First-Order-Delay-Controlled Slip-Angular Frequency for the Dynamic Performance of an Indirect-Field-Orientation-Controlled Induction Motor-Driving Inertial Load

Sophisticated torque-current control is required in inertial load-drive applications, such as in electric vehicles and electric railway vehicles, over a wide speed range. However, the conventional indirect-field-orientation control (FOC) lacks the current response during the transient response because the conventional feedforward slip-angular-frequency control causes secondary flux fluctuation. Therefore, this article proposes FOC with first-order-delay slip-angular-frequency control, which reduces the secondary flux fluctuation and realizes high-performance torque-current control during transient response. The proposed method was verified through numerical simulation and small-scale model experiments with a 750 W induction motor and an inertial load.

Integral value of d-and q-axis current deviation. ω * s Slip-angular-frequency command. ω 1 Primary angular frequency command. ω r Secondary angular frequency command.

I. INTRODUCTION
Field-orientation control (FOC) for an induction motor (IM) is a technology that realizes independent control of excitation and torque currents and a high torque performance by controlling the phase angle of the secondary flux [1], [2], [3]. Speed control is not employed, but torque (torque-current) control is employed for inertial load-drive applications, such as electric vehicles and electric railway vehicles. High-performance first-order-delay torque-current control is desirable for inertial load-drive applications over a wide speed range. However, an overshoot of the torque current may occur owing to the transient coupling of the secondary flux during the step response of the torque current in the FOC. This article proposes a method to realize sophisticated torque-current control by reducing the transient fluctuation of the secondary flux and the overshoot of the torque current with a simple controller configuration. Direct FOC (DFOC) realizes current control by utilizing direct sensing of the air-gap flux [4], [5], [6]. However, DFOC requires sensors for flux measurement or observers/estimators for flux estimation [7], [8], [9], [10]. However, the controller configuration is complex.
In indirect FOC (IFOC), the phase angle of the secondary flux is controlled by an appropriate slip-angular-frequency command based on the motor parameters and current commands; therefore, sensors for flux measurement and estimators for flux estimation are unnecessary [1], [2], [3]. However, transient misalignment cannot be compensated for because the secondary flux is controlled by feedforward control. The slip-angular-frequency command ω * s is given by (1) where R 2 , L 2 , I * 1q , I * 1dF 2 , T 2 , and I * 1d represent the secondary resistance, secondary self-inductance, the q-axis current command, filtered value of the d-axis current command, secondary time constant, and d-axis current command, respectively. In the torque control system, the slip-angular-frequency command changes steeply according to (1), when the torquecurrent command is input. The q-axis secondary flux derived from the voltage equations of the IM, represented in a rotating d-and q-axis reference frame with the primary angular frequency, is expressed as follows: where M, I 1d , and I 1q represent the mutual inductance, d-axis current, and q-axis current, respectively. When the slip-angular-frequency command changes steeply, the q-axis secondary flux fluctuates transiently, and the coupling of the secondary flux occurs [11], [12]. Various control applications, such as autodisturbance-rejection control [13], morel reference adaptive system [14], sliding-mode control [14], [15], and neural networks [16], [17], [18], have been proposed to improve the current response. These advanced controllers can reduce the current overshoot by its higher disturbancerejection capacity. However, the controller configuration is complex, and the controller has many adjustment parameters. A proportional-integral (PI) current controller is easy to implement because of its simple configuration [19], [20], [21]. However, the PI current controller cannot compensate for the coupling of the secondary flux; thus, it lacks a transient current response [22]. A decoupling control improves a current control performance [23]. Decoupling between the d-and q-axis of the primary side of the IM makes the torque-current response better [24], [25], [26], [27]. However, the secondary flux fluctuation occurs; the torque response degrades when the slip-angular-frequency command changes steeply. Previous studies have focused on the primary side of IMs. Thus, the investigations focusing on the secondary flux response during transient response are insufficient.
In this article, a control method with a simple configuration that realizes sophisticated torque-current control over a wide speed range is proposed. In the proposed method, a first-order-delay filter is added to the slip-angular-frequency controller to reduce the transient fluctuation of the q-axis secondary flux and overshoot of the torque current. The proposed first-order-delay slip-angular-frequency control reduces the transient fluctuation of the q-axis secondary flux and realizes sophisticated torque-current control even during transient response. Therefore, the proposed method makes the controller configuration simple because an additional decoupling controller is not required.
The rest of this article is organized as follows. Section II derives the analytical models of the conventional and proposed IFOC systems. Because the magnitude of the IM transfer function changes with the capacity of the IM, an investigation was conducted for both small-capacity and high-capacity motors. The analysis and numerical simulations with a 750 W IM are presented in Section III. The experimental results with a 750 W IM and inertial load are presented in Section IV. The validity of the analytical model is confirmed by simulation and experiments in Sections III and IV. Assuming electric railway vehicle applications, a 150 kW IM model is used for analysis and numerical simulation in Section V. Finally, Section VI concludes this article. Fig. 1 shows the FOC system of IMs, where R 1 , L 1 , σ L 1 , 2d , ω r , ω 1 , V * 1d , V * 1q , K p , K i , and T d represent the primary resistance, primary self-inductance, primary leakage inductance, secondary angular frequency, primary angular frequency, d-axis secondary flux, d-axis voltage command, q-axis voltage command, proportional gain, integral gain, and time constant of the current control, respectively; R 1s , (σ L 1 ) s , L 2s , and M s represent the set values of the motor parameters in the current controller.

A. MODELING OF THE CONVENTIONAL IFOC
The primary angular frequency is expressed as follows: The slip-angular-frequency command is defined again as where R 2s represents the set value of the secondary resistance of the current controller. The PI current-controller gains are expressed as follows: The voltage equations of the IM represented in a rotating d-and q-axis reference frame with primary angular frequency are expressed as follows: The state equations derived from (8) are presented as The d-and q-axis voltage commands are expressed as where the d-and q-axis integrator models are expressed as follows: The d-and q-axis filtered value of current commands are expressed as follows: The state equations of the IM linearized near the operation point (I 1d0 , I 1q0 , 2d0 , 2q0 , I * 1dF 0 , I * 1qF 0 , I * 1dF 20 , ω 10 , ω r0 , ω * s0 , I * 1d0 , I * 1q0 ) based on (4)-(9) are presented as follows: where δ represents the infinitesimal displacement and the secondary angular frequency is assumed to be constant. The transfer functions of the current controller based on (2), (4)- (7), and (10)-(15) are expressed as follows: The state equations of the conventional IFOC system linearized near the operation point (I 1d0 , I 1q0 , 2d0 , 2q0 , X 1d0 , X 1q0 , I * 1dF 0 , I * 1qF 0 , I * 1dF 20 , ω 10 , ω r0 , ω * s0 , I * 1d0 , I * 1q0 ) based on (2) and (4)- (15) are expressed as follows: System matrix A 1 and input matrix B 1 are quite complex; therefore, A 1 and B 1 are shown in Appendix A1.

B. PROPOSED SLIP-ANGULAR-FREQUENCY FIRST-ORDER-DELAY CONTROL
The Taylor expansion of (3) with respect to ω * s near (I 1d0 , I 1q0 , ω * s0 ) is expressed as (26) Therefore, 2q is a second-order-delay system with respect to the input ω * s . The undershoot of 2q and misalignment of the secondary flux occur when the slip-angular-frequency command changes according to (5). In this study, a method that reduces the high-frequency components of the slipangular frequency to reduce the fluctuation of 2q is proposed.
The proposed first-order-delay-controlled slip-angularfrequency command is presented as In the proposed method, a first-order-delay filter is added to the slip-angular-frequency controller to reduce the abrupt change in the slip-angular frequency when the q-axis current command is input. The transient fluctuation of the q-axis secondary flux is reduced and the FOC is realized even during the transient response. Although a high-order-delay filter is assumed, a first-order-delay filter is used to reduce the number of tuning parameters in this study.

III. ANALYSIS AND VERIFICATION OF THE CONVENTIONAL AND PROPOSED CONTROL SYSTEM WITH A 750 W IM A. ANALYSIS OF THE CONVENTIONAL AND PROPOSED IFOC
The conventional and proposed IFOC systems were analyzed based on the analytical models derived in Section II. Table 1 summarizes the IFOC of a 750 W IM. Table 2 lists the operational points of the analysis. The target time constant T d of the IFOC is assumed to be 10 ms in this study. In this section, the time constant T ds of the proposed slip-angularfrequency first-order-delay controller is equivalent to T d . The determination method for T ds is discussed in Section III-B. Fig. 2 shows the closed-loop transfer function between δI * 1q and δI 1q in the conventional and proposed IFOC systems according to (22)- (25) and (35)-(38) when the primary angular frequency is 250 rad/s. In a conventional IFOC system, the magnitude of the transfer function is relatively large and is a second-order system. Therefore, an overshoot of the q-axis  current is expected in the step response. In the proposed IFOC system, the magnitude of the transfer function is -3 dB at 100 rad/s, and the maximum value is 0 dB. Therefore, the proposed IFOC system is expected to be a first-order-delay system of 100 rad/s. Fig. 3 shows the closed-loop transfer function between δI * 1q and δ 2q in the conventional and proposed IFOC systems according to (22)-(25) and (35)-(38), when the primary angular frequency is 250 rad/s. Compared with the conventional IFOC system, the proposed IFOC system reduces the q-axis secondary flux fluctuation because of the first-order-delay slip-angular-frequency control.   frequency is 50 rad/s. In a conventional IFOC system, the magnitude of the transfer function is relatively larger than that of the proposed method, but the peak magnitude is lower than the case of 250 rad/s. In the low-speed region, the response of the conventional IFOC system is expected to be similar to that of the first-order-delay system. Fig. 5 shows the bode plots of the conventional IFOC system. The bode plots of the current controller are based on the transfer function between δI * 1q and δV * 1q according to (16). The bode plots of the IM plant model are based on the transfer function between δV * 1q and δI 1q according to (21). The open-loop transfer function between δI * 1q and δI 1q is based on (22)- (25). The peak magnitude of the IM plant causes the magnitude of the control system to be high in the high-frequency region when the primary angular frequency is high. Therefore, the magnitude of the closed-loop transfer function of the IFOC system tends to increase near the cutoff frequency when the primary angular frequency is high. This tendency is more noticeable in the case of large-capacity IMs, as shown in Section V, because the inductance of a large-capacity IM is small; and the peak magnitude of the IM plant is larger than that shown in Fig. 5.  Fig. 6 shows the closed-loop transfer function of the proposed IFOC when the primary angular frequency is 250 rad/s and T ds changes.

B. DETERMINATION OF THE TIME CONSTANT T ds IN THE SLIP-ANGULAR-FREQUENCY FIRST-ORDER-DELAY CONTROLLER
In the case of a small time constant T ds , the sensitivity of the q-axis secondary flux with respect to the torque-current command increases because the suppression effect of the slipangular-frequency fluctuation by the first-order-delay filter is reduced. The magnitude of the control system at a frequency of approximately 100 rad/s was large. Therefore, the time constant of the torque-current response is expected to be shorter than the target response time constant T d , and an overshoot may occur.
In the case of a large time constant T ds , the magnitude of the control system at a frequency of approximately 100 rad/s is small, and the time constant of the torque-current response is expected to be longer than the target response time constant T d . When T ds is equal to the time constant T d of the current controller, the torque-current response is expected to be equal to the target response time constant T d . The numerator of the transfer function of the q-axis secondary flux based on (3) and (27) is expressed as follows: (39) The constant term in (39) can be made close to zero by setting the time constant of I * 1qF s to T d because the torque current I 1q is expected to be a response of the time constant T d . Therefore, the sensitivity of the q-axis secondary flux with respect to the torque-current command can be suppressed by making the time constant of the slip-angular-frequency firstorder-delay controller equal to that of the current controller.

C. INFLUENCE OF THE PARAMETER ERROR OF THE SECONDARY RESISTANCE SET VALUE R 2s
Assuming a parameter error R 2 with respect to the true secondary resistance value R 2 , the secondary resistance set value is defined as follows: Fig. 7 shows the closed-loop transfer function of the conventional and proposed IFOC systems when the primary angular frequency is 250 rad/s and R 2 changes. Degradation of the current control performance is inevitable for both the conventional and proposed IFOC systems because the error in the secondary resistance set value causes the misalignment of the secondary flux in a steady state. However, compared with the conventional IFOC, the proposed IFOC makes the magnitude of the control system reduced in a high-frequency region; the secondary flux fluctuation and the torque-current overshoot during transient response are expected to be reduced.

D. NUMERICAL SIMULATION OF THE CONVENTIONAL AND PROPOSED IFOC
A numerical simulation based on Table 2 was conducted to verify the proposed method, including its secondary flux response. In this simulation, pulsewidth-modulation control was not considered because the purpose was to verify the proposed method in principle. Fig. 8 shows the simulation results of the torque-current step response when the primary angular frequency was 250 rad/s. In the conventional IFOC, the q-axis secondary flux fluctuates transiently when the step q-axis current command is input, and a vibration of the q-axis current occurs owing to the coupling of the secondary flux. In the proposed IFOC, the misalignment is reduced, and the FOC and first-order-delay responses are realized.  Fig. 9 shows the simulation results of the torque-current step response when the primary angular frequency was 50 rad/s. The torque-current response was close to the firstorder-delay response of 100 rad/s, even in the conventional IFOC; this result corresponds to the tendency of the analysis in Section III-A. Fig. 10 shows the simulation results of the torque-current step response when the primary angular frequency was 250 rad/s and T ds is changed. In the case of a small time constant T ds , the torque-current response is faster than the target response, and an overshoot occurs because the magnitude of the control system at a frequency of approximately 100 rad/s is large. In the case of a large time constant T ds , the torque-current response is slower than the target response because the magnitude of the control system at a frequency of approximately 100 rad/s is small. When T ds is not equal to T d , the suppression effect of the q-axis secondary flux fluctuation degrades, and misalignment of the secondary flux occurs.

F. NUMERICAL SIMULATION OF THE CONVENTIONAL AND PROPOSED IFOC FOR VERIFICATION OF R 2
Figs. 11 and 12 show the simulation results of the torquecurrent step response when the primary angular frequency was 250 rad/s and R 2 changed. When the set value of the secondary resistance has errors, misalignment of the secondary flux occurs in the steady state; the torque-current control performance is degraded in both the conventional and proposed IFOC systems. However, compared with the conventional IFOC, the proposed IFOC reduces the secondary flux fluctuation and torque-current overshoot during a transient response. The slip-angular-frequency command is calculated based on the secondary resistance in IFOC. Therefore, the misalignment due to R 2 is inevitable. Thus, the correct secondary resistance is needed. However, the proposed control method is very simple construction, and the high torque-current control performance is realized by only tuning the secondary resistance.

IV. EXPERIMENTAL VERIFICATION OF THE CONVENTIONAL AND PROPOSED CONTROL SYSTEMS WITH 750 W IM A. EXPERIMENTS OF THE CONVENTIONAL AND PROPOSED IFOC
Experimental verification was conducted because the deadtime of the inverter, forward voltage of the power semiconductor devices, and harmonic components of the current and voltage were not considered in the numerical simulation. An eight-pole 750 W IM with an inertial load was used for the verification. Table 1 summarizes the IM used for the experimental verification. Fig. 13 shows the configuration of the experimental system. The dc link voltage was 300 V, the sampling frequency was 5 kHz, the d-and q-axis current commands were 4 A and 5.7 A, the time constant of the current controller T d was 10 ms, and the time constant of the slip-angular-frequency first-order-delay controller T ds was 10 ms. Fig. 14 shows the experimental results in acceleration and deceleration operation. Since the proposed first-order-delay slip-angular-frequency control is a method to improve the transient characteristics of the torque current, there is no difference between the conventional and proposed methods in the steady-state operation. Fig. 15 shows the experimental waveforms of the torquecurrent step response when the primary angular frequency was 250 rad/s. The proposed IFOC realizes a torque-current response that is closer to the target response than the conventional IFOC. Fig. 16 shows the experimental waveforms of the torquecurrent step response when the primary angular frequency is 50 rad/s, wherein the sixth-order harmonic component of the fundamental frequency owing to the deadtime increases the current ripple. The torque-current response is the first-orderdelay response of 100 rad/s even in the conventional IFOC; this result corresponds to the tendency of the analysis. Fig. 17 shows the experimental waveforms of the torquecurrent step response when the primary angular frequency was 250 rad/s and T ds changes. In the case of a small time constant T ds , the torque-current response is faster than the target response, and an overshoot occurs because the magnitude of the control system around the frequency of 100 rad/s is large. In the case of a large time constant T ds , the torquecurrent response is slower than the target response because the magnitude of the control system at a frequency of approximately 100 rad/s is small. Therefore, the time constant of the slip-angular-frequency first-order-delay controller was determined to be equal to that of the current controller.  Fig. 18 shows the experimental waveforms of the torquecurrent step response when the primary angular frequency was 250 rad/s and R 2 changed. Compared with the conventional IFOC, the proposed IFOC reduces the overshoot of the torque current during the transient response; the torque-current response is close to the target response.

D. VERIFICATION OF THE CONVENTIONAL AND PROPOSED IFOC IN A WIDE SPEED RANGE
The torque-current step response of the conventional and proposed IFOC systems was verified in the primary angular frequency range of 50-300 rad/s. In the verification, the tracking error J te and overshoot rate J ov were calculated according to the definitions in Fig. 19, (41), and (42). In addition, the settling time is verified The torque-current step response of the conventional and proposed IFOC systems was verified in the primary angular frequency range of 50-300 rad/s. Filtering the current waveform affects the cutoff frequency component, especially in the low-speed region. Therefore, the actual current waveform, which includes harmonic components, was used for verification. Fig. 20 shows the verification results. As the primary angular frequency increased, the tracking error and overshoot rate deteriorated in the conventional IFOC system because the magnitude of the transfer function tends to increase in this system. Conversely, the proposed IFOC realized the firstorder-delay response of 100 rad/s over a wide speed range. The settling time in the proposed method is shorter than that of the conventional method because the proposed method removes the current vibration. Therefore, the proposed IFOC achieves high torque-current control performance over a wide speed range.

A. ANALYSIS OF THE CONVENTIONAL AND PROPOSED IFOC
It is confirmed that the analytical results correspond to the simulation and experimental results in Sections III and VI. Finally, the analysis and simulation with a 150 kW IM are conducted in this section. Table 3 summarizes the IFOC of a    Table 4 lists the operational points of the analysis. The target time constant T d of the IFOC is assumed to be 10 ms in this study. Fig. 21 shows the closed-loop transfer function between δI * 1q and δI 1q in the conventional and proposed IFOC systems according to (22)- (25) and (35)-(38) when the primary angular frequency is 200 rad/s. In a conventional IFOC system, the magnitude of the transfer function is large and is a secondorder system. The peak magnitude is also larger than that of Fig. 2 because the inductance of a large-capacity IM is smaller than that of a small-capacity IM; and the peak magnitude of the IM plant is larger than that shown in Fig. 5, as shown in Fig. 23. Therefore, an overshoot of the q-axis current is expected in the step response. In the proposed IFOC system, the magnitude of the transfer function is -3 dB at 100 rad/s,  and the maximum value is 0 dB. Therefore, the proposed IFOC system is expected to be a first-order-delay system of 100 rad/s. Fig. 22 shows the closed-loop transfer function between δI * 1q and δ 2q in the conventional and proposed IFOC systems according to (22)-(25) and (35)-(38), when the primary angular frequency is 200 rad/s. Compared with the conventional IFOC system, the proposed IFOC system reduces the q-axis secondary flux fluctuation because of the first-order-delay slip-angular-frequency control. Fig. 23 shows the bode plots of the conventional IFOC system. The bode plots of the current controller are based on the transfer function between δI * 1q and δV * 1q according to (16). The bode plots of the IM plant model are based on the transfer function between δV * 1q and δI 1q according to (21). The open-loop transfer function between δI * 1q and δI 1q is based on (22)- (25). The peak magnitude of the IM plant causes the magnitude of the control system to be high in the highfrequency region when the primary angular frequency is high. Therefore, the magnitude of the closed-loop transfer function of the IFOC system tends to increase near the cutoff frequency when the primary angular frequency is high. Fig. 24 shows the simulation results of the torque-current step response when the primary angular frequency was 200 rad/s. In the conventional IFOC, the q-axis secondary flux fluctuates transiently when the step q-axis current command is input, and an overshoot of the q-axis current occurs owing to the coupling of the secondary flux. In the proposed IFOC, the misalignment is reduced, and the FOC and first-order-delay responses are realized.

B. VERIFICATION OF THE CONVENTIONAL AND PROPOSED IFOC IN A WIDE SPEED RANGE
The torque-current step response of the conventional and proposed IFOC systems was verified in the primary angular frequency range of 50-300 rad/s. In the verification, the tracking error J te and overshoot rate J ov were calculated. In addition, the settling time is verified. Fig. 25 shows the verification results. As the primary angular frequency increased, the tracking error and overshoot rate deteriorated in the conventional IFOC system because the magnitude of the transfer function tends to increase in this system. Conversely, the proposed IFOC realized the firstorder-delay response of 100 rad/s over a wide speed range. The settling time in the proposed method is shorter than that of the conventional method because the proposed method removes the current vibration. Therefore, the proposed IFOC achieves high torque-current control performance over a wide speed range.

VI. CONCLUSION
An IFOC with a first-order-delay slip-angular-frequency controller, which has a simple configuration, was proposed. In the proposed method, a first-order-delay filter was added to the slip-angular-frequency controller to reduce abrupt changes in the slip-angular frequency when the q-axis current command is input. The proposed method reduces the transient fluctuation of the q-axis secondary flux and realizes sophisticated torque-current control even during a transient response. Therefore, the proposed method makes the controller configuration simple because an additional decoupling controller is not required; and the tuning parameter is only the secondary resistance. The proposed method was verified through analysis, numerical simulation, and experiments. The proposed IFOC realizes a first-order-delay torque-current response to reduce the transient fluctuation of the secondary flux over a wide speed range; the proposed method realizes the high torque-current performance by only tuning the secondary resistance.

APPENDIX A1
Components a ij of system matrix A 1 , except for 0, are presented as follows: Components b i j of input matrix B 1 , except for 0, are presented as follows:

APPENDIX A2
Components α ij of system matrix A 2 , except for 0, are presented as follows: Components β i j of input matrix B 2 , except for 0, are presented as follows: