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This paper presents a sensorless speed control strategy for a permanent-magnet synchronous motor (PMSM) based on an adaptive nonlinear extended state observer (ANLESO). In this paper, an extended state observer (ESO), which takes back-EMF (back electromotive force) as an extended state, is used to estimate the rotor position and the rotor speed because of its simpler structure and higher accuracy. Both linear ESO (LESO) and nonlinear ESO (NLESO) are considered to estimate the back-EMF of PMSM, and NLESO is finally implemented due to its obvious advantage in convergence. The convergence characteristics of the estimation error of the observer are analyzed by the Lyapunov theory. In order to take both stability and steady-state error into consideration, an adaptive NLESO is proposed, which adaptively adjusts the parameters of NLESO to a compromised value. The performance of the proposed method was demonstrated by simulations and experiments.

In recent years, permanent-magnet synchronous motors have increasingly gained lots of applications due to their high efficiency, high dynamic response, and high torque to current ratio [

PMSM drives require a position sensor with high resolution to achieve an efficient vector control, such as a shaft encoder or a resolver. Unfortunately, these sensors are expensive and have a limited lifetime. Besides, the size of the motor assembly will become larger. Therefore, many sensorless control methods have been developed to eliminate these mechanical sensors. In particular in some extreme environments where position sensors may work abnormally, sensorless control methods are more robust than those using position sensors [

By means of injecting high frequency signal into motor, the impedance difference caused by magnetic saturation, which contains the information about the rotor position, can be calculated [

The extended state observer (ESO) first proposed by Huang and Han in [

In this paper, a sensorless speed control scheme for a PMSM based on an ANLESO is proposed. ESO is applied to estimate the rotor position and rotor speed of PMSM for the first time. The information of the rotor position and the rotor speed of PMSM can be calculated by estimating the back-EMF. For this reason, the estimation accuracy of back-EMF affects the accuracy of the position and speed directly. ESO usually treats all the external disturbances and/or system uncertainties as the “total disturbance” and extends it as an additional state variable [

The state equations of the stator current in a stator-fixed reference frame and the back-EMF for each phase in the fixed frame of the PMSM are discussed in Section

The voltages in the three phases can be transformed into the synchronous coordinates in the two phases for vector control. The state equations of the stator current, written in a stator-fixed reference frame

The back EMF for each phase can be represented in the fixed frame as

In the sensorless control system, the back-EMF

The block diagram of the sensorless control system for PMSM is shown in Figure

Block diagram of the sensorless ve r the positively definite prctor control of PMSM.

Note that this system is symmetric, and we only need to design an observer to estimate

Set

The output can be written as

Now

Let

Obviously, the LESO is bounded-input bounded-output (BIBO) stable if

For the sake of reducing the steady-state error, a nonlinear extended state observer is proposed as follows:

Let

When the steady state is reached, we can obtain

Hence, the steady-state errors are

For system (

The goal is to prove that there exists a positive definite energy function whose derivative is always made negative. The following energy function is chosen [

In order to guarantee that

Since

Calculate the partial derivative of

The derivative of the energy function is as follows:

By substituting (

From (

Assume that

Equation (

From ① we have

From ② we have

Then, from ③ we can derive that

Then, the following expression can be obtained

Assume that

Equation (

It can be verified that there exists some meaningful

Therefore,

Hence,

Inequality (

The value of

Since

In the same way, the value of

Consider that

Obviously, the state error would not actually go to zero but enter, and is contained within, a neighborhood of the origin by this method. Therefore, by means of setting appropriate

Using a sigmoid function as the nonlinear function in ESO can reduce chatting in the system [

Although a large

Rewrite inequality (

Thus, the roots of (

This means that the lower bound of

Figure

Flow chart of the algorithm to calculate

Using the estimated back-EMF obtained by the ANLESO, the position and speed signals can be calculated easily. According to (

Similarly, the rotor position can be deduced from (

In order to show the high-speed performance of the proposed ANLESO, it is necessary to compare it with the conventional SMO through the MatLab/Simulink programming environment. The parameters of PMSM used in simulation are listed in Table

The parameters of PMSM.

Parameters | Values |
---|---|

Rated power | 1.5 KW |

Rated speed | 2500 r/min |

Input voltage (DC) | 310 V |

Current | 6 A |

Stator resistance | 1.18 Ω |

Stator inductance | 53.26 mH |

Rotational inertia | 1.33 × 10^{−3} kg·m^{2} |

Flux of permanent magnet | 0.0356 Wb |

Poles | 8 |

Figures

Simulation waveforms of estimated back EMF when speed is 500 r/min and the load torque is 1 Nm: (a) obtained by the proposed ANLESO and (b) obtained by the conventional SMO.

Simulation waveforms of actual and estimated

Simulation waveforms of actual and estimated speeds when speed is 500 r/min and the load torque is 1 Nm: (a) obtained by the proposed ANLESO and (b) obtained by the conventional SMO.

Simulation waveforms of actual rotor position, estimated rotor position, and estimated error when speed is 500 r/min and the load torque is 1 Nm: (a) obtained by the proposed ANLESO and (b) obtained by the conventional SMO.

As can be seen from Figures

Figures

Simulation waveforms of estimated back EMF when speed is 2000 r/min and the load torque is 1 Nm: (a) obtained by the proposed ANLESO and (b) obtained by the conventional SMO.

Simulation waveforms of actual and estimated

Simulation waveforms of actual and estimated speeds when speed is 2000 r/min and the load torque is 1 Nm: (a) obtained by the proposed ANLESO and (b) obtained by the conventional SMO.

Simulation waveforms of actual rotor position, estimated rotor position, and estimated error when speed is 2000 r/min and the load torque is 1 Nm: (a) obtained by the proposed ANLESO and (b) obtained by the conventional SMO.

As can be seen from Figures

To further verify the performance of the new SMO for estimating rotor position and speed, an experimental system has been designed to control the 130SJT-M060D (1.5 kw) sinusoidal PMSM motor made by GSK.

Figure

Platform of 1.5 kw PMSM sensorless control system based on DSP.

Figures

Operating waveforms of estimated back EMF in steady state when speed is 500 r/min and the load torque is 1 Nm: (a) obtained by the proposed ANLESO and (b) obtained by the conventional SMO.

Operating waveforms of estimated

Operating waveforms of estimated rotor speeds when speed is 500 r/min and the load torque is 1 Nm: (a) obtained by the proposed ANLESO and (b) obtained by the conventional SMO.

Operating waveforms of rotor position and estimate error when speed is 500 r/min and the load torque is 1 Nm. (a) Actual and estimated rotor position obtained by the proposed ANLESO. (b) Actual and estimated rotor position obtained by the conventional SMO. (c) Estimated error obtained by the proposed ANLESO. (d) Estimated error obtained by the conventional SMO.

Figures

Figure

Figures

Operating waveforms of estimated back EMF in steady state when speed is 2000 r/min and the load torque is 1 Nm: (a) obtained by the proposed ANLESO and (b) obtained by the conventional SMO.

Operating waveforms of estimated

Operating waveforms of estimated rotor speeds when speed is 2000 r/min and the load torque is 1 Nm: (a) obtained by the proposed ANLESO and (b) obtained by the conventional SMO.

Operating waveforms of rotor position and estimate error when speed is 2000 r/min and the load torque is 1 Nm. (a) Actual and estimated rotor position obtained by the proposed ANLESO. (b) Actual and estimated rotor position obtained by the conventional SMO. (c) Estimated error obtained by the proposed ANLESO. (d) Estimated error obtained by the conventional SMO.

Figures

Figures

As shown in Figure

In this paper, an adaptive nonlinear extended state observer has been designed for the sensorless control of a PMSM. The convergence of this observer has been proved by means of a Lyapunov stability analysis. An adaptive algorithm is adopted to calculate the compromised parameter of ESO in order to take both stability and steady-state error into consideration. The good performance of the proposed sensorless control system was verified by several experimental results. The results show that ANLESO has more advantages than conventional SMO. Compared to conventional SMO, ANLESO has smaller chattering, higher accuracy, and no phase delay.

In future works, we will explore the stability of the ESO under parameter uncertainties in sensorless control system.

The authors declare that there is no conflict of interests regarding the publication of this paper.

This project was supported by the National Natural Science Foundation of China (Grant no. 61304097) and Projects of Major International (Regional) Joint Research Program NSFC (Grant no. 61120106010).