Mathematical Problems in Engineering

Mathematical Problems in Engineering / 2015 / Article

Research Article | Open Access

Volume 2015 |Article ID 807615 | 14 pages | https://doi.org/10.1155/2015/807615

An Adaptive Nonlinear Extended State Observer for the Sensorless Speed Control of a PMSM

Academic Editor: Andrzej Swierniak
Received13 Dec 2014
Accepted05 Mar 2015
Published05 Apr 2015

Abstract

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.

1. Introduction

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 [1, 2]. Compared to DC motors, PMSMs are superior in robustness and overload capability and they do not need mechanical commutation. As the fast switch components and digital signal processors become economical, PMSMs are widely used as an alternative for energy saving applications. Besides, some countries have kept using low-cost induction motors due to their simple structure and easiness in manufacturing. However, induction motors are less efficient than PMSMs in terms of performance [3, 4].

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 [4].

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 [5]. However, injecting high frequency signal brings noise and torque ripples. Because of good standstill performance, the method is usually used within low-speed range [6, 7]. The extended Kalman filter (EKF) was also employed for rotor speed and position estimation due to its good dynamic performance and strong antijamming capability [8]. Algorithm complexity and large calculating quantity are disadvantages of EKF, which limits its practical application. Neural network technology has a good advantage in terms of parameter identification and it has been used in sensorless control system by many scholars [9]. In these papers, the rotor position is identified online by neural network. However, the method needs a long computation time and brings great burden on the digital signal processing, which gives rise to choke points in applications. Zhang and Li proposed the model reference adaptive scheme (MRAS) to implement a sensorless control system. The idea of MRAS is to build a reference model of the PMSM system without containing unknown parameters and to obtain the difference between the reference model outputs and actual system outputs. Next, an appropriate adaptive law will be designed to adjust parameters of reference model in real time according to the difference [10]. But the accuracy of this method depends on the designed reference model, which may prohibit the convergence of the system. Moreover, in [11, 12], the flux linkage is used as state variable and is estimated directly. Some authors used a sliding mode observer (SMO) approach [6, 1316]. The structure of SMO is simple and convenient to be used in practical applications. Unfortunately, the estimation accuracy of SMO would be affected by chattering.

The extended state observer (ESO) first proposed by Huang and Han in [17] is the key step of the active disturbance rejection control (ADRC) that is taking off as a technology after numerous successful applications in engineering. ESO has simple structure and it is able to estimate unmodeled dynamics accurately in many cases. In terms of ADRC, a class of nonlinear ESOs is designed to estimate the sum of both the states and external disturbances [18]. After that, Gao proposed a class of linear ESOs (LESO) and provided guidance on how to choose the optimal parameters in the controller design. In this way, the number of parameters to be configured is reduced [19]. At present, ESO is mainly used in control system to estimate disturbances and to compensate them via a feed-forward cancellation technique [2022]. Besides, ESO can be extended to multi-input–multi-output systems as well [22].

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 [16]. Therefore, system states and unknown parts of system can be estimated by ESO in the meantime without knowing the model of the unknown parts (the derivative of the unknown parts). In this scheme, back-EMF is the unknown part of the system and it would be treated as an additional state variable to be estimated. Back-EMF contains the information of the rotor position and speed that cannot be obtained without sensor. For this reason, back-EMF cannot be modeled directly. As mentioned previously, ESO would treat back-EMF as “disturbance” and estimate the back-EMF accurately without knowing its model. By means of Lyapunov stability analysis, an adaptive control scheme is adopted to balance stability and steady-state error. The adaptive control scheme could adaptively adjust the parameters of ESO to a compromised value, and the steady-state error will converge to a minimum on the premise of system stable. Compared to other sensorless control methods, ANLESO has simpler structure, smaller chattering, and higher accuracy.

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 2. In Section 3, an adaptive nonlinear extended state observer for the sensorless control scheme is designed and discussed. Section 4 gives the simulation results and Experimental results. Finally, conclusions and suggestions for future improvements are given in Section 5.

2. Mathematical Model

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 , take the following form [23]:where , , and represent the current, electromotive force, and voltage for each phase, respectively. and represent the stator resistance and inductance, respectively.

The back EMF for each phase can be represented in the fixed frame aswhere , , and represent the magnetic flux of the PM, the electric angular velocity, and the rotor angle, respectively.

3. Proposed Adaptive Nonlinear ESO

In the sensorless control system, the back-EMF , are usually figured out by the information of control signals , and output currents , . In other words, from the system formulated by (1) and (2), and can be treated as the unknown parts of the system. Thus, the back-EMF would be extended as additional state variables. In this section, linear ESO and nonlinear ESO are described, respectively. For LESO, the bounded-input bounded-output (BIBO) stability can be reached. For NLESO, a Lyapunov stability analysis is shown. At last, an adaptive control scheme is designed in order to improve the estimated accuracy.

The block diagram of the sensorless control system for PMSM is shown in Figure 1. The space vector pulse width modulation (SVPWM) algorithm is used as modulation strategy to generate a sinusoidal current of the stator. To control the PMSM, the three-phase coordinates need to be transformed into synchronous coordinates by Clarke transformation and Park transformation. An ADC (analog to digital converter) module is used to detect phase current and bus voltage. The proportional-integral (PI) control is used in speed loop and current loop. All the sensorless control algorithms are simulated by MatLab and executed on a DSP chip.

3.1. Linear ESO

Note that this system is symmetric, and we only need to design an observer to estimate (the structure and parameters of the observer). In this way, the order of the observer can be reduced.

Set and as augmented states. is the derivative of , which is a variable that we do not need to know. The state equation (1) is written in the following form:

The output can be written aswhere .

Now can be estimated using a linear extended state observer as follows:where and are used to estimate the value of and , respectively. is the observer gain vector, which can be obtained using any known method, for example, the pole placement technique.

Let ; then the error equation can be written aswhere

Obviously, the LESO is bounded-input bounded-output (BIBO) stable if is stable and is bounded [19].

3.2. Nonlinear ESO

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

Let ; then the error equation can be written as

When the steady state is reached, we can obtain

Hence, the steady-state errors are

Theorem 1. For system (4), the state error of the nonlinear extended state observer (9) converges asymptotically if the following conditions are satisfied: (C1)(C2).

Proof. 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 [24]:where , , and are the parameters to be determined. Suitable , , and need to be chosen to let the energy function meet the Lyapunov stability theorem.
In order to guarantee that is positively definite, the following rules need to be followed: (R1)(R2).
Since is bounded, (R2) always can be satisfied by choosing a suitable .
Calculate the partial derivative of as follows:The derivative of the energy function is as follows:By substituting (15) into (16), can be given asFrom (17), the following expression can be deduced:Assume thatEquation (18) can be rewritten aswhere is a quadratic function with regard to and . is a plane function with regard to and . A necessary and sufficient condition for the positively definite property of the quadratic function can be obtained: .
From ① we have (R3).
From ② we have(R4), which means .
Then, from ③ we can derive thatThen, the following expression can be obtainedAssume thatEquation (22) can be rewritten asIt can be verified that there exists some meaningful satisfying inequation (24) and meeting the rules (R1)~(R4) if and only if the following conditions are satisfied:Therefore, , and inequality (24) denotes a parabola going upwards. Due to the existence of some meaningful satisfying inequality (24), the vertex of the parabola must be negative:Hence,Inequality (26) is established between the two real roots of the quadratic equation . In order to separate the two roots as far as possible and make the smaller one very small, a very large needs to be selected, which makes , and the choice of parameter is required to ensure . Finally, choose appropriate to make parameters and satisfy rules (R1)~(R4). In this way, will be a negative definite function with variable in a very wide range and arbitrary . Meanwhile, the area in which does not satisfy negative conditions (the area ) is the part above the intersecting line of the quadratic function and the plane function .
The value of in the intersecting line is the same order as which is the root of the following equation:Since is the same order as , the value of in the intersecting line is the same order as . Therefore, a large enough can guarantee that is stable in a very wide range. The larger we set, the smaller the unstable boundary of , and then it will result in a better convergence.
In the same way, the value of in the intersecting line is the same order as which is the root of the following equation:Consider that , the value of in the intersecting line is the same order as . From inequality (21), the inequality holds. Therefore, a large enough , by setting , can guarantee the existence of appropriate and to make small enough.

Remark 1. 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 and , the performance required in practical engineering can be achieved.

Remark 2. Using a sigmoid function as the nonlinear function in ESO can reduce chatting in the system [15]. Considerwhere

3.3. Proposed Adaptive Nonlinear ESO

Although a large will render stable in a wider range, the steady-state error still has to be considered here. From (13) we know that a large leads to the increase of in steady state. The improved method proposed in this paper is to adaptively adjust the parameters of NLESO to a compromised value which takes both stability and steady-state error into consideration.

Rewrite inequality (27) as the following expression:

Thus, the roots of (29) hold the following equality:

This means that the lower bound of is larger than . is a design parameter, so the size of can be changed by means of adjusting . The main idea of this method is to decrease gradually to reduce steady-state error and increase when the system may become unstable. By several iterations, the value of would approximate the compromised value gradually.

Figure 2 illustrates the algorithm to calculate , where and are adjustment factors. is the error of command velocity and measured velocity.

3.4. Rotor Position and Speed Estimate

Using the estimated back-EMF obtained by the ANLESO, the position and speed signals can be calculated easily. According to (3), the rotor speed can be obtained as follows:

Similarly, the rotor position can be deduced from (3) as follows:

4. Simulation and Experimental Results

4.1. Simulation Results

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 1.


ParametersValues

Rated power1.5 KW
Rated speed2500 r/min
Input voltage (DC)310 V
Current6 A
Stator resistance1.18 Ω
Stator inductance53.26 mH
Rotational inertia1.33 × 10−3 kg·m2
Flux of permanent magnet0.0356 Wb
Poles8

Figures 36 show the simulation waveforms when the reference speed is changed from zero to 500 r/min and the load torque is 1 Nm. In these figures, part (a) gives the simulation waveforms obtained by the proposed ANLESO and part (b) gives the simulation waveforms obtained by the conventional SMO. Estimated back-EMF, actual and estimated phase currents, actual and estimated rotor speed, and actual and estimated rotor position are also shown in these figures.

As can be seen from Figures 36, the accuracy of estimation has been greatly improved with the help of proposed ANLESO. Different from the conventional SMO, the back-EMF in Figure 3(a) almost has no chattering. Compared with Figures 4(c) and 4(d), the estimated results of phase currents in Figures 4(a) and 4(b) are better. Due to the good performance of estimating back-EMF and currents, the chattering phenomenon of the estimated rotor position and speed is reduced, which can be seen from Figures 5(a), 5(b), 6(a), and 6(b). In part (a) of these figures, the estimated speed is almost the same as actual speed; however, the estimated speed in part (b) of these figures exhibits the chattering phenomenon. Besides, it can be seen that the estimated error of the position in part (a) is smaller than that shown in part (b) of these figures.

Figures 710 show the two sets of simulation waveforms when the reference speed is changed from zero to 2000 r/min and the load torque is 1 Nm. In these figures, part (a) gives the simulation waveforms obtained by the proposed ANLESO and part (b) gives the simulation waveforms obtained by the conventional SMO. Estimated back-EMF, actual and estimated phase currents, actual and estimated rotor speed, and actual and estimated rotor position are also shown in these figures.

As can be seen from Figures 710, when the rotor speed is raised to 2000 r/min, advantages of ANLESO become more apparent than conventional SMO. Comparing the estimated back-EMF in Figures 7(a) and 7(b), we can find that the former has smaller chattering. As a result, the estimation of rotor speed becomes more accurate by ANLESO, which can be seen in Figures 9(a) and 9(b). The waveform forms presented in Figures 10(a) and 10(b) show the estimated rotor position. The position estimation error is reduced and the accuracy of rotor position estimation is improved in Figure 10(a).

4.2. Experimental Results

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 11 shows a photograph of the sensorless control system. The High Voltage Digital Motor Control Kit, made by TI, is utilized in the system, which contains power modules, switching devices, current and voltage sensing circuit, ADC modules, and PWM DAC (digital to analog converter) modules. A floating-point TMS320F28335 DSP is employed as the digital controller. The current control cycle is 100 μs and the velocity control cycle is 1 ms. A 2 μs dead time is chosen for the PWM switching. All of the control variables are monitored in real time by an oscilloscope after they are converted to analog signals through the PWM DACs modules, which use an external low pass filter to generate the waveforms. The motor parameters used in experiment are given in Table 1.

Figures 1214 show the operating waveforms when the reference speed is changed from zero to 500 r/min and the load torque is 1 Nm. In these figures, part (a) gives the operating waveforms obtained by the proposed ANLESO and part (b) gives the operating waveforms obtained by the conventional SMO. Estimated back-EMF, estimated phase currents, and estimated rotor speed are shown in these figures. In Figure 15, actual rotor position, estimated rotor position, and their error are shown.

Figures 12(a) and 13(a) show the experimental results of the proposed ANLESO. Compared to the waveforms obtained by SMO (see part (b) in Figures 12 and 13), the waveforms of the estimated back-EMF and current shown in Figures 12(a) and 13(a) are more accurate. The waveforms presented in Figures 14(a) and 14(b) show the differences of rotor speed estimated by two kinds of observers. The former speed waveform has less chattering. In other words, ANLESO has higher estimated precision.

Figure 15 shows the experimental results of position response. The estimated error shown in Figure 15(c) is smaller than that in Figure 15(d), which verifies the high precision of ANLESO.

Figures 1618 show the operating waveforms when the reference speed is changed from zero to 2000 r/min and the load torque is 1 Nm. In these figures, part (a) gives the operating waveforms obtained by the proposed ANLESO and part (b) gives the operating waveforms obtained by the conventional SMO. In these figures, estimated back-EMF, estimated phase currents, and estimated rotor speed are shown. In Figure 19, actual rotor position, estimated rotor position, and their error are shown.

Figures 1215 show the operating waveforms when the reference speed is changed from zero to 2000 r/min. Figures 12 and 13 provide the operating waveforms obtained by the proposed ANLESO. Figures 14 and 15 show the simulation waveforms obtained by the conventional SMO. In these figures, estimated back-EMF, estimated phase currents, estimated rotor speed, and actual and estimated rotor position are shown.

Figures 1618 show the experimental results of estimated back-EMF, current and rotor speed when the rotor speed is raised to 2000 r/min. It can be seen that the waveforms in part (a) of these figures have smaller chattering and higher accuracy. In this way, the advantages of ANLESO have been verified.

As shown in Figure 19, when the rotor speed is raised to 2000 r/min, the advantages of ANLESO to estimate rotor position become more apparent than conventional SMO. The estimation error in Figure 19(c) is significantly less than that in Figure 19(d). That is to say, ANLESO has higher accuracy and smaller chattering when being applied to estimate position than SMO.

5. Conclusions

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.

Conflict of Interests

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

Acknowledgments

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).

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Copyright © 2015 Minggang Gan and Chenyi Wang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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