Abstract
The problem of detecting a rotor speed sensor fault in induction motor applications with load torque and rotor/stator resistances uncertainties is addressed. It is shown that in typical operating conditions involving constant rotor speed and flux modulus and non-zero load torque, a constant non-zero (sufficiently large) difference between the measured speed and the actual speed may be on-line identified by an adaptive flux observer which incorporates a convergent rotor resistance identifier and relies on the measured rotor speed and stator currents/voltages. Simulation and experimental results illustrate the effectiveness of the proposed solution and show satisfactory fault detection performances.
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Notes
- 1.
The terms \(-\omega \hat{i}_{sb}\) and \(\omega \hat{i}_{sa}\) in the first two equations of (1.3) compensate for the rotor back electro-motive forces with the estimates \((\hat{i}_{sa}, \hat{i}_{sb})\) in place of \(({i}_{sa}, {i}_{sb})\) leading to skew-symmetric terms in the estimation error dynamics.
- 2.
Recall that in these operating conditions the rotor resistance cannot be identified by stator currents and rotor speed measurements since the motor equations (1.1) become
$$\begin{aligned}\dot{\omega }&= 0, \ \ \dot{\psi }_{ra}= -\omega \psi _{rb}, \ \ \dot{\psi }_{rb} = \omega \psi _{ra}, \ \ { \mathrm{d} i_{sa} \over \mathrm{d} t } = -\omega i_{sb}, \ \ { \mathrm{d} i_{sb} \over \mathrm{d} t }= \omega i_{sa} \end{aligned}$$and do not depend on the rotor resistance \(R_{r}\).
- 3.
A negative load torque (for regenerative brake actions) is also allowed provided that a non-zero rotor flux vector speed results.
- 4.
It suffices that at least they asymptotically tend to constant values with time derivatives asymptotically converging to zero.
- 5.
When no load torque is applied (or equivalently when a zero slip speed results), we have \(Mi_{sa} = \psi _{ra}\), \(Mi_{sb} = \psi _{rb}\) as preliminarily discussed by footnote 2.
- 6.
Recall from [16] that the proof of convergence is not constrained to the positiveness of the parameter \(\alpha _{e}\).
- 7.
Note that in the considered conditions (non-zero load torque and constant rotor speed and (non-zero) rotor flux modulus) it is possible to only locally identify the uncertain \(R_{r}\), \(R_{s}\), \(T_{L}\) from the measured outputs \((i_{sa}\), \(i_{sb}\), \(\omega )\). In fact, according to Sect. 1.3 of [9]:
-
\(R_{r}\), \(R_{s}\) and \(T_{L}\) can be expressed in terms of the measured outputs and their time derivatives (\({i}_{sa,d}=\mathrm{d} i_{sa}/\mathrm{d}t\), \({i}_{sb,d}=\mathrm{d} i_{sb}/\mathrm{d}t\)) as solutions to the system of nonlinear equations
$$\begin{aligned}\mathcal{P}&= i_{sa}(t_{*}){\sqrt{u_{sa}^{2}(t_{*})+u_{sb}^{2}(t_{*})}} \\ \mathcal{Q}&= i_{sb}(t_{*}){\sqrt{u_{sa}^{2}(t_{*})+u_{sb}^{2}(t_{*})}} \\ \dot{\rho }^{*} - \frac{R_{r}T_{L}}{\psi _{r}^{2}}&= \omega \\ \dot{\rho }^{*}&= \frac{- {i}_{sa,d}(t_{*})i_{sb}(t_{*})+{i}_{sb,d}(t_{*})i_{sa}(t_{*})}{i_{sa}^{2}(t_{*})+i_{sb}^{2}(t_{*})} \\ \mathcal{V}^{2}&= u_{sa}^{2}(t_{*})+u_{sb}^{2}(t_{*}) \end{aligned}$$where: \(\mathcal{P}=u_{sd}i_{sd}+u_{sq}i_{sq}\) and \(\mathcal{Q}=-u_{sq}i_{sd}+u_{sd}i_{sq}\) are proportional to the active and reactive electrical powers, respectively; \(\mathcal{V}=\sqrt{u_{sd}^{2}+u_{sq}^{2}}\) is the modulus of the stator voltage vector; \(\psi _{r}=\sqrt{\psi _{ra}^{2}+\psi _{rb}^{2}}\) is the modulus of the rotor flux vector; the constant \(u_{sd}\), \(u_{sq}\), \(i_{sd}\), \(i_{sq}\) are the \((d,q)\)-components of the stator voltage and current vectors which are known functions of \(\psi _{r},R_{r},R_{s},T_{L}\) (see Sect. 1.3 of [9]); \(t_{*}\) is such that \(u_{sa}(t_{*})=\mathcal{V}\), \(u_{sb}(t_{*})=0\);
-
there may exist two possible solutions \((\psi _{r},R_{r1},R_{s1},T_{L1})\), \((\psi _{r},R_{r2},R_{s2},T_{L2})\) with \(R_{r1}=-R_{r2}\) and \(T_{L1}=-T_{L2}\) to the above system of nonlinear equations to which correspond the same output and input profiles.
-
- 8.
Note that, for constant \(\omega \) and \(\omega _{m}\) and convergent rotor fluxes estimates, exponential convergence to zero of \(\tilde{T}_{L}\) and of \(\omega _{m}-\hat{\omega }\) can be proved by using the quadratic function \(V_{T}\) with \(\omega _{m}-\hat{\omega }\) in place of \(\tilde{\omega }\).
- 9.
When \(\alpha \) is known the two approaches are equivalent.
- 10.
Even though in this case steady-state stator currents estimation errors may appear (as in [10]), the presence in practice of unavoidable measurements noise which forces those steady-state estimationerrors to always be not identically zero makes not reliable the approach of using them as additional residuals.
- 11.
In accordance with the \(\mathcal{P}_{e}\) condition and the related analysis, different transient behaviours result.
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Marino, R., Scalzi, S., Tomei, P., Verrelli, C.M. (2014). Adaptive Flux Observers and Rotor Speed Sensor Fault Detection in Induction Motors. In: Ferrier, JL., Bernard, A., Gusikhin, O., Madani, K. (eds) Informatics in Control, Automation and Robotics. Lecture Notes in Electrical Engineering, vol 283. Springer, Cham. https://doi.org/10.1007/978-3-319-03500-0_1
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