Due to the velocity loop response is faster than the position loop response, that is, the cut-off frequency of the position loop is far less than inverse of the time constant of the velocity loop, so the velocity loop can be equivalent to the first-order inertia link as analytic system . The simplified structure diagram of position control system based the designed SOFSMC controller is described in Fig. 3.
Let x1=, , state space equation of position servo system is:Where T is Ls/Rs, K is the system gain, i is reduction rate.
The position error isSwitching surface is designed as:Where c is set as a positive constant.
The equivalent control method of the sliding mode state space equation is expressed as:According to Eqs. (21) and (18) is obtained as:Based on the developed switching surface, a switching control law satisfies the hitting condition. The switching control function is indicated as:Where is set as a positive constant.
A sliding mode controller can be express as:According to above equations, we obtainBased on the Lyapunov theorem, the sliding surface reaching condition is . Therefore, the system is globally stable. If a control input u can be chose to satisfy this reaching condition, the control system will converge to origin of the phase plane. The relating theory about the convergence and stability of SW033291 process on the basis of the minimization of can be found in Ref. .
Here, a fuzzy logic control is employ to approximate the nonlinear of equivalent control law. The control speed for each step is derived from fuzzy inference and defuzzification calculation instead of the equivalent control law derived form the nominal model at the sliding surface. The proposed SOFSMC scheme does not need the robust term to compensate the system mathematical model uncertainty. Hence, it can eliminate the chattering phenomenon of a traditional sliding mode control. Membership functions of fuzzy input and output variables, and the fuzzy rules of the SOFSMC are shown in Fig. 4a and b, respectively.
The membership functions can be expanded or shrunk by changing the scaling parameters of the membership function. The gain self organizing parameter is used to map the corresponding variables into Silurian Period nominal range. These parameters are specified as gs and gu respectively. Whose nonlinear functions are shown in Table 3. These gain scaling parameters are continuous functions of state control error during the whole control history, this design method can avoid the control law discontinuous jump of a traditional gain scheduling scheme and simplify the trial-and-error effort for designing the fuzzy rules table. The values of these parameters are not important for this gain self organizing fuzzy sliding mode controller. The same gain self organizing functions are suitable for various position responses. They can be determined by simple simulation tests. Then the same values can be employed for different position step value to achieve appropriate speed performance and tracking error.
To give the further comparison, the proposed SOFSMC and fuzzy controller with auto-tuning will be used to control the following PMSM drive system, respectively. The parameters of the motor are given in Table 4. The proposed SOFSMC parameters are chosen as follows:Where and are input and output parameters, respectively.
In this section, a step change is specified to evaluate the control performance of these controllers described in previous section. The gain auto-tuning parameters are listed in Tables 1 and 3. The position response and the speed reference values are shown in Fig. 5. It can be observed that the position response can achieve good response, but the speed change, torque change and current may be complicated for the fuzzy controller.
In electrical drives, the most important parameters affecting the robustness are inertia () and load torque of the motor. Therefore, the robustness of the servo system is tested under inertial variation and unwanted disturbances. First, inertia of the motor is increased approximately six times of the nominal inertia. The simulation result given in Fig. 6 displays the position step performance for the different step value with the increased (six times) inertia between the fuzzy controller and the SOFSMC. It should be noted that the good performance is obtained under the SOFSMC. The overshoot is not seen in low reference and high reference for the SOFSMC.