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Bài báo trình bày phương pháp thiết kế bộ điều khiển bám thích nghi bền vững cho hệ truyền động qua bánh răng trên cơ sở sử dụng nguyên tắc điều khiển trượt và thích nghi giả định rõ. Chất lượng bám ổn định tiệm cận luôn được đảm bảo và không phụ thuộc vào sự xuất hiện của các thành phần bất định trong hệ thống gồm khe hở, ma sát và độ không cứng vững của các bánh răng. Kết quả mô phỏng đã chứng minh khả năng bám thích nghi bền vững rất tốt của hệ kín. “nội dung được trích dẫn từ 123doc.org - cộng đồng mua bán chia sẻ tài liệu hàng đầu Việt Nam”
Tuyển tập công trình Hội nghị Cơ điện tử toàn quốc lần thứ 6 25 Mã bài: 07 Robust and Adaptive Tracking Control of Two-Wheel-Gearing Transmission Systems Điều khiển thích nghi bền vững hệ truyền động qua một cặp bánh răng Le Thi Thu Ha 1) , Nguyen Thi Chinh 2) , Nguyen Doan Phuoc 3) 1) , 2) Thai Nguyen University of Technology; 3) Hanoi University of Science and Technology e-mail: 1) hahien1977@gmail.com, 2) nguyenthichinh-tdh@tnut.edu.vn, 3) phuocnd-ac@mail.hut.edu.vn Abstract This paper proposes a new design procedure of an adaptive and robust tracking controller for gearing mechanical transmission systems by using the sliding mode control technique and the certainty equivalence principle. The asymptotic tracking behavior of the system in the presence of all uncertainties caused by backlash, friction or cogwheel elasticity is proved. The simulation results are provided to illustrate the satisfactory performance of the closed loop system. Keywords: Adaptive control; sliding mode; nonlinear system; backlash; cogwheel elasticity; friction Tóm tắt Bài báo trình bày phương pháp thiết kế bộ điều khiển bám thích nghi bền vững cho hệ truyền động qua bánh răng trên cơ sở sử dụng nguyên tắc điều khiển trượt và thích nghi giả định rõ. Chất lượng bám ổn định tiệm cận luôn được đảm bảo và không phụ thuộc vào sự xuất hiện của các thành phần bất định trong hệ thống gồm khe hở, ma sát và độ không cứng vững của các bánh răng. Kết quả mô phỏng đã chứng minh khả năng bám thích nghi bền vững rất tốt của hệ kín. Từ khóa: Điều khiển thích nghi; điều khiển trượt; hệ phi tuyến; khe hở; bánh răng; ma sát 1. Introduction The uncertainties that usually limit the performance of a gearing transmission control system in many practical applications are mainly caused by immeasurable friction, unpredictable elasticity of shafts and imprecise description of backlash between cogwheels[1], [3], [7], [9]. Those inevitable uncertainties can reduce the lifetime of the whole system or even disturb the system behavior. Therefore, damping the torsional vibration due to the shaft or cogwheel elasticity and suppressing the effect of friction or backlash are the most important control problems of mechanical systems in general and of gearing transmission systems in particular. Conventionally, self-tuning PI controllers are often used to approach these problems [8]. However, only using such PI controllers cannot damp torsional vibrations effectively [1]. Furthermore, desired results in suppression of the effect of the shaft elasticity or backlash between cogwheels at once at damping torsional vibrations cannot be achieved without additional states feedback [7]. Therefore, many attempts of using additional states feedback controller to improve the performance of mechanical systems with shaft elasticity or backlash have been carried out during the last few years (see, for examples, [3] and [10]). Such proposed controllers, however, can only be used either for systems with shaft elasticity or with backlash separately [12]. Moreover, a good tracking performance of systems, in which all uncertainties like immeasurable friction, unpredictable elasticity of shafts and backlash are simultaneous present, cannot be achieved with such nonadaptive states feedback controller. To overcome this problem, the adaptive robust control based on the sliding mode technique (see, for example, [11]) and the certainty equivalence principle (see, for example, [6]) is applied to improve the overall tracking performance of the closed loop system. 26 Le Thi Thu Ha, Nguyen Thi Chinh, Nguyen Doan Phuoc VCM2012 The sliding mode control is one of the robust control theories to suppress the effect of bounded noises or disturbances in systems. In addition, the certainty equivalence is also the most successfully used principle in adaptive controller designs for uncertain nonlinear systems in the presence of unknown constants in the systems' model. In this connection, the paper combines both the sliding mode technique and the certainty equivalence principle for designing an adaptive robust tracking controller for gearing transmission systems, in which the unpredictable elasticity of cogwheels and the imprecise description of backlash between cogwheels are considered as unknown constant parameters, whereas immeasurable shaft friction and the load capacity are regarded as bounded time dependent noises and disturbances in the system. This paper is organized as follows. In section 2 the mathematical model of gearing transmission systems is included. The section 3 describes design procedure of robust adaptive controller for system. In section 4 the experimental results and simulations are inculdes. Finally are included in section 5 some conclutions and commentaries about future research. 2. Model of Gearing Transmission Systems 2.1 Euler-Lagrange model Consider a gearing transmission system with a controller as depicted in Figure 1. The driving motor provides a control torque d M which is transmitted to the load c M through two wheel gears 1 and 2 and two elastic shafts. Let 1 f M and 2 f M denote the friction moment on each shaft. Both shafts have the same elasticity factor denoted by c . Let 1 and 2 be the rotational angles of corresponding shaft and the backlash between cogwheels. The Euler-Lagrange model of this gearing transmission system is given as follows (see, for example, [2]). 2 2 1 1 1 1 12 2 1 2 2 2 2 2 2 21 1 2 cos ( ) cos ( ) d f c f J cr i M M J cr i M M (1) where 1 2 and r r are the outer radii of corresponding wheels 1 and 2, 1 12 21 i i is the transmission rate of the two wheels and 1 2 , , d J J J are the inertia moments of wheel 1, wheel 2 and the driving motor respectively and 1 1 d J J J denotes the sum of inertia moments of wheel 1 and the driving motor. Figure 1. Configuration of a gearing transmission system While 1 2 12 21 1 , , , , J J i i r and 2 r in Euler-Lagrange model (1) can be considered as known parameters, the other parameters such as shaft elasticity , c friction moments 1 2 , , f f M M load moment , c M backlash are all uncertainties or disturbances of the system. 2.2 States space model In the following, all unknown constant parameters of the model will be denoted by k , whereas disturbances by k d . By using 2 2 1 2 2 1 1 2 2 1 1 1 1 1 2 2 2 2 2 cos , cos ( , ) , ( , ) f c f cr cr M b d t M M b d t where 1 2 , b b known constants, 1 2 , unknown constants, ( ) 1 1 1 1 1 , , , , T p , ( ) 2 2 2 2 2 , , , , T q , ( ) k x th k derivative of x , , p q finite positive integers, 1 1 2 2 ( , ) , ( , ) d t d t unknown disturbances, the Euler-Lagrange model (1) becomes 1 1 1 1 12 2 1 1 1 1 1 2 2 2 2 12 1 2 2 2 ( ) ( ) d J i M b d J i b d (2) From the second equation of (2), it is easy to see that 1 12 2 2 2 2 2 2 2 3 2 4 2 12 2 3 i J b d i d (3) with 3 12 2 2 3 2 2 4 12 2 2 , , d i d J i b and 1 3 2 4 2 12 2 4 (4) 1 3 4 2 12 2 5 2 i d i d (4) where 4 3 5 4 , d d d d . From (2), (3) and (4), it follows that 1 f M c M Load c c d M 1 2 2 f M M Controller 1 2 1 2 Tuyển tập công trình Hội nghị Cơ điện tử toàn quốc lần thứ 6 27 Mã bài: 07 (4) 1 3 1 4 1 3 2 1 4 1 3 1 12 2 2 1 4 1 12 2 1 5 1 3 1 4 1 d M J J b b J i b i J d d b d d Next, let states vector x , truncated states vector x , input control signal u , vector of unknown constants f , unknown constant g and unknown disturbance ( , ) d t x be defined as follows 1 2 2 2 3 2 4 2 x x x x x , 2 2 3 2 4 2 x x x x , d u M 1 4 1 12 1 4 1 3 1 12 1 3 1 4 1 3 1 f b i b J i J J b , 1 3 g J 1 5 1 3 1 4 1 1 3 1 ( , ) d t J d d b d d J x The Euler-Lagrange model (2) of the gearing transmission system, can now be rewritten in the form of uncertain states model (5) 1 4 if 1 3 ( , ) k k T f g x x k x d t u x x (5) with ( , ) d t x being bounded by a number 0 , that is ( , ) d t x for all , t x . (6) 3. Robust and Adaptive Tracking Controller 3.1 Sliding mode controller Let ( ) w t be the reference signal, so the reference trajectory for system (5) will be , , , T w w w w w and the vector of reference error is , , , T e e e e e where 1 2 e w x w . To control the states vector ( ) t x of (5) to asymptotically track the reference trajectory ( ) t w based on sliding mode control, first the following sliding surface is used 1 2 3 ( ) T s e a e a e a e e a e (7) where all elements 1 2 3 , , a a a of vector 1 2 3 , , , 1 T T a a aa are chosen such that the following polynomial 2 3 1 2 3 ( )p a a a (8) will be Hurwitz. Note that, by using sliding surface (7), in order to ensure the asymptotic tracking performance 0 e and e the nesecessary and sufficient condition is ( ) 0 s e . Thus, the initial tracking control aim can now be replaced with ( ) 0 s e and ( ) for 0 s e t . Now consider the following candidate control Lyapunov function (CLF) 2 1 ( ) 2 V s e (9) with its derivative being given by (4) 1 2 3 4 3 ( ) (4) 1 ( , ) . i T i f g i V ss s a e a e a e w x s a e w d t u x x Therefore, if the following controller is used 3 1 ( ) (4) 1 sgn( ) k T g k f k u a e w s x , (10) then 3 ( ) (4) 1 ( , ) sgn ( ) ( , ) sgn ( ) 0, i T i f g i V s a e w d u sd t s s e s d t s s e s x x x which sufficiently ensures the boundedness of ( ) s e as well as the asymptotic decay to zero of ( ) s e . 3.2 Adaptive Parameters Adjustment In practice, the controller (10), however, cannot be used because of the unknown parameters f and g . To overcome this limitation, the certainty equivalence principle will be employed. First, the unknown constants f and g in (10) are replaced by time functions ( ) f t and ( ) g t , respectively, yielding 3 1 ( ) (4) 1 sgn( ) k T g k f k u a e w s x (11) where is any chosen constant . 28 Le Thi Thu Ha, Nguyen Thi Chinh, Nguyen Doan Phuoc VCM2012 With this replacement, the derivative of the sliding surface (7) is now given by (4) 1 2 3 (4) 1 2 3 4 3 ( ) (4) 1 3 ( ) (4) 1 3 ( ) (4) 1 3 ( ) 1 ( , ) ( , ) ( , ) k T k f g k k k k T f g g g k T k f g g k k k k s a e a e a e e a e a e a e w x a e w d t u a e w d t u u a e w d t u a e x x x x x x (4) sgn( ) ( , ) sgn( ) ( , ) sgn( ) T f T f f g g T f g w s u d t s u d t s x x x x x (12) where f f f and g g g . It can be noted further that f f and g g . (13) because of constancy of f and g . Second, by using an adaptive CLF candidate 1 2 2 1 2 1 1 ( ) ( ) ( ) 2 2 1 1 1 2 2 2 T f f f f g g T f f g V V s F F where 3 3 F R is any symmetric positive definite matrix and is an arbitrary positive constant. Figure 2. Configuration of the closed loop system By using (12) and (13), one subsequently obtains: 1 1 1 1 1 ( , ) sgn( ) 1 ( , ) sgn( ) 1 ( , ) sgn( ) T f f g g T f g T f f g g T f g T f f g g T f f V ss s u d t s s u d t s sd t s s s F F F F x x x x x x 1 g g su (14) Now, by using the following adaptive adjustments for the time functions ( ) f t and ( ) g t of controller (11) ( ) ( ) f g s e s e u F x (15) the derivative V becomes negative definite ( , ) sgn( ) ( , ) 0 V sd t s s s d t s s x x which is sufficient for ensuring that ( ) s e and ( ) 0 s e . 3.3 Controller Design Procedure Figure 2. shows the main configuration of the closed loop system, in which the designed controller, including sliding mode controller (11) and adaptive parameters laws (15), always drives the output 1 2 y x to asymptotically converge to any four times differentiable desired trajectory ( ) w t . To obtain this closed loop system’s tracking performance, in summary, the following steps should be executed. Estimate of according to (6) Choose three constants 1 2 3 , , a a a so that the polynomial (8) is Hurwitz Construct the sliding surface ( ) s e according to (7) Choose any symmetric positive defined matrix 3 3 F R and a positive constant . Construct the adaptive adjustor according to (15) Construct the sliding mode controller according to (11) u w x , f g Plant ( 2 ) Controller ( 11 ) Adjustor (15) Tuyển tập công trình Hội nghị Cơ điện tử toàn quốc lần thứ 6 29 Mã bài: 07 4. Numerical Example Consider a gearing transmission system as described in Figure 1. where ( , ) d t x is a white noise with 0.1 d and the reference signal is given by ( ) sin(0.1 ) w t t . Let design parameters be chosen as follows: 3 50 F I , where 3 I is the unity matrix in 3 3 R 0.1 1 2 3 125, 75, 15 a a a , sliding surface constants 0.5 is infinite norm of disturbance 1 is parameter for controller (11) The tracking error and the system output are shown in Figure 3. and Figure 4. ; three elements of the vector f and g from the adaptive adjustors, are also given in Figure 5. and Figure 6. , respectively. From the simulation results, it can be seen that the system output asymptotically converges to the desired trajectory even in the presence of the unknown parameters f , g and the bounded disturbance ( , ) d t x . Figure 3. The tracking error Figure 4. Desired trajectory and system output 0 10 20 30 40 50 60 70 -80 -60 -40 -20 0 20 40 60 80 100 Figure 5. Adjusted parameters f 0 10 20 30 40 50 60 70 0 0.2 0.4 0.6 0.8 1 1.2 1.4 Figure 6. Adjusted parameter g 0 10 20 30 40 50 60 70 -25 -20 -15 -10 -5 0 5 10 15 20 25 Figure 7. Desired trajectory and system output by time dependent uncertainties 0 10 20 30 40 50 60 70 80 90 100 -1.6 -1.4 -1.2 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 Figure 8. Adjusted parameters [1] f compared with [1] ( ) f t 0 10 20 30 40 50 60 70 -1.5 -1 -0.5 0 0.5 1 1.5 f ( ) w t 2 ( ) t 0 10 20 30 40 50 60 70 -0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 2 e w 30 Le Thi Thu Ha, Nguyen Thi Chinh, Nguyen Doan Phuoc VCM2012 0 10 20 30 40 50 60 70 -2 -1.5 -1 -0.5 0 0.5 1 Figure 9. Adjusted parameters [2] f compared with [2] ( ) f t It should be noted that the adjusted parameters f and g do not tend to the actual values of unknown parameters f and g . In fact, in this example, the plant (5) was simulated with 1 2 3 T f and 1 g . However, this does not affect the tracking performance of the system. In addition, although the asymptotic tracking convergence of system is theoretical proved under assumtion that uncertainties , f g are constants, this performance is still keeping even in the case of time dependent uncertain vector ( ), ( ) f g t t as the experimental simulation results has shown in Figure 7. Figure 10. , whichs are carried out for system (5) with the time dependent functions of three uncertainties: [1] 1 0.4sin(0.5 ) f t [2] 1 0.2sin(0.5 ) f t [3] 1 0.2sin(0.5 ) f t 0 10 20 30 40 50 60 70 80 90 100 -0.2 0 0.2 0.4 0.6 0.8 1 1.2 Figure 10. Adjusted parameters [3] f compared with [3] ( ) f t 5. Conclusion The adaptive and robust controller, which is designed by using the procedure proposed in this paper, obviously satisfies the tracking requirement of the system. This satisfaction has been proved theoretically and numerically. In fact, the controller can effectively attenuate the disturbance and suppess the effect of parameter uncertainties. Note that although the tracking error is guaranteed to be zero at its steady state, its value during the transient period cannot be constrained in a predetermined range. This limitation can be avoided by using a barrier CLF instead of (9) and choosing 1 2 3 , , a a a of the sliding surface (7) appropriately. Furthermore, as a consequence of using sliding mode control, there still exists the chattering in the system. In oder to damp this undesired behavior, the constant should be chosen as small as possible but not less than . In the case, that the constant has to be choosen less than , the controller (11) can be revised as 3 ( ) (4) 1 ( , ) sgn( ) k T k f k g a e w d t s u x x where ( , ) d t x is an estimate of ( , ) d t x such that , sup ( , ) ( , ) t d t d t x x x The function ( , ) d t x can be obtained easily by using, for example, a neural network. References [1] Eutebach, T. and Pacas, J.M.: Damping of torsional vibration in high dynamic drivers. 8. European Conference on Power Electronics and Applications EPE 99, 1999. [2] Ha,L.T.T.: Modelling of transmission two- weel gearing System. Reaserch report, TNUT, 2012. [3] Hara, K.; Hashimoto, S.; Funato, H and Kamiyama, K.: Robust comparison between feedback based speed control system without states observer in resonant motor drivers. Power Electronics and Applications, 1997. [4] Hori, Y.; Sawada, H. and Chun, Y.: Slow Resonance Ratio Control for Suppression and Disturcances Rejection in Torsional Systems. IEEE trans. on Industial Electronics, Vol.46, No.1, pp. 162-168, 1999. [5] Kraftmueller, M: Adaptive Fuzzy Controller Design. Atca Polytechnica Hungaria, Vol.6, No.4, 2009. [6] Krstic,M.; Kanellakopoulos,I.; Kokotovic,P.: Nonlinear and Adaptive Control Design. John Wiley & Sohn Inc., 1995. Tuyển tập công trình Hội nghị Cơ điện tử toàn quốc lần thứ 6 31 Mã bài: 07 [7] Menon, K. and Krishnamurty: Control of low friction and gear backlash in machine tool feed drive systems Mechatronics 9, pp.33-52, 1999. [8] Sugiura, K. and Hori, Y.: Vibration Suppenssion in 2- and 3-Mass System based on Feedback of Imperfect Derivative of the Estimated Torsional Torque. IEEE trans. on Industial Electronics, Vol.43, No.1, pp. 56-64, 1996. [9] Szabat,K. and Orlowska,K.T.: Vibration suppenssion in two mass drive system using PI speed controller and additional feedbacks - comparative study. IEEE trans. on Industial Electronics, Vol.54, No.2, pp. 1193-1206, 2007. [10] Szabat,K. and Orlowska,K.T.: Performance Improvement of the Indusrial Drivers with mechanical Elasticity using nonlinear adaptive Kalman Filter. IEEE trans. on Industial Electronics, Vol.55, No.3, pp. 1075- 1084, 2008. [11] Utkin, V.: Sliding Modes in Optimization and Control. Springer Verlag New York, 1992. [12] Walha, L.; Fakhfakh, T. and Haddar, M.: Nonlinear dynamic of two stage gear system with mesh stiffness fluctuation, bearing flexibility and backlash. Mechanism and Machine 44, pp.1058-1069, 2009. Le Thi Thu Ha received B.S. and M.S. degrees from Thai Nguyen University of Technology in 1999 and 2003 respectively, all in automation technology. Since 2000 she has been with Electrical Engineering Department at TNUT Viet Nam, where she is nominated as head of department in 2008. Her research interests include modeling of mechanical systems and controller design for Euler-Lagrange systems Nguyen Thi Chinh received her B.S. degree from Hanoi University of Science and Technology in 2003 and M.S. degree from Thai Nguyen University of Technology in 2007. Since 2003 she is working as Uni. lecture in Industrial Automation Department of TNUT Viet Nam. Her research interests are Fuzzy Control and Neural Networking. Nguyen Doan Phưoc received his Dipl Ing. and Dr Ing. degree from Institut für Steuerungs- und Regelungstheorie, TU Dresden, Germany in 1982 and 1994. From 1994 to 1996 he has worked with Fraunhofer Institut Dresden on Modelling and Simulation. Since 1997 he has been with Automatic Control Department at HUST Viet Nam, where he is nominated as associate professor in 2003. His research interests are adaptive and robust control, optimization and optimal control. . 40 50 60 70 -0.2 -0.15 -0.1 -0 .05 0 0 .05 0.1 0.15 2 e w 30 Le Thi Thu Ha, Nguyen Thi Chinh, Nguyen Doan Phuoc VCM2 012 0 10 20 30 40 50 60 70 -2 -1.5 -1 -0.5 0 0.5 1 . where is any chosen constant . 28 Le Thi Thu Ha, Nguyen Thi Chinh, Nguyen Doan Phuoc VCM2 012 With this replacement, the derivative of the sliding surface (7) is now given by . performance of the closed loop system. 26 Le Thi Thu Ha, Nguyen Thi Chinh, Nguyen Doan Phuoc VCM2 012 The sliding mode control is one of the robust control theories to suppress the effect of