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

In high-performance drive systems, wide bandwidth and reference tracking accuracy of current control loop are fundamental requirements. The conventional PI controller provides robustness against the machine parameter mismatching and zero steady-state error, but its dynamic performance degrades at high speed due to the bandwidth limitation. In this article, a new control structure of discrete PI controller with a deadbeat response is proposed, which combines the advantage of conventional PI controller with deadbeat characteristics. The proposed controller shows a decoupled tracking performance of up to 15% of the switching frequency while also providing extra control freedom of the disturbance rejection, which effectively improves the system stability. Experiments show a reduction of oscillation by 30% compared to the conventional PI and the validity and applicability of the proposed control method for high-speed applications with low sampling-to-fundamental (S2F) frequency ratios.

performance. A fast transient response and satisfactory steadystate tracking performance is required [1].
However, the bandwidth of a conventional PI regulator is limited by the time delay and discretization in digital implementation, which inhibits its use in high-performance drives [2], [13], [14], [15], [16]. In high-speed applications, where a low ratio of sampling-to-fundamental (S2F) frequency is applied, high oscillation and even instability can occur if the design of the current regulator does not take these effects into account [14]. In transient states such as during sudden reference change and load disturbances, fast dynamic response is very important for high-speed drives. A slow transient characteristic not only degrades the performance of the current control but also causes further degradation of the outer speed control loop. Moreover, since conventional PI controllers are usually implemented in an SRF, the cross coupling of the states caused by the reference frame transformation deteriorates the dynamic performance of the current controller. Thus, although the SRF PI regulator provides the desirable performance in steady state, it is still of interest to improve its dynamic response.
Previous studies developed several control schemes for the digital inverter in order to improve the current regulating and dynamic performance and a summary is presented in Table I. For the digital implementation of continuous timedomain design methods (i.e., SPI, FC-SPI, and CPI), time delay compensation methods are suggested for further improving the dynamic response at low r S2F [13], [15], and the active damping methods are introduced to improve the system 2332-7782 © 2022 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission.
Authorized licensed use limited to the terms of the applicable license agreement with IEEE. Restrictions apply.  [17], [18], [19], [20], [21]. While the discrete-time-domain design methods provide a more straightforward way of decoupling design [14], [22], [23], [24], the proposal does not provide for full control of the closed-loop plant, which limits the controller capability in cases of disturbance rejection. Moreover, the constrained freedom of the control scheme restricts the ability of searching the optimal control voltage to force the current to the reference value in the minimum time possible, which causes a limited bandwidth and slower the dynamic response of the controller. The cross-coupling effects in the current control system have been analyzed, and an accurate discrete-time-domain plant model has been developed in the previous work [25]. In this article, a new control structure with both the satisfactory steady-state characteristics and fast transient response is proposed. The novelty of this article lies in the design of the new structure of two-degree-of-freedom (2-DOF) decoupled discrete PI (DDPI) control for low S2F ratios. Considering the discretization and time delay in a digital control system, the controller is designed to eliminate the cross-coupling effects in SRF and ensure a decoupled tracking performance independent of the machine speed and S2F ratio. The proposed controller also features the deadbeat characteristic, which provides a wide-bandwidth and low harmonic distortion solution for high-speed applications. This article is organized as follows. In Section II, an accurate system model considering the time delay and rotational transformation is built, and the cross coupling in the current control system is analyzed. Section III analyzes the digital implementation and side effect of the existing design for low S2F applications, and subsequently, an improved 2-DOF discrete PI controller is proposed. The latter is then verified by simulations and experiments, which are illustrated in Section IV. To conclude, the matching results are shown in Section V, where the benefits and applicability of the proposed decoupled discrete current control are highlighted.

II. DYNAMIC MODEL OF PM MACHINE, CURRENT REGULATION, AND BACK EMF DECOUPLING
The nonlinear state equations governing the electrical and electromagnetic behavior of a PM machine using complex vector notation, considering the armature voltage space vector u dq Conv as the input and armature current i dq Conv as output variable of the motor, can be described by ψ f is a space vector that represents the rotor flux linkage, τ σ is the transient stator time constant, r σ is the stator resistance, and ω k is the electrical angular frequency. Ignoring the back EMF influence and taking the electrical and electromagnetic behavior of a PM machine as well as the sampling, calculation, and D/A transfer characteristics into account [14], [22], a simplified current transfer loop can be illustrated as shown in Fig. 1 with the complex-valued transfer function of system dynamic described in [25] F dq τ s is the sampling period and τ d is the time delay due to the sampling and calculating process. By introducing the transformation law to (2), the accurate discrete plant model considering the time delay and the transformation from stationary reference frame to rotational reference frame can be obtained It can be seen from (3) that the plant model includes two poles in the complex-valued transfer function: the first pole ρ 1 = 0 is located at the origin and the second pole  system gain with ι 1 = (1 − e −τ s /τ σ )/r σ , ι 2 = e − j 2ω k τ s . Z and L −1 represent the z-transformation and the inverse Laplacetransformation, respectively. The pole map of the accurate plant model in a discrete-time domain is shown in Fig. 2. As can be seen, the first pole ρ 1 is fixed to the coordinate origin of z-domain, while the second pole ρ 2 varies accordingly with the ratio of S2F frequency (r S2F = f s / f e = ω s /ω k = 2π/ω k τ s ). As this ratio reduces, the system pole ρ 2 steps into the left half-plane and the system gain K s also changes.

A. Conventional PI Current Control
The schematic of the widely separated synchronous reference PI control with feedforward compensation decoupling method (hereinafter referred to as FC-SPI) is shown in Fig. 3(a). To improve the rejection of disturbance u dq dis (s), in [26], it was suggested to implement an active resistance R a in the digital controller, as shown in Fig. 3(b). This technique is named "active damping method," which has the same effect as increasing the resistance of the controlled machine, if the time delay and zero-order-hold characteristic are neglected [12].
However, the tracking performance and disturbance rejection performance of these controllers degrade at low S2F ratios where the practical issues caused by the digital implementation process, such as inverter and sampling delay, cannot be ignored. As shown in Fig. 4, the poles of the closed-loop system vary with S2F ratios, which indicate the incomplete decoupling and speed-dependent control performance.

B. Proposed 2-DOF DDPI
Due to the aforementioned limitations of conventional PI and active damping method in low S2F ratios, in this section,  based on a complex vector design method, a novel 2-DOF DDPI and its decoupling tuning method are proposed. The proposed method shows a frequency-independent dynamic response and improved stability. The scheme of the proposed 2-DOF DDPI is shown in Fig. 5. A transfer function F dq f and a feedforward loop with an active damping gain K f 3 are introduced to achieve the decoupling between the axes and the repositioning of the poles of plant (3) Then, the inner loop transfer function can be described as Authorized licensed use limited to the terms of the applicable license agreement with IEEE. Restrictions apply. By properly choosing the values of gains K f 1 , K f 2 , and K f 3 , the resulting system is completely decoupled since all the coefficients involved in (5) are real numbers with ρ 2 ∈ R, and |ρ 2 | < 1 is the constant parameter of the controller, which is selected by the designer. With this choice, the closed loop of the inner plant system results The poles are relocated to coordinate origin and ρ 2 along the real axis in the z-plane, as shown in Fig. 6. Comparing to (3) and Fig. 2, the poles of the revised inner loop do not change with different values of the mechanical angular velocity ω k . This verifies that the internal loop is capable of decoupling axes d and q and relocating the poles. The outer loop of DDPI controller (10) is designed to regulate the current tracking performance, with the zero z 0 used to compensate for the pole ρ 2 of F dq C_inner (z), and with controller gain K c equal to the real-valued factor γ , γ > 0 is introduced to shape the command response of the current controller, where γ ∈ R is a constant The closed-loop transfer function of the system without disturbance results While considering the disturbance in the control system, the closed-loop transfer function of system is obtained It can be noticed that the disturbance transfer function F dq dis_B (z) in (14) has the pole located at ρ 2 , which is a constant tuning parameter chosen by designers and does not change with the rotating frequency ω k . Designers can use the real-valued factor γ to shape the command response of the current controller and use ρ 2 to improve the disturbance rejection dynamics and system stability, as shown in Fig. 7.

C. PDPI (High-Bandwidth Tuning Methods of 2-DOF DDPI)
In this section, the predictive decoupled PI (PDPI) controller is proposed, featuring the same structure but with different tuning method compared to that presented earlier in Section III-B. It presents the deadbeat characteristic, provides high bandwidth, and further improves the dynamic response. The conventional PI controllers have a limited bandwidth   because there is only one degree of freedom (1-DOF) and, as a result, this scheme cannot find the optimal control voltage, which will force the current to the reference value in the minimum possible time. By changing the tuning method of 2-DOF DDPI, a wide-bandwidth solution with deadbeat  characteristics can be achieved with The inner loop transfer function is then obtained as By employing z 0 = ρ 2 , K c = 1 to (10), and ρ 3 = −1 to (18), the closed-loop transfer function of the system is obtained It can be seen from (19) that the transfer function shows a deadbeat characteristic, which allows for a fast-tracking performance. Considering the disturbance in the control system, the closed-loop transfer function of the system is obtained Comparing the closed-loop pole maps of Figs. 7 and 8 with Fig. 4, it can be seen that the proposed 2-DOF DDPI controllers present an S2F ratio-independent characteristic. In addition, it provides an adjusted disturbance rejection ability without affecting a controller tracking performance.

A. Simulation and Experimental Setup
To verify the proposed 2-DOF DDPI current regulator, simulations are performed within MATLAB/Simulink environment, where a continuous time-domain PMSM model (parameters obtained from experimental test results), an average model of two-level inverter (with one-step delay of output voltage), and a discrete controller are used. The inductance values of test machine are shown in Fig. 9, and the other parameters are shown in Table II.
The validation of the proposed control strategy is also performed on a 120-krpm high-speed dynamometer with the SiC inverter-fed high-speed SPM prototype, as shown in Fig. 10. The SiC-based two-level inverter is designed for the high-speed application, where the maximum switching frequency is equal to 80 kHz, and the dead time is equal to 500 ns. During the tests, the switching frequency is set as 10 kHz to emphasize control performances with reduced samples per fundamental period. The controllers for the experimental tests are implemented on the designed DSP + FPGA controller board. During the experimental test, the dyno works in speed mode and the test motor works in load mode to validate the proposed current control strategy. The same PWM frequency and control frequency are used in all simulations and experiments presented in this article.
B. Performance Analysis of the Proposed DDPI 1) Simulation Results: First, the dynamic response of the conventional PI controller (FC-SPI) and the proposed 2-DOF DDPI controllers are compared in simulation. Fig. 11 presents the transient response of different regulators at a high S2F frequency ratio, i.e., r S2F = 50. From top to bottom, Fig. 11(a) presents the performance of conventional decoupling method FC-SPI, which shows the longest settling time (t s = 50T s ) and around 10% overshoot of d-axis current (which indicates the incomplete decoupling). The performance of the proposed 2-DOF DDPI shows a fasttracking performance (t s = 10T s ) and zero overshoot of i d [see Fig. 11(b)]. In addition, with the proposed predictive tuning method, 2-DOF DDPI presents decoupled control performance with the deadbeat characteristic, which tracks the reference value in two steps (t s = 2T s ), as shown in Fig. 11(c). The decoupling performance of the proposed 2-DOF DDPI with decreased S2F ratios is tested, as shown in Fig. 12. It can be seen from Fig. 12(a) to (c), as the S2F ratio decreases, conventional decoupling method FC-SPI shows a degrading tracking performance with increased setting time (from 50T s to 150T s ) and overshoot of d-axis current (from 10% to 22%). On the other hand, the proposed 2-DOF DDPI shows an S2F ratio-independent dynamic response and tracks the reference value in 2T s , as shown in Fig. 12(d)-(f). It is also worth mentioning that at an extremely low S2F ratio, i.e., below 10, the control system with FC-SPI becomes unstable, while the proposed 2-DOF DDPI can keep the same decoupled deadbeat control performance, as shown in Fig. 12.
2) Improved Dynamic Performance and Decoupled Control at Different Frequency/Ratios: The decoupled control performance and the high bandwidth characteristic of the proposed 2-DOF DDPI have been verified in the experimental test. First, the tracking performances of FC-SPI and the proposed controller have been compared at a relatively high S2F ratio (r S2F = 33), as shown in Fig. 13. The settling time (t s ) is marked, and the transient behavior of each method is highlighted by the red dashed circle in the figure. It can be seen from Fig. 13(a) that, during the sudden load change, the conventional PI presents a slow tracking performance (t s > 50T s ), while the proposed 2-DOF DDPI presents high bandwidth characteristics, which tracks the reference value in two steps (t s = 2T s ), as shown in Fig. 13(b). The decoupled control performance of the proposed 2-DOF DDPI controller with varied S2F ratios has been verified both in the d-q domain and the a-b-c domain, as shown in Figs. 14 and 15-17, respectively. In Fig. 14(a), the dynamic response of the proposed control with a q-axis current reference step change (10-5 A) at 9000 rpm is presented. It can be seen that there is no cross coupling between d-and q-axis currents, and the controller presents a fast-tracking performance without overshoot; this matches with the simulation results shown in Fig. 12(e). The same decoupled control performance at a low S2F ratio (r S2F = 25) is shown in Fig. 14(b), which experimentally verified the simulation results shown in Fig. 12(f).
The S2F ratio-independent decoupled tracking performance with minimum settling time of the proposed 2-DOF DDPI has been verified at a stationary reference domain. The three-phase currents tracking performance with different S2F ratios, during sudden change of load, are shown in Figs. [15][16][17]. The S2F ratio is reduced as the speed increases, but the current control settling time keeps the same. The proposed controller presents a high bandwidth characteristic, which tracks the reference value in two steps (t s = 2T s ) regardless of the S2F ratio. The same conclusion can also be drawn from Fig. 12(d) to (f).   3) Improved Steady-State Performance and THD%: The steady-state performance is compared and shown in Fig. 18.   This experimental benchmark has confirmed the effectiveness of the proposed 2-DOF DDPI controller over the PI controller in terms of an overall low THD. The harmonic mitigation capability of the proposed DDPI controller and the PI controller is compared, as shown in Figs. 19 and 20. As shown in Fig. 19, the PI controller can maintain lower harmonics in a wide range of frequencies (e.g., 4 kHz). The low-frequency harmonics close to the fundamental frequency are reduced (see Fig. 20) and the fundamental component increases by 2.91%, while the total THD% is reduced by 30% with respect to the conventional PI controller (see Fig. 19). These results are confirming the benefits of the method introduced and its applicability to high-speed drives where the S2F frequency ratios are low.

4) Performance Under Parameter Mismatching:
The effect of the parameter variation on the controllers has been analyzed using simulation, for two different tuning methods, as shown in Figs. 21 and 22. Here, L is the machine real inductance value and L is the machine parameter used in the control. It can be noticed that both tuning methods keep the same tracking performance with fixed settling time, and the decoupled control performance is not affected by the parameter mismatch.

V. CONCLUSION
In this article, a novel 2-DOF DDPI method is proposed, which aims to eliminate the cross coupling between d-and q-axes and improves the dynamic response of conventional PI controllers. The proposed 2-DOF scheme provides an extra control freedom of the disturbance rejection, which effectively improves the system stability without affecting the tracking performance.
Furthermore, two decoupled tuning methods for the proposed 2-DOF DDPI have been designed. One is for decoupled control with improved stability, while the other is for decoupled control with deadbeat characteristics.
The proposed method is performed experimentally on a 5-kW high-speed SPMSM, with the matching simulation and practical test results validating the theoretical analysis. Without current and flux observers, the proposed discrete time-domain current controller allows to guarantee the decoupled tracking performance of up to 15% of the switching frequency with respect to the state-of-the-art discrete current control of 8.3% [14]. The total THD% is reduced by a significant 30% compared to the conventional controller.

APPENDIX
The tuning method of conventional PI controller used in this article has been explained in detail in the previous work [25]. To realize the maximum bandwidth, k max ≈ (9.3/100)·2π f s is selected as the controller gain in this article. The discretization process of the conventional controller is illustrated as follows.
Synchronous PI (SPI) controller in the s-domain 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) can be canceled, i.e., it replaces the plant transfer function by F dq P (s − j ω k ) = 1/r σ (1 + sτ σ ), which is used in FC-SPI. The resulting pole of F dq P (s − j ω k ) is compensated by the zero of controller (21), and the open-loop transfer function of the current control system can be presented as Considering the time delay and the characteristic of D/A and using the Tustin transformation, the PI current controller in the discrete-time domain can be described in the following equation: (23) where A = k c (1 + (1/2)τ s τ −1 i ), B = k c ((1/2)τ s τ −1 i − 1), and the closed-loop transfer function is given as follows: