# Observer Design for AC Motors

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Part of the Advances in Industrial Control book series (AIC)

## Abstract

Assuming that for AC electrical machines, the only measurable variables are the currents and the voltages, the mathematical models of the synchronous and induction motors are used to study their respective observability property. If this property is satisfied and under some necessary conditions introduced later, observers are designed to estimate the non-measurable variables of the electric machines. First, some definitions and an introduction to the Nonlinear Observers design are developed. Next, a classification in terms of the convergence rate of two classes of observers is studied: (1) Observers with an asymptotic convergence. (2) Observers with a finite-time convergence. Furthermore, as for nonlinear systems there are no canonical forms, several observer structures are introduced to be next applied to AC machines. From the mathematical model of the PMSM, rewritten in the form of two interconnected subsystems, an adaptive interconnected observer can be designed to estimate the rotor speed, rotor position, and load torque. Some assumptions are considered in order to ensure its asymptotic convergence of the observer. Because the stator resistance depends on the temperature which introduces a variation with respect to its nominal value, then in order to determine its real value, an adaptive interconnected observer is designed to estimate the stator resistance and simultaneously the rotor speed, rotor position, and non-measured load torque. Sufficient conditions are obtained to ensure the asymptotic convergence. Then, a super-twisting observer for a class of nonlinear systems is considered. The advantages of this observer are robustness with respect to parametric uncertainties and finite-time convergence which allows to guarantee that the separation principle can be satisfied when a controller is next applied. Similarly, from the IMPSM mathematical model, an adaptive interconnected observer is designed to estimate the rotor position, rotor speed, load torque, and stator resistance. Finally, for the induction motor, an adaptive interconnected observer is designed to simultaneously estimate the rotor speed, the fluxes, and the load torque. To guarantee the robustness property, an extension of the above observer under parametric uncertainties is developed and, by using practical stability concepts, the practical stability of the estimation error is ensured.

### Keywords

• Interconnected Observer
• Finite-time Convergence
• High Order Sliding Mode Observer (HOSMO)
• Permanent Magnet Synchronous Motor

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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1. 1.

This chapter includes excerpts of , originally published in the proceedings of IFAC world Congress, Milano, Italy, IFAC-PapersOnLine IFAC 2011.

2. 2.

This chapter includes excerpts reprinted from Journal of the Franklin Institute, 349(5):1734–1757, Hamida M, Glumineau A, De Leon J (2012) Robust integral backstepping control for sensorless IPM synchronous motor controller. Copyright (2012), with permission from Elsevier.

3. 3.

This chapter includes excerpts of , (2010) IEEE. Reprinted, with permission, from Ezzat M, De Leon J, Gonzalez N, Glumineau A, Observer-controller scheme using high order sliding mode techniques for sensorless speed control of permanent magnet synchronous motor. In: Decision and Control (CDC), 49th IEEE Conference on Decision Control.

4. 4.

This chapter includes an excerpt of : “Hamida M, De Leon J, Glumineau A (2013) High order sliding-mode observers and integral backstepping sensorless control of IPMS motor. International Journal of Control DOI: 10.1080/00207179.2014.904523” reprinted by permission of the publisher Taylor & Francis Ltd, http://www.tandf.co.uk/journals.

## Author information

Authors

### Corresponding author

Correspondence to Alain Glumineau .

## Appendix: Practical Stability Definitions

### Appendix: Practical Stability Definitions

This part is devoted to introduce some concepts and results of practical stability properties using in terms of Lyapunov functions .

Theory of stability is the basis of the control systems study. Moreover, the concept of practical stability allows to study the properties of a nonlinear system when the state of this system is bringing close to a set instead of the equilibrium point. From a practical point of view, a system will be considered stable if the deviations remain bounded around the equilibrium point. This is clear that, in practice, asymptotic stability towards a domain whose the size has to be determined, for performance checking, is sufficient. If the behavior of the system can be bounded by certain bounds, the notion of practical stability becomes useful. For example, for AC machines the stability towards a domain with specified bounds during a fixed time interval is a concept equivalent to a finite time stability.

Now, we introduce definitions which are useful for to guarantee the practical stability, in terms of Lyapunov functions.

Define the following class of function:

\begin{aligned} \mathbf W&= \{d_{1}\in C[\mathfrak {R}^{+},\mathfrak {R}^{+}]:\\&\quad \,\, d_{1}(l) \text{ is } \text{ strictly } \text{ increasing } \text{ in } l \text { and }\\&\quad \,\, d_{1}(l)\rightarrow \infty \text { as } l\rightarrow \infty \}. \end{aligned}

Let $$B_{r}=\{e \in \mathfrak {R}^n:\left\| e\right\| \le r\}$$. Consider the dynamical system

\begin{aligned} \dot{e}=f(t,e), e(t_{0})=e_{0}, t_{0}\ge 0, \end{aligned}
(3.114)

System (3.114) is said to be

### Definition 3.8

UPS Uniformly practically stable if, given $$(\hbar _{1}, \hbar _{2})$$ with $$0<\hbar _{1}<\hbar _{2}$$, one has

\begin{aligned} \left\| e_{0}\right\| \le \hbar _{1} \Rightarrow \left\| e(t)\right\| \le \hbar _{2}, \qquad \forall t\ge t_{0}. \end{aligned}
(3.115)

### Definition 3.9

UPQS Uniformly practically quasi-stable if, given $$\hbar _{1}> 0$$, $$\mathfrak {I}> 0$$, $$T>0$$ and $$\forall t_{0}\in \mathfrak {R}^{+}$$, one has

\begin{aligned} \left\| e_{0}\right\| \le \hbar _{1} \Rightarrow \left\| e(t)\right\| \le \mathfrak {I}, \qquad t\ge t_{0}+T. \end{aligned}
(3.116)

### Definition 3.10

SUPS Strongly uniformly practically stable, if (UPS) and (UPQS) hold together.

A result of the practical stability in terms of Lyapunov-like functions is presented

### Theorem 3.8

 Assume that

1. (i)

$$\hbar _{1}$$ and $$\hbar _{2}$$ are given such that $$0<\hbar _{1}<\hbar _{2}$$,

2. (ii)

$$V\in C[\mathfrak {R}^{+}\times \mathfrak {R}^n,{R}^{+}]$$ and $$V(t,e)$$ is locally Lipschitz in $$e$$,

3. (iii)

for $$(t,e)\in \mathfrak {R}^{+}\times B_{\hbar _{2}}, d_{1}(\left\| e\right\| )\le V(t,e)\le d_{2}(\left\| e\right\| )$$ and

\begin{aligned} \dot{V}(t,e) \le \wp (t,V(t,e)) \end{aligned}
(3.117)

where $$d_{1}$$, $$d_{2}\in \mathbf W$$ and $$\wp \in C[\mathfrak {R}^{+,2},\mathfrak {R}]$$,

4. (iv)

$$d_{2}(\hbar _{1}) < d_{1}(\hbar _{2})$$ holds.

Consequently, the practical stability properties of

\begin{aligned} \dot{l} =\wp (t,l), l(t_{0})=l_{0} \ge 0, \end{aligned}
(3.118)

implies the corresponding practical stability properties of system (3.114).

From the above theorem, the following criteria can be established

### Corollary 3.1

 In Theorem 3.8, if $$\wp (t,l) = - \alpha _{1} l + \alpha _{2}$$, with $$\alpha _{1}$$ and $$\alpha _{2} > 0$$, it implies strong uniform practical stability (SUPS) of system (3.114).

We can see that the solution of equation

\begin{aligned} \dot{l}(t) = - \alpha _{1} l(t) + \alpha _{2} \end{aligned}
(3.119)

is of the form

\begin{aligned} l(t)= l_{0}e^{-\alpha _{1} t-t_{0}}+\frac{\alpha _{2}}{\alpha _{1}}\left[ 1-e^{-\alpha _{1} (t-t_{0})}\right] , t \ge t_{0} \end{aligned}
(3.120)

The strong uniform practical stability of (3.119) is obtained.

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### Cite this chapter

Glumineau, A., de León Morales, J. (2015). Observer Design for AC Motors. In: Sensorless AC Electric Motor Control. Advances in Industrial Control. Springer, Cham. https://doi.org/10.1007/978-3-319-14586-0_3