Adaptive Backstepping Control of Electrical Transmission Drives with Elastic, Unknown Backlash and Coulomb Friction Nonlinearity

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.


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.

Velocity sensor
Position sensor Fig. 1.A schematic diagram of the nonlinear electrical transmission drives with PID controller ; In equation (1), 12 () Where, 12 , ( ) q q rad   are angular of shaft motor and load; q rad s   are the motor and load angular speeds;  (.) signsign function of (.).We can rewrite (2) in form as: x q q  (5) We obtain: In [3] and [4], we can approximate (6) by smooth function as: In [6] and [8], we can approximate (3) as: In ( 7) and ( 8), , ab are positive numbers, which can be chosen when designing (in figure 2a, choose   , we can rewrite (1) as: , we obtain: ; ; ; a a a a are known parameters (can be measured); unknown parameters are: 1  -width of backlash, 2  -including Coulomb friction.We can rewrite (9) as: 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 12 yx  , because this speed can not be measured directly when variation of elastic is included, name its asymptotic value is d y , adjusting error 1 z can be calculated as: Because 12 ,  are unknown parameters, we denote their corresponding estimated parameters are 12 ˆ, , tracking errors are: We choose Lyapunov function for 1 z is: Where, ,  are adaptation gains.Differentiating of 1 V as: We choose the first virtual control 1  is: Step 2: Expanding the 12 () zz  term: , , , , )  ( , , , , ) From ( 14), we can write: Substituting ( 19)-(23) into equation (17), we obtain: We choose: (25) Step 3: Again expanding the 23 () zz  term: , , , ) We calculate the partial derivatives of 2  : Substituting (31)-(35) into equation ( 27), we obtain: , , , ) Choose: Substituting (38) into (36), then (36) into (26), we obtain:  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.
Fig 2a.Model of backlash and smooth approximation , we obtain: