The control u x t with its respective entry x

The control u(x,t) with its respective entry (x,t) has the formwhere , and are continuous functions. is an (n−1) dimensional switching function. Since (x,) undergoes discontinuity on the surface is called a switching surface or switching hyperplane as in Fig. 1.
With sliding condition
Since it buy Flavopiridol hydrochloride is required to produce a control input (x,) that applied to the system to obtain the desired output. i.e., it is required to determine the PMSM wave currents (,) and voltages (,) in order to obtain the desired output speed (ω).
Due to the nonlinearity and the complexity of the motor model, it is needed to decouple the system into two subsystems that are the electrical and the mechanical systems that shown in Fig. 2.

Sliding mode control design
The main idea behind the decoupled strategy is to decouple a nonlinear system appearing in the form of Eq. (7) into two subsystems as electrical and mechanical in the form of Eqs. (11) and (12). The electrical subsystem is chosen as a primary target while the mechanical subsystem is used as a secondary target.
However, the selection of the primary and the secondary subsystems is problem dependent. Here, the control objective is to devise a control strategy that would move the states of both subsystems toward their sliding surfaces S1=0 and S2=0. The electrical subsystem involves knowledge from mechanical subsystem, and the mechanical subsystem is driven from the electrical subsystem.
Let the sliding surface function S1 be defined aswhere c1 and c2 are the sliding surface S1 constants for dq-axis, for simplified calculations use current transform to convert from dq-axis to abc-axis, so Eq. (13) can be expressed as:where c1, c2 and c3 are the sliding surface S1 constants for abc-axis , and are the references three phases currents. , and are the actual three phases currents.Let the sliding surface function S2 is defined aswhere c4 is the sliding surface S2 constant, is the reference motor speed. ω is the actual motor speed.
In the design of decoupled sliding-mode controller, an equivalent control is first given so that the states can stay on sliding surface. Thus, in sliding motion, the system dynamic is independent of the original system and a stable equivalent control system is achieved. So the decoupled sliding-mode controller u can be divided into an equivalent control input and a hitting control input if cytoplasm has the following control law:where M is a positive constant, is the equivalent.
The sliding mode controller block diagram can be shown in Fig. 3 after determining the desired position and calculating the reference currents , , . Sliding mode control produces on/off signals g for the switching inverter which in turn operates the synchronous machine. Observing the motor current position and angular velocity to adapt the sliding mode for reaching the sliding surfaces.

SMC controller
The SMC speed controller is shown in Fig. 4.
The model contains PI block which is used to calculate the reference signal needed in dq-axis. A dq-abc-transform block transfers the dq-currents to abc-currents format. The SMC2 starts to calculate the tracking error in speed Eq. (15), hence produces the reference currents needed for the motor. SMC1 calculates the tracking error in currents Eq. (14), then produces output voltages to force the system toward the sliding surface first (reaching) , then SMCs try to keep the system slides on the switching surfaces till reaching the desired speed . The SMC1 contents are shown in Fig. 5.
The contents of compare, compare2 and 3 are shown in Figs. 6–8, respectively.
The transfer function is a filter to reduce the harmonics with the measured current, then satisfy the sliding surface (S1) to obtain the control output. Selecting c1=c2=c3 because of the contribution of each phase is similar to the others. Large constants required for faster response. But not too large, otherwise the motor will suffer from torque ripples. When the S1 moves toward the origin (the reference currents are near the actual currents) and with motor current ranges from +1.72 to −1.72A, larger constants c1, c2 and c3 are required specially with small current differences. If these constants are too large then S1 may change its sign and may induce torque ripples.