Weight-Adaptable Disturbance Observer for Continuous-Control-Set Model Predictive Control of NPC-3L-Fed PMSMs

Abstract

This paper presents a cascaded control strategy for neutral-point-clamped three-level (NPC-3L) inverter-fed permanent magnet synchronous motors (PMSMs), integrating continuous-control-set model-predictive control (CCS-MPC) with mid-point voltage regulation and an online Lyapunov-stable neural-network (NN) disturbance observer. The outer CCS-MPC loop optimizes voltage vector application for accurate current tracking and harmonic suppression, while the inner loop balances mid-point voltage by adjusting the dwell times of P/N small-voltage vectors (VVs). The NN-based disturbance observer compensates parameter mismatches in real time, reducing steady-state dq-axis current errors. To validate the effectiveness of the proposed strategy, experiments are conducted using a three-phase PMSM fed by three-phase NPC-3L inverters. Experimental results demonstrate substantial improvements in mid-point voltage balance, current quality, and robustness against model uncertainties.

1. Introduction

Permanent magnet synchronous motors (PMSMs) have been increasingly adopted in high-power applications due to their high efficiency, high power density, low torque ripples, and small volume and weight [1,2,3,4]. In parallel, multilevel inverters, particularly the neutral-point-clamped three-level (NPC-3L) topology, have emerged as a cornerstone technology for medium-voltage, high-power motor drives, as they reduce both current and voltage stress on power devices, mitigate torque ripple, and enhance fault-tolerant capability [5]. Nevertheless, the NPC-3L inherently suffers from mid-point voltage imbalance in its dual-capacitor DClink, which significantly constrains the inverter’s ability to generate a symmetric AC voltage and thus degrades overall drive performance and reliability. To address the strong nonlinearities of PMSM systems, model-predictive control (MPC) has garnered substantial academic and industrial interest because of its intuitive control structure, design flexibility, and fast transient response [6,7,8,9]. Within the MPC framework, finite-control-set MPC (FCS-MPC) and continuous-control-set MPC (CCS-MPC) each offer distinct benefits. FCS-MPC is simpler to implement but generates considerable current harmonics [10], whereas CCS-MPC can synthesize a complete voltage vector, resulting in superior harmonic suppression and thereby enhancing steady-state performance [11]. Despite these advantages, both FCS-MPC and CCS-MPC depend heavily on accurate motor model parameters. In practice, PMSM parameters such as stator inductance and flux linkage vary with temperature and operating conditions, and cannot be measured directly in real time [12,13]. Consequently, parameter mismatches degrade MPC drive performance, causing higher steady-state current error or torque ripple [14]. Therefore, mitigating or compensating for MPC’s sensitivity to model inaccuracies remains the primary technical challenge for practical CCS-MPC implementation [15,16].

Prior research has addressed key challenges in NPC-3L-fed PMSM drives, yet significant limitations remain [17]. Refs. [18,19] implemented a model-predictive torque control (MPTC) scheme incorporating constraints for torque tracking and mid-point voltage balancing within a cost function; however, the adjustment of weighting coefficients in this approach proves cumbersome. While carrier-based PWM methods effectively inject zero-sequence voltage to counteract mid-point current fluctuations and achieve voltage balance [20,21], they nonetheless risk introducing unbalanced line voltages that can degrade overall system performance. Regarding parametric sensitivity, disturbance observers, such as Luenberger [22] and sliding-mode observers [23], compensate for motor parameter mismatches by estimating disturbances, and an Extended Kalman Filter observer [24] has been shown to reduce steady-state tracking errors. Although these observers enhance robustness to parameter variations through flexible estimation, they critically rely entirely on precise knowledge of the system dynamics and require manual gain tuning.

To overcome these limitations, this paper proposes a cascaded CCS-MPC strategy augmented with an online Lyapunov-stable neural-network disturbance observer and mid-point voltage regulation for NPC-3L-fed PMSMs. More precisely, the contribution consists of the following key elements:

(1)

Cascaded CCS-MPC with Mid-Point Voltage Control—The outer CCS-MPC loop formulates a quadratic program based solely on voltage–vector dwell times to minimize torque ripple and achieve precise current tracking. The inner loop dynamically balances the NPC-3L DC-link mid-point by adjusting P- and N-type small-vector dwell times.

(2)

Neural-Network Disturbance Observer—An online weight-tuning law derived via Lyapunov stability theory enables the NN-based disturbance observer to estimate and compensate parametric mismatches in real time. Experimental results demonstrate that this adaptive approach achieves substantially improved current tracking accuracy while requiring no offline gain tuning.

2. Mathematical Models

2.1. Model of Three-Phase PMSMs

Figure 1 illustrates the configuration of the NPC-3L inverter-fed three-phase PMSM drive system. By decomposing the 3-D space into the αβ subspace, which is subsequently transformed into the synchronous dq frame, the dynamics of the three-phase PMSM are described as

$${\displaystyle \frac{d}{dt}}i=A_{0}i+B_{0}u+D_0+E_0,$$

(1)

where i, u, and D₀ are the vector of stator current, voltage, and back electromotive force, respectively. It is important to highlight that E₀ from parameter mismatches and other disturbances can be estimated with an NN-based disturbance observer, proposed in Section 4. The vectors above are described as

$$i={[\begin{array}{cc}{i}_d& i_q\end{array}]}^{\mathrm{T}}, u={[\begin{array}{cc}{u}_d& u_q\end{array}]}^{\mathrm{T}}, D_0={[\begin{array}{cc}0& -{\displaystyle \frac{{\omega }_e{\psi }_f}{L_q}}\end{array}]}^{\mathrm{T}}, E_0={[\begin{array}{cc}{f}_d& f_q\end{array}]}^{\mathrm{T}},$$

(2)

where i_d and i_q are the dq-axis stator currents, u_d and u_q are the dq-axis stator voltages, ω_e is the electrical angular velocity, and ψ_f is the stator flux. The state matrix F and the input matrix G in Equation (1) are given by

$$A_0=\left[\begin{array}{cc}-{\displaystyle \frac{R_s}{L_d}}& {\omega }_e\\ -{\omega }_e& -{\displaystyle \frac{R_s}{L_q}}\end{array}\right], B_0=\left[\begin{array}{cc}{\displaystyle \frac{1}{L_d}}& 0\\ 0& {\displaystyle \frac{1}{L_q}}\end{array}\right],$$

(3)

where R_s is the stator resistance, L_d is the d-axis inductance, and L_q is the q-axis inductance. Based on Equations (1) and (3), the discrete-time state-space model of the PMSM is deduced as

$$\left[\begin{array}{c}{i}_d(k+1)\\ i_q(k+1)\end{array}\right]=A\left[\begin{array}{c}{i}_d(k)\\ i_q(k)\end{array}\right]+B\left[\begin{array}{c}{u}_d(k)\\ u_q(k)\end{array}\right]+D+E,$$

(4)

where the matrices are calculated via forward Euler of the form $A=I - A_0{T}_s$, $B=B_0{T}_s$, $D=D_0{T}_s$, and $E={\displaystyle {\int }_t^{t+T_s}{E}_0}\mathrm{d}t$, with T_s as the sampling interval. The state matrix A, input matrix B, and the back electromotive force matrix D are given by

$$A=\left[\begin{array}{cc}1-{\displaystyle \frac{R_s}{L_d}}{T}_s& {\displaystyle \frac{{\omega }_e{L}_q}{L_d}}{T}_s\\ -{\displaystyle \frac{{\omega }_e{L}_d}{L_q}}{T}_s& 1-{\displaystyle \frac{R_s}{L_q}}{T}_s\end{array}\right], B=\left[\begin{array}{cc}{\displaystyle \frac{T_s}{L_d}}& 0\\ 0& {\displaystyle \frac{T_s}{L_q}}\end{array}\right], D=\left[\begin{array}{c}0\\ -{\displaystyle \frac{{\omega }_e{\psi }_f}{L_q}}{T}_s\end{array}\right]$$

(5)

2.2. NPC-3L Inverters and Output Voltage Vectors

The NPC-3L inverter circuit studied in this paper is shown in Figure 1. The DC side of the inverter is divided by capacitors C₁ and C₂ and clamped by diodes. Compared with the neutral point, each of phase A, B, and C can output three voltages V_dc/2, 0, − V_dc/2, namely P-, O-, and N-level operation.

As shown in Figure 1, each phase of this inverter has three switching states, and can generate 3³ = 27 VVs, as shown in Figure 2. The switching function is accordingly defined as

$$S_x=\left\{\begin{array}{cc}1,v_{\mathrm{out}}=V_{\mathrm{dc}}/2\hfill & \left(S_{x_1},S_{x_2}\mathrm{ON}{S}_{x_3},S_{x_4}\mathrm{OFF}\right)\hfill \\ 0,v_{\mathrm{out}}=0\hfill & \left(S_{x_2},S_{x_3}\mathrm{ON}{S}_{x_1},S_{x_4}\mathrm{OFF}\right)\hfill \\ -1,v_{\mathrm{out}}=-V_{\mathrm{dc}}/2\hfill & \left(S_{x_3},S_{x_4}\mathrm{ON}{S}_{x_1},S_{x_2}\mathrm{OFF}\right)\hfill \end{array}x\in \{a,b,c\},\right.$$

(6)

where S_x is the switch function of phase x, S₁, S₂, S₃, and S₄ are the four switches in one leg of the NPC-3L inverter, respectively, as shown in Figure 2.

The output voltage vector v_α−β in fundamental subspace can be presented as

$$v_{\alpha -\beta }={\displaystyle \frac{1}{6}}{V}_{\mathrm{dc}}\left(S_a+S_b{e}^{j{\displaystyle \frac{2\pi }{3}}}+S_c{e}^{j{\displaystyle \frac{4\pi }{3}}}\right)$$

(7)

All the 27 vectors can be categorized into four layers based on their magnitudes: zero vectors, small vectors, medium vectors, and large vectors, as shown in

Table 1. Four basic voltage vector magnitudes.

Voltage Vector Types	Magnitudes
Zero VVs	0
Small VVs	Vdc/3
Middle VVs	$\sqrt{3}$Vdc/3
Large VVs	2Vdc/3

.

The α-β space is divided by these VVs into six large sectors, with each large sector further subdivided into six smaller sectors (marked by red numbers), as shown in Figure 2.

It is important to highlight that the mid-point voltage balancing issues should be considered during the modulation for this NPC-3-L inverter. The zero vectors and large vectors do not affect the mid-point voltage. The impact of the medium vectors on the mid-point voltage is uncertain, and the small vectors can be further divided into P-type and N-type small vectors, which exert opposite effects on the mid-point voltage deviation. Therefore, in the subsequent switching sequence allocation, the effect of the mid-point voltage deviation can be controlled by adjusting the dwell time of P-type and N-type small vectors.

The definition of the mid-point voltage V_np in the DC-link is shown in Equation (8):

$$V_{\mathrm{np}}={\displaystyle \frac{U_{C2}-U_{C1}}{2}},$$

(8)

where U_C1 and U_C2 are the voltages of capacitors C₁ and C₂, separately. The equation of V_np is presented in Equation (9):

$${\displaystyle \frac{d{V}_{\mathrm{np}}}{dt}}=-{\displaystyle \frac{1}{2C}}{i}_o=-{\displaystyle \frac{1}{2C}}{\displaystyle \sum _x\left(1-\left|S_x\right|\right)}{i}_x,x\in \left\{a,b,c\right\},$$

(9)

where C is the capacitance of C₁ and C₂, i_x is the stator current in phase x, and i_o is the current flowing into or out of the neutral point. The positive direction of stator current is specified to be from the inverter side toward the machine side.

3. Cascaded Structure with CCS-MPC and Mid-Point Voltage Control

As shown in Figure 3, the proposed control method is a cascaded structure, with the outer loop being CCS-MPC and the inner loop for mid-point voltage control. The reference of i_q is given by the speed controller which regulates the angular speed ω to its reference ω^*.

3.1. Outer CCS-MPC

The outer loop CCS-MPC consists of two main steps: sector selection, and the formulation and solution of the optimization problem to determine the (constrained) application time of the VVs.

• Sector Selection

This process employs deadbeat (DB) control to achieve zero current tracking error at the k + 1 step, as

$$v_{\mathrm{ref},\mathrm{dq}}(k)=B^{-1}\left[i_{\mathrm{dq}}^{*}(k+1)-A{i}_{\mathrm{dq}}(k)-D\right]$$

(10)

By determining the rotational angle of v_ref from Equation (10), the corresponding large sector can be identified and subsequently, the small sector can be determined according to the specified boundary lines. Then, the three basic VVs corresponding to the vertices of the triangular sector can be determined to synthesize v_ref.

• Objective Function

In the proposed CCS-MPC method, the control variable is the duration of the switching states, an additional transformation matrix T_svm which represents the magnitude matrix of VVs, given by

Table 2. Magnitude matrix T_svm of VVs.

Sector	1	2	3	4	5	6
I	$\left[\begin{array}{ccc}0.333& 0& 0.167\\ 0& 0& 0.289\end{array}\right]$	$\left[\begin{array}{ccc}0.167& 0& 0.333\\ 0.289& 0& 0\end{array}\right]$	$\left[\begin{array}{ccc}0.333& 0.167& 0.5\\ 0& 0.289& 0.289\end{array}\right]$	$\left[\begin{array}{ccc}0.167& 0.333& 0.5\\ 0.289& 0& 0.289\end{array}\right]$	$\left[\begin{array}{ccc}0.333& 0.667& 0.5\\ 0& 0& 0.289\end{array}\right]$	$\left[\begin{array}{ccc}0.167& 0.5& 0.333\\ 0.289& 0.577& 0.289\end{array}\right]$
Ⅱ	$\left[\begin{array}{ccc}0.167& 0& -0.167\\ 0.289& 0& 0.289\end{array}\right]$	$\left[\begin{array}{ccc}-0.167& 0& 0.167\\ 0.289& 0& 0.289\end{array}\right]$	$\left[\begin{array}{ccc}0.167& -0.167& 0\\ 0.289& 0.289& 0.577\end{array}\right]$	$\left[\begin{array}{ccc}-0.167& 0.167& 0\\ 0.289& 0.289& 0.577\end{array}\right]$	$\left[\begin{array}{ccc}0.167& 0.333& 0\\ 0.289& 0.577& 0.577\end{array}\right]$	$\left[\begin{array}{ccc}-0.167& 0& -0.333\\ 0.289& 0.577& 0.577\end{array}\right]$
Ⅲ	$\left[\begin{array}{ccc}-0.167& 0& -0.333\\ 0.289& 0& 0\end{array}\right]$	$\left[\begin{array}{ccc}-0.333& 0& -0.167\\ 0& 0& 0.289\end{array}\right]$	$\left[\begin{array}{ccc}-0.167& -0.333& -0.5\\ 0.289& 0& 0.289\end{array}\right]$	$\left[\begin{array}{ccc}0.333& 0.167& 0.5\\ 0& -0.289& -0.289\end{array}\right]$	$\left[\begin{array}{ccc}-0.167& -0.333& -0.5\\ 0.289& 0.577& 0.289\end{array}\right]$	$\left[\begin{array}{ccc}-0.333& -0.5& -0.667\\ 0& 0.289& 0\end{array}\right]$
Ⅳ	$\left[\begin{array}{ccc}-0.333& 0& -0.167\\ 0& 0& -0.289\end{array}\right]$	$\left[\begin{array}{ccc}-0.167& 0& -0.333\\ -0.289& 0& 0\end{array}\right]$	$\left[\begin{array}{ccc}-0.333& -0.167& -0.5\\ 0& -0.289& -0.289\end{array}\right]$	$\left[\begin{array}{ccc}-0.167& -0.333& -0.5\\ -0.289& 0& -0.289\end{array}\right]$	$\left[\begin{array}{ccc}-0.333& -0.667& -0.5\\ 0& 0& -0.289\end{array}\right]$	$\left[\begin{array}{ccc}-0.167& -0.333& -0.5\\ -0.289& -0.577& -0.289\end{array}\right]$
Ⅴ	$\left[\begin{array}{ccc}-0.167& 0& 0.167\\ -0.289& 0& -0.289\end{array}\right]$	$\left[\begin{array}{ccc}0.167& 0& -0.167\\ -0.289& 0& -0.289\end{array}\right]$	$\left[\begin{array}{ccc}-0.167& 0.167& 0\\ -0.289& -0.289& -0.577\end{array}\right]$	$\left[\begin{array}{ccc}0.167& -0.167& 0\\ -0.289& -0.289& -0.577\end{array}\right]$	$\left[\begin{array}{ccc}-0.167& -0.333& 0\\ -0.289& -0.577& -0.577\end{array}\right]$	$\left[\begin{array}{ccc}0.167& 0.333& 0\\ -0.289& -0.577& -0.289\end{array}\right]$
Ⅵ	$\left[\begin{array}{ccc}0.167& 0& 0.333\\ -0.289& 0& 0\end{array}\right]$	$\left[\begin{array}{ccc}0.333& 0& 0.167\\ 0& 0& -0.289\end{array}\right]$	$\left[\begin{array}{ccc}0.167& 0.333& 0.5\\ -0.289& 0& -0.289\end{array}\right]$	$\left[\begin{array}{ccc}0.333& 0.167& 0.5\\ 0& -0.289& -0.289\end{array}\right]$	$\left[\begin{array}{ccc}0.167& 0.333& 0.5\\ -0.289& -0.577& -0.289\end{array}\right]$	$\left[\begin{array}{ccc}0.333& 0.5& 0.667\\ 0& -0.289& 0\end{array}\right]$

. To calculate the application time t(k) of the VVs within the selected sector, the prediction model is modified accordingly, as

$$i(k+1)=Ai(k)+B{T}_{\mathrm{park}}{T}_{\mathrm{svm}}t(k)+D,$$

(11)

where T_park is the matrix of Park transformation.

Given the new prediction model, Equation (11), the objective function that accounts for the stator current tracking error is defined as

$$J={‖i(k+1)-i^{*}(k+1)‖}_2^2$$

(12)

Considering the formulation of i(k + 1), function Equation (12) can be described as

$$J={‖Ai(k)+D-i^{*}(k+1)+B{T}_{\mathrm{park}}{T}_{\mathrm{svm}}t(k)‖}_2^2$$

(13)

This can further be transformed into solving an optimization problem, as

$$\begin{array}{l}{\begin{array}{cc}\underset{t\in {\mathbb{R}}^3}{\mathrm{minimize}}& ‖r+Mt(k)‖\end{array}}_2^2\\ \begin{array}{cc}\mathrm{s}.\mathrm{t}. t_{\mathrm{i}}\ge 0& (\mathrm{i}=0,1,2)\end{array}\\ {\displaystyle \sum _{\mathrm{i}=0}^3{t}_{\mathrm{i}}}=T_s\end{array},$$

(14)

where r is Ai(k) + D − i*(k + 1), and M is BT_parkT_svm.

The solution to this optimization problem is the activation times t₁, t_2, and t₃ of the three VVs.

3.2. Inner Mid-Point Voltage Control

The inner loop aims to regulate the mid-point voltage deviation. As mentioned in Section 2.2, the P-type and N-type small vectors exert opposite effects on the mid-point voltage deviation. Therefore, the mid-point voltage deviation can be controlled by adjusting the dwell time of P-type and N-type small vectors in the subsequent switching sequence allocation. The inner loop achieves this regulation by solving λ, which adjusts the dwell time of the small vector.

Based on Equations (8) and (9), the discrete-time state-space equation for the mid-point voltage V_np(k + 1) of the NPC-3L is shown in Equation (15)

$$\begin{array}{ll}{V}_{\mathrm{np}}(k+1)& ={\displaystyle \frac{U_{C_2}(k+1)-U_{C_1}(k+1)}{2}}\\ & ={\displaystyle \frac{U_{C_2}(k)-U_{C_1}(k)}{2}}-{\displaystyle \frac{T_s}{2C}}{\displaystyle \sum _x\left(1-\left|S_x\right|\right)}{i}_x,x\in \left\{a,b,c\right\}\\ & =U_{\mathrm{np}}-{\displaystyle \frac{T_s}{2C}}{i}_o\end{array},$$

(15)

where U_np is the measured mid-point voltage.

According to the above equation, by setting the mid-point voltage V_np(k + 1) at the next time step to zero, it can be obtained that i_o = 2U_np×C/T_s. For each sector, there always exists a λ such that the current i_o flowing into the neutral point satisfies the above equation, thereby ensuring that the mid-point voltage at the next time step is zero, i.e.:

$$i_{\mathrm{o}}=\lambda t_1{i}_1^{+}+(1-\lambda )t_1{i}_1^{-}+t_2{i}_2+t_3{i}_3={\displaystyle \frac{2U_{\mathrm{np}}C}{T_s}},$$

(16)

where i₁⁺, i₁⁻, i₂ and i₃ are the currents generated by the corresponding VVs utilized to synthesize v_ref. Here, i₁⁺ and i₁⁻ represent the currents flowing into or out of the neutral point under the influence of the P-type and N-type small vectors, respectively, and it holds that i₁⁺ = i₁⁻. Meanwhile, i₂ and i₃ represent the influence of the other two vectors.

Thus, λ can be determined as

$$\lambda ={\displaystyle \frac{2U_{\mathrm{np}}\mathrm{C}-T_s(-t_1{i}_1^{+}+t_2{i}_2+t_3{i}_3)}{2T_s{t}_1{i}_1^{+}}}$$

(17)

where λ = 0.5 indicates equal dwell times for P- and N-type small vectors, while λ ≠ 0.5 introduces asymmetry to compensate for voltage deviation.

For example, when the v_ref falls into Sector I-1, the participating vectors are the zero vector OOO, and small vectors POO, ONN, and OON, where POO and ONN are redundant small vectors. Their effects on the mid-point voltage are shown in Figure 4.

The λ that ensures the mid-point voltage is zero in Sector I-1 is given by

$$\lambda ={\displaystyle \frac{2U_{\mathrm{np}}\mathrm{C}-T_s\left[-t_1(i_b+i_c)+t_3(i_a+i_b)\right]}{-2T_s{t}_1{i}_a}},$$

(18)

where i_a, i_b and i_c are the phase current measured by the current sensor.

Then, the dwell times of the vectors POO, ONN, OON, and OOO are λt₁, (1 − λ)t₁, t₂ and t₃, respectively. The pattern of switch positions in sector I-1 is shown in Figure 5.

4. Adaptive Weight NN-Based Disturbance Observer

The aforementioned CCS-MPC is a model-based method, and as seen from Equations (4) and (10), its control performance heavily relies on the accuracy of the motor model, particularly the precision of the model parameters. However, there inevitably exists a deviation between the parameters in the prediction model and the actual motor parameters. The parameters that most significantly affect the accuracy of the prediction model include the stator resistance R_s, stator inductance L, and stator flux ψ_f.

The motor model including parameter mismatch can be expressed as Equation (4). Where E represents the lumped error term, which can be specifically expressed as

$$E=\left[\begin{array}{c}{f}_d(k)\\ f_q(k)\end{array}\right]=\left[\begin{array}{cc}1-{\displaystyle \frac{\Delta R_s}{\Delta L_d}}{T}_s& {\displaystyle \frac{{\omega }_e\Delta L_q}{\Delta L_d}}{T}_s\\ -{\displaystyle \frac{{\omega }_e\Delta L_d}{\Delta L_q}}{T}_s& 1-{\displaystyle \frac{\Delta R_s}{\Delta L_q}}{T}_s\end{array}\right]\left[\begin{array}{c}{i}_d(k)\\ i_q(k)\end{array}\right]+\left[\begin{array}{cc}{\displaystyle \frac{T_s}{\Delta L_d}}& 0\\ 0& {\displaystyle \frac{T_s}{\Delta L_q}}\end{array}\right]\left[\begin{array}{c}{u}_d(k)\\ u_q(k)\end{array}\right]+\left[\begin{array}{c}0\\ -{\displaystyle \frac{{\omega }_e\Delta {\psi }_f}{\Delta L_q}}{T}_s\end{array}\right]+\left[\begin{array}{c}{\xi }_d\\ {\xi }_q\end{array}\right],$$

(19)

where $\Delta R_s=R_{s0}-R_s$, $\Delta L_d=L_{d0}-L_d$, $\Delta L_q=L_{q0}-L_q$, $\Delta {\psi }_f={\psi }_{f0}-{\psi }_f$, and R_s0, L_d0, L_q0 and ${\psi }_{f0}$ are the actual motor stator resistance, dq-axis inductance and stator flux, respectively. ${\xi }_d$, ${\xi }_q$ are the unmodeled disturbance excluding parameter mismatch.

The NN-based disturbance observer model for Equation (1) is given by

$$\dot{\widehat{x}}=A_0\widehat{x} + B_{0}u + D_0 + K(x-\widehat{x}) + E_0,$$

(20)

where $\dot{\widehat{x}}$ and $\widehat{x}$ are ${[\begin{array}{cc}{\dot{\widehat{i}}}_d& {\dot{\widehat{i}}}_q\end{array}]}^{\mathrm{T}}$ and ${[\begin{array}{cc}{\widehat{i}}_d& {\widehat{i}}_q\end{array}]}^{\mathrm{T}}$, respectively. K is the observer gain. According to the Weierstrass higher-order approximation theorem, the lumped error term E can be approximated by the following NN:

$$E_0=W\sigma (Vx),$$

(21)

where $x={\left[\begin{array}{ccc}{i}_d& i_q& i_{const}\end{array}\right]}^{\mathrm{T}}$ is the input of the NN, and $W\in R^{2\times n}$, $V\in R^{n\times 2}$ are the unknown ideal weight matrix of the NN, and n is the number of neurons in the hidden layer neurons. And $\sigma (\begin{array}{c}\cdot \end{array})$ is the NN activation function. According to NN approximation property, $W$ and $\sigma (x)$ are bounded such that $‖W‖\le b_W$, $‖\sigma (x)‖\le b_{\sigma }$ for some positive constants $b_W$, $b_{\sigma }$.

To enhance the expressive and approximation capability of the NN, the constant term i_const is introduced into the input vector. This constant input acts similarly to a bias neuron, allowing the network to approximate not only the nonlinear mapping between the system states (i_d, i_q) and the lumped disturbances but also to capture static biases and unmodeled constant disturbances that are independent of the state variables.

According to the Weierstrass higher-order approximation theorem, there exists an ideal weight matrix for the NN. Since the ideal weight matrix is unknown, Equation (21) can be rewritten as

$$\widehat{E}=\widehat{W}\sigma (\widehat{x}),$$

(22)

where $\widehat{E}$ and $\widehat{W}$ are the estimates of the lumped error term and the actual weights of the NN, respectively. The NN-based disturbance observer dynamic of Equation (20) can be rewritten as

$$\dot{\widehat{x}}=A_0\widehat{x} + B_{0}u + D_0 + K(x-\widehat{x}) + \widehat{E},$$

(23)

where K is the observer gain.

Then, the observer estimation error can be obtained by subtracting Equations (23) and (1)

$$\dot{\tilde{x}}=(A_0-K{C}_0)\tilde{x}+W\sigma (\widehat{x})+W\tilde{\sigma }(x),$$

(24)

where $\tilde{x}=x-\widehat{x}$, $\tilde{W}=W-\widehat{W}$ and $\tilde{\sigma }(x)=\sigma (x)-\sigma (\widehat{x})$, respectively.

To guarantee $\widehat{W}$ → $W$, the NN weight-tuning law has to be designed. Furthermore, the stability of the NN-observer needs to be ensured in the online update process. We propose a structural diagram of the NN-observer scheme in Figure 6. Then, we design and analysis the stability of an online NN parameter update law for the scheme.

Theorem 1.

Consider the PMSM model Equation (4) and the NN-based disturbance observer model Equation (23). Let the NN weight-tuning law be designed as

$$\dot{\tilde{W}}=-\delta ‖P\tilde{x}‖\widehat{W}+\phi P\tilde{x}{\sigma }^T(\widehat{x}),$$

(25)

where$\delta ,\phi >0$ are the update rate and P is a positive definite solution of the following Lyapunov equation for some positive definite matrices Q:

$${(A_0-K{C}_0)}^{T}P+P(A_0-K{C}_0)=-Q,$$

(26)

Then, according to Lyapunov theorem, the state estimation error and the weight estimation error are ultimately uniformly bounded under the designed weight adaptation law, ensuring the stability of the observer.

Proof of Theorem 1.

Select the Lyapunov function candidate:

$$V={\displaystyle \frac{1}{2}}{\tilde{x}}^{T}P\tilde{x}+{\displaystyle \frac{1}{2}}\mathrm{tr}\left({\tilde{W}}^T{\phi }^{-1} \tilde{W}\right)$$

(27)

By differentiating the above Equation (27) and using Equation (25), one will obtain

$$\dot{V}={\displaystyle \frac{1}{2}}{\tilde{x}}^T\left[{(A-KC)}^{T}P+P(A-KC)\right]\tilde{x}+{\tilde{x}}^{T}P\left[\tilde{W}\sigma (\widehat{x})+W\tilde{\sigma }(x)\right]-{\tilde{y}}^{T}P\tilde{W}\sigma (\widehat{x})+{\phi }^{-1}\delta ‖P\tilde{y}‖\mathrm{tr}\left({\tilde{W}}^T\widehat{W}\right)$$

(28)

Equation (28) can be simplified to yield

$$\dot{V}=-{\displaystyle \frac{1}{2}}{\tilde{x}}^{T}Q\tilde{x}+{\tilde{x}}^{T}PW\tilde{\sigma }(x)+{\phi }^{-1}\delta ‖P\tilde{y}‖\mathrm{tr}\left({\tilde{W}}^T\widehat{W}\right)$$

(29)

Considering $\mathrm{tr}\left({\tilde{W}}^T\widehat{W}\right)=\mathrm{tr}\left[{\tilde{W}}^T(W-\tilde{W})\right]=\mathrm{tr}\left({\tilde{W}}^{T}W\right)-\mathrm{tr}\left({\tilde{W}}^T\tilde{W}\right)<{\tilde{W}}^{T}W-\Vert \tilde{W}{\Vert }^2$, Equation (29) can be rewritten as follows:

$$\dot{V}\le -{\displaystyle \frac{1}{2}}{\lambda }_{\min }(Q){‖\tilde{x}‖}^2+‖\tilde{x}‖‖P‖‖W‖‖\tilde{\sigma }(x)‖+{\phi }^{-1}\delta ‖\tilde{x}‖‖P‖\left(‖\tilde{W}‖‖W‖-\Vert \tilde{W}{\Vert }^2\right),$$

(30)

where ${\lambda }_{\min }(Q)$ is the minimum eigenvalue of the matrix Q.

Considering $‖W‖\le b_W$, $‖\sigma (x)‖\le b_{\sigma }$, and $‖\tilde{W}‖‖W‖-{‖\tilde{W}‖}^2={\displaystyle \frac{1}{2}}{‖W‖}^2-{\displaystyle \frac{1}{2}}{‖\tilde{W}‖}^2$, Equation (30) can be rewritten as follows:

$$\dot{V}\le -{\displaystyle \frac{1}{2}}{\lambda }_{\min }(Q){‖\tilde{x}‖}^2-{\displaystyle \frac{{\phi }^{-1}\delta }{2}}‖\tilde{x}‖‖P‖{‖\tilde{W}‖}^2+‖\tilde{x}‖‖P‖\left(b_W{b}_{\sigma }+{\displaystyle \frac{{\phi }^{-1}\delta }{2}}{b}_W^2\right)$$

(31)

The negative semidefiniteness of $\dot{V}$ is guaranteed if either of the following conditions holds:

$$‖\tilde{x}‖\ge {\displaystyle \frac{2‖P‖}{{\lambda }_{\min }(Q)}}\left(b_W{b}_{\sigma }+{\displaystyle \frac{{\phi }^{-1}\delta }{2}}{b}_W^2\right)$$

(32)

or

$$‖\tilde{W}‖\ge \sqrt{{\displaystyle \frac{2b_W{b}_{\sigma }}{{\phi }^{-1}\delta }}+b_W^2}.$$

(33)

It can be concluded that there exists a compact set such that when the state estimation error or the weight estimation error lies outside this set, the Lyapunov function is negative semidefinite. Consequently, the state estimation error will continuously decrease until it is attracted into the set. Under the constraints of Equation (32) or Equation (33), both the state estimation error and the weight estimation error are ultimately uniformly bounded, indicating that the adaptive weight NN-based disturbance observer is stable. □

In real-time implementation, the optimal update rates δ and φ are selected as 0.005 and 1000, respectively. These values are acquired by trial and error. The initial weight matrix is

$$W_0=\left[\begin{array}{ccc}0.3500& 0.1966& 0.2511\\ 0.6160& 0.4733& 0.3517\end{array}\right]$$

(34)

which is generated randomly.

Figure 7 illustrates the model-predictive control structure incorporating the adaptive weight NN-based disturbance observer. The CCS-MPC and mid-point voltage control components are consistent with those described in the previous section.

5. Experiment

The proposed scheme is implemented on a three-phase PMSM supplied by NPC-3L VSIs to examine the steady and transient state performances. The experimental setup is shown in Figure 8. In the experiment, the NPC-3L inverter circuit is composed of the Imperix PEN8018 modules. All of these are powered by a DC source with a DC-link voltage of 100 V. The B-Board PRO of imperix is adopted to carry out the control algorithm and to generate the PWM signals. A magnetic powder brake provides the electrical load. The switching frequency of the inverter is kept as f_sw = 10 kHz. The parameters of the PMSM are given in

Table 3. Parameters of the PMSM.

Parameter	Symbol	Value
Rated phase voltage	UN	220 V
Rated phase current	IN	12 A
Rated speed	ωmN	2000 rpm
Permanent flux linkage	ψm	0.096 Wb
Phase resistance	Rs	0.78 Ω
Phase inductance	Ls	3.4 mH

.

5.1. Performance of Cascaded Structure with CCS-MPC and Mid-Point Voltage Control

Figure 9 compares the steady-state performance of CCS-MPC with active (with dynamic calculation of λ) and passive (with λ fixed at 0.5) mid-point voltage control at motor speeds of 500 rpm and 800 rpm, with a load torque of 4 N·m. The figure presents the waveforms of phase current, dq-axis currents, and mid-point voltage.

As shown in Figure 9, under CCS-MPC cascade control of the mid-point voltage, the total harmonic distortion (THD) values of currents in phases A at motor speeds of 500 rpm and 800 rpm are 2.69% and 2.53%, respectively. The d-axis current ripples amplitudes are 0.620 A and 0.635 A, while the q-axis current ripples amplitudes are 0.486 A and 0.489 A, respectively. Compared with the passive control (with λ fixed at 0.5) of the mid-point voltage, the mid-point voltage ripple is significantly reduced under active control, with peak-to-peak ripple values of 0.9 V and 1.8 V at 500 rpm and 800 rpm, respectively. In contrast, passive control leads to a substantial deviation in the mid-point voltage, which subsequently increases the phase current THD and the dq-axis current ripples to a certain extent.

Figure 10 illustrates the transient performance of cascaded structure with CCS-MPC and mid-point voltage control under a step change in motor speed from 500 rpm to 800 rpm, exhibiting a dynamic adjustment time of 70 ms, during which the q-axis current rapidly reaches its peak to deliver maximum acceleration.

In summary, both the steady-state and transient experimental results confirm that the proposed cascaded structure with CCS-MPC and mid-point voltage control effectively suppresses mid-point voltage deviation and reduces current distortion.

5.2. Performance of Adaptive Weight NN-Based Disturbance Observer

To verify the influence of parameter mismatch on CCS-MPC, Figure 11 presents steady-state experiments under different parameter mismatch conditions. The parameter mismatch cases include no mismatch, L = 0.6L₀, L = 1.4L₀, ψ_f = 0.6ψ_f0, and ψ_f = 1.4ψ_f0.

Figure 11 illustrates the impact of inductance and flux linkage parameter mismatch on the steady-state performance of the motor. Under no parameter mismatch, the d-axis current remains stable around 0 A with a ripple of 0.68 A, while the q-axis current stabilizes around 7.2 A with a ripple of 0.41 A. The dq-axis currents closely track their reference values, indicating that CCS-MPC achieves excellent steady-state performance. Inductance mismatch primarily affects the d-axis current, while the q-axis current remains largely unaffected. When the inductance in the prediction model is set to 0.6 and 1.4 times the actual motor inductance, the resulting d-axis steady-state current errors are approximately 0.789 A and 0.47 A, respectively, and flux linkage mismatch predominantly affects the q-axis current, with minimal impact on the d-axis. When the flux linkage in the prediction model is set to 0.6 and 1.4 times the actual motor flux linkage, the resulting q-axis steady-state current errors are 0.688 A and 0.643 A, respectively, while the d-axis current error remains negligible.

To verify the influence of parameter mismatch on CCS-MPC with adaptive weight NN-based disturbance observer, Figure 12 presents steady-state experiments under different parameter mismatch conditions. The parameter mismatch cases include L = 0.6L₀, L = 1.4L₀, ψ_f = 0.6ψ_f0, and ψ_f = 1.4ψ_f0. The NN weights were initialized randomly and updated online via the Lyapunov-based adaptation law Equation (25).

As shown in Figure 12, when L = 0.6L₀, the steady-state error of the d-axis current decreases from 0.789 A to 0.265 A, with the lumped error terms f_d and f_q stabilizing at −7.8 V and −17.4 V, respectively. When L = 1.4L₀, the steady-state error of the d-axis current reduces from 0.470 A to 0.247 A, and f_d and f_q stabilize at −11 V and −18.1 V, respectively. When ψ_f = 0.6ψ_f0, the steady-state error of the q-axis current decreases from 0.688 A to 0.264 A, with f_d and f_q stabilizing at −18.7 V and −22.8 V, respectively. When ψ_f = 1.4ψ_f0, the steady-state error of the q-axis current decreases from 0.643 A to 0.282 A, with f_d and f_q stabilizing at −2.9 V and −12.5 V, respectively.

Overall, the results in Figure 12 demonstrate that the proposed adaptive weight NN-based disturbance observer exhibits superior capability in compensating for steady-state errors in dq-axis currents, achieving significantly better current tracking ability under parameter mismatch compared to uncompensated cases. This highlights the effectiveness and robustness of the proposed approach in enhancing the steady-state performance of the system.

As a comparison, the steady-state performance of the sliding mode disturbance observer is depicted in Figure 13. In the inductance mismatch cases of L = 0.6L₀ and L = 1.4L₀, the sliding mode observer yields the steady-state errors of the q-axis current of 0.759A and 0.727A, respectively. As for the flux mismatch cases, the steady-state errors of the q-axis current are 0.803A and 0.282A, respectively. These results validate that the proposed adaptive weight NN-based disturbance observer outperforms the sliding mode disturbance observer by more accurate current tracking in such a NPC-fed drive system.

6. Conclusions

This paper proposes a cascaded control strategy that integrates CCS-MPC with mid-point voltage regulation, in conjunction with an adaptive weight NN-based disturbance observer, for NPC-3L inverter-fed PMSMs. The fluctuation in mid-point voltage keeps within 2% of the DC-link voltage while the dynamic performance of CCS-MPC is not weakened. In the cases of parameter mismatch, the Lyapunov-stable observer reduces the current tracking errors by 60%. Accordingly, the proposed weight-adaptive observer effectively improves the robustness of the cascaded CCS-MPC strategy. The comparative experimental results show that the intelligent disturbance observer is more suitable to be applied in NPC-3L inverter-fed systems.