Voltage Control of IPMSM Servo Drive in Constant Power Region With Intelligent Parameter Estimation

A novel voltage control scheme for an interior permanent magnet synchronous motor (IPMSM) servo drive in the constant power region with intelligent estimation of the motor parameter is proposed in this study. In the novel voltage control scheme, a feedforward voltage angle controller is proposed where an intelligent parameter estimation method by using a wavelet fuzzy neural network (WFNN) is developed to estimate the <inline-formula> <tex-math notation="LaTeX">$q$ </tex-math></inline-formula>-axis inductance online. In this study, in order to minimize the copper loss, a flux-weakening (FW) control scheme under maximum phase voltage is developed first. Then, an adaptive backstepping based nonlinear controller (ABNC) considering nonzero <inline-formula> <tex-math notation="LaTeX">$d$ </tex-math></inline-formula>-axis current is developed to improve the robustness of the speed control. The Lyapunov stability theorem is used to derive the adaptive law of the online estimation of the lumped uncertainty to ensure the asymptotical stability of the ABNC. Moreover, a feedforward voltage angle controller is developed for the voltage control where the <inline-formula> <tex-math notation="LaTeX">$q$ </tex-math></inline-formula>-axis current controller is retained in order to ensure the steady-state performance of the control system. Furthermore, the WFNN is adopted to estimate the actual <inline-formula> <tex-math notation="LaTeX">$q$ </tex-math></inline-formula>-axis inductance value online for the feedforward voltage angle controller to improve the dynamic response. In addition, some experimental results are demonstrated to verify the effectiveness of the proposed voltage control scheme with ABNC in the constant power region.


I. INTRODUCTION
There are many attractive characteristics of the interior per- 19 manent magnet synchronous motors (IPMSMs) including 20 wide speed operating range, superior power density, high 21 efficiency, and high torque-to-inertia ratio. These features 22 permit the IPMSMs to be operated not only in the constant 23 torque region but also in the constant power region up to 24 a high speed by using flux weakening. Thus, IPMSMs has 25 been adopted in many industrial applications [1], [2], [3], 26 [4]. Moreover, to improve the control performance and effi-27 ciency of the IPMSM servo drives, optimal control methods 28 such as maximum torque per ampere (MTPA) control and 29 The associate editor coordinating the review of this manuscript and approving it for publication was Haibin Sun . flux-weakening (FW) control have been proposed [1], [2], 30 [3], [4]. In order to utilize the advantages of the reluctance 31 torque term of the IPMSMs in the constant torque region, 32 the MTPA control has been developed to improve the torque 33 output. Furthermore, the FW control is an important issue in 34 the range of high speed with the back electromotive force 35 (EMF) increasing along with the rising speed, which will 36 reach the limit of dc-link voltage in the constant power region. 37 There are mainly two strategies to achieve the FW control 38 which are the current control methods [4], [5], [6], the voltage 39 control methods [7], [8], [9], [10], [11]. In [4], to limit the 40 inverter output voltage to the maximum phase voltage of the 41 inverter at high speed, a voltage control loop was designed for 42 the current control in the constant power region. A FW control 43 was achieved by using current control with the flux level 44 a popular research topic in the past two decades [13], [14], 86 [15], [16], [17], [18]. In [13], an online parameter estimation 87 method based on a discrete-time dynamic model for the 88 IPMSMs was proposed. The proposed method consists of two 89 affine projection algorithms and has adopted the difference 90 in dynamics of motor parameters. Moreover, an estimation 91 method of the spatial inductance map by spatially scanning 92 the motor using the sinusoidal voltage injection was devel-93 oped in [14]. In [15], a real-time method to estimate the an IPMSM drive was developed in [16]. In [17], the Adaline 100 NN algorithm was employed to design the estimators for the 101 rotor flux linkage and stator winding resistance. In addition, 102 an online parameter estimation methodology using d-axis 103 current injection, which can estimate the distortion voltage 104 of the current-controlled voltage source inverter (CCVSI), the 105 varying dq-axis inductances, and the rotor flux, was proposed 106 in [18]. However, most of the parameter estimation methods 107 mentioned above were only developed for the control of the 108 IPMSMs in the constant torque region.

109
The PI speed controller is largely adopted in many control 110 applications due to its simplicity. Nevertheless, the disad-111 vantages of the PI controller, such as sensitive to parameter 112 variations and external disturbances, is well known. On the 113 other hand, the backstepping control, as a systematic and non-114 linear recursive design method, has attracted much attention 115 for the nonlinear feedback control [19], [20], [21], [22], [23]. 116 Moreover, the backstepping control is based on Lyapunov 117 stability theory. Its control law is derived by constructing 118 the Lyapunov function, and the global asymptotic stability is 119 ensured. However, the sign function in the backstepping con-120 trol may cause undesired chattering phenomena. Therefore, 121 to reduce the chattering phenomena and improve the con-122 trol performance, some control methods such as adaptive 123 control [19], [20], intelligent control [21], and sliding mode 124 control [22], [23], have been proposed to merge with the 125 backstepping control. Therefore, one of the objectives of this 126 study is to replace the conventional PI speed controller by an 127 adaptive backstepping based nonlinear controller (ABNC).

128
To improve the control performance of an IPMSM servo 129 drive in the constant power region, a novel voltage con-130 trol scheme with a feedforward voltage angle controller is 131 developed in this study. Moreover, an intelligent parame-132 ter estimation method by using a wavelet fuzzy neural net-133 work (WFNN) [24], [25], [26] is proposed to estimate the 134 q-axis inductance online for the feedforward voltage angle 135 controller. This study is organized into six sections. The 136 dynamic analysis of a field-oriented control (FOC) IPMSM 137 servo drive with traditional FW control is studied in Sec. II. 138 Then, an ABNC speed controller considering nonzero d-axis 139 current with adaptive online estimation of the lumped uncer-140 tainty is discussed in Sec. III. Furthermore, the proposed 141 voltage control by using a feedforward voltage angle control 142 with the q-axis current controller is introduced in Sec. IV. 143 In addition, the experimentation based on a TMS320F28075 144 32-bit floating-point digital signal processor (DSP) with 145 some experimental results to verify the effectiveness of the 146 proposed voltage control scheme is presented in Sec. V. 147 Finally, some conclusions are addressed in Sec. VI.

148
The main contributions of this study are listed as follows: 149 (1) An ABNC speed controller considering nonzero d-axis 150 current is proposed to improve the robustness of the speed 151 control. (2) A feedforward voltage angle controller, in which 152 an intelligent parameter estimation method using WFNN is 153 adopted to estimate the q-axis inductance online, is devel-154 oped.
(3) The current control and voltage control modes with 155 ABNC speed controller to achieve the current and voltage 156 control of an IPMSM servo drive at different speeds and load 157 torque conditions are successfully implemented.

159
The voltage model at steady state of an IPMSM in the dq 160 reference frame can be expressed as follows: is defined by the machine rated current i rated . The following 173 two constraints must be fulfilled for the control of an IPMSM: For the speeds above the rated speed, where the stator resis- IPMSM can be obtained as follows: According to (4) and (7), the operation limit of IPMSM   the mechanical speed command ω * rm to obtain the mechanical 204 speed errore 1 . Then, e 1 is inputted into PI speed controller to 205 get the q-axis current commandi * q . Moreover, i * q is substituted 206 into the MTPA formula to derive the d-axis current command 207 i * d,MTPA for the MTPA control. While the FW control is 208 proceeded, the motor speed will be increased above the rated 209 speed. The input of the MTPA block shown in Fig. 2 will be 210 switched to FW and the i * d,MTPA will be kept constant during 211 the FW control. Furthermore, to make sure that the stator 212 voltage command v * s will not exceed the phase voltage limit 213 V s_max of the inverter, V s_max − v * s is inputted into a PI con-214 troller to generate the variation of d-axis current command 215 i * d which is a negative value. Only when the inverter output 216 voltage v * s exceeds the maximum phase voltage V s_max of 217 the inverter, the input of the PI controller will be nonzero. 218 As shown in Fig. 2, the three-phase currents i a , i b and i c of the 219 CCVSI are transformed to the corresponding q-axis current i q 220 and d-axis current i d by using the coordinate transformation. 221 In addition, i q and i d are subtracted respectively from i * q and 222 i * d , and then the dq-axis voltage commands v * d and v * q are 223 obtained through the PI controllers of the current loop with 224 the decouple control shown in Fig. 2. After v * d and v * q are 225 obtained, v * α and v * β are derived by using coordinate transfor-226 mation. Additionally, the switching signals of the insulated-227 gate bipolar transistors (IGBTs) of the CCVSI are generated 228  q is used to obtain the feedforward voltage angle command 250 θ * vf as shown in the block of voltage control mode of Fig. 3. 251 Furthermore, the q-axis current error e q is inputted into PI 252 controller to generate the compensating value of voltage 253 angle θ * v . Owing to the voltage magnitude is fixed in voltage 254 control mode, the dq-axis voltage commands can be obtained 255 from the voltage angle command θ * v .

256
The mechanical dynamic equation of the IPMSM servo 257 drive system can be represented as follows: where J is the inertia coefficient; B is the damping coefficient; 260 T L is the load torque. By neglecting the load torque, (8) can 261 be modified as The developed electromagnetic torque T e can be represented 264 by the following equation: The IPMSM servo drive system can be formulated by rewrit-267 ing (9) and (10) as follows: And E m is assumed to be bounded where ρ is a given positive constant.

288
Define the speed tracking error and its derivative term as 289 follows: whereω rm can be viewed as a virtual control input. Then, 293 define the following stabilizing function as: where λ 1 is a positive constant. Moreover, a virtual control 296 error is defined as: Furthermore, a Lyapunov function is chosen as follows: is the estimated value of E m ; δ is a positive constant. Taking 302 the time derivative of the Lyapunov function and using (12) 303 and (18), one can obtain: Therefore, according to (20), the ABNC control law U ABNC 316 and adaptive lawĖ m are designed as follows: Substituting (21) and (22) into (20), the following equation 321 can be obtained: SinceV (t) ≤ 0 is negative semidefinite and V (t) > 0, 324 it implies that e 1 , e 2 , andẼ m are nonincreasing and bounded. 325 According to Lyapunov Theorem and Barbalat's Lemma, 326 e 1 and e 2 will converge to zero as t → ∞. Thus, considering 327 the dynamic equation of the IPMSM servo drive system 328 represented by (12), if the ABNC control law and adaptive 329 law are designed as (21) and (22), the asymptotically stable 330 of the IPMSM servo drive system using the ABNC speed 331 controller can be guaranteed.

333
To achieve wide speed range operation, two control modes 334 are proposed for the IPMSM servo drive and the switch 335 signal is obtained by using the mode selector as shown in 336 Fig. 3  be expressed as: Based on formula (21), (22) and (27), the feedforward voltage 371 angle command is designed as: where ω * e is the command of the electrical speed and can 374 be obtained by multiplying ω * rm with the pole pairs P/2.

375
Considering the steady state stability of the proposed control 376 system, the q-axis current regulator is still retained. Besides, 377 q-axis current error can be represented as: A PI controller is adopted to generate the compensating value 380 of voltage angle θ * v by using e q . Then, the voltage angle 381 command can be expressed as: However, since the q-axis inductance valueL q shown in (30) 384 varies significantly in FW region, a WFNN [24], [25], [26] is 385 adopted to estimate the q-axis inductance valueL q online as 386 shown in Fig. 3 to improve the dynamic response of voltage 387 control.

389
Owing to the nonlinearity and time delay of the CCVSI, 390 the distortion voltage errors between the dq-axis voltages 391 command and the real dq-axis voltages of the IPMSM servo 392 drive are inevitable. Thus, considering the distortion voltage 393 errors, the dq-axis voltages command can be expressed as 394 follows in the FW operating region: where V dead is the distortion voltage and D d , D q are the 398 distorted coefficients [18]. To estimate the distortion voltage 399 V dead , the instantaneous measurement of the electrical speed, 400 dq-axis voltages and currents are required. However, when 401 the motor is operated at high speed, these signals will be con-402 taminated by various noises. Therefore, the smoothing values 403 of the measured signals are adopted to improve the estimation 404 accuracy. Eq. (33), (34) can be rewritten as follows: where the smoothing values of dq-axis voltage commands 408 v * dq , dq-axis currentī dq and electrical speedω e are averaged 409 every ten times. Moreover, the smoothing values of distorted 410 coefficientsD d ,D q are defined as follows [18]: The coupling terms of both the real and estimated q-axis 460 voltage model are required as shown in Fig. 5. Therefore, after 461 some mathematical manipulations using (3), (33) and (34), 462 one can obtain: SubstitutingD d ,D q ,V dead , ω * e and i * q into (44), the coupling 466 term of the estimated voltage model can be expressed as: Moreover, the output of the WFNN is the adaptation value 470 of the q-axis inductance L q . Thus, the estimated value of 471 the q-axis inductance can be obtained by using the following 472 equation: The adaptation is processed recursively untilî q = i q , which is 475 the same as the traditional model following control scheme. 476 Then, the steady-state estimation value of the q-axis induc-477 tanceL q at a specific high-speed operating condition can be 478 obtained. Furthermore, since the coupling term shown in (45) 479 is replaced by the right hand side of the equality, the variations 480 of L d and λ m are effectively taken into account during the 481 adjustment of the q-axis inductance.

483
The experimental setup of the IPMSM servo drive system 484 are shown in Fig. 6. The IPMSM test platform is composed 485 VOLUME 10, 2022

531
For the comparison of the control performance, the exper-532 imental results of PI speed controller with proportional gain 533 2 and integral gain 0.14 are also given. The gains of the PI 534 speed controller are obtained by trial and error to achieve 535 acceptable transient and steady-state responses. To demon-536 strate the control performance of the PI and ABNC speed 537 controllers, the IPMSM is operated at rated speed 2000 rpm 538 and rated torque 9.5 Nm with 100 rpm step command, and 539 the results are shown in Fig. 8. Figs. 8(a)    Furthermore, the estimated distortion voltageV dead can be 594 obtained by using Eq. (43), which is larger at higher speed 595 with lower torque as shown in Figs. 9(d), 10(d), 11(d). The 596 same phenomenon can also be found in [18]. In addition, 597 the estimatedL q are 8.23mH, 7.33mH and 6.43mH, respec-598 tively, by using the proposed intelligent q-axis inductance 599 estimation scheme at three FW operating conditions as shown 600 in Figs    and the compensating voltage angle command θ * v . More-623 over, the estimated distortion voltageV dead , the q-axis current 624 i q , the estimated q-axis currentî q , and the estimated q-axis 625 inductanceL q are also 8.23mH, 7.33mH and 6.43mH as 626 shown in Figs