305-Article Text-594-1-10-20171124.Pdf

Adaptive Backstepping Control of Electrical Transmission Drives with Elastic, Unknown Backlash and Coulomb Friction Nonlinearity Huỳnh Văn Đông*, Trần Xuân Kiên**, Nguyễn Công Định*** Abstract: In this paper, we present a new scheme to design an adaptive controller for uncertain nonlinear systems with unknown backlash, Coulomb friction nonlinearity The control design is achieved by introducing a smooth approximate backlash model and certain well defined functions and by using backstepping technique It is shown that the proposed controller can guarantee that the system is global asymptotic stable Keywords: Adaptive control, backstepping, backlash, Coulomb friction, dead-zone, nonlinear systems, stability I Introduction Electrical transmission drives is an important part of a control system, which pass the control command from the controller to the objects Conventionally, for convenience in designing the controller, the effects of nonlinear backlash, deadzone and friction are usually ignored However, very often, the mentioned above parameters exist in many devices such as gearbox, transmission shaft, valve (hydraulic), DC servo motor, and so on They are nonlinear elements, and can change from time to time, causing different limitations of quality of the whole system Research on Electrical Transmission Drives, which includes nonlinear backlash, dead-zone and friction, is a hot topic The target is to improve the quality of the system based on looking at the useful nonlinear characteristic of the system Current researches on two-mass systems can be referred to in [2]-[18] The researches and estimations about the systems, where exist backlash and friction, can be seen in [10], [11], [12], [13] The controller based on sliding mode for two-mass systems is introduced in [2], robust control is used in [3], [5] Other methods based on PI control are shown in [17], PD/PI associated with Fuzzy is in[18], Fuzzy based on Takagi-Sugeno model is in [7], [17], Kalman filter is shown in [15], accurate linearization is in [4], reference model building with parameter adjustment is in [13], linearization is in [14], and backstepping is introduced in [9] In [9] and [18], model of the plant is built, taking into consideration the parameter resilience, ignoring dead-zone and friction moment In [14], the nonlinear elements, such as dead-zone and friction, are linearized by secants method This paper shows the study of common nonlinear class, as in [8] Backlash and friction are in two differential equations of the system The existence of backlash and friction causes difficulties for the development of the controller A new model which smoothes backlash is chosen, and the controller is built based on recursive backstepping design Nonlinear parameters are smoothed, continued and can be differentiated In this paper, instead of concerning the effects of nonlinear backlash, resilience and friction as limited noises (as in [10], [11], [12], [13]), they are included in controller design Research on system, which includes nonlinear parameters, improves the quality and stability of the system The backstepping controller, which is designed with two adapt laws for unknown parameters, is shown and it guarantees that the system is global asymptotic stable Model of Nonlinear Electrical Transmission Drives ur J1 ref _ Proportional u0  u p _ –  Velocity sensor J2 Position sensor C ua Derivative Mm Motor Amplyfier ka Load q1 Ir 1 2 q2 Mf Kd Kp Fig.1 A schematic diagram of the nonlinear electrical transmission drives with PID controller 2  J 21 (Ts  M f ); M y  C (1  2 );    J11 ( M m  Ts );   (1)  ka ua ke   M m  k m I r ; ur  k a ua ; I r  R R   q1  1 ; q2  2 In equation (1), M y  C (q1  q2 ) is elastic moment [ Nm] , when elastic connection is without backlash; Ts is elastic moment [ Nm] , with backlash 2 [rad ] in elastic connection and is nonlinear function (undifferentiable), which have the following form:  M y  C , if M y  C  Ts  0, if M y  C   M y  C , if  M y  C Where, q1 ,  q2 (rad ) are (2) angular of shaft motor and load; 1  q1 , 2  q2 [rad / s] are the motor and load angular speeds; J1 , J [kgm2 ] are the motor and load moments of inertia; C[ Nm / rad ] is the spring constant; M m [ Nm] is the motor torque, ke [Vs / rad ] is the motor’s torque constant; km is constant; R[] is the armature coil resistance; K p , Kd are proportional and derivative gains; u p [V ] is output voltage of proportional controller; ua [V ] is output voltage of derivative controller; ur [V ] is the motor armature voltage; I r [ A] is the armature current; ref [rad ] is reference angular; u0  u p [V ] is signal control which follows reference program (for speed loop, it is output signal of positional controller); M f [ Nm] – Coulomb friction, from [8], we obtain: (3) M f   sign(2 )  - positive constant; sign(.) – sign function of (.) We can rewrite (2) in form as: C  (q1  q2 )    , if C (q1  q2 )  C  Ts  0, if C (q1  q2 )  C  C  (q1  q2 )    , if C (q1  q2 )  C (4) Backlash Ts(x2) Approximation of backlash -1 -2 -3 -4 -5 -4 -3 -2 -1 x2 Fig 2a Model of backlash and smooth approximation 1.5 Coulomb friction Mf(w2) 0.5 Approximation of Coulomb friction -0.5 -1 -1.5 -2 -5 -4 -3 -2 -1 w2 Fig 2b Model of Coulomb friction and smooth approximation Set: x2  q1  q2 (5) We obtain: C ( x2   ), if x2    Ts  0, if x2   C ( x   ), if x    2 (6) In [3] and [4], we can approximate (6) by smooth function as: Ts  C  x2   tanh(ax2 ) (7) In [6] and [8], we can approximate (3) as: M ms   sign(2 )   tanh(b2 ) (8) In (7) and (8), a, b are positive numbers, which can be chosen when designing (in figure 2a, choose a  1, 25 ; in figure 2b, choose b  ) Set x1  2 ; x2  q1  q2 ; x3  1  2 , we can rewrite (1) as: x1  C   x2   tanh(ax2 )  tanh(bx1 ) J2 J2 x2  x3 (9) C C  kk k k  x3     tanh(bx1 )  e m ( x1  x3 )  m a ua   x2   tanh(ax2 )   J2 J1 R J1 R  J1 J  y  x1 Set: a1  C C  kk k k C  ;   ; a2      ; a3  e m ; a4  m a ;   1 , J2 J2 J1 R J1 R  J1 J  we obtain: a1; a2 ; a3 ; a4 are known parameters (can be measured); unknown parameters are: 1 - width of backlash,  - including Coulomb friction We can rewrite (9) as: x1  a1  x2  1 tanh(ax2 )    tanh(bx1 ) x2  x3 x3  a2  x2  1 tanh(ax2 )    tanh(bx1 )  a3 ( x1  x3 )  a4ua (10) y  x1 For system described by (10), we can design adaptive backstepping controller for system (1) based on theory introduced in [1] Design of Adaptive Backstepping Controller: Step 1: Set the system’s final output y  x1  2 , because this speed can not be measured directly when variation of elastic is included, name its asymptotic value is yd , adjusting error z1 can be calculated as: z1  y  yd  x1  yd Assume that yd  , we obtain: z1  x1  a1  x2  1 tanh(ax2 )  2 tanh(bx1 ) (11) Because 1 ,2 are unknown parameters, we denote their corresponding estimated parameters are ˆ1 ,ˆ2 , tracking errors are: (12) 1  1  ˆ1 or 1  ˆ1  1 (13) 2  2  ˆ2 or 2  ˆ2  2 We choose Lyapunov function for z1 is: V1  2 z1  1  2 2a1 2 2 Where,  ,  are adaptation gains Differentiating of V1 as: V1   1 z1 z1  11   2  a1   1 z1 a1  x2  1 tanh(ax2 )   tanh(bx1 )  11   2 a1     1  z1   x2  (1  ˆ1 ) tanh(ax2 )   (2  ˆ2 ) tanh(bx1 )   1 (ˆ1 )  2 (ˆ2 ) a1       1 z  z1   x2  ˆ1 tanh(ax2 )   ˆ2 tanh(bx1 )   z11 tanh(ax2 )  1 (ˆ1 )  2 tanh(bx1 )  2 (ˆ2 ) a1  a1      z  1    z1 ( x2  1 )  1  ˆ1 tanh(ax2 )  ˆ2 tanh(bx1 )   1  z1 tanh(ax2 )  ˆ1   2  tanh(bx1 )  ˆ2  a1         a1 We choose the first virtual control 1 is: ˆ 2 tanh(bx1 ) a1 (14) z    V1  c1 z12  z1 z2  1  z1 tanh(ax2 )  ˆ1   2  tanh(bx1 )  ˆ2        a1 (15) 1  c1 z1  ˆ1 tanh(ax2 )  Step 2: V2  V1  z2 or z    V2  V1  z2 z2  c1 z12  z2 ( z1  z2 )  1  z1 tanh(ax2 )  ˆ1   2  tanh(bx1 )  ˆ2        a1 (16) Expanding the ( z1  z2 ) term: z1  z2  z1  x2  yd  1 ( x1 , yd ,ˆ1 ,ˆ2 , x )  z1  x3  1 ( x1 , yd ,ˆ1 ,ˆ2 , x ) (17) From (14), we can write: 1 1  c1 z1  ˆ1 tanh(ax2 )  ˆ2 tanh(bx1 )  c1 x1  c1 yd  ˆ1 tanh(ax2 )  ˆ2 tanh(bx1 ) (18) a1 a1 1 b  c1  ˆ2 1  (bx1 )  (19) x1 a1 1 1  tanh(ax2 ) (20) ; (21) 0 yd ˆ1 1  tanh(bx1 ) ˆ2 a1 (22); 1  aˆ1 1  (ax2 )  x2 (23) Substituting (19)-(23) into equation (17), we obtain: z1  z2  z1  ( x3   )    c1   z1  z3    c1  b ˆ 2 1  (bx1 )   tanh(ax2 )  tanh(bx1 )  aˆ1 1  (ax2 )  a1 a1 b ˆ  1  (bx1 )   tanh(ax2 )  tanh(bx1 )  aˆ1 1  (ax2 )  a1 a1 We choose:   b   c2 z2   z1  c1  ˆ2 1  (bx1 )   tanh(ax2 )  tanh(bx1 )  aˆ1 1  (ax2 )  (24) a1 a1   z    V2  V1  z2 z2  c1 z12  c2 z22  z2 z3  1  z1 tanh(ax2 )  ˆ1   2  tanh(bx1 )  ˆ2        a1 (25) Step 3: z3 V3  V2  z3 z3 V3  V2  or z     c1 z12  c2 z22  z3 ( z2  z3 )  1  z1 tanh(ax2 )  ˆ1     tanh(bx1 )  ˆ2        a1 Again expanding the ( z2  z3 ) term: z  z  z  x   ( x , y ,ˆ ,ˆ , x ) 3 d 2  z2  a2  x2  1 tanh(ax2 )  2 tanh(bx1 )  a3 ( x1  x3 )  a4ua   ( x1 , yd ,ˆ1 ,ˆ2 , x ) (26) (27)   c2 ( x2  yd  1 )   (28) b   z1  c1  ˆ2 1  (bx1 )   tanh(ax2 )  tanh(bx1 )  aˆ1 1  (ax2 )   a1 a1     c2 x2  c1c2 z1  c2ˆ1 tanh(ax2 )  c2 ˆ2 tanh(bx1 ) a1   b   z1  c1  ˆ2 1  (bx1 )   tanh(ax2 )  tanh(bx1 )  aˆ1 1  (ax2 )   a1 a1     c2 x2  c1c2 ( x1  yd )  c2ˆ1 tanh(ax2 )  c2 ˆ2 tanh(bx1 ) a1   b  ( x1  yd )  c1  ˆ2 1  (bx1 )   tanh(ax2 )  tanh(bx1 )  aˆ1 1  (ax2 )   a1 a1   (29) (30) We calculate the partial derivatives of  :  bc 2b2 ˆ b  c1c2  ˆ2 1  (bx1 )    2 tanh(bx1 ) 1  (bx1 )  1  (bx1 ) (31) x1 a1 a1 a1  (32) 0 yd   c2  c2 aˆ1 1  (ax2 )   a 1  (ax2 )   2a 2ˆ1 tanh(ax2 ) 1  (ax2 )  (33) x2   c2 tanh(ax2 )  a 1  (ax2 )  ˆ 1  c b   tanh(bx1 )  1  (bx1 )  ˆ a1 a1  (34) (35) Substituting (31)-(35) into equation (27), we obtain: z2  z3  z2  a2  x2  (1  ˆ1 ) tanh(ax2 )  (2  ˆ2 ) tanh(bx1 )  a3 ( x1  x3 )  a4ua   ( x1 , yd ,ˆ1 ,ˆ2 , x )  z2  a2 x2  a2ˆ1 tanh(ax2 )  a21 tanh(ax2 )  ˆ2 tanh(bx1 )  2 tanh(bx1 )  a3 ( x1  x3 )  a4ua bc2 ˆ 2b2 ˆ b 2 1  (bx1 )    2 tanh(bx1 ) 1  (bx1 )   1  (bx1 )  a1 a1 a1  c2  c2 aˆ1 1  (ax2 )   a 1  (ax2 )   2a 2ˆ1 tanh(ax2 ) 1  (ax2 )  c b (36)  c2 tanh(ax2 )  a 1  (ax2 )   tanh(bx1 )  1  (bx1 )  a1 a1  c1c2  Choose:  a4ua  c3 z3  z2  a2 x2  a2ˆ1 tanh(ax2 )  ˆ2 tanh(bx1 )  a3 ( x1  x3 ) bc2 ˆ 2b2 ˆ 2b 2 1  (bx1 )    2 tanh(bx1 ) 1  (bx1 )   1  (bx1 ) a1 a1 a1  c2  c2 aˆ1 1  (ax2 )   2a 1  (ax2 )   2a2ˆ1 tanh(ax2 ) 1  (ax2 )   c (37)  c2 tanh(ax2 )  tanh(bx1 )  a1   c1c2  ua   c3 z3  z2  a2 x2  a2ˆ1 tanh(ax2 )  ˆ2 tanh(bx1 )  a3 ( x1  x3 ) a4 bc2 ˆ 2b2 ˆ 2b    c1c2  2 1  (bx1 )    2 tanh(bx1 ) 1  (bx1 )   1  (bx1 )  a1 a1 a1 2  c2  c2 aˆ1 1  (ax2 )   2a 1  (ax2 )   2a ˆ1 tanh(ax2 ) 1  (ax2 )   c (38)  c2 tanh(ax2 )  tanh(bx1 )  a1  Substituting (38) into (36), then (36) into (26), we obtain: V3  c1 z12  c2 z22  z3  c3 z3  a21 tanh(ax2 )  2 tanh(bx1 ) z     1  z1 tanh(ax2 )  ˆ1   2  tanh(bx1 )  ˆ2        a1 V3  c1 z12  c2 z22  c3 z32 (39)    z  (40)  1  z3a2 tanh(ax2 )  z1 tanh(ax2 )  ˆ1   2   z3 tanh(bx1 )  tanh(bx1 )  ˆ2    a1     ˆ1   z3a2 tanh(ax2 )   z1 tanh(ax2 ) z ˆ2  z3  tanh(bx1 )  tanh(bx1 ) a1 V3  c1 z12  c2 z22  c3 z32  with c1  0, c2  0, c3  (41) To conclude, when c1  0, c2  0, c3  , with control law (38) and adaptive law (41), system (1) becomes GAS Simulation: a) Simulation in Matlab-Simulink: Sign -KTs Usum Betap Betad Ka 1/R Ua Km Ir Ur Repeating Sequence W1 s 1/J1 Mm C My s W2 s 1/J2 Ea Ke Kd q2 s Switch simout Usum Repeating Sequence1 To Workspace ADAPTIVE BACKSTEPPING CONTROLLER W2 My W1 Fig 3a Simulating in Matlab-Simulink backstepping control (38) and adaptive law (41) for nonlinear electrical transmission drives (1) z1 Usum c1 X1 My Z2 -K- X2 w1 a U gama 1/s teta1 -KX1 b -K- c2 -K- c2 c2 c2 teta2 -K1/s X3 a33 -KZ3 1/a44 2*b/a11 2*a alpha2 c1 c1 1/2 c2 1/2 1/2 -Kc3 c2 1/2 c2 2*a 2*b c2 -K-K- -K-K- c2 -K- -K-K-K-K- -K- -K-K- Fig 3b Adaptive backstepping controller (38) -K- 20 40 15 Position of load (rad) Speed of the motor (rad/s) 60 20 -20 -40 -60 10 20 30 40 50 60 70 80 90 10 -5 -10 -15 100 Time(s) -20 10 20 30 40 50 60 70 80 90 100 Time (s) Fig 3c Speed of motor Fig 3d Position of load b) Simulation on real model in lab: Speed of motor (degree/ sec) Fig 4a Experimental model of nonlinear electrical transmission drives 1- DC motor, 2- Velocity sensor 1 , 3- Pulse width modulation (PWM) and power amplifier, 4- Torsion spring connecting between two masses, 5- The first mass, 6- The second mass, 7- Position sensor, 8- Card PCI 1711 Advantech, 9- Embedded computer, 10- Controling software in Matlab-Simulink 300 200 100 -100 -200 -300 10 20 30 40 50 60 70 80 Time (s) Fig 4b Speed of motor in model 4a 90 100 Position of load (degree) 200 125 100 -100 -125 -200 10 20 30 40 50 60 70 80 90 100 Time (s) Fig 4c Position of load in model 4a Looking on the figures 3c, 3d, 4b, 4c, during the first 50 seconds, the velocity signal is driven by the PID control, this value is fluctuated During the next 50s, the speed is driven by the adaptive backstepping control, the speed signal is steady and the speed of motor and load follows the reference command accurately The comparison of the simulating results in Matlab-Simulink and on real model can conclude about the truth of the designed control algorithm Conclusion: In fact, backlash, elastic and friction always exist in electro-mechanic systems Backlash and Coulomb friction are typical nonlinear elements They cause bad effects on system’s operation quality This can not be overcome by using the traditional controllers By using adaptive backstepping technique, the bad effects from backlash, elastic and friction are solved The controller has designed for the electro-mechanic object class, which includes two nonlinear masses The controller drives the system in a “calmer” operation, also gains “good” nonlinear characteristics Especially, it always keeps the system in global asymptote stability References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] Miroslav Krstic’, Loannis Kanellakopoulos, Petar Kokotovic’(1995), Nonlinear 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Vietnam email: hdongkh@yahoo.com ** Central of Science and Technology, Vietnam email: txkien2003@yahoo.com *** Le Quy Don Technical University, Vietnam email: ncdinh63@yahoo.com