Improved ANFIS based MRAC observer for sensorless control of PMSM

Abstract

Detection of the rotor position is an important prerequisite for controlling the speed and developed torque in permanent magnet synchronous motor (PMSM). Even though use of incremental encoder and resolver is one of the popular schemes for sensing the rotor position in a PMSM drive, it increases the size and weight of the drive and reduces its reliability. Dynamic modeling of the motor and control algorithms are often used in sensor-less control of PMSM to estimate rotor position and motor speed. Most sensor-less control algorithms use machine parameters like torque constant, stator inductances and stator resistance for estimating the rotor position and speed. However, with accuracy of such estimation and the performance of the motor degrades with variation in motor parameters. Model reference adaptive control (MRAC) provides a simple solution to this issue. An improved Adaptive neuro-fuzzy inference system (ANFIS) based MRAC observer for speed control of PMSM drive is presented in this paper. In the proposed method adaptive model and adaptive mechanism are replaced by an improved ANFIS controller, which neutralize the effect of parametric variation and results in improved performance of the drive. The modeling equations of PMSM are used to estimate the rotor position for speed and torque control of the drive. Simulation studies have been carried out under various operating condition using MATLAB/Simulink. In addition, a comparative analysis of the conventional MRAC based observer and improved ANFIS based MRAC observer is carried out. It is observed that the proposed method results in better performance of the PMSM drive.

Keywords

PMSM space vector PWM (SVPWM)model reference adaptive control PI controller adaptive neuro-fuzzy inference system

1 Introduction

Nowadays, alternating current (AC) motors are widely used for industrial, robotic and automobile applications in comparison to direct current (DC) motors due to their better efficiency, small size for the same power output and low maintenance. However, the control of AC motor is more complex than that of DC motor. Induction motor (IM) and PMSM are the most widely used AC motors. PMSMs are preferred in industrial applications over IM, due to their better reliability, compactness, higher power density and higher efficiency [1 –5]. However, for efficient control of PMSM it is essential to know the rotor position, which can either be obtained using resolvers / encoders or estimated using the measured stator currents and back emf. Position sensors like resolver and encoder are mounted on the motor shaft, which results in reduced reliability due to sensitivity to vibration, high temperature, and noise. This also increases the volume, weight and the machine cost. These issues can be addressed by estimation of the rotor position rather than using resolvers / encoders. The reported methods for sensor-less control of PMSM drives include: 1) flux linkage and back-emf based estimation of rotor position [6, 7]; 2) high frequency (HF) signal injection for tracking the rotor saliency [8, 9] and 3) nonlinear state observer-based estimation of rotor position and speed [10]. Based on these categories many established algorithms are used to estimate the rotor position and motor speed in PMSM drives [11 –16].

The back-EMF based method is generally employed for medium to high speed operation of the PMSM drive; while HF signal injection method is used for low operation. In HF signal injection method, the position of the rotor is obtained by analyzing the injected HF signal. The observer-based sliding mode observer techniques are easy to implement and immune to parametric variations but often results in the chattering problem. MRAC based observer is a direct control approach [17], which can effectively manage any systems with parametric variations using the models, viz- reference model and adjustable model. An adaptive mechanism is implemented to adjust the motor parameters continuously. While adjustable model depends on unknown parameter, reference model is not parameter dependent. The error signal of two models is applied to an adaptive mechanism to estimate unknown quantities which tunes the adjustable model. Adaptive control is widely used for both the linear and nonlinear systems. However, the estimation of unknown quantities of the motor in nonlinear system is complex to implement.

This paper proposes an improved ANFIS-based MRAC technique to solve these problems related to uncertain parameters and input saturation [18, 19]. In the proposed method, the adaptive model and mechanism to minimize the stator current error are replaced by ANFIS controller. In the architecture of ANFIS, fuzzy has the ability to handle uncertainties and artificial neural network (ANN) has the ability to learn from the process. An adaptive model of PMSM is developed using ANFIS at undefined operating conditions, which automatically compensates for variations in resistance, inductance etc. The PMSM drive is modeled using the rotating reference frame. This mitigates the issues associated with adaptive control in estimating PMSM’s rotor position with unknown parameters. The novelties of the proposed improved ANFIS based MRAC observer include: -

Performance of the proposed observer is improved by using adaptive normalization method to normalize the inputs and output data extracted from the Proportional-plus-integral (PI) controller, which is further applied to reduce the error of two models to fine tune the membership function near the desired speed.

The PI controller is replaced by an improved ANFIS controller.

This paper focuses on the effect of load variation on the speed response in terms of settling time, rise time and overshoot.

The proposed improved ANFIS based MRAC observer is investigated through simulation studies using MATLAB/Simulink for sensor-less speed control of PMSM drive over a wide varying operating condition. The performance of drive with the proposed MRAC is also demonstrated for low-speed operation at low load torque. This paper is organized into six sections with Introduction in Section 1, Sections 2 to 6 cover Modeling of PMSM in d-q Coordinates, Design of Improved ANFIS based MRAC Observer, Principal of SVPWM, Results and Discussions followed by the Conclusion.

2 Dynamic modeling of PMSM in d-q coordinates

The PMSM is modelled in the d-q rotating reference frame as $[\begin{matrix} v_{qs}^{r} \\ v_{ds}^{r} \end{matrix}] = [\begin{matrix} R_{s} & 0 \\ 0 & R_{s} \end{matrix}] [\begin{matrix} i_{qs}^{r} \\ i_{ds}^{r} \end{matrix}] + [\begin{matrix} ρ & ω_{r} \\ - ω_{r} & ρ \end{matrix}] [\begin{matrix} λ_{qs}^{r} \\ λ_{ds}^{r} \end{matrix}]$ (1) $[\begin{matrix} λ_{qs}^{r} \\ λ_{ds}^{r} \end{matrix}] = [\begin{matrix} L_{q} & 0 \\ 0 & L_{d} \end{matrix}] [\begin{matrix} i_{qs}^{r} \\ i_{ds}^{r} \end{matrix}] + [\begin{matrix} 0 \\ λ_{pm} \end{matrix}]$ (2)

Where $v_{qs}^{r}$ , $v_{ds}^{r}$ , $i_{qs}^{r}$ , $i_{ds}^{r}$ , L_q, L_d and $λ_{qs}^{r}$ , $λ_{ds}^{r}$ are defined as the stator voltages, currents, inductances and flux in the d-q coordinates respectively. R_s, ω_r and λ_pm are stator winding resistance, electrical rotor speed and flux linkage of permanent magnet respectively. ρ is the differential function and P is number of poles. The electromagnetic motor torque, T_e is derived as: $T_{e} = (\frac{3}{2}) (\frac{P}{2}) [λ_{ds}^{r} i_{qs}^{r} - λ_{qs}^{r} i_{ds}^{r}]$ (3)

The load torque, T_L is defined in terms of T_e, mechanical speed (ω_m), inertia constant (J) and damping coefficient (B) as: $T_{L} = T_{e} - B ω_{m} - J \frac{d ω_{m}}{dt}$ (4)

The mechanical motor speed of given by $ω_{m} = \frac{1}{J} \int (T_{e} - T_{L} - B ω_{m}) dt$ (5)

The rotor electrical speed, ω_r and rotor position, θ are: $ω_{r} = \frac{P}{2} ω_{m}$ (6) $θ = \int ω_{m} dt$ (7)

Figure 1 shows the block diagram of PMSM Drive with improved ANFIS based MRAC observer. It includes an improved ANFIS based MRAC observer with SVPWM and a speed controller in outer loop and two current controllers in inner loop. The rotor position and motor speed and are assessed through the proposed observer using (8), (9) and (10). The measured terminal voltage and current are transformed into the d-q axes using vector transformation. The PWM pulses are generated by SVPWM and fed to voltage source inverter (VSI) which produces the voltage signal for the operation of PMSM.

Fig. 1

A Block diagram of PMSM Drive with Improved ANFIS based MRAC Observer.

3 Design of improved ANFIS based MRAC observer

The rotor position and speed of the PMSM are effectively estimated through MRAC based observer. Figures 2(a) and 2(b) show the conventional MRAC based observer and improved ANFIS based MRAC observer used for sensor less control of the PMSM. In MRAC based observers the reference model is derived from PMSM model and adjustable model is derived from stator winding currents, which is regulated by estimated value of speed [20 –22]. In the conventional MRAC adjustable model is regulated by a PI controller used in the adaptive mechanism; while in the proposed method, the adjustable model is regulated by the improved ANFIS based adaptive mechanism.

Fig. 2

A Block diagram of: (a) Conventional MRAC, (b) Improved ANFIS based MRAC.

Adjustable model is derived with the help of stator current equation. Speed error is continuously monitored for negative feedback to ensure system stability.

The stator current equation of PMSM is given as $\frac{{di}_{d}}{dt} = - \frac{R_{s}}{L_{s}} i_{d} + ω_{r} i_{q} + \frac{v_{d}}{L_{s}}$ (8) $\frac{{di}_{q}}{dt} = - \frac{R_{s}}{L_{s}} i_{q} - ω_{r} i_{d} - \frac{ψ_{r}}{L_{s}} ω_{r} + \frac{v_{q}}{L_{s}}$ (9) here L_s is defined as stator inductance and ψ_r is rotor flux

The Equations (8) and (9) can be expressed as $ρ [\begin{matrix} i_{d}^{'} \\ i_{q}^{'} \end{matrix}] = [\begin{matrix} - \frac{R_{s}}{L_{s}} & ω_{r} \\ - ω_{r} & - \frac{R_{s}}{L_{s}} \end{matrix}] [\begin{matrix} i_{d}^{'} \\ i_{q}^{'} \end{matrix}] + \frac{1}{L_{s}} [\begin{matrix} v_{d}^{'} \\ v_{q}^{'} \end{matrix}]$ (10)

Where $i_{d}^{'} = i_{d} + \frac{ψ_{r}}{L_{s}}$ , $i_{q}^{'} = i_{q}$ , $v_{d}^{'} = v_{d} + \frac{R_{s} ψ_{r}}{L_{s}}$ and $v_{q}^{'} = v_{q}$ .

Replacing the conventional adjustable model and adaptive mechanism by improved adjustable model and ANFIS based adaptive mechanism mitigates parameter uncertainties. The ANFIS strategy is based of Takagi-Sugeno (TS) system. The rules are defined in linguistic forms hence transitional results can be evaluated and understood easily [23]. The rules can be changed during training and optimization process.

The data set required for training of the ANFIS learning is the I/O data pairs of the defined system. The broad outline of the architecture of the ANFIS controller involves the i) defining of fuzzy inference systems; ii) defining of training data set and checking the data sets; iii) defining number of data pairs; iv) defining number of epochs; and v) Learning from the results.

The architecture of ANFIS is given in Fig. 3. It is similar to the fuzzy inference system, expect for the neural network block. It is structured in five layers linked to each other and termed as follows:

Fig. 3

Architecture of (a) Fuzzy inference system (b) Corresponding ANFIS.

Layer 1: It is defined as the input layer consisting input variables. In this layer triangular membership function are used and input variables are delivered to the next layer at every node.

Layer 2: It is defined as membership function (MF) layer in which the weights of every membership function can be checked. It accepts the input variables from the input layer and determined the MFs for the corresponding input variable’s fuzzy sets. It also calculates the membership values in order to identify the degree of the respective input value that is transferred to the next layer to determine the input of that layer.

Layer 3: Every neuron matches the required necessary conditions of fuzzy rules in this layer. The number of layers is equal to number of fuzzy rules and the normalized weights are estimated by each node of these layers. It is called rule layer.

Layer 4: This layer is termed as the defuzzification layer, which generates output values based on application of rules. Links between layers 3 and layer 4 is weighted by the fuzzy singletons that characterizes another set of parameters for the neuro-fuzzy network.

Layer 5: It combines all deliverable inputs from defuzzification layer and converts the fuzzy sets into a crisp value. It is called the output layer.

Figure 4 shows the flow chart of proposed MRAC observer, which summarizes how the proposed observer is designed to estimate the motor speed and rotor position.

Fig. 4

Flow chart of proposed Improved ANFIS based MRAC Observer.

4 Principal of SVPWM

SVPWM method of PWM generation is the most prevalent among all the methods of PWM generation as it provides high DC voltage, V_dc and is easy for implementation in the digital domain [24 –27]. The output voltages of the inverters are represented as space vectors. The spatial position of the three phase voltages (v_an, v_bn and v_cn) fed to the stator windings, determines the magnitude and position of the space vector, v. The axis and the length of their coordinates define the voltage vectors direction. $v = v_{an} + v_{bn} + v_{cn}$ (11)

Figure 5 shows the block diagram of the VSI. The SVPWM consists of 6 active inverter switching states which are defined as (100) $\vec{V_{1}}$ , (110) $\vec{V_{2}}$ , (010) $\vec{V_{3}}$ , (001) $\vec{V_{4}}$ , (011) $\vec{V_{5}}$ , (101) $\vec{V_{6}}$ and two null voltage vectors which are defined as (000) $\vec{V_{0}}$ , (111) $\vec{V_{7}}$ . When the status is ’1’, the switch is on and when the status is ’0’, the switch is off. Figure 6 shows the corresponding space vectors.

Fig. 5

Block diagram of VSI.

Fig. 6

Space voltage vectors.

The active voltage vectors are spaced apart with a phase angle of 60⁰ and null vectors are situated at the zero. The desired output voltage vector can be situated in any of the six sectors, which make a hexagon.

The reference voltage vector. V_ref, is created by null vector at operating time T_K and adjacent vector at operating times, T_K+1, in the sector where V_ref resides. At any given sampling instance, the duration of the active and null vectors should follow the voltage-seconds balance, T_S. The SVPWM input should be in the α-β reference coordinates, so that the voltage components V_dref and V_qref can converted into V_∝ and V_β by inverse park transformation. This is executed in four stages as given below:

Step-I. Sector selection

The Sector, N, is selected based on the relationship between V_∝ and V_β in each sector as: ${\begin{matrix} V_{ref 1} = V_{β} \\ V_{ref 2} = \frac{(- V_{β} + \sqrt{3} V_{α})}{2} \\ V_{ref 3} = \frac{(- V_{β} - \sqrt{3} V_{α})}{2} \end{matrix}$ (12)

If ${\begin{matrix} V_{ref 1} > 0; A = 1 \\ V_{ref 2} > 0; B = 1 \\ V_{ref 3} > 0; C = 1 \end{matrix}$ (13)

The sector, N in which the voltage vector lies is determined by the relationship: $N = A + 2 B + 4 C$ (14)

Step-II. Calculation of vector switching time

The operating time, X, Y and Z of respective voltage in current sector T_K and T_K+1 are calculated as follows: ${\begin{matrix} X = \frac{\sqrt{3}}{V_{dc}} {TV}_{β} \\ Y = \frac{\sqrt{3}}{2 V_{dc}} T (\sqrt{3} V_{α} + V_{β}) \\ Z = \frac{\sqrt{3}}{2 V_{dc}} T (- \sqrt{3} V_{α} + V_{β}) \end{matrix}$ (15) where, T is the absolute period of switching. Now the functioning time can be found from Table 1.

Table 1

Evolution of T_K and T_K+1

Sector	I	II	III	IV	V	VI
T _K	-z	y	x	z	-y	-x
T _K+1	x	z	-y	-x	-z	y

The mathematical addition of T_K and T_K+1 should be less than the total switching period T. If the summation is more than zero, it is revised as: $T_{K} = \frac{T_{K} T}{T_{K} + T_{K + 1}}$ (16) $T_{K + 1} = \frac{T_{K + 1} T}{T_{K} + T_{K + 1}}$ (17)

Step-III. Identifying the vector switching points

By using symmetrical PWM arrangement the voltage vectors are switched and the switching points can be evaluated as below:

${\begin{matrix} T_{a} = \frac{1}{4} (T - T_{K} - T_{K + 1}) \\ T_{b} = T_{a} + \frac{1}{2} T_{K} \\ T_{c} = T_{b} + \frac{1}{2} T_{K + 1} \end{matrix}$ (18)

Table 2 defines the switching points T_sp1, T_sp2 and T_sp3.

Table 2

Vector Switching Points

Vector switching points	Sector
	I	II	III	IV	V	VI
T _sp1	T _a	T _b	T _c	T _c	T _b	T _a
T _sp2	T _b	T _a	T _a	T _b	T _c	T _c
T _sp3	T _c	T _c	T _b	T _a	T _a	T _b

Step-IV. Generation of the PWM signal

T_sp1, T_sp2 and T_sp3 are compared with repeating sequence wave with different frequencies to generate the PWM signals PWM_1,3,5 and PWM_4,6,2 are the compliments of PWM_1,3,5,

5 Results and discussions

A 3-phase 3.4 kW, 314rad/s vector controlled PMSM drive is modeled and simulated in MATLAB/Simulink.

The drive performance is studied and examined under various operating conditions using the conventional MRAC observer and the improved ANFIS based MRAC observer. For simulation studies the sampling time is set at 10μ sec. The parameters of the PMSM is tabulated in Table 3.

Table 3
Rated parameter of PMSM

Parameter Symbols Ratings

Power P_in 3.4 kW

Voltage V 380 V

Stator current i_s 6.9 A

Rated Speed ω _m 314 rad/s

Rated Torque T_e 11 Nm

d-q axes inductances L_d, L_q 0.0114 H

Stator Resistance R_S 1.93 Ω

Pair of poles p 4

PM Flux linkage λ _pm 0.865 Wb

Inertia of rotor J 0.11 kgm²

Parameter	Symbols	Ratings
Power	P_in	3.4 kW
Voltage	V	380 V
Stator current	i_s	6.9 A
Rated Speed	ω _m	314 rad/s
Rated Torque	T_e	11 Nm
d-q axes inductances	L_d, L_q	0.0114 H
Stator Resistance	R_S	1.93 Ω
Pair of poles	p	4
PM Flux linkage	λ _pm	0.865 Wb
Inertia of rotor	J	0.11 kgm²

5.1 Dynamic characteristics of PMSM at 314rad/s

Figure 7(a) and 7(b) show the dynamic characteristics of sensor less PMSM drive at a rated speed operation with the conventional MRAC observer and the improved ANFIS based MRAC observer. The motor is started without any load torque at rated speed of 314 rad/s.

Fig. 7

Dynamic characteristics of sensor less PMSM drive at rated speed with: (a) conventional MRAC, (b) Improved ANFIS based MRAC.

At 0.03 sec, when full load is applied to the motor, there is a mild dip in the speed during this transition, however, the motor immediately recovers and follow the commanded speed smoothly. During the starting, the stator current is high, as expected, to provide the required dynamic torque needed to overcome the motor inertia. No variation is observed in the estimated position of the rotor with change in load torque as observed from

Figure 7(a) and 7(b). It is also noted that the settling time and time rise in MRAC based on adaptive ANFIS is less than the conventional MRAC, hence PMSM response with MRAC based on adaptive ANFIS is faster.

5.2 Dynamic characteristics of PMSM under speed variation

Figure 8(a) and 8(b) show the dynamic characteristics of sensor less PMSM drive with conventional the MRAC observer and the improved ANFIS based MRAC observer for step change in speed and torque. The motor is started with no-load at half of the rated speed i.e. 157rad/s. At 0.03 sec rated load torque of 11 Nm is applied to the motor and the motor is observed to follow the commanded speed smoothly. At 0.05 sec the motor speed is increased to rated speed of 314rad/s. It is observed that motor follows the commanded speed smoothly.

Fig. 8

Dynamic characteristics of sensor less PMSM drive under speed variation with: (a) conventional MRAC, (b) Improved ANFIS based MRAC.

In the case of the conventional MRAC a mild overshoot in torque is observed during the transition in load, while in the improved ANFIS based MRAC no such overshoot is observed. Thus, improved dynamic torque is achieved with proposed method.

5.3 Dynamic characteristics of PMSM during speed reversal

Figure 9(a) and 9(b) show the dynamic characteristics of sensor less PMSM drive with the conventional MRAC observe and the improved ANFIS based MRAC observer during speed reversal.

Fig. 9

Dynamic characteristics of sensorless PMSM drive in reverse speed with: (a) conventional MRAC, (b) Improved ANFIS based MRAC.

The motor is started at rated speed of 314 rad/s under no load condition and the rated load of 11 Nm is applied at 0.03 sec. At 0.05 s a speed of the motor is reversed.

Some oscillations are observed in the torque response when load torque is applied with the conventional MRAC, while with the proposed method, the load transition is smooth without any oscillations. However, no change is speed is observed during this load transition. During speed reversal, the motor tracks the changes commanded speed smoothly. It is also observed that while achieving steady state speed, there is an overshoot in torque with the conventional MRAC, which is not there in the improved MRAC based on ANFIS.

5.4 Dynamic characteristics of PMSM under variations in load

Figure 10(a) and 10(b) illustrate the dynamic characteristics of the sensor less PMSM drive with conventional MRAC observer and improved ANFIS based MRAC observer under load variations.

Fig. 10

Dynamic characteristics of sensorless PMSM drive under load variation with: (a) Conventional MRAC, (b) Improved ANFIS based MRAC.

The motor is running at no load rated speed condition in starting. At t = 0.03 sec a load torque of 5 Nm is applied to the rotor and the load torque is increased to 11 Nm at t = 0.05 sec. The load is reduced to 5 Nm at t = 0.07 sec followed by a change to no load operation at t = 0.09 sec.

During the load transition, some overshoots are observed with the conventional MRAC observer. However, no overshoots are observed with the proposed method and the motor develops the required torque smoothly.

5.5 Dynamic characteristics of PMSM at low speed and low load torque operation

Figure 11(a) and 11(b) illustrate the dynamic characteristics of the sensor less PMSM drive with the conventional MRAC observer and the improved ANFIS based MRAC observer under low speed operation of 10rad/s with a load torque of 5 Nm. The motor is started at a commanded speed of 10 rad/sec under no load condition and a load torque of 5 Nm is applied at 0.1 sec. It is noted that the motor speed follows the commanded value and the required torque is developed smoothly. The settling and rising times of the speed and torque responses of the improved ANFIS based MRAC observer are observed to be lesser than the conventional MRAC observer.

Fig. 11

Dynamic characteristics of sensor less PMSM drive at low speed and low load torque with: (a) Conventional MRAC, (b) Improved ANFIS based MRAC.

6 Conclusion

In this paper an improved ANFIS based MRAC observer is proposed for speed and position estimation in sensor less control of PMSM drive and its performance compared with the conventional MRAC observer. The robustness of the proposed observer is demonstrated successfully under various operating conditions like speed change and torque variation in steps, speed reversal, low speed operation etc. In proposed MRAC observer, an improved ANFIS based adaptation technique is employed which estimates the speed of the rotor by minimizing the error of the reference model and adjustable model. The simulation results obtained confirms the improved performance of proposed observer for all operating speeds and different load conditions. The proposed ANFIS based MRAC observer provides improved steady state and dynamic characteristics of the motor at different speed and load change operation as compared to the conventional MRAC observer.

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, Performance Analysis of FOC and DTC for PMSM Drives using SVPWM Technique, International Conference on Sciences and Techniques of Automatic Control and Computer Engineering- STA (2015), 228–233. DOI: 10.1109/STA.2015.7505167

27.

Singh

and Tiwari

A.N.

, Analysis and simulation of vector controlled PMSM drive using SVPWM inverter, 2nd International Conference for Convergence in Technology (I2CT) (2017), Mumbai, India, 709–714. DOI: 10.1109/I2CT.2017.8226221