Elsevier

Automatica

Volume 40, Issue 6, June 2004, Pages 1079-1085
Automatica

Brief paper
Global observability analysis of sensorless induction motors

https://doi.org/10.1016/j.automatica.2004.01.020Get rights and content

Abstract

The current problems to successfully apply sensorless controllers for induction motors are the existence of operation regimes for which the performance is remarkably deteriorated, due to the difficulties in estimating correctly motor speed and flux, and the lack of a theoretical explanation for this kind of behavior. In this paper a global observability analysis for these machines is carried out. It is first shown that all indistinguishable trajectories of the system, i.e. pairs of state trajectories with the same input/output behavior, can be described by a differential equation on a manifold, named here the indistinguishable dynamics. Studying the stability properties of this latter system it can be shown that the induction motor is not completely observable nor detectable in a local or in a global sense, and for every set of parameters. This implies that it is impossible to construct a state observer for the motor that converges for every trajectory of the system. Moreover, the indistinguishable dynamics provides a systematic method to study, understand and explain particular operation regimes, and this is illustrated by some case studies of practical relevant operating conditions.

Introduction

In the last years many efforts have been made to reduce the complexity of induction motor (IM) based drives. The main motivations are, from an economic perspective, dropping costs in the production process and, from a technical viewpoint, minimizing sensor failure probability. As a result the technique called (shaft) sensorless control has been proposed (Holtz & Quan, 2001; Lorenz, 1999; Rasmussen, Vadstrup, & Børsting, 2002). With it high-performance operation is pursued without sensing mechanical variables (both position and speed).

Since many excellent sensored controllers are available, the main research stream has been focused on searching for reliable speed estimation methods, with the aim to replace the estimated by the actual speed in the controller structure. Currently it is possible to (roughly) classify the contributions into four approaches: rotor slot ripple method, high-frequency current injection method, extended Kalman filter technique and model reference adaptive control (Rajashekara, Kuwamura, & Matsue, 1996). However, the usefulness of the proposed schemes has been validated only in an experimental setting. The existence of operating conditions where the output feedback controller, and in particular the estimation scheme, fails, has been always recognized. It has been pointed out, for example, that operation of the control system at low speed can lead to instability (or at least to very poor performances (Asher, 2000)), and also that constant (or low frequency) flux operation leads to control difficulties (Montanari, Peresada, Tilli, & Tonielli, 2000). Operation at nominal speed with a field oriented control can also exhibit undesirable behavior (Harnefors, 2000). Several remedies have been proposed to alleviate these disadvantages (Asher, 2000), apparently related to the observability properties of the induction motor. However, no formal analysis has been yet done to clearly understand the nature of such undesirable phenomena, except the local observability analysis reported in (Canudas, Youssef, Barbot, Martin, & Malrait, 2000), where the problem of working at low frequencies is studied.

The aim of this paper is to understand the theoretical possibilities and limitations to achieve reliable speed estimation schemes for sensorless induction motors. As in the preliminary paper (Moreno, Espinosa-Pérez, & Ibarra, 2002) this is reached by making a complete and global observability analysis of this kind of motors. The cornerstone of this result is the indistinguishable trajectories concept (Hermann & Krener, 1977), i.e. internal trajectories of a system that are different under the same input/output behavior. The main result is the characterization of the complete set of the indistinguishable trajectories for the sensorless IM by means of a set of differential equations on a manifold, named indistinguishable dynamics (ID). The observability/detectability of the IM is then related to the stability properties of the ID.

The main contributions are: (1) It is formally shown that, for every set of physically meaningful parameters of the IM, there are diverging indistinguishable trajectories. The IM is then neither observable nor detectable, globally or locally. (2) There are therefore always operating regimes for which any observer fails to converge. (3) A complete characterization of the indistinguishable trajectories is given in terms of the ID system. (4) The knowledge obtained from the study of the ID can be used to understand the operation of the industrial schemes, and to establish a solid background for the design of new flux and speed estimators.

The paper is organized as follows: In Section 2, the problem is stated, and the ID derived. In Section 3, the main observability properties of the motor, and their consequences for the existence of observers are obtained. For illustration some operating regimes of the IM are analyzed in Section 4.

Section snippets

Indistinguishable dynamics of the induction motor

Consider the (equivalent 2φ) unsaturated IM model given by (Meisel, 1966)Σ:ω̇=−fω+αψTJi−TLJ,ψ̇=−aψ−npωJψ+Mai,didt=β[aψ+npωJψ−(Ma+b)i+cu],where rotor speed ω, rotor fluxes ψ=[ψa,ψb]T and stator currents i=[ia,ib]T are the states; stator voltages u=[ua,ub]T and load torque TL are the external signals applied to the motor; rotor inertia J>0, stator and rotor inductances Ls,Lr>0, mutual inductance M>0, stator and rotor resistances Rs⩾0, Rr>0, rotor friction f⩾0 and the number of pole pairs np>0 are

Observability and existence of observers

(Local) observability and (local) detectability of the IM, under sensorless conditions, can be easily characterized in terms of the ID (3) or (5).

Proposition 4

The IM under sensorless operation is:

  • (a)

    Observable if and only if the ID has no solutions.

  • (b)

    Locally observable if and only if the ID has no solutions on a neighborhood of the point (ε,Δ)=0.

  • (c)

    Detectable if and only if the point (ε,Δ)=0 of the ID is globally attractive, for all the trajectories that are defined for all the time, i.e.t∈[0,∞).

  • (d)

    Locally detectable

Examples

Some cases of indistinguishable behavior are illustrated.

Concluding remarks

The impossibility of constructing observers for the IM that converge for every trajectory shows that only observers that (always) converge for a proper subset of the IM trajectories can exist. Diverging indistinguishable trajectories are those for which any observer fails, but other trajectories may also lead to divergence of a specific observer. This is analogous to the persistence of excitation conditions known in the parameter identification, and the adaptive control literature. If the load

Sebastian Ibarra-Rojas received the B.S. and M.I. in electrical engineering in 1994 and 2001, respectively, from Mexico National University, Mexico, where he is currently working towards the Ph.D. degree. His fields of interest are nonlinear systems and applications of modern control theory in electromechanical systems.

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Sebastian Ibarra-Rojas received the B.S. and M.I. in electrical engineering in 1994 and 2001, respectively, from Mexico National University, Mexico, where he is currently working towards the Ph.D. degree. His fields of interest are nonlinear systems and applications of modern control theory in electromechanical systems.

Jaime Moreno was born in Medellı́n, Colombia, in 1961. He received his Ph.D. degree in Electrical Engineering (Automatic Control) from the Universität der Bundeswehr-Hamburg, Hamburg, Germany in 1995. The Diplom-Degree in Electrical Engineering (Automatic Control) from the Universität zu Karlsruhe, Karlsruhe, Germany in 1990. He is presently titular Professor at the Automation and Control Department at the Institute of Engineering, National University of Mexico (UNAM) in Mexico City. His current research interests include robust and nonlinear control, in particular with applications to (bio)chemical processes (wastewater treatment processes); and design of nonlinear observers for nonlinear systems.

Gerardo Espinosa-Pérez was born in México in 1961. He received the B.Sc. degree in Mechanical and Electrical Engineering from the National University of México in 1987, the M.Sc. degree in Electrical Engineering from the Centre of Research and Advanced Studies of the Polytechnical Institute of México in 1989 and the Ph.D. in Automatic Control from the National University of México in 1993. Since 1986 he has given several lectures related with Automatic Control at the University of México and currently he is a Full Professor at the Enginnering Faculty of this University. His main research interests are in the field of nonlinear systems with application to electrical machines, power electronics and power systems.

This paper was presented at 15th IFAC World Congress, Barcelona, Spain, 2002. Work supported by Conacyt, Mexico, as a scholarship for the first author, and through projects 41298, 34934A, and PAPIIT, UNAM 106901. This paper was recommended for publication in revised form by Associate Editor Dragan Nesic under the direction of Editor Hassan Khalil.

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