A fuzzy adaptive sliding mode controller for uncertain nonlinear multi motor systems

. Multi-motor drive systems are nonlinear, multi-input multi-output (MIMO) and strong-coupling complicated system, including the effect of friction and elastic, backlash. They have been widely used in many modern industries. The control law for this dive system much depend on the determining of the tension being hard to obtain this tension in practice based on a load cell or a pressure meter due to the accuracy of sensors or external disturbance. An emerging proposed technique in the control law is the use of adaptive sliding mode control scheme to stabilize closed system. However, the control system would be affected by chattering phenomenon. In order to eliminate this term, fuzzy technique is proposed by adjusting equivalent coefficients. The theory analysis and simulation results point out the good performance of the proposed fuzzy adaptive sliding mode control for the drive system.


Introduction
Multi-motor drive systems have been investigated by many researchers in the recent times. The neural network technique based control law have been proposed by Yaoji Me et al. (2013) (see [1][2][3][4][5][6][7][8][9]). However, it is hard to find the equivalent networks as well as corresponding learning rules. Besides, the model of this system is approximately described as a linear system to use the transfer function to design the control law. Furthermore, the tracking ability or the stabilization of the whole system are not still solved under the effects of neural network based observer. In the multi-motor drive control systems, it is necessary to obtain the belt tension to design the suitable state feedback control law. However, it is hard to measure this belt tension based on sensors, … and the high gain technique based observer is proposed in our work. Besides, the state feedback control design based on sliding mode control technique ensure that it is easy to remove efficient of disturbance and uncertainties. Therefore, an adaptive sliding mode controller is proposed to obtain tracking effectiveness. Moreover, fuzzy technique is considered to eliminate the chattering phenomenon disappearing by sliding mode control. The stability of closed system is obtained and verified by theory analysis, simulations.

Problem statements
Due to the effects by backlash and elastic ( Fig. 1) and parameters (Table 1), we extend the model in [1] to obtain the following dynamic equation (2,3) and the corresponding transfer function diagram (Fig. 2 are components including backlash and elastic to obtain the following equations: The model is described by eq. (3) belongs to the class of nonlinear systems as follows:

Remark 1:
The dynamic equations (1,2, 3) and Figures 1, 2 are described by the effect of friction, backlash, elastic and pointed out the nonlinear property of multi-motor systems. Fig. 2

Fuzzy adaptive sliding mode control design
In this section, the main work is to find a state feedback control law based on the adaptive sliding mode control technique for the class of multi motor systems. The proposed control law based on the following theorems as follows: Theorem 1: The adjusting mechanism (5): with s e α ξ = + , ensure some results as follows: 1. All signals of closed loop will be bounded and , The Lyapunov candidate function is selected as follows: and: However, we have the relation: Therefore, we obtain: On the other side, x B x and we obtain:

Theorem 2:
The control input s e α ξ = + ensure the nonlinear system (6) stability in finite time.

Proof:
We obtain the result: In order to eliminate the effect of chattering phenomenon, fuzzy technique (by Tagaki -Sugeno -Kang) would be proposed to adjust the coefficient α depending on the sliding surfaces s and s  , table 2, 3 and figure 3: Table 2. Rule Matrix of control.

Simulation results
In this section, we consider several simulation results to demonstrate the effectiveness of the proposed sliding mode control law based on the two-motor system having parameters as follows:

Conclusion
This paper described a fuzzy adaptive sliding mode control law the two-motor system in presence of elastic and backlash, friction. The effectiveness of the proposed control scheme was pointed out by theoretical analysis and simulation results.