# Optimal Lyapunov-Based Sliding Mode Control for Slotless-Self Bearing Motor System
## 4 An Illustrative Example  
To shown effectiveness of the OLB-SMC, the experimental verifications was implemented in two condition operation. The detail parameters of the SSBM system are referred to the previous paper [8]. The fully hardware setup in practice is displayed in the Figure 2.  
The experimental system is listed as follows: a PC computer system supported by the Windows XP, a dSPACE DS1104 controller board, two displacement sensors, a six-phases power amplifier system, the SSBM motor and a rotary encoder. The sampling frequency is 10 kHz. The configuration of the control system will be shown in the Figure 3 below. Figure 2: The experimental system. The block \(\pi/4\) is used to present that the initial position of SSBM is \(\pi/4\). The initial angular position of the SSBM is \(\pi/4\), the detail analysis of the block \(\pi/4\) was discussed in reference paper [8]. From Equation (5), the torque can be controlled by and the bearing force can be controlled by \(i_{d}\), \(i_{q}\). These values are calculated after the block controller OLB-SMC. The block \(1/3\), that is used to transform the amplitude to the phases torque current, is known as the current transfer block. The phases bearing current is obtained after transferring id, iq via the \(dq/abc\) block. Then, the total current of each phase is obtained. These reference currents is sent to the amplifier system via the DAC channels of the dSPACE 1104 controller board. Then, the phases stator current was supplied through the power amplifier. In this research, the SSBM system was considered operating in the two conditions: free and sudden load. The movements were measured by the displacement sensors. The speed was measured by the rotary encoder. These sensors convert the measured values to voltages. And then, the ADC channels of dSPACE board read the values. The control parameters are listed as follows: \(\lambda_{\omega}=0.18\), \(k_{1\omega}=0.001\), \(k_{2\omega}=0.015\), \(\lambda_{x}=15\), \(k_{1x}=10\), \(k_{2x}=150\), \(\lambda_{y}=15\), \(k_{1x}=10\), \(k_{2x}=150\). The experimental test was completed with two cases study, the results are shown as follows:  
**Case 1:**\(\omega=1000\) (rpm)  
In this case, the motor speed was 1000 (rpm) with free-load. The experimental results are displayed as follows: in Figure 4: The displacements on \(x\)-axes and \(y\)-axes, in Figure 5: the rotor speed, in Figure 6: the rotor orbit, in Figure 7: the bearing current \(i_{d}\) and \(i_{q}\), and in Figure 8: the torque current \(A_{m}\). Figure 3: The experimental system. In the Figure 4, the overshoots on \(x\)-, and \(y\)-axes are 0.05 (mm) and 0.13 (mm), the response time is 1 s. The rotor speed shows in the Figure 5 below. The motor speed can reach 1000 (rpm) in 0.5 s. In the Figure 6, the rotor orbit is displayed. The rotor of SSBM motor is moved around the stable point. The bearing currents \(i_{d,q}\) are shown in the Figure 7.  
Figure 4: The displacement on \(x\)-, and \(y\)-axes. Figure 2:0 The rotor speed. Figure 2:1 The rotor orbit. The range of the bearing currents is \(-1\) (A) \(<i_{dq}<1\) (A). The torque current \(A_{m}\) is shown in the Figure 2:2  
The peak of torque current is \(-0.5\) (A), and the range of \(A_{m}\) is \(-0.7<A_{m}<\)0.3 (A). The first step to test the system is as follows: the SSBM system is in the steady-state working operation at the initial speed is 0 (rpm). At 0.5 (s), the referenced speed is increased from 0 (rpm) to 1000 (rpm). The overshoots of the displacements are \(O_{x}=0.05\) (mm), and \(O_{y}=0.13\) (mm). The respond time of the displacements is 1 (s). The rotor speed can catch the referenced speed in 0.5 (s). In the steady-state operation, the displacements errors are small, the range of error is \(|ex|<0.02\) (mm), and \(|ey|<0.02\) (mm). The rotor orbit is converged to the original point. The range of torque current is \(-0.7<A_{m}<0.3\) (A), when the motor is accelerating, the peak of \(A_{m}\) is \(-0.5\) (A). The range of bearing current is \(-1<i_{dq}<\)1.