 Proceedings of Engineering and Technology Innovation, vol. 8, 2018, pp. 15 -22 A Direct Lyapunov-Backstepping Approach for Stabilizing Gantry Systems with Flexible Cable T. L. Nguyen 1,2,* , M. D. Duong 1 , and T. H. Do 1 1 School of Electrical Engineering, Hanoi University of Science and Technology , Vietnam. 2 Institute for Control Engineering and Automation, Hanoi University of Science and Technology , Vietnam. Received 11 January 2018; received in revised form 10 February 2018; accept ed 06 March 2018 Abstract Trolley positioning and payload swinging control problem of a fle xible cable gantry crane system are addressed in this paper. The system’s equations of motion that couple the crane’s cable and actuators dynamics are derived via e xtended Ha milton’s principle. The control signal is designed based on the Lyapunov direct meth od to derive control force and backstepping technique is e mployed to determine input signal for the actuator. The stability of the c losed loop system is proven analytically. Nu merical simulat ions are included to demonstrate the effectiveness and robustness of the closed-loop system. Keywor ds: flexible systems, overhead crane, field oriented control, Lyapunov direct method 1. Introduction Nowadays, gantry crane systems are widely used in industrial and logistic applicat ions because of their fle xibility in load handling. However, swinging payload phenomenon causes slowing down goods handling operations and can be a potential threat to human and surrounding devices. Certain types of payload s can ignite multi-modes or double-lin k pendulum effects [1-4]. In addition, characterized as a class of under-actuated systems, precisely controlling trolley position and suppressing payload vibration simultaneously pose many challenges for control engineers. In order to overco me the a fore mentioned control proble m, various approaches are considered. A conventional robust linear control la w is proposed to control the overhead crane [5]. Since crane system is a nonlinear coupling system, instead of linear control, many researchers focus on nonlinear control approaches. A decoupling control law is proposed to asymptotically stabilize tro lley position and swing angle of the payload [6]. However, the designed control only guarantees bounded swing angle. An improve ment [7] is made with varying crane rope length that is meaningful in practice is considered. A switching control action is derived based on feedback linearization technique. Position control and vibration suppression o f gantry crane with coupling e ffect between trolley and payload mot ions are ta ken int o account [8]. However, the obtained results are relatively limited in practice because of system’s parameters and actuator’s dynamics variat ions. In order to dea l with system uncertainty, an adaptive mechanis m is integrated in proposed control la w suggested [9]. Well-known with its robustness against system uncertainties and disturbances, sliding mode control is applied in gantry control [10]. Ho wever, it is need to cooperate with a pre-shape input to gain better performances [11]. Severa l adaptive schemes [12-14] fo r gantry control also presented. Intelligent control schemes are also considered to control the crane system such as fuzzy control [15] or neural network [16]. Instead of feedback controls, some other researches consider feed -forwa rd control approaches where control actions from operators are modified befo re sending to the gantry actuators as shown [17-18]. The advantage of pre-shape input technique * Corresponding author. E-mail address: lam.nguyentung@hust.edu.vn Proceedings of Engineering and Technology Innovation, vol. 8, 2018, pp. 15 -22 Copyright © TAETI 16 over feedback control is that measure ment of system states is not required but a fu ll knowledge of the system must be availab le. To rectify this drawback, pre-shape input method can be hybridized with a robust control as indicated [19-20]. The limitation of afore mentioned studies is an assumption of pendulum mot io ns for the payload. The assumption results in a system of ordinary differentia l equations govern system motions. However, practica l applications have shown that it is not the case, and gantry cable actually considered as a fle xible system whose motions are modeled as a system of pa rtial diffe rential equations. Boundary controllers for stabilizing the fle xib le rope crane system based on Lyapunov’s direct approach are developed [21-23]. The fle xib le rope where coupled longitudinal-t ransverse, transverse-transverse motions and 3D model are investigated [24-27]. Dyna mics of the rope without model truncation is investigated, however the dynamics of the actuator are totally ignored. The ignorance of the actuator dynamics might to system instability . This paper directly designs a gantry control in consideration with fle xib le rope but in other direction. We con struct a distributed model of the overhead crane in which the mass and the fle xible of payload suspending cable are fully taken into account. The analytical mechanics inc luding Ha milton ’s principle is used to construct the crane model. Based on the obtaine d model, the controller is then designed systematically with the help of backstepping control. The rest of the paper is organized as fo llo ws. The mathe matica l mode l of fle xible overhead crane system is presented in Section 2. In section 3, control design is developed based on the Lyapunov direct method and backstepping technique. The numerical simu lation is performed in Section 4 to show the efficiency of the proposed control design. Conclusions and further studies are presented in Section 5. 2. Mathematical Model An overhead crane system is illustrated in Fig. 1, where Fig. 1 A gantry crane system ( , )y x t and dy are the transverse motions of the crane’s cable and the target position of the payload, respectively. 0P is the cable tension, is the mass per unit length of the cable , and L is the length of the cable. M and m are tro lley’s and payload’s masses, respectively. The force (t)F is generally generated by an induction motor. The kinetic energy of the system is given as 2 2 2 0 0 ( ) ( ) ( ) 2 2 2 L y y LM y m T dx t t t (1) Proceedings of Engineering and Technology Innovation, vol. 8, 2018, pp. 15 -22 Copyright © TAETI 17 In Eq. (1) and fro m now onward the argu ment ( , )x t is o mitted for neat representation. In addit ion , y(0) and y(L) are used to denote y(0,t) and y(L,t), respectively. The potential energy can be expressed as 2 0 0 ( ) 2 LP y P dx x (2) where 0 P is the tension of the cable. The work done by the external control force is given as W ( ) ( )F t y L (3) Re mark 1. Bending stiffness of the cable is considerably s mall so that potential energy due to bending stiffness can be ignored. The cable is assumed to be inextensible, and the cable deforms in Oxy plane. The extended Hamilton’s principle is expressed as follows 2 1 ( W) t t T P dx (4) Substituting Eqs . (1), (2) and (3) into Eq. (9) results in 2 1 2 2 2 2 0 02 2 2 0 (0) (0) (0) (0) ( ) (0) 0 0 t L t t Ly y y y y P dx P M y dt m y F t y dt xt x t t (5) Using integration by parts, the equations of motions and boundary conditions of the crane system can be given as 0 0 tt xx y P y (6) 0 x (0) (0) ( ) tt My P y F t (7) tt 0 x ( ) ( ) 0my L P y L (8) The force acting on the trolley (t)F can be calculated in term of the motor torque as (t) M b i F m R (9) where b R is the radius of the dru m, i is the transmission ratio of the gearbo x and is the e ffic iency of the t ransmission system. Mathematical model of the asynchronous motor can be written as follows : 1 1 1 1 1 sd sd s sq rd rd sd r r s i i i u T T L (10) 1 1 1 1 1 sd s sd sd rd rd sq s r r s i i i u T T T L (11) 1 1 rd sd rd s rd r r i T T (12) 1 1 rq sq s rd rq r r i T T (13) 3 2 M m p rd sq m K z i (14) where sdi and sqi are d irect and quadrature co mponents of stator current. 𝑇𝑟 and 𝑇𝑠 are rotor and stator time constants, pz is number of pole pairs, and is total magnetic leakage factor. 𝐾𝑚 = 𝐿𝑚 𝐿𝑟 , where 𝐿𝑚 and 𝐿𝑟 are mutual and rotor inductance. Proceedings of Engineering and Technology Innovation, vol. 8, 2018, pp. 15 -22 Copyright © TAETI 18 rd rd m L and , rq rq m L where rd and rq are dq components of the rotor flu x. The coupled electrical -mechanica l sys tem is rewritten as follow: 0 0 tt xx y P y (15) 0 x (0) P y (0) tt sq My i (16) 1 2 3 4sq sd sq sq i i i u (17) 0 x (L) P y (L) 0 tt my (18) where 1 2 3 1 1 1 , , s rd s r T T (19) and 4 1 3 , 2 m p rd s b i K z L R (20) Re mark 2. Eqs. (15)-(18) is derived under a condition that the rotor flu x o rientation is obtained, i.e., 0. rq Moreover, it is assumed that sd i and rd are kept constants by current and flu x controllers and their values are available for feedback. In addition, the current controller has the ability of decoupling 𝑖 𝑠𝑑 and 𝑖 𝑠𝑞 . 3. Control Design The control objective is to simu ltaneously stabilize the t rolley and the payload at the desired position. An investigation of the system g iven in Eqs. (15)-(18) shows that the system is of strict-feedback form. Hence, in this paper, backstepping technique will be e mployed to design the control input.𝑢𝑠𝑞 The choice of back-stepping as a design tools make it ready if system para meters adaptation is needed. The control design process comprises of two steps. In order to satisfy the control objective, at first we take sq i as a control and define. sq z i (21) where is a virtual control. Consider the following Lyapunov candidate function 2 2 2 20 d 0 0 W (L) (0) [ (0) y ] 2 2 2 2 2 L L t x t t P m M y dx y dx y y y (22) where D is a strictly positive constant. It is straightforward to show that V can be lower and upper bounded as below 2 2 2 2 2 1 0 0 W [ (L) y (0) [ (0) ] L L t x t t d y dx y dx y y y (23) and 2 2 2 2 2 2 0 0 W [ (L) y (0) [ (0) ] L L t x t t d y dx y dx y y y (24) where 2 2 2 2 2 1 0 0 1 min( , , (L), (0), [ (0) ] ) 2 L L t x t t d y dx y dx y y y y (25) Proceedings of Engineering and Technology Innovation, vol. 8, 2018, pp. 15 -22 Copyright © TAETI 19 and 2 2 2 2 2 2 0 0 1 min( , , (L), (0), [ (0) ] ) 2 L L t x t t d y dx y dx y y y y (26) Taking time derivative with respect to Eq.(26),then 0 t x 0 0 1 1 0 W = P (L) (L) (0)[ (0)] ( (0) ) (0) (0){ [ (0) ]} L t x t x d t t d y y P y y y z P y y y y y z y y (27) Eq. (27) suggests that virtual control 𝑎1 can be chosen as follows (0) [ (0) ] t d ky y y (28) where k is a strictly positive constant. In the second step, the actual control input 𝑢𝑠𝑞 is designed to regulate 𝑧1 at the origin. To archive this target, we consider a Lyapunov candidate function as follows 21 W 2 V z (29) Taking time derivative with respect to Eq. (29), it yields 2 1 1 1 2 3 4 (0) [ (0) ( 1) (0)] 1 t sd sq sq tt t V k y z i i u k y y (30) The actual control input 𝑢𝑠𝑞 can be derived as 34 1 2 1 1 (0) ( 1) (0) sq sd sq tt t u i i k y y (31) where 2 k is positive constant. Remark 3: The control input sd u can be derived based on backstepping control method as follows 2 1 1 2 2 1 2 22 ˆ ˆ1 1 1 1 1ˆ ˆ( )( ) rd rd sd sd s sq rd sd rd r r r r d d u i i c i c T T c z z d z s T r T dt Tdt (32) where 2 2 2 1 1 ( ) ( ) r T (33) 1 c and 2 c are strictly positive constant, 1 z and 2 z are errors between desired and virtual control when designing the flu x controller. The control design is co mpleted and it is straightforwa rd to show that with the selected control input sq u render the first time derivative of the Lyapunov candidate function V as 1 2 (0) 0 t V k y k z (34) Eq. (34) shows that (t)V is upper bounded by (0)V . This consequently implies that ( )ty L , (0)ty and (0)t dy y are bounded. Further investigation of Eq. (34) can prove exponential convergence to dy of (0)y and ( ).y L 4. Numerical Simulations In order to verify the effectiveness of the proposed feedback control. Simu lations are carried out using an induction motor of 7.5kW (other motor power ranges can be applied with no loss of generality). The closed loop system is simu lat ed in Proceedings of Engineering and Technology Innovation, vol. 8, 2018, pp. 15 -22 Copyright © TAETI 20 Matlab/Simulink environment. Simulation parameters are given in Table 1. Table 1 Induction motor parameters Parameter Value Nominal power PN = 2,5kW Pairs of pole Pp = 2 Nominal current UN = 340V Nominal speed n = 1400 rpm Stator resistance RS = 2.521Ω Stator inductance LS= 0.1825 H Rotor resistance RS = 0.976 Rotor inductance LR = 0.1858 H Inertial moment J = 0,117 kg.m2 The simulat ion is carried out with the mass of the trolley is of 100kg, payload mass and cable length are of 400kg and 5m, respectively. Simu lation scenario is to regulate trolley and payload position to a desired position of 5m fro m the init ial condition. It is assumed that initially the positions of the trolley and payload are coinciding. Fig. 2 System response of the trolley and payload without control Fig. 3 System response of the trolley and payload with control Fig. 2 clearly represents large fluctuation of the trolley and the payload. This phenomenon is undesirable in practice. Fig. 4 Velocity response of the trolley with control Fig. 5 Control input It can be seen from the simulation results the effect of control action to the system. Without control, the payload variation with the ma ximu m va lue of appro ximate 1m around the desired position. When the control is activated, the swing angle of the payload is reduced considerably. Finally, the payload mass and cable length are assigned to 600kg and 7m, respectively. Due to heavier load and longer cable, higher payload fluctuation is expected in comparison with the previous results. Nu merical simu lation indicates that the effectiveness of the proposed control design. Trolley position is regulated at the Proceedings of Engineering and Technology Innovation, vol. 8, 2018, pp. 15 -22 Copyright © TAETI 21 desired value after 20s, and the payload also tracks the target after a few oscillat ions. In addition, the control input is of the applicable practice range. Fig. 6 System response of the trolley and payload without control Fig. 7 System response of the trolley and payload with control Fig. 8 Velocity response of the trolley with control Fig. 9 Control input 5. Conclusion A design of a position and vibration suppression control of a gantry crane system is demonstrated in this paper. Based on energy approach, a system of partia l differentia l and ordinary differentia l equations that govern the system’s mot ions includ ing cable and actuator dynamics are derived. The system dynamics represent fle xib ility of the gantry cable. The Lyapunov direct method and backstepping technique are employed to design the controller. Stability and the effectiveness of the closed -loop system are verified ana lytically and illustrated numerically. The direct e xtension of the paper is to consider the motion of the system in three-dimensional space. References [1] R. 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