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

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## Abstract

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.

## Keywords

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

- Ω
Rotor speed

*θ*Rotor position

- Φ
_{f} Permeant magnets flux

*I*_{d},*I*_{q}The (d, q) stator currents

*u*_{d},*u*_{q}The (d, q) stator voltages

*L*_{d},*L*_{q}The (d, q) stator inductances

*R*_{s}Stator resistance

*J*Moment of inertia

*f*Friction coefficient

*p*Pole pair number

*T*_{e}Electromagnetic torque

*T*_{L}Load torque

*T*_{e}Electromagnetic torque

*L*_{0}Observer gain

*K*_{0}Proportional controller gain

- \(\bar{K}\)
Proportional integral controller gain

## Abbreviations

- MVT
Mean value theorem

- PMSM
Permanent magnet synchronous machine

- LPV
Linear parameter varying

- LMI’s
Linear matrix inequalities

- FOC
Field oriented control

- DTC
Direct torque control

- MRAC
Model reference adaptive control

- P
Proportional

- PI
Proportional integral

- PWM
Pulse width modulation

## Notes

## References

- Allag A, Benakcha A, Allag M, Zein I, Ayad MY (2015) Classical state feedback controller for nonlinear systems using mean value theorem: closed loop-FOC of PMSM motor application. Front Energy 9(4):413CrossRefGoogle Scholar
- Azza HB, Moujahed M, Jemli M, Boussak M (2017) Implementation of improved sliding mode observer and fault tolerant control for a PMSM drive. J Circuits Syst Comput 26(02):1750032CrossRefGoogle Scholar
- Benchabane F, Titaouine A, Bennis O, Yahia K, Taibi D (2012) Sensorless fuzzy sliding mode control for permanent magnet synchronous motor fed by AC/DC/AC converter. Int J Syst Assur Eng Manag 3(3):221–229CrossRefGoogle Scholar
- Hammoudi MY, Allag A, Becherif M, Benbouzid M, Alloui H (2014) Observer design for induction motor: an approach based on the mean value theorem. Front Energy 8(4):426–433CrossRefGoogle Scholar
- Ichalal D, Marx B, Mammar S, Maquin D, Ragot J (2018) How to cope with unmeasurable premise variables in Takagi–Sugeno observer design: dynamic extension approach. Eng Appl Artif Intell 67:430–435CrossRefGoogle Scholar
- Lan Y-H (2018) Backstepping control with disturbance observer for permanent magnet synchronous motor. J Control Sci Eng 2018:1–8MathSciNetCrossRefGoogle Scholar
- Liang D, Li J, Qu R, Kong W (2018) Adaptive second-order sliding-mode observer for PMSM sensorless control considering VSI nonlinearity. IEEE Trans Power Electron 33(10):8994–9004CrossRefGoogle Scholar
- Mani P, Rajan R, Shanmugam L, Joo Y-H (2018) Adaptive fractional fuzzy integral sliding mode control for PMSM model. IEEE Trans Fuzzy Syst 27(8):1674–1686CrossRefGoogle Scholar
- Niu F, Wang B, Babel AS, Li K, Strangas EG (2016) Comparative evaluation of direct torque control strategies for permanent magnet synchronous machines. IEEE Trans Power Electron 31(2):1408–1424CrossRefGoogle Scholar
- Phanomchoeng G, Rajamani R, Piyabongkarn D (2011) Nonlinear observer for bounded Jacobian systems, with applications to automotive slip angle estimation. IEEE Trans Autom Control 56(5):1163–1170MathSciNetCrossRefGoogle Scholar
- Ren J-J, Liu Y-C, Wang N, Liu S-Y (2015) Sensorless control of ship propulsion interior permanent magnet synchronous motor based on a new sliding mode observer. ISA Trans 54:15–26CrossRefGoogle Scholar
- Tanaka K, Wang HO (2004) Fuzzy control systems design and analysis: a linear matrix inequality approach. Wiley, New YorkGoogle Scholar
- Wu Y-J, Li G-F (2018) Adaptive disturbance compensation finite control set optimal control for PMSM systems based on sliding mode extended state observer. Mech Syst Signal Process 98:402–414CrossRefGoogle Scholar
- Yoneyama J (2014) Nonlinear control design based on generalized Takagi–Sugeno fuzzy systems. J Frankl Inst 351(7):3524–3535MathSciNetCrossRefGoogle Scholar
- Zhang X, Li Z (2016) Sliding-mode observer-based mechanical parameter estimation for permanent magnet synchronous motor. IEEE Trans Power Electron 31(8):5732–5745CrossRefGoogle Scholar
- Zhong C, Lin Y (2017) Model reference adaptive control (MRAC)-based parameter identification applied to surface-mounted permanent magnet synchronous motor. Int J Electron 104(11):1854–1873CrossRefGoogle Scholar