TIDUBE5A January 2022 – October 2022
The conventional PLL integrated into the SMO is shown in Figure 2-18.
The traditional reduced-order sliding mode observer is constructed, which mathematical model is shown in Equation 51 and the block diagram is shown in Figure 2-19.
where and are sliding mode feedback components and are defined as:
Where and are the constant sliding mode gain designed by Lyapunov stability analysis. If and are positive and significant enough to guarantee the stable operation of the SMO, the and should be large enough to hold and .
The estimated value of EEMF in α-β axes ( , ) can be obtained by low-pass filter from the discontinuous switching signals and :
Where is the cutoff angular frequency of the LPF, which is usually selected according to the fundamental frequency of the stator current.
Therefore, the rotor position can be directly calculated from arc-tangent the back EMF, defined as follow
Low pass filter removes the high-frequency term of the sliding mode function, which leads to occur phase delay resulting. It can be compensated by the relationship between the cut-off frequency and back EMF frequency , which is defined as:
And then the estimated rotor position by using SMO method is:
In a digital control application, a time discrete equation of the SMO is needed. The Euler method is the appropriate way to transform to a time discrete observer. The time discrete system matrix of Equation 51 in α-β coordinates is given by Equation 57 as:
Where the matrix and are given by Equation 58 and Equation 59 as:
The time discrete form of Equation 53 is given by Equation 60 as: