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Corresponding author: yqj Research on Synchronization Control Strategy for Hydraulic Transmission of Tracked Vehicle Yang Qingjun*, Zhang Zhao, Zhang Shenghui School of Mechatronics Engineering Harbin Institute of Technology Harbin, China e-mail: yqj Lv Qingjun, Yang Lei, Xiong Qinghui Science and Technology on Vehicle Transmission Lab China North Vehicle Research institute Beijing, China AbstractSynchronization problem was the principal problem and technological challenge for hydrostatic driving tracked vehicle in straight working conditions. The asymmetry caused by parameter perturbations, friction resistance coefficient, leakage coefficient and pump displacement are taken into consideration. These factors were treated as disturbance inputs, and were suppressed by control algorithm. A simulation model of system was developed by Matlab/Simulink and synchronization control strategies for hydraulic transmission of tracked vehicle were designed. The coupling system was decoupled with matrix theory and unilateral closed loop correction was completed based on lag-lead corrector. Then equalizing synchronous circuit and cross-coupling synchronous circuit were applied to reduce the synchronous error. Finally, auto-disturbance rejection controller (ADRC) was proposed. The simulation results showed that the cross-coupling synchronous circuit had better resistance to parameters asymmetry than equalizing synchronous circuit. ADRC improved the synchronous performance further and had a better robustness to the time-varying system. Keywordshydraulic transmission; tracked vehicle; synchronous circuit; Auto-Disturbance Rejection Controller I. INTRODUCTION Recently, with the development of technology, tracked vehicles are applied more and more widely. They gradually extended to military vehicles, engineering vehicles, agricultural machinery, and many other fields. As a typical transmission way of tracked vehicle, hydrostatic transmission can implement infinitely variable speed and steering by variety of regulation and control 1. Yang Lei 3 studied on the steering of hydrostatic driving tracked vehicle mainly, which was worth learning for straight driving. The hydraulic transmission of tracked vehicle generally adopts two sets of variable- displacement pump driving variable-displacement motor. Wang Yan6 improved the robustness of the system by means of bang-bang controller. K.Dasgupta et al7 had a further study on hydrostatic transmission, establishing foundation for application to tracked vehicle. Although hydraulic synchronization control circuits11 have been developed systematically, the adaptability to hydraulic transmission of tracked vehicle remains to be analyzed. This paper focused on the synchronization of hydrostatic driving tracked vehicle caused by asymmetry of pump and motor parameters and running conditions of the two driving units. The linear model was build, decoupled and corrected. Simulation comparison was made among different circuits and ADRC to achieve the excepted robustness and adaptability. II. MODELING AND LINEARIZATION A.Variable-displacement pump driving variable- displacement motor Generally, the variable-displacement pump driving variable- displacement motor works in normal working conditions when tracked vehicle goes straight. The inside and outside motors both provide tractive effort and output power. According to hydraulic theory, the system can be represented by a set of equations. 0 1 1 () (1,2) () () ii piippipirepi i piimiremiimmi e iimir au Ts qa DCppC p i V dp qCppCpb D dt Tb Dpp ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? where i a is the normalized displacements of pump,Tis the time constant of the variable-displacement pump, i uis the control signal, pi q is the output flow rate of pump, p Dis the full displacement of motor; p ?denotes the speeds of the pump; ip C, ep C, im Cand em Care the inside and outside leakage coefficients of pump and motor; i pand r pare the pressure of the high-pressure cavity and the low-pressure cavity respectively; i b are the normalized displacements of motor, m Dare the full displacement of motor; mi ?is the speed of motor; 0 Vis the total volume of high-pressure chamber ; e ?is the hydraulic elastic modulus; i Tis the motor load; 12i? ?, representing the different unit. B.Tracked vehicle The force analysis diagram of tracked vehicle is shown in Fig. 1. F1 and F2 are the tractive efforts; FR1 and FR2 are the friction resistance force; w Fis air resistance; Bis the distance of track centers; Lis the length of crawler contacting ground; ? 2015 International Conference on Fluid Power and Mechatronics 978-1-4799-8770-2/15/$31.00 2015 IEEE August 5-7, 2015 Harbin, China M?is the steering resisting torque; Ris turning radius; 1 Vand 2 V are the velocity of inside and outside caterpillars; c Vis the speed of the center of vehicle ; ?is the steering angular velocity. w 1R F 2R F B O C 1 C 2 C 1 V 2 V c V M? R L 1 F 2 F w F Fig.1 Force analysis diagram of tracked vehicle According to the tracked vehicle dynamics, the model can be represented as follows: 1212 2112 12 max 2 ()2 2 0.50.5 (1,2 0.5 1 0.1544 0.925 1 2 cRRw RR RR i i ii c wDc mVFFFFF JFFFFBM fmg FF Ti mgmg r Fi TiTi mg rr GL MGL V B FC A V ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? where m is the mass of tracked vehicle; J is the moment of inertia of tracked vehicle;fis the frictional resistance coefficient;?is the ground adhesion coefficient; ?is the transmission efficiency; iis the transmission ratio ;ris the radius of driving wheel; max ?is the maximum steering resistance coefficient. A is windward area of vehicle; CD is the coefficient of air resistance; ?is air density; Because of the asymmetry of those system parameters, two motors speeds differ from each other and synchronous error occurs. Friction resistance coefficient, leakage coefficient and pump displacement are treated as disturbance inputs. Hence, assuming f, Ct and a are variable, the Laplace transform of the above equation can be established. C.Linearization ,analysis and correction The hydraulic transmission of tracked vehicle is a system with dua1-input dual-output. The total working process of variable-displacement pump driving variable-displacement motor is divided into two stages: variable-displacement pump driving fixed-displacement motor at low speed and fixed-displacement pump driving variable- displacement motor at high speed. The displacements of motor and pump are set to be the maximal value for the two stages respectively. Taking the variable-displacement pump driving fixed- displacement motor as an example, u1and u2 are the control signals of main channel, n1and n2 are selected as the output signals. The system is linearized around certain constant speed, and the Bode diagram of the system is shown in Fig 2. It can be seen from Fig. 2 that there is strong coupling effect between the inside and outside of caterpillar vehicles. In order to control the system accurately, decoupling is essential. -40 -20 0 20 40 60 Magnitude (dB) 10 -1 10 0 10 1 10 2 -360 -180 0 Phase (deg) Bode Diagram Frequency (rad/sec) u1-n1 u1-n2 a) u1-n1 and u1-n2 -40 -20 0 20 40 60 Magnitude (dB) 10 -1 10 0 10 1 10 2 -360 -180 0 Phase (deg) Bode Diagram Frequency (rad/sec) u2-n2 u2-n1 b) u2-n1 and u2-n2 Fig 2 Bode diagram Introducing a new control vector A by equation ? ? ? 32 32 1111 22 1111 22 GG BB A sA s GG BB ? ? ? ? ? ? ? ? ? ? , The system is then converted to a pseudo linear system between output V and pseudo input Aas follows. ? ? ? ? 1 2 0 2 0 d V sA s d B ? ? ? ? ? where pp m D w i D r ? , 2 2 2 m i D r ? ? 10Dc cC A V? max0 2 2 00 0.15 4 0.15 0.925 c c mgLV c B wV B ? ? ? ? ? ? ? 1 0 1 2 2 c t e d V mscCs ? ? ? ? ? ? ? ? ? 2 0 2 2 2 c t e d V JscCs B ? ? ? ? ? ? ? ? ? ? 0 1 2 0 2 2 2 c t e c t e V mscCs G V JscCsB B ? ? ? ? ? ? ? ? ? ? ? ? ? , ? ? 0 2 3 0 1 2 2 c t e c t e V JscCsB B G V mscCs ? ? ? ? ? ? ? ? ? ? ? ? ? ? The results of decoupling show that the transfer matrix and coupling matrix are related to speed. Hence, the system is time-varying. Lead-lag compensators are designed as follow. 1 0.21 0.171 ( )0.39 10 +10.011 ss G s ss ? ? ? 2 0.21 0.291 ( )0.04 101 0.011 ss G s ss ? ? ? where 1 G is matched with x1-n1 channel; 2 G is applied to x2- n2 channel. The Bode diagrams of the correction system are shown in Fig. 3. The figure shows that after correction the system is stable and the dynamic characteristics are close. III.ANALYSIS OF SYNCHRONOUS CIRCUIT On the basis of lead-lag compensator, the control effect is compared and analyzed in equalizing synchronous circuit and cross-coupling synchronous circuit 11. -40 -20 0 20 40 60 Magnitude (dB) 10 -2 10 -1 10 0 10 1 10 2 -270 -225 -180 -135 -90 -45 0 Phase (deg) Bode Diagram Frequency (rad/sec) Corrected Original a) u1-n1and u1-n2 -40 -20 0 20 40 60 Magnitude (dB) 10 -2 10 -1 10 0 10 1 10 2 -270 -180 -90 0 Phase (deg) Bode Diagram Frequency (rad/sec) Corrected Original b) u2-n1and u2-n2 Fig 3 Bode diagram The principle of synchronous circuit is shown in Fig 4. 1 C ( ) s 1( ) K s 1( ) G s 2 C ( ) s 2( ) K s 2( ) G s r 1 y 2 y y 1( )f Gs 2( )f Gs 1f 2 f 1 e 2 e 1 u 2 u ? ? ? ? a) Equalizing synchronous circuit 1 C ( ) s 1( ) K s 1( ) G s 2 C ( ) s 2( ) K s 2( ) G s r 1y 2 y y 1( )f Gs 2( )f Gs 1f 2f 1e 2 e 1 u 2 u ? ? ? ? ? ? ( ) f Gs 3 K 4 K ? ? ? ? b) Cross-coupling synchronous circuit Fig 4 Synchronous circuits ? According to control theory, the controller of cross- coupling circuit isdesigned to 0 0.21 0.451 ( )0.039 101 0.011 ss G s ss ? ? ? , whose cut-off frequency, phase margin and gain margin match the main channel. Given the same variation of system parameters, the control effects of different synchronizing strategies are shown in Fig. 5. 0123 0 0.005 0.01 0.015 0.02 t(s) w(rad/s) Original Equalizing Cross-coupling a) Turning velocity 0100200300 0 5 10 15 20 25 30 x(m) y(m) Original Equalizing Cross-coupling b) Vehicle locus Fig 5 Comparison of synchronization performances The simulation results show that compared with equalizing synchronous circuit, the cross-coupling synchronous circuit has better synchronous performance. IV.AUTO-DISTURBANCESREJECTION CONTROLLER (ADRC) The lead-lag compensator is designed based on the accurate model while the system is time-varying after decoupling. So, when the speed of linearization point changes, the anti-disturbance capacity of cross-coupling synchronous circuit cant achieve the ideal effect. The hydraulic transmission of tracked vehicle is 3 order system, so the ADRC is as shown in figure 6. v 1 v 2 v ? ? 1 e 2 e 0 u u w y b 1 b ? 3 v ? 3 e 1 z 2 z 3 z 4 z Fig. 6 ADRC The auto-disturbance rejection controller is composed of three parts: tracking-differentiator, extended state observer and nonlinear state error feedback control law14. According to the separation theorem, the three parts of ADRC can be designed separately. A.Tracking differentiator Because of the tracked vehicle is a third order system, assume that the input is applied to the system through a third order system. The transfer function is 3 1 3 ( ) () k vw s vv sk ? ? . The state variable equations is: 12 23 31023 ( ( ()3)3 ) vv vv vk k k vvvv ? ? ? ? ? ? ? ? ? ? ? where h is the simulation step and kis the speed factor which decides the tracking speed. Based on the system, the parameters are set as 10k ?0.001h ? B.Extended state observer(ESO) The ESO provides the estimation of the unmeasured systems state and the real time action of the unknown disturbances. Then it improves the performance of the system by compensating for the estimated equivalent disturbance. It regards the total effects of variations of parameters as an equivalent disturbance acting at the system control input point. The state equation of system is: 12 2 34 1 3 4 ( ) xx xx xxbu xr t yx ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? The state equation of ESO is: 11 121 1 232 1 343 1 44 1 ezy zzl e zzl e zzl ebu zl e ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? where 1 x represents the speed of motor, 4 x is disturbance,( )r t is the rate of disturbance change,? 123 Llll? is the state feedback matrix. The observer is linear, whose characteristic equation is 432 1234 sl sl sl sl?. Pole assignment method is used to locate all the poles at -w. Based on the system, the parameters are set as 200000b?.120w ?.Then,L can be calculated. ? C.Feedback composition Generally, in the feedback composition of ADRC, nonlinear functions are used to improve the performance of system. But it is difficult to adjust parameters. Combined with the three state feedback method of hydraulic system, control signal is generated by linear feedback. The equation of state feedback is: 111 222 333 01 1223 3 evz evz evz uf ef ef e ? ? ? ? ? ? ? ? ? ? where, 123 Ffff?is the state feedback matrix. The control signal is compensated with disturbance observation by 4 0 z uu b ?. Using the principle of state feedback and pole assignment, the state feedback matrix is given as: 0.18690.01750.0004F ? Compare the control effect of ADRC controller with cross coupling synchronization control. 0123 -0.01 -0.005 0 0.005 0.01 t(s) w(rad/s) ADRC Cross-coupling a) turning velocity 0100200300 -1 0 1 2 3 4 5 x(m) y(m) ADRC Cross-coupling b) offset Fig 7 Comparison between cross-coupling control and ADRC The figure shows that compared with the cross coupling control mode, ADRC controller has a better effect since the offset is only about 20%. Change the speed of tracked vehicle to observe the Robustness of ADRC to time-varying system. Considering that variable-displacement pump driving fixed-displacement motor works in low speed condition, v=3m/s, v=5m/s and v=7m/s are taken as examples. The results are shown in Fig8. 0123 -8 -6 -4 -2 0 2 4x 10 -3 t(s) w(rad/s) v=3m/s v=5m/s v=7m/s a) turning velocity 0100200300 -1.5 -1 -0.5 0 0.5 x(m) y(m) v=3m/s v=5m/s v=7m/s b) offset Fig 8 Robustness of ADRC at different speeds The results show that ADRC does not depend on the model. When the vehicles speed deviates from linearization point, it still can maintain good ability of keeping synchronism. ADRC is not influenced by model parameters and the controller has a good adaptability for the system. ADRC views the precise factors in the model as comprehensive disturbance. It is not designed for the precise model. Hence, ADRC can adapt the time-varying system. V. CONCLUSIONS The synchronization performance of a dual variable displacement pump driving variable displacement motors has been studied. Several control strategies has been investigated and good performance has been achieved. 1)The system has strong coupling effect and hence a decoupling controller has been developed as a basis of other control strategies. 2)The cross-coupling controller based on lag-lead controller for each side has better performance compared with equalizing synchronization control. 3)Adaptive disturbance rejection control has the best performance since it observes all unideal factors as disturbance injecting into the system at the control input, and cancel the equivalent disturbance accordingly. It also performs well at different speed. ? REFERENCES 1?ang Likun et al. Research on Classification and Evolvement Law for the Driveline of VehicleJ. Machine Design & Research, 2013(04): 42- 47 2Yang Lei et al. Analysis of hydrostatic transmission technology of tracked vehicle C National equipment lubrication and hydraulic Academic Conference,2007. 3Liu Bin et al.? ?Overview of Power System Development of Tracked Vehicle J Journal of Sichuan Ordnance,2014(01):68-72 4Yang Lei et al.?Theoretical Study on the Torque Control Project for Hydrostatic Drive Tracked Vehicles J China Mechanical Engineering,2009(21): 2632-2637 5Yang Lei et al. Steering Control Strategy of High-speed Hydrostatic Drive Tracked Vehicle J Transactio