3.2 Sliding Mode Observer

The Sliding Mode Observers (SMO) belong to a class of non-linear observers used to estimate the internal state of an observable system based on the measured input and output. In this application, the SMO is used to estimate the back-EMFs of the PMSM. The major advantage of using SMO over a conventional linear back-EMF based rotor position and speed estimation is its robustness in the presence of unknown signals and uncertainties.

The SMO estimates the BEMFs using a PMSM system model, voltage, and current vector input. The following figure shows a simplified block diagram of the SMO-based rotor position and speed estimation method.

The following set of equations represent the state space model of a PMSM motor:

${\displaystyle [\begin{array}{c}{i}_{\alpha }\\ i_{\beta }\end{array}{]}^{\prime }=-\frac{R_s}{L_s}[\begin{array}{c}{i}_{\alpha }\\ i_{\beta }\end{array}]-\frac{1}{L_s}[\begin{array}{c}{e}_{\alpha }\\ e_{\beta }\end{array}]+\frac{1}{L_s}[\begin{array}{c}{U}_{\alpha }\\ U_{\beta }\end{array}]}$

${\displaystyle [\begin{array}{c}{e}_{\alpha }\\ e_{\beta }\end{array}{]}^{\prime }=w_e\mathbf{J}[\begin{array}{c}{e}_{\alpha }\\ e_{\beta }\end{array}]}$

The observer can be expressed as follows:

${\displaystyle [\begin{array}{c}{\hat{i}}_a\\ {\hat{i}}_{\beta }\end{array}{]}^{\prime }=-\frac{R_s}{L_s}[\begin{array}{c}{\hat{i}}_a\\ {\hat{i}}_{\beta }\end{array}]-\frac{1}{L_s}[\begin{array}{c}{\hat{e}}_a\\ {\hat{e}}_{\beta }\end{array}]+\frac{1}{L_s}[\begin{array}{c}{\hat{U}}_a\\ {\hat{U}}_{\beta }\end{array}]+[\begin{array}{c}\sigma \\ \sigma \end{array}\begin{array}{c}(i_a-{\hat{i}}_a)\\ (i_{\beta }-{\hat{i}}_{\beta })\end{array}]}$

${\displaystyle [\begin{array}{c}{\hat{e}}_a\\ {\hat{e}}_{\beta }\end{array}{]}^{\prime }=w_e\mathbf{J}[\begin{array}{c}{\hat{e}}_a\\ {\hat{e}}_{\beta }\end{array}]-\mathbf{L}[\begin{array}{c}\sigma \\ \sigma \end{array}\begin{array}{c}(i_a-{\hat{i}}_a)\\ (i_{\beta }-{\hat{i}}_{\beta })\end{array}]}$

Where:

• ${\displaystyle \sigma }$ is the sliding function, and L is the back-EMF observer pole placement matrix

From the equations, the estimation error can be expressed as follows:

${\displaystyle [\begin{array}{c}{\tilde{i}}_a\\ {\tilde{i}}_{\beta }\end{array}{]}^{\prime }=-\frac{R_s}{L_s}[\begin{array}{c}{\tilde{i}}_a\\ {\tilde{i}}_{\beta }\end{array}]-\frac{1}{L_s}[\begin{array}{c}{\tilde{e}}_a\\ {\tilde{e}}_{\beta }\end{array}]-[\begin{array}{c}\sigma \\ \sigma \end{array}\begin{array}{c}(i_a-{\hat{i}}_a)\\ (i_{\beta }-{\hat{i}}_{\beta })\end{array}]}$

${\displaystyle [\begin{array}{c}{\tilde{e}}_a\\ {\tilde{e}}_{\beta }\end{array}{]}^{\prime }=w_e\mathbf{J}[\begin{array}{c}{\tilde{e}}_a\\ {\tilde{e}}_{\beta }\end{array}]+\mathbf{L}[\begin{array}{c}\sigma \\ \sigma \end{array}\begin{array}{c}(i_a-{\hat{i}}_a)\\ (i_{\beta }-{\hat{i}}_{\beta })\end{array}]}$

Where:

• ${\displaystyle \left[\begin{array}{c}{\tilde{i}}_a\\ {\tilde{i}}_{\beta }\end{array}\right]=\left[\begin{array}{c}{i}_a-\\ i_{\beta }-\end{array}\begin{array}{c}{\hat{i}}_a\\ {\hat{i}}_{\beta }\end{array}\right]}$ is the current estimation error
• ${\displaystyle \left[\begin{array}{c}{\tilde{e}}_a\\ {\tilde{e}}_{\beta }\end{array}\right]=\left[\begin{array}{c}{e}_a-\\ e_{\beta }-\end{array}\begin{array}{c}{\hat{e}}_a\\ {\hat{e}}_{\beta }\end{array}\right]}$ is the back-EMF estimation error

The sliding function σ is designed to be a discontinuous function of estimation errors such that the sliding condition is assured, and the system state is forced to zero in finite time. In this document, the following sliding mode function is used:

${\displaystyle \sigma \left(x\right)=\{\begin{array}{ccc}\frac{mx}{\phi }& if& \left|x\right|<\phi \\ m,& if& x\ge \phi \\ -m& if& x\le -\phi \end{array}}$

If the value of m in the sliding mode function is large enough, a sliding condition is achieved and therefore the error dynamics evolves to zero. This implies:

Therefore, from the equations, the error dynamics of back-EMF observer can be expressed as:

${\displaystyle [\begin{array}{c}{\tilde{e}}_a\\ {\tilde{e}}_{\beta }\end{array}{]}^{\prime }=(w_e\mathbf{J}-\frac{1}{L_s}\mathbf{L}\left)\right[\begin{array}{c}{\tilde{e}}_a\\ {\tilde{e}}_{\beta }\end{array}]}$

For a stable back-EMF observer, the matrix ${\displaystyle (w_e\mathbf{J}-\frac{1}{L_s}\mathbf{L})}$ should have negative eigenvalues. In other words, for an eigenvalue of λ, the matrix L can be determined as follows:

or,

${\displaystyle \mathbf{L}=-L_s(\lambda \mathbf{I}-w_e\mathbf{J})}$