# Optimal Lyapunov-Based Sliding Mode Control for Slotless-Self Bearing Motor System
## 1 Introduction  
Disturbance and uncertainty are two main factors affecting to the control output of the system. These factors are mainly souced by the outside effect and inside variations. The disturbance of the slotless self-bearing motor (SSBM) comes from the self vibration due to the reason of the length of the rotor. To reduce the vibration, the controller need to be a robustness controller or the disturbance observer need to be equipped. Furthermore, the uncertainties comes form the incorrect winding, incorrect fabrication, and so on are also the inversed problem of not only SSBM. To achieve the expected values, a powerful controller is required to refuse these inversed factors. In particular, the external perturbations affect on the rotor of SSBM and its self-vibration. A sliding mode control (SMC) will be proposed in this paper to decrease the unknown parameters for SSBM. First, the general concepts of the proposed motor can be found in [1; 2; 3; 4; 5; 6]. The disturbance and uncertainty are considered as unknown and unexpected values, and the discussion of these values was presented in [6]. Herein, the mathematics of SSBM are expressed for reader easy to understand the ideas of the work. Generally, the disturbance parameter is the outside factors such as wind blowing, etc, and the uncertainty parameter is the inside factors such as thermal change of the system in the working operation. Besides, the fabrication errors are also considered as hard factors, that strongly affect to the precision of the control system. Moreover, the fabrication errors will lead to the winding error. In detail, the reason of the perturbationsfactors of the SSBM motor was shown in the [7]. Herein, an optimal control algorithm for the SSBM is required. The basic concepts of modeling and control for the bearingsless motor was discussed in the paper [8]. The analysis and control of the bearing system was presented in [9; 10]. Furthermore, an adaptive placement control for commissioning of a active magnetic bearing system (AMBs) was disscussed in the [11], the disturbance observer-based control methods for the AMBs can be found in [12; 13; 14]. Moreover, the advanced control methods for the bearingsless system can be found in [15; 16]. In additional, the sensorless speed control of bearingsless motor was proposed in the paper [17; 18; 19; 20], the design and analysis of hardware was presented in [21; 22] and the force analysis of SSBM in paper [23]. To reject the perturbations of SSBM was discussed in the previous paper [7]. In this paper, an optimal control will be proposed for SSBM system. In recent years, the magnetic bearing motor has been interesting topic. The configure of the magnetic bearing actuator was discussed in the paper [24; 25; 26] but it is large. To reduce the size of the system, the reduction size of the magnetic bearing systems was discussed in [5; 8; 10]. The proposed SSBM was presented with its levitating and rotation capability. To control the SSBM, the movements and rotation must be controlled. In this paper, an OLB-SMC method will be proposed to complete the expected target. SMC method is a class of robust control method, which gives robust power for a drive system with uncertainty and disturbance factors. SMC is also a nonlinear control technique, which can be used to control a linear or nonlinear object. The detail idea and theory of SMC method can be found in paper [27]. SMC method consists of an equivalent and a switching values. Featuring the switching control constant comes up with the chattering problem, the elimination of the switching function can be found in [28]. In the paper [29; 30], a robust SMC method was presented. A robust SMC for SSBM system was investigated in previous paper [7]. The Lyapunov stability analysis can be found in [31; 32]. In recent years, the new stability for SMC and disturbance observer are investigated such as [33; 34; 35].