An extended observer based-controller applied to sensorless PMSM drive using the mean value theorem and Lipchitz models

  • Bilal HamidaniEmail author
  • Abdelkrim Allag
  • Okba Zeghib
  • Abrar Allag
Original Article


The focus of this paper is on the notion of an extended observer based on a state feedback controller. The mean value theorem and sector non-linearity approaches were implemented for a class of the Lipchitz model of permanent magnet synchronous machine (PMSM). Based on these mathematical approaches, the proposed design allows the expression of non-linear error dynamics for the state control and the extended observer becomes a convex combination of known matrices that have time varying coefficients. This is similar to a linear parameter varying systems after the non-linear system of the PMSM is represented in the Lipchitz form. Through the Lyapunov theory, one can obtain and express the stability conditions in a term of linear matrix inequalities. The extended observer and the controller gains are separately determined based on the separation principles and have gotten by the exploitation of YALMIP software computer. Furthermore, these gains have completely independent from the PMSM states. As a proof of the efficacy of the proposed approach, one applies an illustrative simulation to the sensorless field oriented control of the PMSM drive by utilizing the MATLAB/SIMULINK environment.


Extended observer State feedback controller Mean value theorem Lipchitz model Linear matrix inequalities Permanent magnet synchronous machine Field oriented control 

List of symbols

\(x\left( t \right)\)

State vector

\(x_{e} \left( t \right)\)

Extended state vector

\(\hat{x}\left( t \right)\)

Estimated state vector

\(\hat{x}_{e} \left( t \right)\)

Extended estimated state vector

\(x_{r} \left( t \right)\)

Reference state vector

\(e_{o} \left( t \right)\)

State estimation error

\(u\left( t \right)\)

Input vector

\(y\left( t \right)\)

Output vector


Rotor speed


Rotor position


Permeant magnets flux

Id, Iq

The (d, q) stator currents

ud, uq

The (d, q) stator voltages

Ld, Lq

The (d, q) stator inductances


Stator resistance


Moment of inertia


Friction coefficient


Pole pair number


Electromagnetic torque


Load torque


Electromagnetic torque


Observer gain


Proportional controller gain


Proportional integral controller gain



Mean value theorem


Permanent magnet synchronous machine


Linear parameter varying


Linear matrix inequalities


Field oriented control


Direct torque control


Model reference adaptive control




Proportional integral


Pulse width modulation



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Copyright information

© The Society for Reliability Engineering, Quality and Operations Management (SREQOM), India and The Division of Operation and Maintenance, Lulea University of Technology, Sweden 2019

Authors and Affiliations

  1. 1.LEVRES LaboratoryEl Oued UniversityEl OuedAlgeria
  2. 2.LABTHOP LaboratoryEl Oued UniversityEl OuedAlgeria
  3. 3.LGEB LaboratoryBiskra UniversityBiskraAlgeria

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