Neuro-Fuzzy Control of Interior Permanent Magnet Synchronous Motors: Stability Analysis and Implementation

Abstract

This paper investigates a robust neuro-fuzzy control (NFC) method which can accurately follow the speed reference of an interior permanent magnet synchronous motor (IPMSM) in the existence of nonlinearities and system uncertainties. A neuro-fuzzy control term is proposed to estimate these nonlinear and uncertain factors, therefore, this difficulty is completely solved. To make the global stability analysis simple and systematic, the time derivative of the quadratic Lyapunov function is selected as the cost function to be minimized. Moreover, the design procedure of the online self-tuning algorithm is comparatively simplified to reduce a computational burden of the NFC. Next, a rotor angular acceleration is obtained through the disturbance observer. The proposed observer-based NFC strategy can achieve better control performance (i.e., less steady-state error, less sensitivity) than the feedback linearization control method even when there exist some uncertainties in the electrical and mechanical parameters. Finally, the validity of the proposed neuro-fuzzy speed controller is confirmed through simulation and experimental studies on a prototype IPMSM drive system with a TMS320F28335 DSP.

1. Introduction

The permanent magnet synchronous motor (PMSM) has been extensively used in variable-speed motor drives such as electric vehicles, home appliances, military or medical equipment, machine tools, and industrial robots because of its wide speed range operation and high power density [1-4]. Depending on the position of the permanent magnet on the rotor, the PMSM can be mainly classified into two types: surface-mounted PMSM (SPMSM) and interior PMSM (IPMSM). It is well-known that the IPMSM is more difficult to control due to more complex dynamic model, but has better performance (i.e., a wider speed range operation capability due to flux weakening control and a higher torque generation capability due to inherent saliency) than the SPMSM. Thanks to these attractive advantages, the IPMSM is gaining more and more attention in industrial and home appliance application areas. However, it is complicated to precisely control the IPMSM because of the nonlinearities due to nonlinear properties resulting from the magnets and cross-coupling between the state variables (i.e., the dq-axis currents and speed) in dynamic model equations. Also, there always exist system uncertainties such as motor parameter variations and unknown external disturbances. Therefore, the robust control design requirements should be satisfied, which are insensitive to the uncertainties mentioned previously. Consequently, the classical linear control schemes based on time-invariant system model, e.g., the PI controller [5], cannot achieve a good tracking performance.

To solve these difficulties, many advanced control approaches have been presented such as feedback linearization control [6], adaptive backstepping control [7], optimal control [8], predictive control [9-10], etc. In [6], the control performance of the feedback linearization algorithm is not satisfactory because its property is sensitive to parameter uncertainties and external disturbances. In [7], the design procedure of the adaptive backstepping control methods looks complicated since many parameters are adapted. In [8-10], the control algorithms of the IPMSMs are comparatively simple because the d-axis current reference is set to zero. However, the torque of the IPMSMs cannot be maximized by making the reluctance torque be zero. Also, the fuzzy control schemes [11-12] have been introduced to make up for the nonlinearities of IPMSM system. It is easy to see that the control scheme becomes complex and difficult to be designed as the number of fuzzy rules increases. The sliding mode control approach [13-14] has its favorable advantage which is insensitive to parameter uncertainties and external disturbances. However, the robustness of this control method can be guaranteed only within the bounds of the uncertainties, and it still suffers from a chattering problem. Beside the above control methods, many papers have concentrated to investigate the effects of both the mechanical and electrical parameter variations on the servo IPMSM drives [15-16]. Some outstanding methods have been used to estimate the motor parameters, such as an online method [15] and a terminal sliding-mode observer method [16]. These papers implied that the control scheme can significantly improve the system performance, if the motor parameters are accurately estimated and then directly used to design the controllers.

Recently, the intelligent control methods such as fuzzy logic control, neural network control, neuro-fuzzy control (NFC) have received a lot of attentions since these controllers do not need an exact mathematical model of the system and can achieve high performance. Especially, the NFC is a combination of the advantages of artificial neural network control and fuzzy logic control. That means the artificial neural network has learning ability that can acquire the appropriate information based on the data while the fuzzy logic control can reasonably characterize the input/output behaviors of an uncertain system. In [17-18], the NFC is applied to tune the PI gains for IPMSM drives. However, its parameter training is optimized offline. In [19-21], the NFC faces a burdensome computation because of using a huge number of membership functions and rules. This is a major constraint for industrial applications. Furthermore, the NFC algorithms of [22-23] are quite complex when the reference models of the online selftuning algorithm are utilized.

This paper introduces a robust neuro-fuzzy speed control strategy that can accurately track the speed reference trajectory of IPMSM in spite of system uncertainties. The proposed NFC method contains a state feedback control term and a NFC term. The former stabilizes the system error dynamics and the latter makes up for nonlinearities and uncertain factors. Furthermore, the maximum torque per ampere (MTPA) control is combined with the proposed controller to maximize the torque generation in the constant torque region. Unlike the previous online selftuning algorithms [17-23] in which the quadratic cost function to be minimized includes only the speed error, the proposed control technique chooses the time derivative of the quadratic Lyapunov function as the cost function to be minimized. Thus, the global stability analysis can be simple and systematic. Furthermore, the design procedure of the online self-tuning algorithm is comparatively simplified to reduce a computational burden of the NFC. A rotor angular acceleration is attained through the disturbance observer. To prove the robustness of the proposed NFC scheme and the effects of two kinds of parameter variations, in both simulation and experimental studies, the results of the proposed NFC and the feedback linearization control (FLC) methods are presented under the mechanical parameters variations as well as the electrical parameters variations using Matlab/Simulink software and a prototype IPMSM drive system with a TMS320F28335 DSP, respectively. The results show that the proposed observer-based NFC scheme can attain better tracking control performance (i.e., less steady-state error, more robustness) than the observer-based FLC method even when there exist some uncertainties in the electrical and mechanical parameters.

 

2. Neuro-fuzzy Speed Controller Design and Stability Analysis

2.1 Mathematical model of IPMSM

In the synchronously rotating dq reference frame, where the d-axis is aligned with the rotor flux vector and the q-axis is always 90° ahead of the d-axis, a three-phase IPMSM can be modeled by the dynamic equations including system uncertainties [9] as

where

𝜔 is the electrical rotor speed; ids and iqs are the d-axis and q-axis currents; Vds and Vqs are the d-axis and q-axis voltages; TL is the load torque; p is the number of poles; Rs, Lds, Lqs, J, B, and λm are the nominal values of the stator resistance, d-axis inductance, q-axis inductance, rotor inertia, viscous friction coefficient, and magnetic flux, respectively; the disturbances di(t) defined in [9] represent motor parameter uncertainties and unknown external load torque.

In this work, the speed controller design will be based on the assumptions and definitions described below:

Assumption 1: 𝜔, ids, and iqs are measurable.

Assumption 2: The desired speed 𝜔d and the disturbances di(t) vary slowly for a sampling period [9-10].

Definition 1: The electrical angular acceleration is represented as

Definition 2: To produce the maximum torque per ampere (MTPA) of the stator current, the d-axis reference current is calculated [12-13] by

Definition 3: The speed error, the d-axis current error, and the error vector are introduced as 𝜔e = 𝜔 - 𝜔d, idse = ids - idsd, and x = [𝜔e 𝛽 idse]T, respectively.

From the above assumptions and definitions, the dynamic Eq. (1) can be converted to the following form:

where

Note that the above state-space model (4) is considered to construct the proposed control law and requires the rotor angular acceleration information 𝛽 which is usually immeasurable. Moreover, the functions fq and fd include the nonlinear terms and the uncertainty terms. Assume that there exists a constant parameter matrix W*2r×1 such that

where H2×2r is a known function matrix, and r is a positive integer that denotes the number of fuzzy rules in the following subsection.

2.2 Neuro-fuzzy speed controller design

Let the control law u be decomposed as the state feedback control term ufb and the adaptive compensating control term unf

First, the state feedback term design is based on a linear matrix inequality (LMI) condition. Assume that there is an existing pair of matrix solution (X, Y) that satisfies the following inequality

where X∈R3×3 and Y∈R2×3 are decision variables.

The gain matrix K of the state feedback term ufb is obtained by

Fig. 1.Five-layer NFC with weight normalization

Then the NFC is applied to construct the adaptive compensating term that deals with fq and fd. Fig. 1 shows the structure of the NFC with weight normalization which consists of five basic layers. The first layer which is an input layer distributing the input variables to each of the nodes in the second layer is indicated. For the aim of reducing a computational burden, only three input signals zj (j = 1, 2, 3) are considered: z1 = 𝜔, z2 = iqs, and z3 = ids. Layer 2 is a fuzzification layer and every node in this layer acts as the membership function. In this work, for easy implementation, the following Gaussian membership functions can be selected as

where r is the number of fuzzy rules, εji and 𝜎ji denote the center and the width of the membership function mji, respectively. Next, layer 3 is a rule layer and the output of every node in this rule layer is defined as

Layer 4 is a normalizing layer with r nodes and every node in this layer can be regarded as the normalized weight of each rule.

Layer 5 is an output layer containing two nodes and eachoutput unfk (k =1, 2) of the kth node is denoted as

where wik are the adjustable parameter vectors. Finally, the adaptive compensating term is denoted as

Then, the online self-tuning algorithm is utilized to update all parameters in real-time, and the object function to be minimized is defined as

where P=X-1. Based on the back-propagation learning rule that is computed recursively from the output layer backward to the input layer, the weights in the output layer are updated by

Then, the adaptive law can be expressed by

where Ew=Diag(𝜂11, 𝜂21,…, 𝜂r1, 𝜂12, 𝜂22,…, 𝜂r2).

2.3 Stability analysis

Theorem 1: The following controller (17) enforces the error dynamics x to converge to zero.

Proof: Assume that (7) is feasible, and then there exists a matrix Q> 0 such that

Let the Lyapunov function be chosen as

where We = W - W*. The time derivative of (19) along the error dynamics (4) is given by

Integrating both sides of (20) yields the following equation

This implies x∈L2∩L∞ and W∈L∞. Using the above results and Barbalat’s lemma, x asymptotically converges to zero as time approaches infinity.

Remark 1: Note that the LMI parameterization of the K (8) can be combined with various useful convex performance criteria (i.e., α-stability, quadratic performance, generalized H2/H∞ performance, etc). For instance, if the K is given by (8) satisfying for some α > 0, then

It implies that x converges to zero with a minimum decay rate α.

Remark 2: Note that the NFC algorithm consists of layers, nodes in each layer, and online self-tuning algorithm. In general, the control performance can get better as the number of layers and nodes increases. However, this leads to a more complicated control system structure. Additionally, the updated laws for the parameters of all layers bring about a burdensome computation because of a large series of training data. Therefore, the structure and online self-tuning algorithm of the NFC should be designed simply and reasonably. For the proposed speed controller, the updated laws can be applied only to extract the parameters of the output layer. Besides, the parameters of the inner layer such as the standard deviation parameters of the membership functions can be selected as fixed parameters according to the control engineering knowledge. This solution still guarantees the stability criteria of the control system (20).

Remark 3: This remark discusses how the controller gains are chosen. The proposed control law in (17) consists of the control term ufb and the control term unf. The weights wik are integrated into the control term unf in (12). To attain the fast convergence and transient response, the weights wik are tuned to large values. As indicated in (15), these weights wik are proportional to the parameters 𝜂ik, so the large values of 𝜂ik lead to those of wik. Meanwhile, the gain matrix K of the control term ufb is achieved by solving the LMIs (7) or (22). Finally, the design parameters K and 𝜂ik can be systematically tuned as follows:

Step 1: Solving the LMIs (7) or (22) yields the gain matrix K;

Step 2: Set 𝜂ik to quite small values and then increase 𝜂ik by a small amount;

Step 3: If the transient and steady-state performances are satisfactory, then this process is completed. Otherwise, return to Step 2 above.

 

3. Disturbance observer design

The proposed control law in (17) requires the rotor angular acceleration information 𝛽 which is usually unavailable. Based on Definition 2, the rotor angular acceleration information 𝛽 can be directly obtained from the time derivative of the speed. However, this calculation method can be affected by its high-frequency noises. To avoid directly calculating the time derivative of the speed, the rotor angular acceleration 𝛽 needs the knowledge of the disturbance d1(t). Thus, a simple disturbance observer is designed in this paper.

From the (1) and assumption 2, the disturbance observer can be established as

where L∈R2×1 is an observer gain matrix, and

From (23) the error dynamics is expressed by

where

Theorem 2: Assume that the following LMI condition is feasible

where Po∈R2×2 and Yo∈R2×1 are decision variables. Also, assume that the observer gain matrix L is calculated by

Then, exponentially goes to zero.

Proof: Assume that (25) is feasible, and then there exists a matrix Qo > 0 such that

Let the Lyapunov function be defined as

The time derivative of (28) along the error dynamics (24) is represented by

It indicates that is asymptotically stable.

Remark 4: Note that the LMI parameterization of the L (26) can be combined with various convex performance criteria (i.e., α-stability, quadratic performance, generalized H2/H∞ performance, etc). For example, if the L is computed by (26) satisfying with some α > 0, then

It means that converges to zero with a minimum decay rate α.

Using the NFC and the disturbance observer above, a disturbance observer-based control law can be designed, thus the control inputs (Vqs and Vds) can be expressed as

Remark 5: The design procedure of the proposed observer-based NFC method can be generalized as follows:

Step 1: Solve the LMIs (25) or (30). Then obtain the observer gain (26) and construct the observer (23).

Step 2: Solve the LMIs (7) or (22). Then obtain the controller gain matrix K using the formula (8).

Step 3: Choose the membership functions of the speed, q-axis current, and d-axis current. Then, construct the matrix H using (10) and (11).

Step 4: Using Remark 3, choose the parameters 𝜂ik, calculate the updated law (34), and construct the observer based neuro-fuzzy control term (33).

Step 5: Construct the observer-based control inputs (31) based on (32) and (33).

 

4. Performance Investigation by Simulation and Experimental Results

To demonstrate the effectiveness of the proposed observer-based NFC algorithm, simulation and experiment studies are realized on a prototype IPMSM with the following specifications: rated power Prated = 390 W; p = 4; Rs = 2.48 Ω; Lqs = 114 mH; Lds = 75 mH; λm = 0.193 V.sec/rad; J = 0.00015 kg.m2; B = 0.0001 N.m.sec/rad.

Let us design a disturbance observer that guarantees the minimum decay rate α = 300. By solving the LMI condition (30) and (26), the gain matrix L is calculated as

Next, let us design a speed controller that guarantees the minimum decay rate α = 70. By solving the LMI condition (22) and (8), the gain matrix K is achieved as

As described in Remark 2, considering a trade-off between a simple implementation and a satisfactory performance, the fuzzy rules are selected as r = 3 × 2× 2 = 12. That is, the membership functions for three input variables (z1 = 𝜔, z2 = iqs, and z3 = ids) are adopted as m11 =m12 =m13 =m14 =e-(ω-300)2/3002,m15 =m16 =m17 =m18 =e-ω2/3002,m19 =m110 =m111 =m112 =e-(ω+300)2/3002,m21 =m22 =m25 =m26 =m29 =m210 =e-(iqs -2)2/22,m23 =m24 =m27 =m28 =m211 =m212 =e-(iqs +2)2/22,m31 =m33 =m35 =m37 =m39 =m311 =e-(ids -1)2/22,m32 =m34 =m36 =m38 =m310 =m312 =e-(ids +1)2/22,

Then, this leads to the control term unf with

Fig. 2 shows the overall schematic diagram of a laboratory prototype IPMSM drive system which contains an IPMSM, a three-phase PWM inverter, an encoder, a control board with a TMS320F28335 DSP and load motor. The rotor position angle is measured through an encoder RIA-40-2500ZO, and the two stator currents (ia, ib) are detected via hall-effect current sensors. The three-phase stator current signals (ia, ib, ic) are transformed to twophase signals (iqs, ids) in the dq reference frame. Meanwhile, the rotor angular acceleration information 𝛽 is calculated via the disturbance observer, and then is provided for the proposed neuro-fuzzy speed controller. Considering the system efficiency, control performance and current ripples, the sampling and switching frequencies are chosen as 5 [kHz], and a space vector PWM (SVPWM) technique is employed. Furthermore, a servo IPMSM drive is used as a load motor to apply the load torque.

Fig. 2.Overall schematic diagram of a prototype IPMSM drive system.

For a fair comparison, the observer-based FLC scheme in [2], which consists of the control structure similar to the proposed observer-based NFC algorithm, is adopted in this paper. The control inputs (Vqs and Vds) are represented by

where ufb1 is the state feedback term, uff is the feed forward term, and is an estimate of TL. The information of is attained by constructing the load torque observer in [6]. For a fair comparison like [11], the gains of the proposed NFC and the FLC methods are tuned to reach the similar speed response such as overshoot, settling time and error in steady-state under the nominal motor parameters. The proposed NFC scheme consists of the control terms ufb and unf, while the FLC method consists of the control terms ufb1 and uff. It notes that the performance of the proposed NFC scheme depends on the gain matrix K of the ufb and the parameters 𝜂w of the unf, whereas that of the FLC scheme mainly depends on the gain matrix K1 of the ufb1. Additionally, the K and 𝜂w can be easily tuned as in Remark 3. Meanwhile, the K1 can be easily designed by using the pole placement technique. Consequently, the 𝜂w of the proposed NFC strategy is adopted as 10,000, whereas the K1 of the FLC method is elected as k11 = 62,500, k12 = 500, and k23 = 3,000.

In this work, four case studies summarized in Table 1 are simulated using MATLAB/SIMULINK and tested to verify the feasibility of the proposed observer-based NFC scheme and the observer-based FLC scheme. Cases 1 to 3 show the speed dynamic behaviors after a sudden change in the desired speed (𝜔d), i.e., when the 𝜔d abruptly decreases from 209.4 [rad/s] to −209.4 [rad/s], but the TL keeps 0.75 [N⋅m]. Next, Case 4 shows the torque dynamic behaviors after a sudden change in the load torque (TL), i.e., when the TL suddenly changes from 0.5 [N⋅m] to 1.5 [N⋅m], but the 𝜔d holds 104.7 [rad/s]. As presented in [24-25], the stator resistance (Rs) of the IPMSM varies as a function of the temperature, while the λm, Lds and Lqs vary as a function of the operating current. The Rs increases as the temperature rises [24], and the λm and Lqs decrease as the iqs increases [25]. However, the Lds slightly increases as the ids is negative [25]. Furthermore, the J and B may strongly increase as the external mechanical load is applied to the IPMSM drive. Thus, to demonstrate the robustness of the proposed observer-based NFC scheme, two kinds of parameter variations are given: Case 2 is carried out under the electrical parameters variations (ΔRs = +0.5Rs, ΔLqs = -0.3Lqs, ΔLds = +0.1Lqs, Δλm = -0.2λm) based on [24-25], whereas Cases 3 and 4 are executed under the mechanical parameters variations (ΔJ = +2.0J, ΔB = +1.0B). It is wellknown that the motor parameters can be easily adjusted in the simulation studies, but it is not an easy task to directly modify the motor parameters in the experiments. Generally, it can be an alternative solution to indirectly change the motor parameters in a real IPMSM drive by simply changing the motor parameters in the control scheme [9-10]. Thus, to conduct an experiment on the proposed observer-based control scheme under the variations of the motor parameters (Rs, Lqs, Lds, λm, J, B), the motor parameters in the controller are indirectly changed rather than those in the real IPMSM drive.

Table 1.Case studies for simulations and experiments

Figs. 3(a)-5(a) show the speed step responses of the proposed NFC method under Cases 1−3, respectively, while Figs. 3(b)-5(b) show the speed step responses of the FLC method under Cases 1−3, respectively. Figs. 6(a) and 6(b) show the torque transient responses of both control methods under Case 4, respectively. In Figs. 3(a) and 3(b), the overshoot, the settling time and the steady-state error of the proposed NFC method and the FLC method are obtained under nominal motor parameters as (0.0%, 66 ms, 0.0%) and (0.0%, 65 ms, 0.03%), respectively. These results imply that the comparative evaluation was satisfactorily conducted and the FLC scheme can precisely track the speed reference trajectory of the IPMSM in case of the nominal motor parameters. In Figs. 4(a) and 4(b), it is observed that the speed errors in steady-state of both methods are almost zero and 4.67%, and the settling times are 57 [ms] and 69 [ms], respectively. Figs. 5(a) and 5(b) show that the undershoots of both control methods are observed as 1.45% and 6.66%, and the settling times are 66 [ms] and 82 [ms], respectively. Figs. 6(a) and 6(b) show that the speed errors in steady-state are negligible, but the motor speed of Fig. 6(a) keeps much more stable than that of Fig. 6(b) during an abrupt load change.

Fig. 3.Simulation results of the proposed observer-based NFC method and the observer-based FLC method under Case 1.

Fig. 4.Simulation results of the proposed observer-based NFC method and the observer-based FLC method under Case 2.

Fig. 5.Simulation results of the proposed observer-based NFC method and the observer-based FLC method under Case 3.

Fig. 6.Simulation results of the proposed observer-based NFC method and the observer-based FLC method under Case 4.

In this work, Cases 3 and 4, which show the speed and torque transient responses under the mechanical parameters variations (ΔJ = +2.0J, ΔB = +1.0B), are chosen to experiment because of limited space. Figs. 7 and 8 show the experimental results of the proposed NFC method under Cases 3 and 4, respectively. Meanwhile, Figs. 9 and 10 show the test results of the FLC method under Cases 3 and 4, respectively. Table 2 summarizes the control performance of two control strategies during the transient and steady-state based on the simulation and experimental results.

Fig. 7.Experimental results of the proposed observer-based NFC method under Case 3.

Fig. 8.Experimental results of the proposed observer-based NFC method under Case 4.

Fig. 9.Experimental results of the observer-based FLC method under Case 3.

Table 2.Performance summaries of two control strategies during transient and steady-state based on simulation and experimental results

From Figs. 3-10, it can be seen that the variations of the mechanical parameters (J, B) mainly affect the transient response. In fact, the variations of the electrical parameters (Rs, Lqs, Lds, λm) slightly affect the steady-state response. Also, it is obvious that the proposed observer-based NFC scheme can obtain a better control performance (i.e., less steady-state error and more robustness) than the observerbased FLC scheme in case that some uncertainties in electrical and mechanical parameters exist.

Fig. 10.Experimental results of the observer-based FLC method under Case 4.

 

5. Conclusion

In this work, a disturbance observer-based NFC methodology of a servo IPMSM drive system has been proposed. The proposed observer-based speed controller is insensitive to uncertain factors such as motor parameter variations and load torque disturbances. Additionally, the maximum torque per ampere (MTPA) control was incorporated to improve the torque generator in the constant torque region. In this study, the global stability analysis is comparatively simple and systematic since the time derivative of the quadratic Lyapunov function is elected as the cost function to be minimized. Moreover, the design procedure of the online self-tuning algorithm is simplified to reduce a computational burden of the NFC. Simulation and experimental results surely show that the proposed observer-based NFC scheme has a better speed tracking performance such as less steady-state error, more robustness than the observer-based FLC method in the existence of the uncertainties in electrical parameters and mechanical parameters.