Current decoupling control of permanent magnet synchronous motor based on improved nonlinear extended state observer

Abstract

Precise current decoupling is required to achieve good control performance in the vector control of permanent magnet synchronous motor (PMSM), but traditional decoupling methods are unable to solve the problem of poor current decoupling effectiveness when the system inductance parameters are mismatched. Therefore, a decoupling control method of PMSM based on improved nonlinear extended state observer (INESO) is proposed in this paper. Firstly, a smooth, differentiable nonlinear function is used to replace the traditional fal function, which overcomes the problem of discontinuous gain changes. Secondly, a time-varying gain is designed to address the problem of reduced observer performance due to differential peak values in traditional NESO. Finally, the stability of INESO and the current loop closed-loop system is proven, followed by simulation and experimental verification. The simulation and experimental results show the proposed method effectively observes and compensates for system uncertainties, including parameter perturbations, external disturbances, and unmodeled dynamics. This method achieves complete decoupling of d and q-axes currents, while suppressing interference and enhancing the control performance of PMSM. The proposed INESO decoupling method reduces the d-axis current fluctuation about 96% and 90% when the load changes abruptly under motor inductance parameter mismatch, and can reduce q-axis current fluctuation about 88% and 78% when the d-axis current steps, respectively, compared with the conventional Current Feedback Decoupling Control (CFDC) and CDDC decoupling methods.

Keywords

Permanent magnet synchronous motor current decoupling improved nonlinear extended state observer time-varying gain stability analysis

1. Introduction

Permanent magnet synchronous motor (PMSM) has gradually become the mainstream driving motor in the industrial sector, due to its unique advantages (Kommuri et al., 2016; NP and K, 2015). The vector control technology is applied in PMSM control, which can realize simple and efficient direct independent control of torque and flux by decomposing the stator current of the motor into the excitation component and the torque component through the rotation coordinate transformation. However, there is a cross-coupling phenomenon between the d and q-axes currents of PMSM in the synchronous rotating coordinate system, especially under high speed or heavy load conditions, which may cause torque oscillation and seriously affect the dynamic and steady-state performance (Yepes et al., 2014).

In recent years, researchers have developed several methods to solve the problem of coupling between d and q-axes currents in PMSM, which can be mainly divided into three categories (Fu et al., 2022): diagonalization decoupling (Hussain, 2021; Xia et al., 2012), inverse system decoupling (Sun et al., 2016; Xie and Xie, 2020; Zhang et al., 2021), and anti-interference decoupling (Chen et al., 2021; Xu et al., 2022; Yang et al., 2020). Current Feedback Decoupling Control (CFDC) was proposed earlier (Morimoto et al., 1994), which can improve the current coupling problem under certain conditions by calculating the compensation coupling amount from current and speed information of PMSM. However, when the system parameters of PMSM change, it is difficult to achieve complete decoupling. Harnefors and Nee (1998) propose an internal decoupling model control method, which has a better performance compared with feedback decoupling, but with underdamped oscillation at the working frequency of PMSM. Zhu et al. (2010) propose a Current Deviation Decoupling Control (CDDC) method, which is essentially similar to internal model decoupling control with same limitations. Reverse system decoupling (Li et al., 2020) has low dependence on the motor model and strong algorithm robustness, but the implementation of reverse system decoupling requires the inverse system model of the system, and the modeling algorithm is too complex to achieve practical engineering applications.

Anti-interference decoupling, which regards the coupling of d and q axes as disturbance for observation and compensation, has been widely studied. Li et al. (2015) incorporate the Luenberger Observer based on CDDC, which can solve the problem of incomplete decoupling when inductance parameters change. However, there is a problem of insufficient stability and robustness in the observer. In the work of Liu et al. (2020), a combination of sliding mode observer and CDDC is introduced to achieve dynamic decoupling. It maintains good current decoupling even when there are changes to the system's inductance parameters. However, the design of the whole decoupling control system is more complicated with multiple parameters to be adjusted. Zeng et al. (2017) introduced a nonlinear extended state observer (NESO) to replace the PI controller for observing and compensating the coupling between the current loop's d and q-axes, which improves anti-interference performance. However, the traditional NESO's fal function is a piecewise function, and the sudden change of the switch point gain can easily cause tremors, affecting the system's dynamic performance, and there are relatively many parameters that need to be adjusted.

An anti-interference PMSM current decoupling method based on an improved nonlinear extended state observer (INESO) is proposed in this paper, which can improve the decoupling performance of d and q-axes currents. Continuous variation of error feedback correction gain is achieved through adopting a smooth and continuously differentiable feedback nonlinear function in INESO. Moreover, a larger error gain is obtained in the “small error interval” to speed up error convergence, while a smaller error gain is achieved in the “large error interval” to avoid overshoot. The problem of differential peak values that commonly occur when the initial state of the ESO status and the system's initial state differ significantly is overcome, through introducing of the hyperbolic tangent function. Finally, the performance of the proposed anti-interference decoupling method in this paper is verified through simulations and experiments.

2. PMSM current decoupling control method

2.1. Mathematical model of PMSM

The equations for the d and q-axes currents in the synchronous rotating coordinate system are

{\begin{cases} \frac{d i_{d}}{d t} = - \frac{R_{s}}{L_{d}} i_{d} + \frac{ω_{e} L_{q}}{L_{d}} i_{q} + \frac{u_{d}}{L_{d}} \\ \frac{d i_{q}}{d t} = - \frac{R_{s}}{L_{q}} i_{q} - \frac{ω_{e} L_{d}}{L_{q}} i_{d} - \frac{ω_{e} ψ_{f}}{L_{q}} + \frac{u_{d}}{L_{q}} \end{cases}

(1)

In equation (1), i_d and i_q represent the d and q-axes currents respectively. u_d and u_q represent the d and q-axes voltages separately. L_d and L_q represent the d and q-axes inductances respectively. ω_e is the rotor electrical angular velocity, R_s is the stator resistance, and ψ_f is the permanent magnet flux linkage.

The Laplace transform of equation (1) yields

{\begin{cases} I_{d} (s) = \frac{1}{s L_{d} + R_{s}} [U_{d} (s) + ω_{e} L_{q} I_{q} (s)] \\ I_{q} (s) = \frac{1}{s L_{q} + R_{s}} [U_{q} (s) - ω_{e} L_{d} I_{d} (s) - ω_{e} ψ_{f}] \end{cases}

(2)

As shown in equation (2), there is a cross-coupling term exists between the d and q-axes that is proportional to the electrical angular velocity. This coupling effect becomes more pronounced as the motor speed increases. Without effective decoupling of the cross-coupling term, it can significantly impact the dynamic performance of the control system.

2.2. Current feedback decoupling control

Current Feedback Decoupling Control diagram is shown in Figure 1, where G(s) is the transfer function of the PI controller, G(s) = K_p+K_i/s, where 1/(R_s+Ls) is the actual model of the controlled object, L_d = L_q = L, $\hat{L}$ is the actual inductance value in operation, which may not be equal to the nominal inductance value L.

According to the block diagram of CFDC in Figure 1, it is obtained that

{\begin{cases} G {i_{d}}^{*} - ω_{e} (\hat{L} - L) i_{q} = (R_{s} + L s + G) i_{d} \\ G {i_{q}}^{*} + ω_{e} (\hat{L} - L) i_{d} = (R_{s} + L s + G) i_{q} \end{cases}

(3)

From equation (3), the d and q-axes current expressions can be obtained as follows:

[\begin{array}{l} i_{d} \\ i_{q} \end{array}] = \frac{1}{Δ_{1}} [\begin{array}{l} g_{d d 1} & g_{d q 1} \\ g_{q d 1} & g_{q q 1} \end{array}] [\begin{array}{l} {i_{d}}^{*} \\ {i_{q}}^{*} \end{array}]

(4)

Among them:

{\begin{cases} Δ_{1} = {(G + R_{s} + L s)}^{2} + {ω_{e}}^{2} {(L - \hat{L})}^{2} \\ g_{d d 1} = g_{q q 1} = G (G + R_{s} + L s) \\ g_{d q 1} = - g_{q d 1} = ω_{e} (L - \hat{L}) G \end{cases}

(5)

g_dd1/Δ₁ and g_qq1/Δ₁ are transfer functions of d and q-axes current controllers respectively, g_dq1/Δ₁ and g_qd1/Δ₁ are cross-coupling transfer functions between the two systems. By analyzing the two coupling transfer functions, it can be deduced that complete decoupling can only be achieved when the $\hat{L}$ is equal to the $L$ . However, in practical situations, variations in operating conditions and measurement errors can lead to mismatches in inductance, resulting in $\hat{L} \neq L$ . Meanwhile, the decoupling compensation cannot fully counteract the actual coupling amount of the system, and the dynamic performance of the system will be adversely affected. Figure 2 shows the Bode diagram of the CFDC coupling transfer function when there is inductance mismatch. The parameters of the tested motor are given in the simulation part, and the coefficients of the PI regulator are K_p = 5 and K_i = 10.

As shown in Figure 2, when the inductor parameters are mismatched, the amplitude-frequency gain is lower and decreases continuously in the high frequency band, and the decoupling effect between the d and q-axes is better. When in the low and middle frequency bands, the amplitude-frequency gain has a larger amplitude, leading to larger current coupling between the two subsystems, which indicates that CFDC cannot effectively solve the problem of current coupling between the d and q-axes.

2.3. Current deviation decoupling control

The CDDC takes the difference between the reference current and the feedback current as the control variable, which is superimposed with the controller generated by the PI controller to realize the decoupling control of the current. The CDDC block diagram is shown in Figure 3.

From Figure 3:

{\begin{cases} G_{1} {i_{d}}^{*} + G_{4} {i_{q}}^{*} - (G_{4} - ω_{e} L) i_{q} = i_{d} (G_{1} + R_{s} + L s) \\ G_{2} {i_{q}}^{*} + G_{3} {i_{d}}^{*} - (G_{3} + ω_{e} L) i_{d} = i_{q} (G_{2} + R_{s} + L s) \end{cases}

(6)

The equation (6) can be written as follows:

[\begin{array}{l} i_{d} \\ i_{q} \end{array}] = \frac{1}{Δ_{2}} [\begin{array}{l} g_{d d 2} & g_{d q 2} \\ g_{q d 2} & g_{q q 2} \end{array}] [\begin{array}{l} {i_{d}}^{*} \\ {i_{q}}^{*} \end{array}]

(7)

Of which:

{\begin{cases} g_{d d 2} = G_{1} (G_{2} + R_{s} + L s) + ω_{e} L G_{3} - G_{3} G_{4} \\ g_{q q 2} = G_{2} (G_{1} + R_{s} + L s) - ω_{e} L G_{4} - G_{4} G_{3} \\ g_{d q 2} = G_{4} (R_{s} + L s) + ω_{e} L G_{2} \\ g_{q d 2} = G_{3} (R_{s} + L s) - ω_{e} L G_{1} \\ Δ_{2} = (G_{1} + R_{s} + L s) (G_{2} + R_{s} + L s) - (G_{3} + ω_{e} L) (G_{4} - ω_{e} L) \\ G_{1} = G_{2} = G \end{cases}

(8)

when the system is completely decoupled, g_dq2 = g_qd2 = 0, it can be concluded that

G_{3} (s) = - G_{4} (s) = \frac{ω_{e} L}{R_{s} + L s} G (s)

(9)

Using $\hat{L}$ to replace L in equation (9), and adding the decoupling compensation input signal point to the PI regulator, the effect of stator resistance changes on decoupling implementation is ignored in actual situations, the equivalent CDDC block diagram is shown in Figure 4.

From Figure 4, the parameters in equation (8) can be written as

{\begin{cases} g_{d d 2} = g_{q q 2} = G (G + R_{s} + L s) + G_{3} (G_{3} + ω_{e} L) \\ g_{d q 2} = - g_{q d 2} = ω_{e} R_{s} G \frac{L - \hat{L}}{R_{s} + \hat{L} s} \\ G_{3} = \frac{ω_{e} \hat{L}}{R_{s} + \hat{L} s} G \\ Δ_{2} = {(G + R_{s} + L s)}^{2} + {(G_{3} + ω_{e} L)}^{2} \end{cases}

(10)

As shown equation (10), the decoupling transfer function is 0 when $\hat{L} = L$ , which can achieve complete decoupling between the two systems. When $\hat{L} \neq L$ , Δ₂ is always greater than Δ₁, and |g_dq2/Δ₂|<|g_dq1/Δ₁|. The decoupling effect of bias decoupling is less sensitive to changes in inductance parameters, which has a better decoupling effect compared with the current feedback decoupling. Figure 5 shows the Bode diagram of the CDDC decoupling transfer function.

Compared with CFDC as shown in Figure 2, CDDC in Figure 5 has a significantly better decoupling effect. However, the transfer function of CDDC exhibits resonance at operating frequency of motor, which causes the amplitude frequency increasing rapidly with a deteriorating decoupling effect and underdamped oscillations. Both CFDC and CDDC cannot effectively solve the problem of severe current coupling between the d and q-axes when there is a mismatch in inductance. It is necessary to develop a better decoupling method for PMSM control to achieve complete decoupling of currents and improve the motor’s dynamic performance.

3. Design of nonlinear extended state observer

3.1. Nonlinear extended state observer

It is evident that traditional current decoupling methods exhibit good decoupling effects when the motor operates under ideal conditions. However, parameter perturbations are inevitable during the practical operation of PMSM. Unmodeled dynamics and external disturbances during the motor’s operation can significantly impact the system’s dynamic response and robustness. In such cases, traditional decoupling methods often struggle to provide effective solutions. A nonlinear extended state observer (NESO) can be introduced in PMSM control, which should improve the current decoupling effect and enhance the control performance.

The PMSM model that considers the influence of parameter perturbations, external disturbances, and unmodeled dynamics is shown as below.

{\begin{cases} \frac{d i_{d}}{d t} = (u_{d} - R_{s} i_{d} + ω_{e} L_{q} i_{q} - D_{d}) / L_{d} \\ \frac{d i_{q}}{d t} = (u_{q} - R_{s} i_{q} - ω_{e} L_{d} i_{d} - ω_{e} ψ_{f} - D_{q}) / L_{q} \\ D_{d} = Δ R_{s} i_{d} + Δ L_{d} \frac{d i_{d}}{d t} - Δ L_{q} ω_{e} i_{q} + d_{d} \\ D_{q} = Δ R_{s} i_{q} + Δ L_{q} \frac{d i_{q}}{d t} + Δ L_{d} ω_{e} i_{d} + Δ ψ_{f} ω_{e} + d_{q} \end{cases}

(11)

where ΔR_s, ΔL_d, ΔL_q, Δψ_f are the deviations of the actual and nominal values of the system, while d_d, d_q represent the sum of the unmodeled dynamics and external disturbances of the system, and D_d, D_q represents the total sum of the unknown uncertainties of the system. Take the q-axis current loop as an example to design NESO, and the d-axes current loop can be designed similarly.

\frac{d i_{q}}{d t} = b_{q} u_{q} + f_{q}

(12)

where

{\begin{cases} f_{q} = - \frac{D_{q}}{L_{q}} - \frac{R_{s}}{L_{q}} i_{q} - \frac{ω_{e} L_{d}}{L_{q}} i_{d} - \frac{ω_{e} ψ_{f}}{L_{q}} \\ b_{q} = \frac{1}{L_{q}} \end{cases}

(13)

Considering the total disturbance f_q as an expansion state variable of the system and setting the state variables x₁ = i_q, x₂ = f_q, then

{\begin{cases} y_{q} = x_{1} \\ \dot{x_{1}} = f_{q} + b_{q} u_{q} \\ {\dot{x}}_{2} = {\dot{f}}_{q} = ζ_{q} \end{cases}

(14)

To obtain observations of x₁, x₂, the NESO can be designed as

{\begin{cases} e = {\hat{x}}_{1} - x_{1} \\ {\dot{x}}_{1} = {\hat{x}}_{2} - β_{1} f a l (e, α_{1}, δ_{1}) + b_{q} u_{q} \\ {\dot{x}}_{2} = - β_{2} f a l (e, α_{2}, δ_{2}) \end{cases}

(15)

Among of them:

f a l (e, α, δ) = {\begin{cases} {| e |}^{α} s i g n (e), | e | > δ \\ \frac{e}{δ^{1 - α}}, | e | \leq δ \end{cases}

(16)

{\hat{x}}_{1}

is the estimate of x₁,

{\hat{x}}_{2}

is the estimate of x₂, β₁, β₂ are the gain coefficients of the observer, α is the nonlinear factor, and δ is the length of the interval of the linear segment.

As shown in equation (16), the nonlinear error feedback function for NESO should switch between two expressions based on different errors. The non-differentiability at the switching point causes sudden changes in gain, which easily leads to oscillation problems and affects the performance of the observer. Additionally, there are many parameters, which increase the difficulty of tuning.

3.2. The INESO design

The selection of nonlinear error gain function in NESO directly affects the observation accuracy of NESO and has an important influence on the decoupling performance. The Lfal function is designed, as shown in equation (17).

L f a l = \frac{e}{| e | + K}, K > 0

(17)

where e is the difference between the observed and actual values of the current, K is a constant greater than 0. As shown in equation (17), The Lfal function is a smooth, continuous and differentiable odd function without piecewise point, which is no need to switch between multiple expressions. The error gain of the Lfal function is stable and continuous. The single parameter K in Lfal function is needed to adjust, which simplifies adjustment process. Figure 6 shows the curves of Lfal and fal. Where α = 0.25, δ = 0.2, K = 0.1.

As shown in Figure 6, compared with fal, Lfal has a greater gain in the small error interval while having a smaller gain in the large error interval. Additionally, Lfal maintains a continuous gain characteristic, which realizes the design concept of “small gain for large errors” and “large gain for small errors.”

As shown in equation (15), when the difference between the initial values of the system state x₁ and ESO state ${\hat{x}}_{1}$ is relatively large, a differential peak phenomenon will appear. A hyperbolic tangent function with saturation characteristics is introduced to overcome this problem. As shown in equation (18), variable gain is achieved in q-axis INESO in combination with equation (17):

{\begin{cases} e_{q} = {\hat{i}}_{q} - i_{q} \\ {\dot{\hat{i}}}_{q} = {\hat{f}}_{q} - a_{1} L f a l (e_{q}, K_{1}) + b_{q} u_{q} \\ {\dot{\hat{f}}}_{q} = - a_{2} L f a l (e_{q}, K_{2}) \end{cases}

(18)

where

{\hat{i}}_{q}

and

{\hat{f}}_{q}

are the observed values of

i_{q}

and

f_{q}

, respectively, and the time-varying gains a₁ and a₂ are

{\begin{cases} a_{1} = β_{1} \tanh (λ_{1} t) \\ a_{2} = β_{2} \tanh (λ_{2} t) \end{cases}

(19)

t indicates the time, t > 0. λ₁ and λ₂ are adjustment factors greater than 0. The time-varying gain a₁ and a₂ values are small in the initial state, for the purpose of suppressing the observer's initial state differential peaks due to the inherent characteristics of the hyperbolic tangent function. a₁ and a₂ are gradually approach β₁ and β₂ after the initial state, respectively. The design of the d-axis INESO follows the same steps. Figure 7 shows the decoupling principle diagram corresponding to the expression for q-axis INESO.

3.3. INESO stability analysis

The INESO parameter settings are as follows: according to the bandwidth method (Gao, 2003), A large enough bandwidth ω₀ ensures that the dynamic response of the INESO is rapid enough to track disturbance changes, which is limited by noise and cannot be too large. The choice of ω₀ requires a compromise between system performance and resistance to sampling noise. ω₀ = 3000 rad/s is chosen, β₁ = 2ω₀, β₂ = ω₀². A larger K can speed up the error convergence speed and improve the system immunity, but also make the system chattering, integrated immunity and chattering influence. K₁ = K₂ = 0.1 is selected. The adjustment coefficient λ affects the difference between the initial state value of the system and the initial state value of INESO, and the differential peak at the initial moment is suppressed by adjusting the magnitude of λ. In this paper, λ₁ = λ₂ = 20.

Stability analysis is conducted using the q-axis INESO as an example, and the d-axis can be proven in the same way. Equivalent transformation of Lfal functions in equation (18).

{\begin{cases} L f a l (e_{q}, K_{1}) = \frac{L f a l (e_{q}, K_{1})}{e_{q}} e_{q} = ρ_{1} (e_{q}) e_{q} \\ L f a l (e_{q}, K_{2}) = \frac{L f a l (e_{q}, K_{2})}{e_{q}} e_{q} = ρ_{2} (e_{q}) e_{q} \end{cases}

(20)

ρ₁ (e_q), ρ₂ (e_q) are the error scaling factors. Because K₁ = K₂, ρ₁ (e_q) = ρ₂ (e_q) = ρ(e_q), and equation (18) can be rewritten as

{\begin{cases} e_{q} = {\hat{i}}_{q} - i_{q} \\ {\dot{\hat{i}}}_{q} = {\hat{f}}_{q} - a_{1} ρ (e_{q}) e_{q} + b_{q} u_{q} \\ {\dot{\hat{f}}}_{q} = - a_{2} ρ (e_{q}) e_{q} \end{cases}

(21)

Equation (21) can be regarded as a variable gain linear system, which can be written as follows.

{\begin{cases} {\dot{\hat{i}}}_{q} = - a_{1} ρ (e_{q}) {\hat{i}}_{q} + {\hat{f}}_{q} + a_{1} ρ (e_{q}) i_{q} + b_{q} u_{q} \\ {\dot{\hat{f}}}_{q} = - a_{2} ρ (e_{q}) {\hat{i}}_{q} + a_{2} ρ (e_{q}) i_{q} \end{cases}

(22)

Both sides of equation (22) are integrated at the same time, and then Laplace transform of equation (22)is performed to obtain:

{\begin{cases} {\hat{i}}_{q} = \frac{- a_{1} ρ (e_{q})}{s} {\hat{i}}_{q} + \frac{{\hat{f}}_{q}}{s} + \frac{a_{1} ρ (e_{q})}{s} i_{q} + \frac{b_{q}}{s} u_{q} \\ {\hat{f}}_{q} = \frac{- a_{2} ρ (e_{q})}{s} {\hat{i}}_{q} + \frac{a_{2} ρ (e_{q})}{s} i_{q} \end{cases}

(23)

After sorting out equation (23), we can get:

{\begin{cases} {\hat{i}}_{q} = \frac{a_{1} ρ (e_{q}) s + a_{2} ρ (e_{q})}{s^{2} + a_{1} ρ (e_{q}) s + a_{2} ρ (e_{q})} i_{q} \\ + \frac{b_{q} s}{s^{2} + a_{1} ρ (e_{q}) s + a_{2} ρ (e_{q})} u_{q} \\ {\hat{f}}_{q} = \frac{a_{2} ρ (e_{q}) s}{s^{2} + a_{1} ρ (e_{q}) s + a_{2} ρ (e_{q})} i_{q} \\ - \frac{b_{q} a_{2} ρ (e_{q})}{s^{2} + a_{1} ρ (e_{q}) s + a_{2} ρ (e_{q})} u_{q} \end{cases}

(24)

Choose K₁ = K₂ = 0.1, then

ρ (e_{q}) = \frac{1}{| e_{q} | + 0.1}

(25)

It is easy to know that ρ (e_q) is bounded within the global range of errors, then a₁ρ (e_q), a₂ρ (e_q) are varying within a certain range, and its stability can be proved using the root trajectory method according to Li et al. (2016).

From equation (24), the characteristic equation is obtained as

s^{2} + a_{1} ρ (e_{q}) s + a_{2} ρ (e_{q}) = 0

(26)

The Routh table for equation (26) is

\begin{array}{l} s^{2} 1 a_{2} ρ (e_{q}) \\ s^{1} a_{1} ρ (e_{q}) \\ s^{0} a_{2} ρ (e_{q}) \end{array}

(27)

Then, according to the Routh stability criterion, the system stability condition is

{\begin{cases} a_{1} ρ (e_{q}) > 0 \\ a_{2} ρ (e_{q}) > 0 \end{cases}

(28)

From equations (25) and (19), ρ(e_q) > 0, a₁, a₂ are positive, so the INESO designed in this paper is stable.

3.4. Analysis of INESO observation errors

Firstly, the q-axis observation error equation is obtained by equation (18) minus equation (14):

{\begin{cases} e_{q 1} = {\hat{i}}_{q} - i_{q}, e_{q 2} = {\hat{f}}_{q} - f_{q} \\ {\dot{e}}_{q 1} = e_{q 2} - a_{1} L f a l (e_{q 1}, 0.1) \\ {\dot{e}}_{q 2} = - a_{2} L f a l (e_{q 1}, 0.1) - ζ_{q} \end{cases}

(29)

The observation error in equation (29) converges to zero when the system reaches the steady state, and $L f a l (e_{q 1}, 0.1) = \frac{e_{q 1}}{| e_{q 1} | + 0.1}$ , available in

| e_{q 1} | = | \frac{0.1 ζ_{q}}{a_{2} - ζ_{q}} | | e_{q 2} | = | a_{1} \frac{ζ_{q}}{a_{2}} |

(30)

Obviously, as long as a₂ is much larger than $ζ_{q}$ , the INESO observation error will be small enough to satisfy the observation requirement.

3.5. System stability analysis

The expression of the q-axis closed-loop control system can be obtained from Figure 7:

({i_{q}}^{*} - {\hat{i}}_{q}) K_{p} - \frac{{\hat{f}}_{q}}{b_{q}} = - \frac{f_{q}}{b_{q}} + \frac{i_{q}}{b_{q}} s

(31)

Substituting equation (24) into equation (31) to obtain the transfer function of the q-axis closed-loop system:

\begin{array}{l} i_{q} = K_{p} b_{q} (s^{2} + a_{1} ρ s + a_{2} ρ) {i_{q}}^{*} / χ \\ - ({b_{q}}^{2} K_{p} s - b_{q} a_{2} ρ) u_{q} / χ \\ + f_{q} (s^{2} + a_{1} ρ s + a_{2} ρ) / χ \end{array}

(32)

where ρ = ρ(e_q), χ = s³ + a₁ρs² + (2a₂ρ + b_qa₁ρK_p) s + b_qa₂ρK_p.

From equation (32), the characteristic equation of the system is

s^{3} + a_{1} ρ s^{2} + (2 a_{2} ρ + b_{q} a_{1} ρ K_{p}) s + b_{q} a_{2} ρ K_{p} = 0

(33)

List its Routh table:

\begin{array}{l} s^{3} 1 2 a_{2} ρ + b_{q} a_{1} ρ K_{p} \\ s^{2} a_{1} ρ b_{q} a_{2} ρ K_{p} \\ s^{1} 2 a_{2} ρ + b_{q} a_{1} ρ K_{p} - (b_{q} a_{2} K_{p}) / a_{1} \\ s^{0} b_{q} a_{2} ρ K_{p} \end{array}

(34)

The Routh stability conditions are:

{\begin{cases} a_{1} ρ > 0 \\ 2 a_{2} ρ + b_{q} a_{1} ρ K_{p} - (b_{q} a_{2} K_{p}) / a_{1} > 0 \\ b_{q} a_{2} ρ K_{p} > 0 \end{cases}

(35)

In this paper, a₁, a₂, K_p, and b_q are positive, then it is sufficient to prove that

2 a_{2} ρ + b_{q} a_{1} ρ K_{p} - (b_{q} a_{2} K_{p}) / a_{1} > 0

(36)

As shown in equation (30), the system will be in steady state when |e_q| < .5, from equation (25), ρ > 1. β₁ = 2ω₀, β₂ = ω₀², ω₀ = 3000 rad/s, so β₂<β₁². Because λ₁ = λ₂ = 20, from equation (19), we can see that a₂<a₁². Therefore, it can be concluded that

\begin{array}{l} 2 a_{2} ρ + b_{q} a_{1} ρ K_{p} - (b_{q} a_{2} K_{p}) / a_{1} > \\ 2 a_{2} ρ + b_{q} a_{1} K_{p} - (b_{q} a_{2} K_{p}) / a_{1} = 2 a_{2} ρ + b_{q} K_{p} (a_{1} - \frac{a_{2}}{a_{1}}) > 0 \end{array}

(37)

The q-axis closed-loop system is stable from equations (35) and (37), and the d-axis closed-loop system is also stable.

4. Simulation verification

A PMSM model is built in MATLAB/Simulink to verify the performance of the INESO current decoupling method, and compare it with the traditional decoupling method. The DC bus voltage of the inverter is 24 V, and the switching frequency is 10 kHz. The tested PMSM motor parameters are listed in TABLE 1.

Table 1.

Parameters of permanent magnet synchronous motor.

Parameters	Value
Stator resistance R_s (Ω)	0.44
Inductance L_d = L_q (H)	4.5 × 10⁻⁴
Flux linkage of permanents Ψ_f (Wb)	9.8 × 10⁻³
Rotational inertia J (kg·m²)	2.8 × 10⁻⁵
Viscous friction coefficient B(N·m·s)	5 × 10⁻⁵
Number of pole pairs p	5

4.1. Current response to sudden load torque change

To better compare the performance of the three current decoupling schemes, the speed control loop adopts traditional PI controllers with proportional integral gains of 0.02 and 0.5, respectively. The inductance of PMSM is set to 1.2 times and 0.8 times the nominal value to simulate the situation of inductance mismatch. Figure 8 shows the current response waveforms for CFDC, CDDC, and INESO separately, when the PMSM is subjected to sudden increase and decrease of 0.3 N·m load. Table 2 is the numerical results of motor current fluctuation under three decoupling strategies.

Table 2.

The simulation results of three current decoupling methods when the load changes suddenly.

Decoupling method	The d-axis current fluctuation when $\hat{L} = 1.2 L$ (A)	The d-axis current fluctuation when $\hat{L} = 0.8 L$ (A)
CFDC	0.114	0.135
CDDC	0.064	0.070
INESO	0.033	0.040

As shown in Figure 8(a) and (b), the CFDC and CDDC decoupling schemes are greatly affected by variation of inductance. When the load torque of PMSM change, the step response of the q-axis current shows an obvious overshoot, and the d-axis current changes greatly. However, for the proposed INESO decoupling strategy, the step response the of q-axis current does not show significant overshoot, and the d-axis current is not significantly affected by the q-axis current overshoot when the load changes, which indicates that INESO decoupling has strong robustness and good decoupling characteristics.

From Tables 2, it can be seen that the decoupling effect of INESO is significantly better than that of CFDC and CDDC under the two working conditions.

4.2. Current response during sudden changes in the d-axis current

Figure 9 shows the q-axis current waveform under CFDC, CDDC and INESO when there is a sudden change in the d-axis current of PMSM. As shown in Figure 9(a), there is a significant instantaneous error in the CDDC decoupling method in the q-axis current during the mismatch of inductance in PMSM. The coupling between d and q-axes is reduced with significant current fluctuations. The q-axis current remains stable when there is a sudden change in the d-axis current of PMSM with INESO control, which indicates a better decoupling ability.

As shown in Table 3, When the d-axis current changes abruptly, the q-axis current fluctuations under INESO decoupling are 0.011 A and 0.010 A, which are significantly smaller than CFDC decoupling and CDDC decoupling.

Table 3.

The simulation results of three current decoupling methods when the d-axis current suddenly changes.

Decoupling method	The q-axis current fluctuation when $\hat{L} = 1.2 L$ (A)	The q-axis current fluctuation when $\hat{L} = 0.8 L$ (A)
CFDC	0.086	0.090
CDDC	0.062	0.053
INESO	0.011	0.010

4.3. Current response during sudden changes in q-axis current in current loop mode

Keep the PMSM operating at a constant speed of 1000 r/min, with a given value d-axis current i_d = 0. The response waveforms of the d-axis current under CFDC, CDDC, and INESO are verified in the case of sudden changes in the q-axis current. Figure 10 shows the waveforms of the d-axis current variation corresponding to the sudden change of the q-axis current in the current loop mode under CFDC, CDDC and INESO.

In Figure 10(a), the sudden change in the q-axis current caused a large transient change in d-axis current of PMSM under CFDC control. Subsequently, due to the presence of coupling, the current amplitude remained larger than before. In Figure 10(b), although the fluctuation in d-axis current reduced, oscillations occurred after the sudden change in q-axis current of PMSM under CDDC control. In Figure 10(c), different changes in q-axis current step under different inductance parameters did not have a significant effect on d-axis current of PMSM under INESO control, which indicates that the cross-coupling problem between the two axes has been properly solved.

In Table 4, the d-axis current fluctuation under INESO decoupling is 0.014 A/0.013 A, which is 91% and 68% lower than that of CFDC and CDDC, respectively.

Table 4.

The simulation results of three current decoupling methods when the q-axis current changes abruptly.

Decoupling method	The d-axis current fluctuation when $\hat{L} = 1.2 L$ (A)	The d-axis current fluctuation when $\hat{L} = 0.8 L$ (A)
CFDC	0.151	0.149
CDDC	0.039	0.041
INESO	0.014	0.013

Through analysis of the simulation results under the three different conditions mentioned above, it can be easily observed that the proposed INESO decoupling has strong decoupling ability, and has good decoupling effect under different parameter conditions and operating conditions. The d and q-axes current are stable, and the dynamic performance of the PMSM control system is improved.

5. Experimental verification

As shown in Figure 11, an experimental platform based on dSPACE1202 was set up to verify the decoupling effect of INESO. The parameters of the PMSM and the switching frequency of the inverter used in the experiment were the same as those in the simulation, and the voltage of the DC bus was also the same. The sampling time for both the current control loop and the speed control loop was 0.0001 s.

5.1. Experiment on sudden change of load torque

As shown in Figure 12(a), the q-axis current of PMSM has obvious chattering with CFDC control. When the load torque changes suddenly, the q-axis current chattering is more severe, and the d-axis current also changes significantly. As shown in Figure 12(b), the chattering and coupling of d and q-axes current of PMSM are improved with CDDC control. However, the d and q-axes currents still have obvious coupling. As shown in Figure 10(c), the q-axis current of PMSM is stable without obvious chattering with INESO control, and the d-axis current is almost unaffected when the q-axis current suddenly changes.

From Table 5, it can be clearly seen that when the motor load changes suddenly, the INESO decoupling current fluctuation under the two working conditions is almost 0, indicating that the INESO decoupling effect is excellent.

Table 5.

The experimental results of three current decoupling methods when the load changes suddenly.

Decoupling method	The d-axis current fluctuation when $\hat{L} = 1.2 L$ (A)	The d-axis current fluctuation when $\hat{L} = 0.8 L$ (A)
CFDC	0.025	0.023
CDDC	0.010	0.009
INESO	0.001	0.001

5.2. Current response during d-axis current abrupt changes

Figure 13(a) shows the experimental results of d and q-axes current of PMSM with CFDC decoupling when there is a sudden change in the d-axis current. Both the sudden increase and decrease occur in the d-axis current, with a momentary error of around 0.1 A due to variations of inductance of PMSM. There is still a significant coupling between the d- and q-axes. As shown in Figure 13(b), the cross-coupling between the d- and q-axes current of PMSM is reduced with CDDC control. However, the change in the d-axis current results in oscillation of q-axis current.

As shown in Figure 13(c), the step change of the d-axis current with different inductance parameters, the q-axis current of PMSM does not change significantly with the INESO decoupling strategy, The complete decoupling between d and q-axes can be realized in PMSM control with INESO, which can suppress the torque fluctuation.

As can be seen from Table 6, the decoupling effect of INESO is significantly better than that of CFDC and CDDC under both operating conditions. The q-axis current fluctuation under INESO decoupling is 0.010 A, which is 88% and 78% lower than that of CFDC and CDDC, respectively.

Table 6.

The experimental results of three current decoupling methods when the d-axis current changes abruptly.

Decoupling method	The q-axis current fluctuation when $\hat{L} = 1.2 L$ (A)	The q-axis current fluctuation when $\hat{L} = 0.8 L$ (A)
CFDC	0.085	0.083
CDDC	0.047	0.045
INESO	0.010	0.010

6. Conclusions

An INESO decoupling control method for PMSM is proposed in this paper, which has some unique features compared with CFDC and CDDC. The conclusions are summarized as follows

(1) CFDC effect is sensitive to inductance changes, and the dynamic decoupling performance is poor. When the inductance parameters are mismatched, complete decoupling of the d and q-axes cannot be achieved.

(2) CDDC, similar to CFDC, does not get rid of the limitation of static decoupling. Compared with CFDC, the decoupling performance is improved, but the phenomenon of underdamped shocks appears.

(3) INESO can realize the dynamic and static decoupling of the d and q-axes of PMSM, overcome the defects of CFDC and CDDC, solve the problem of coupling damage in PMSM vector control, and improve the dynamic and steady-state performance of PMSM.

In addition, the decoupling effect of INESO is affected by multiple parameters, and the parameter setting principle that takes into account the decoupling effect and anti-interference needs to be further explored to simplify the difficulty of parameter tuning and improve the performance of the current loop.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This research was funded by “The overseas study visit and training program for out-standing young backbone talents of Anhui Province, grant number gxgwfx2021035,” “The innovation team of Anhui Polytechnic University,” “Graduate Student Innovation Project of Anhui Province, grant number 2022xscx097,” “Anhui Polytechnic University-Jiujiang District Industrial Collaborative Innovation Special Fund Project, grant number 2022cyxtb4” and Anhui Future Technology Research Institute Enterprise Cooperation Project, grant number 2023qyhz16.

ORCID iDs

Shuai Li

Hai Wang

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