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Flux Observer

System model

The governing equation of a salient-pole permanent magnet synchronous motor is expressed as
vdq=Ridq+ddtϕi+ωreJϕi+ωreJϕmωm=P−1ωreτm=P(Φiq+(Ld−Lq)idiq),\begin{align} \bm{v}_{\rm dq}&=\bm{R}\bm{i}_{\rm dq}+\frac{d}{dt}\bm{\phi}_{\rm i}+\omega_{\rm re}\bm{J}\bm{\phi}_{\rm i}+\omega_{\rm re}\bm{J}\bm{\phi}_{\rm m}\\ \omega_{\rm m}&=P^{-1}\omega_{\rm re}\\ \tau_{\rm m}&=P\left(\Phi i_{\rm q}+ (L_{\rm d}-L_{\rm q })i_{\rm d}i_{\rm q}\right), \end{align}
where R\bm{R}, Ldq\bm{L}_{\rm dq}, vdq\bm{v}_{\rm dq}, idq\bm{i}_{\rm dq}, ϕi\bm{\phi}_{\rm i}, ϕm\bm{\phi}_{\rm m}, ωre\omega_{\rm re}, ωm\omega_{\rm m}, τm\tau_{\rm m}, and PP stand for an armature resistance, inductance on the dq-axis, voltage and current on dq-axis, armature reaction flux, rotor flux, electric and mechanical angular speed of a motor, motor torque, and the number of motor poles, expressed as
R=RI, Ldq=diag(Ld, Lq),vdq=[vdvq], idq=[idiq], ϕi=Ldqidq, ϕm=[Φ0].\begin{align} \bm{R}&=R\bm{I},\ \bm{L}_{\rm dq}={\rm diag}(L_{\rm d},\ L_{\rm q}), \notag\\ \bm{v}_{\rm dq}&=\begin{bmatrix} v_{\rm d} \\ v_{\rm q} \end{bmatrix},\ \bm{i}_{\rm dq}=\begin{bmatrix} i_{\rm d} \\ i_{\rm q} \end{bmatrix},\ \bm{\phi}_{\rm i}=\bm{L}_{\rm dq}\bm{i}_{\rm dq},\ \bm{\phi}_{\rm m}=\begin{bmatrix} \Phi \\ 0\end{bmatrix}. \notag \end{align}
A magnetic flux Φ\Phi is generated by a permanent magnet and gets constant value. The matrices I\bm{I} and J\bm{J} denote an identity matrix and rotation operator expressed as
J=[0−110].\begin{align} \bm{J}&=\begin{bmatrix} 0 & -1 \\ 1 & 0 \end{bmatrix}. \end{align}

Flux observer

Now, let us consider a plant model on the γδ\gamma\delta-axis
vγδ=Riγδ+(sI+ωγJ)ϕiγδ+(sI+ωγJ)ϕmγδ=Riγδ+(sI+ωγJ)ϕiγδ+ωreJϕmγδ,\begin{align} \bm{v}_{\gamma\delta}&=\bm{R}\bm{i}_{\gamma\delta}+(s\bm{I}+\omega_{\gamma}\bm{J})\bm{\phi}_{\rm i\gamma\delta}+(s\bm{I}+\omega_{\gamma}\bm{J})\bm{\phi}_{\rm m\gamma\delta}\\ &=\bm{R}\bm{i}_{\gamma\delta}+(s\bm{I}+\omega_{\gamma}\bm{J})\bm{\phi}_{\rm i\gamma\delta}+\omega_{\rm re}\bm{J}\bm{\phi}_{\rm m\gamma\delta}, \end{align}
where vγδ\bm{v}_{\gamma\delta}, iγδ\bm{i}_{\gamma\delta}, and ωγ\omega_{\gamma} stand for an voltage and current on dq-axis and angular velocity of γδ\gamma\delta-axis. Assuming that the γδ\gamma\delta-axis closes to the dq-axis, ϕiγδ=Lγδiγδ≈Ldqiγδ\bm{\phi}_{\rm i\gamma\delta}=\bm{L}_{\gamma\delta}\bm{i}_{\gamma\delta}\approx \bm{L}_{\rm dq}\bm{i}_{\gamma\delta} holds. It means that an armature reaction flux is indirectly observable. Then, we introduces a model that is expressed as
ddt[ϕmγδϕiγδ]=[(ωre−ωγ)J0ωreJ−RLdq−1−ωγJ][ϕmγδϕiγδ]+[0I]vγδ.\begin{align} \frac{d}{dt} \begin{bmatrix} \bm{\phi}_{\rm m\gamma\delta} \\ \bm{\phi}_{\rm i\gamma\delta} \end{bmatrix} = \begin{bmatrix} (\omega_{\rm re} - \omega_{\gamma})\bm{J} & \bm{0} \\ \omega_{\rm re}\bm{J} & -\bm{R}\bm{L}_{\rm dq}^{-1} - \omega_{\gamma}\bm{J} \end{bmatrix} \begin{bmatrix} \bm{\phi}_{\rm m\gamma\delta} \\ \bm{\phi}_{\rm i\gamma\delta} \end{bmatrix} + \begin{bmatrix} \bm{0} \\ \bm{I} \end{bmatrix} \bm{v}_{\gamma\delta}. \end{align}
By constructing a minimal-order state observer, a rotor flux can be estimated as
ϕ^mγδ=(sI−ωreJ+ωγJ+ωreKJ)−1⋅(Kvγδ−[K(sI+ωγJ)+KRLdq−1]ϕiγδ),\begin{align} \hat{\bm{\phi}}_{\rm m\gamma\delta}=&(s\bm{I}-\omega_{\rm re}\bm{J}+\omega_{\gamma}\bm{J}+\omega_{\rm re}\bm{K}\bm{J})^{-1} \notag \\ &\cdot(\bm{K}\bm{v}_{\gamma\delta} - [\bm{K}(s\bm{I}+\omega_{\gamma}\bm{J})+\bm{K}\bm{R}\bm{L}_{\rm dq}^{-1}]\bm{\phi}_{\rm i\gamma\delta}), \end{align}
where K\bm{K} denotes the observer gain. In the paper, we pick the following gain:
K=I−sgn(ωre)J.\begin{align} \bm{K} &= \bm{I} - {\rm sgn}(\omega_{\rm re})\bm{J}. \end{align}
Then, we get the following observation equation:
ϕ^mγδ=(sI+ωγJ+∣ωre∣I)−1⋅K(vγδ−Riγδ−(sI+ωγJ)ϕiγδ).\begin{align} \hat{\bm{\phi}}_{\rm m\gamma\delta} =&(s\bm{I}+\omega_{\gamma}\bm{J}+|\omega_{\rm re}|\bm{I})^{-1} \notag\\ &\cdot\bm{K}(\bm{v}_{\gamma\delta} -\bm{R}\bm{i}_{\gamma\delta} - (s\bm{I}+\omega_{\gamma}\bm{J})\bm{\phi}_{\rm i\gamma\delta}). \end{align}

Electric angle estimation

This estimated value provides an angle error as
θ^γ=tan⁡−1(ϕ^mδ/ϕ^mγ),\begin{align} \hat{\theta}_{\gamma} = \tan^{-1}\left(\hat{\phi}_{{\rm m}\delta}/\hat{\phi}_{{\rm m}\gamma}\right), \end{align}
where θ^γ\hat{\theta}_{\gamma} represents an argument of d-axis and γ\gamma-axis. Then, we can estimate a rotor angle and velocity as
ω^re=(ωθ+0.25s−1ωθ2)θ^γθ^re=s−1ω^re,\begin{align} \hat{\omega}_{\rm re} &= \left(\omega_{\theta} + 0.25s^{-1}\omega_{\theta}^2\right)\hat{\theta}_{\gamma}\\ \hat{\theta}_{\rm re} &= s^{-1}\hat{\omega}_{\rm re}, \end{align}
where ωθ\omega_{\theta} stands for the gain of a phase lock loop.

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