback in sheet metal forming. Journal of Material Processing Technology, 26, [11] T. Zhao, T. Virvalo. Fuzzy control of a hydraulic position servo with unknown load. IEEE, 1995, WJMS for subscription:

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