Decoupled Discrete Current Control for AC Drives at Low Sampling-to-Fundamental Frequency Ratios

Implementation of proportional–integral (PI) controllers in the synchronous reference frame (SRF) is a well-established current control solution for electric drives. It is a general and effective method in digital control as long as the ratio of sampling-to-fundamental (S2F) frequency ratio, <inline-formula> <tex-math notation="LaTeX">$r_{\mathrm {S2F}}$ </tex-math></inline-formula>, remains sufficiently large. When the aforesaid condition is violated, such as operations in high-speed or high-power drives, the performance of the closed-loop system becomes incrementally poor or even unstable. This is due to the cross-coupling of the signal flow between <inline-formula> <tex-math notation="LaTeX">$d$ </tex-math></inline-formula>- and <inline-formula> <tex-math notation="LaTeX">$q$ </tex-math></inline-formula>-axes, which is introduced by the SRF. In this article, an accurate model of current dynamics, which captures the computational delay and PWM characteristics in the discrete-time domain, is developed. This motivates the investigation of eliminating cross-coupling effects in permanent magnet synchronous motor (PMSM) drive systems. A new current control structure in the discrete-time domain is proposed targeting full compensation of cross-coupling effects of SRF while improving dynamic stiffness at low S2F ratios. The matching simulation and experimental results carried out on a 5-kW high-speed drive corroborate the theoretical analysis.

Decoupled Discrete Current Control for AC Drives at Low Sampling-to-Fundamental Frequency Ratios decade with an increased technology uptake [1]. High-speed electric machines are ones that operate at higher fundamental frequencies. This leads to a lower value of the sampling-tofundamental (S2F) frequency ratios in digital drive systems, causing substantial dynamic and stability challenges in control systems [2]- [6].
The design of current regulators in the continuous-time domain, with subsequent discretization to get the resulting digital current regulators, has been widely used and proven adequate for most applications. It is usually assumed that the machine fundamental frequency is much lower than the drive sampling and switching frequency so that the influence of computational delay and discretization errors in digital implementation can be ignored in the design process [2]. However, in high-speed applications, the demand for high fundamental frequencies can lead to significant negative effects of discretization on digital drive system dynamic performance [3] and highly oscillatory response, and even instability may occur if the design of the current regulator does not aptly incorporate the effects of the discretization of the controllers [4]. In addition, it is well known that the digital implementation of control systems introduces delays, whose negative effects on dynamic performance also increase with the lower ratio of S2F frequency [5]. Traditional current controllers are implemented in the synchronous reference frame (SRF); the transformation during the delay time introduces additional cross-coupling components to the plant model, which is usually ignored and further reduces the system stability [6].
Many researchers have attempted to improve the drive performance at low S2F ratios. State feedback decoupling has been widely used to improve system dynamic performance, but it is not sufficient to guarantee system stability at a low ratio of S2F frequency. The internal model control (IMC) proposed in [7] and the complex vector design method introduced in [8] and [9] prompt the robustness of current regulators by implementing zero-pole cancellation to the converted singleinput-single-output (SISO) systems, thus enabling higher fundamental frequency operation. Deadbeat and predictive control have been proposed, which provides a fast dynamic response with negligible error at a steady state [10], [11]. However, even with these attractive attributes, the inherent delay and discretization error of these methods degrade the robustness 2168-6777 © 2022 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission.
See https://www.ieee.org/publications/rights/index.html for more information. of the control system and limit the performance at low S2F ratios. Some research focused on the time delay in the sampling and PWM process and analyzed the bandwidth limitations and compensation methods [4]- [6], [12]- [14]. In [12], a delay compensation method in the continuous-time domain has been proposed, which extended the operation range to an S2F ratio below 10 [13]. A very low S2F ratio control was achieved in further research [13], where an additional model-based error estimator was designed to compensate for the current sampling delay. Several researchers focused on current controllers in the discrete-time domain to eliminate the cross-coupling effects [15]- [18]. A very low S2F ratio control was achieved in further research [13], where an additional model-based error estimator was designed to compensate for the current sampling delay. Several researchers focused on current controllers in the discrete-time domain to eliminate the cross-coupling effects [15]- [18]. In [16], the effects of different discretization methods have been analyzed and compared; a discrete current regulator based on a symmetric machine model has been proposed with superior dynamic response compared to the continuous-time domain design method being reported. In later research [18], a more comprehensive discrete-timedomain system model has been proposed, and discrete current controllers considering different sampling methods have been designed, which showed excellent decoupling capabilities at high S2F ratios (r S2F ≥ 50); however, the decoupling performance degrades at lower S2F ratios (r S2F ≤ 30). Furthermore, recent publications, e.g., [19], demonstrate that the current control in the rotating reference frame and the associated cross-coupling dynamics are of high scientific and practical relevance, and are not yet fully investigated. Although extensive research has been carried out investigating and developing alternative controllers, the current control technique based on proportional-integral (PI) regulators in the SRF is most used, and the design and tuning method for low S2F ratio operations is still considered an interesting research theme and is inadequately reported [20].
This article aims to improve the performance of the current control in the dq reference for low S2F ratio operations. A comprehensive theoretical analysis is provided on the tuning considering the cross-coupling effects in the SRF drive system. The focus is on the discrete-time domain and complex vector modeling of the variable frequency system. A discrete-timedomain system model is developed, which captures the behavior at low S2F ratios and the delays associated with PWM. Compared to the literature that already presents such analysis, i.e., [18], where the stator-voltage output of the inverter is modeled in stator coordinates, this article provides a more precise discrete-time-domain model of the system under study by using the rotor coordinates modeling method.
The detailed design procedure is illustrated, by implementing the proposed decoupled discrete PI (DDPI) to the drive system; the cross-coupling between the flux and torque components is eliminated in a transient state. Theoretically, the parameters of DDPI regulators are automatically tuned along with the machines' operational speed and frequency; thus, the controller tracking performance is independent of the speed and the S2F ratio. Matching simulation and experimental results carried out on a 5-kW high-speed drive corroborate the theoretical analysis.

A. Complex Transfer Function
Even though the performance of the SRF-based PI current regulators may seem intuitive, the multiple-input-multipleoutput (MIMO) nature of the system makes its performance evaluation difficult. The complex vectors are introduced to simplify the model of an ac machine to an equivalent SISO complex vector system. The simplified complex model of the ac machine current loop is shown in Fig. 1.
The space vector u dq Conv is considered as the input, and i dq Conv is the output variable of the motor. The electromagnetic subsystem can then be described by Conv and ψ f are space vectors that represent the stator current and the rotor flux linkage, respectively, u dq Conv is the stator voltage vector,τ σ is the transient stator time constant, r σ is the stator resistance, and ω k is the angular stator frequency, with all variables normalized by their respective nominal values.
With the feedback cross-coupling decoupling method of back electromotive force (EMF), the plant can be simplified as an inductive-resistive circuit. The transfer function of the permanent magnet motor is derived from (1) as follows: Conv (t)} are the respective Laplace transforms. In the digital control system, the computation and modulation imply an additional delay in the stationary frame F αβ d (s) = e −sτ d , where the time constant τ d is the time delay due to the sampling and calculating process [5]. Due to the frequency shift property of the Laplace transform, a generic complex valued vector x αβ (s) from theαβ stationary reference frame can be transformed into the dq rotating coordinates x dq (s) by the Park transformation x dq (s) = x αβ (s + j ω), and then, the delay resulting from digital signal processing is observed in the SRF as The typical sample-and-hold characteristic of digital-toanalog (D/A) conversion for all regular-sampled PWM schemes is the fact that the PWM reference voltage is updated only once per sampling period τ s , which could be addressed by a zero-order-hold (ZOH) element as F αβ ZOH (s) = 1 − e −sτ s /s in the stationary frame. Then, the generic ZOH frequency-domain model at the rotating frame is described as A complete description of all relevant large-signal system dynamics in the continuous-time domain is obtained by taking the inductive-resistive current dynamics in (2) and the sampling, calculation, and D/A transfer characteristics in (3) and (4) into account. Thus, the overall complex-valued transfer function of the system dynamics is To investigate the cross-coupling effect in the discrete-time domain, a discrete-time-domain equivalent description of a continuous-time-domain system transfer function is calculated via the transformation. By introducing the transformation law to (5), an accurate discrete plant model considering the time delay and the transformation from the stationary reference frame to rotation reference frame could be obtained Here, ρ 1 = δ 1 δ 2 is one of the plant poles (p 1 ) in a complexvalued transfer function with δ 1 = e −τ s /τ σ , δ 2 = e − j ω k τ s , K s = ι 1 ι 2 is the system gain with ι 1 = (1−e −τ s /τ σ )/r σ , ι 2 = e −j2ω k τ s , and Z and L −1 represent the z-transformation and the inverse Laplace-transformation, respectively. Fig. 2illustrates the pole map of the accurate plant model in the discrete-time domain. As it can be seen, the first pole ρ 0 is fixed to the coordinate original of the z-domain, while the second pole ρ 1 is varied with the ratios of sampling-to-fundamental frequency (r S2F = f s / f e = ω s /ω k = 2π/ω k τ s ). As the ratio reduces, the system pole ρ 1 steps into the left half-plane, and the system gain K s also changes. This matches with the investigation in [12] that the converter output errors caused by the rotation in SRF during the time delay are not only a τ s -related phase delay but also a magnitude error of the output voltage.

B. Cross-Coupling Effects
It can be noticed from the continuous transfer function in (5) that the source of cross-coupling is the imaginary coefficients j, which interchanges the signal flow between the real and imaginary parts of the controlled system, i.e., i q and i d in a current control system. Moreover, the back EMF generated in the armature winding has an imaginary coefficient, which contributes to the cross-coupling effects, as described in (1).
Considering all the above, Fig. 3 illustrates the crosscoupling elements in the current control system with variable values of the electrical angular velocity ω k . Both the delay and ZOH introduce S2F ratio-related cross-coupling elements into the SRF control system, and these cross-coupling elements' negative influence increases as the ratio of S2F frequency reduces. As shown in (6), the system pole ρ 1 contains the cross-coupling factor caused by the machine inductance dynamics, while the delay and ZOH caused cross-coupling factor are accommodated in the system gain K s . As the operation speed increases (r S2F : ∞ → 1), both system pole and gain move to the left complex plane, with an increased magnitude of imaginary parts, which refers to the increasing cross-coupling effects between d-and q-axes.
Conventional controllers providing real-valued zero and gain cannot compensate for these dynamics of controlled systems. With a high sampling frequency, the bandwidth of the current controller is much higher than the machine fundamental frequency. Thus, the transient-state error is eliminated fast. Even though the cross-coupling exists, it is then compensated after the transient state has faded away. However, in high-performance low-S2F-ratio drive systems, where controller bandwidth is limited and fast reference tracking ability together with a high demand of disturbance rejection is required, the decoupling capability of the current controller becomes the key factor driving the overall controller design process. In the following, a novel approach to compensate for the cross-coupling in the current control system is presented.

A. Complex Transfer Function
The synchronous-frame PI current controller, hereinafter referred to as SPI, is conventionally used for independent adjustment of the respective current components i d and i q in ac dives and grid-tied converters, which achieves reference tracking and disturbance rejection with zero steady-state error.
According to the internal model laws [7], [9], it can be defined as A common measure to eliminate the cross-coupling effect in the stator winding is to add a feedforward compensation signal to the SPI controller that is mathematically the same as the state feedback control. The compensation signal is intended to cancel the internal motion-induced voltage in the stator winding [with imaginary coefficients j in (2)]. By combining the current feedback with gain j ω k L σ added at the output of F SPI (s), the imaginary part of the plant pole in F dq P (s) could be canceled, i.e., it replaces the plant transfer function by F dq P (s − j ω k ) = 1/r σ (1 + sτ σ ). Then, the resulting pole of F dq P (s − j ω k ) is compensated by the zero of the controller in the following equation: This method works well for high sampling/switching frequency applications, where the practical issues caused by the digital implementation process, such as the inverter and sampling delay in (3) and (4), can be neglected. Considering the time delay and the characteristic of D/A, using the Tustin transformation to convert synchronous frame PI current regulator to the discrete domain results in (9), as shown at the bottom of the page, and the close-loop transfer function being (10), as shown at the bottom of the page, where . It can be seen from (11) that the poles of the closed-loop transfer function contain ρ 1 and K s , which changes with the S2F ratio.

B. Tuning Methods for SPI Current Controller
It is usually suggested to tune the SPI controller as k c = kl σ and τ i = τ σ = l σ /r σ , where the controller gain is selected as k = 1/2r σ τ s and the controller time constant τ i is selected to compensate for the transient stator time constant τ σ [6]. In the case of a low ratio of S2F, high bandwidth is required to avoid significant oscillation. Researchers established that time delay is the main limitation for an optimized tuning of the current controllers in terms of the desired large bandwidth [5], [14], [21], [22]. In paper [21], the optimized gain tuning method has been suggested to tune the PI controller and, analogously, all forms of linearized ac current regulators by setting k is the controller gain, as well as the bandwidth for acceptable damped response; taking into account the 1.5τ s delay, the value has been suggested in [21] for the maximum bandwidth as k max ≈(9.3/100)·2π f s , where a slightly greater value of 10% is known as a classic rule of thumb for generic digital control applications, while, in [20], a simple rule of thumb was proposed by setting the open-loop crossover frequency to 4% of the sampling frequency, as k opt ≈(3.9/100)·2π f s , so that nearly the minimum achievable settling time is achieved in combination with a negligible overshoot in the command [20]. The successful results provided by both methods in motor drives have been proved in later works, while the performance at low r S2F is rarely reported. In this article, both tuning methods are employed and compared with the proposed DDPI method.

C. DDPI Current Controller Design
One should expect to realize the fully decoupled control of machine currents by designing the current controller according to the accurate plant model F dq PL (s), as established in Section II. However, the sampling and calculation delay and ZOH models further complicate the controller structure in the s-domain, which leads to the challenge of digital implementation in digital signal processors. Moreover, the approximation used in the controller discretization process leads to incomplete transformation, which degrades the performance of the designed controller.
Instead of designing the controller in the s-domain and implementing it in the z-domain, a potential method is a directly z-domain design. Based on the accurate discretetime-domain plant model presented in (6), the DDPI current controller implementing zero-pole cancellation principle is proposed. The diagram of the proposed discrete-time-domain current regulator is presented in Fig. 4, where the complex and the scalar representation are, respectively, shown in Fig. 4

(a) and (b).
Similar to the SPI regulators, the basic structure of DDPI should contain both integral and proportional control laws, but it is designed to directly cancel the cross-coupling effects of plant F dq PL (z) in the discrete-time domain. Therefore, the structure of the discrete-time-domain current controller F dq The controller's zero z 0 is chosen to compensate for the highest system response time and the cross-coupling effects caused by the inductor dynamics The complex-valued controller proportional gain K c is used to compensate for the system gain K s for both steady and transient states Here, an additional real-valued factor γ > 0 is introduced to shape the command response of the current controller, where γ ∈ R is a constant. By implementing the zero-pole cancellation, the cross-coupling terms of the imaginary coefficients j in the transfer function of (6) have been compensated by the current controller F dq DDPI (z). With all single-complex poles of the plant model in the z-domain being canceled, the open-loop transfer function without considering the disturbance resulting from (13) to (15) can be derived It shows no more complex coefficients; thus, the crosscoupling is eliminated theoretically. The closed-loop transfer function can be obtained as Comparing (17) with (10), it can be seen that the proposed discrete PI regulator realizes the zero-pole cancellation directly in the z-domain. Both S2F-related elements, the plant gain K s and the plant pole ρ 1 are compensated by the controller. The comparison of close-loop zero-pole cancellation of different methods is shown in Fig. 5. As shown in Fig. 5(a), the conventional SPI without compensation gets oscillatory as the machine speed increases and becomes unstable (pole adjacent to the boundary of the unit circle migrates toward outside of the unit, point A to B) if the fundamental frequency is bigger than f s /12, whereas, with the feedforward compensation, it can remain stable with fundamental frequency up to f s /8, as shown in Fig. 5(b). However, there is still a clear tendency of the three single-complex poles to show unbalanced imaginary parts as the angular mechanical velocity increases (A→B, C→D, and E→F). As the frequency gets higher, the system pole cancellation zero varies from the pole location (E→F), which causes high-frequency oscillation. It is worth mentioning that both SPI and feedforward compensated SPI controllers are tuned by k opt for the above analyses, while a similar conclusion is obtained with k max , and a detailed comparison between these two tuning methods will be analyzed in Section IV. As shown in Fig. 5(c), the proposed controller realizes the fully zero-pole cancellation by setting the cancellation complex zero to the machine's complex pole (E→F). It can realize the fully crosscoupling decoupled control if the back EMF distortion can be ignored or fully compensated, i.e., back EMF feedforward compensation is implemented in this article.

A. Simulation and Experimental Setup
To verify the proposed DDPI current regulator, simulations are performed within a MATLAB/Simulink environment, where a continuous-time-domain permanent magnet synchronous motor (PMSM) model (parameters obtained from experimental test results), an average model of a two-level inverter (with a one-step delay of output voltage), and a discrete controller have been used. The parameters of the test machine are shown in Table I. The validation of the proposed control strategy is also performed on a 120-krpm    [23], [24]. The setup consists of a highspeed surface-mounted permanent magnet (SPM) machine and a two-level inverter; the switching frequency of the latter is set as 10 kHz. The controllers for the experimental tests have been implemented on the designed DSP + FPGA controller board. The single update mode PWM modulation strategy is adopted, and the same PWM frequency and control frequency are used in all simulations and experiments presented in this article.

B. Results
The overshoot versus settling time trajectories of different methods has been analyzed. A tolerance band of δ = 1% [20] is defined for the settling time. The q-axis current step response associated with each of the methods at low sampling-to-fundamental frequency ratios is represented in Fig. 6. It can be noticed on closely inspecting Fig. 6(b) and (c) that both SPI methods show a degrading tracking performance (setting time of more than 5 ms and overshoot of more than 10%) with decreased S2F ratios (green→red→yellow: 50→20→15), where the solid line presents SPI with the tuning method k opt and the dashed line presents SPI with the tuning method k max . It is worth mentioning that, for the SPI method without the feedforward compensation, the k max tuning method (SPI-k max ) shows a shorter settling time and smaller overshoot than the k opt method (SPI-k ept ) for all frequency cases, while, in paper [20], the performance of FC-SPI-k opt and FC-SPI-k max has been analyzed at high ratio (≥50) applications, where results show that FC-SPI-k opt presents a better tracking performance with almost zero overshoot, which is verified as shown in Fig. 7(a). The performances of FC-SPI and SPI with both tuning methods (k max and k opt ) have been compared at low S2F ratios, as shown in Fig. 7(b) and (c).
It is worth mentioning that the feedforward compensation method degrades the controller performance rather than improving it with the tuning method k max , where FC-SPI-k max shows a larger overshoot and a settling time than SPI-k max . The reason is that the intended compensation is counteracted by the inverter and the sampling delay at higher angular mechanical velocity.   The settling time and the overshoot of different methods with reduced S2F ratios have been compared, as shown in Figs. 8 and 9. Among these four existing PI design methods, SPI-k max shows the fastest tracking performance, while SPI-k opt has the worst tracking performance with the largest overshoot and the longest settling time at all operation frequencies; FC-SPI-k opt shows the smallest overshoot with the improved settling time at higher S2F ratios, while the performance degrades at extremely low S2F ratio (i.e., r S2F = 15).
In conclusion, the performance of SPI and FC-SPI controllers varies with the sampling-to-fundamental frequency. Due to the uncompleted decoupling, SPI and FC-SPI controllers face the degraded frequency-dependent tracking performance with the increased overshoot and settling time as the S2F ratio reduces, whereas a frequency-   independent tracking performance with the minimum settling time and zero overshoot of the proposed DDPI is shown in Figs. 6, 8, and 9.
The decoupling performance of the proposed discrete PI controller has been verified both in simulation and experimental tests. As shown in Figs. 10-12, the current step response of the proposed discrete-time-domain current regulator is compared to the discrete-time implementation of the SPI. The q-axis current reference is changed rapidly, while the d-axis reference is fixed as zero. Both SPI methods show overshoots of the d-axis current during the transient state. With the fundamental frequency rising from 200 Hz to 1 kHz (r S2F changes from 50 to 10), the overshoot of the d-axis grows fast, which leads to high oscillation of controller output currents. Specifically, SPI-k pot shows a significant overshoot of over  50% at 15 krpm (r S2F equals 20), while SPI-k max shows an improved performance but still reaches the target at 20 krpm (r S2F equals 15).
The proposed DDPI presents a decoupled control performance (without any i d ripple and i q overshoot during the sudden load change) independent of the ratio of sampling-to-fundamental frequency, as shown in result (c) of Figs. 10-12. It is worth mentioning that, due to the nonlinearity of the inverter, the degraded decoupling performance of DDPI occurs at an extremely low ratio of 6.67 (operation speed 45 krpm), but the current tracking performance is still acceptable. Moreover, the cross-coupling effects (see Fig. 13) of different control schemes under different S2F ratios show that the current space vector of the proposed DDPI is moving along the shortest path between the d-and q-axis current commands, which implies the fully decoupled control of the torque and field current components.
The experimental dynamic responses of the proposed DDPI are shown in Figs. 14 and 15. Figs. 14(a)  . Due to the limitation of the converter cooling system, a scaled-down i qref is applied in the experimental test. At low-frequency verification, current steps from 5 to 10 and then 5 A are used, while smaller current references (from 3 to 6 to 3 A) are applied in high-frequency validations. Since the current is smaller (1/10th of the rated current), a relatively higher THD can be observed in the output three-phase current during the high-speed test, as shown in Fig. 15(b). However, the fully decoupled current control is still realized and experimentally demonstrated at these reduced current values.   Moreover, the comparison of the tracking performance of SPI-k max and the proposed DDPI (see Fig. 16) indicates the fast-tracking performance of DDPI; the same conclusion can be also drawn from Fig. 8. Fig. 17 demonstrates the effects of stator resistance mismatches on the step responses of the proposed DDPI at a low S2F ratio of 15. In Fig. 17(b), the normal condition with actual resistance R =R has been illustrated as a reference, where the system presents decoupled control performance with no oscillation in the d-axis current. The same decoupled control performance can be seen in Fig. 17(a) with actual resistance R = 2R and Fig. 17(c) with actual resistance R = 0.5R.

C. Robustness
The effects of stator inductance mismatch on the controlled system are illustrated in Fig. 18 (the S2F ratio of 15 is also considered here). The normal condition with actual inductance  L =L is presented as a reference, as shown in Fig. 18(b). It can be seen from Fig. 18(a)-(c) that the decoupled control performance is not affected by the inductance mismatch.

D. Gain Factor Impact
As shown in Fig. 19, by changing the value of gain factor γ , the influence of the controller proportional gain K c to the pole-zero location of the dominant pole-pair is manipulated without affecting the pole-zero locations of other system  dynamics. The corresponding damping ratio and the natural frequency are presented on the right side of the polar map in Fig. 19. It is worth mentioning that a higher value of γ leads to a reduced damping ratio and natural frequency, which indicates a fast dynamic response of the system. Fig. 20 shows the step response of the closed-loop control system and the reference for the controller parameter γ selection, in which the controller shows no overshoot current tracking with γ = 0.25 and a faster control with 0.25 < γ < 0.5.

V. CONCLUSION
In this article, the influence of sampling and time delay, as well as the D/A characteristic of PWM in digital motor drive systems, is analyzed, and an accurate discrete plant model capturing the cross-coupling effects in the rotation reference frame is proposed. The accurate model is used to evaluate two existing current control methods and the proposed decoupled discrete current controller. Different design methodologies are analyzed in detail and compared. As a result, some important reference guidelines can be drawn, as summarized in Table II.
The proposed DDPI targets to cancel the r S2F -dependent cross-coupling effect of the rotational frame current control system. Without current and flux observers, the proposed discrete-time-domain current controller allows to guarantee the tracking performance at the fundamental frequency of up to 15% of the switching frequency with respect to the state-ofthe-art current control of about 10% [12].
The theoretical analysis presented and simulated in MATLAB/Simulink is validated by means of experimental measurements performed on a 5-kW high-speed drive with the results showing a very close match. The work presented introduces a decoupled PI current control design method that enables improved tracking and dynamic performances with respect to the conventional design methods.