Dual-Vector Model Predictive Current Control with Entire-Time-Domain Current Harmonic Optimization and Robust Control Strategy

Abstract

This article proposes a robust dual-vector model predictive control (RDVMPC) strategy for high-power three-level inverters, specifically designed to optimize current harmonics. By extending the concept of total harmonic distortion (THD) to the entire time domain, the current deviation is redefined, and a novel cost function is formulated. The proposed strategy incorporates dual-vector control to maintain a fixed switching frequency while optimizing the sequences and duty cycles of the vectors to further mitigate THD. A novel ultra-local model is employed to eliminate dependency on precise system parameters, with unknown components accurately estimated through an extended state observer (ESO) equipped with an adaptive bandwidth gain mechanism. This mechanism dynamically adjusts to enhance disturbance rejection under parameter variations and suppress noise under steady-state conditions. The efficacy of the proposed method is substantiated through simulations and experimental validation on a 10 kW T-type inverter, demonstrating significant improvements in harmonic reduction and overall system performance.

1. Introduction

In recent years, the trend of grid-connected new energy generation has shifted towards high voltage and high power. Traditional two-level inverters can no longer meet these demands and have been replaced by three-level inverters, which offer reduced output current harmonics and lower voltage stress and switching losses on power devices [1,2]. Among these, the Neutral Point Clamped (NPC) inverter is the most extensively studied and has become the standard topology for high-performance medium-voltage applications [3]. The T-type inverter, an evolution of the NPC topology, is particularly well-suited for new energy generation systems due to its reduced device count and improved loss distribution [4]. This paper focuses on the T-type inverter as the subject of investigation.

Finite control set model predictive control (FCS-MPC) is particularly effective in addressing both linear and nonlinear problems with multiple constraints and control objectives [5], making it a suitable choice for T-type inverter control. FCS-MPC operates by selecting the optimal switching state in each control cycle. However, the traditional single-vector FCS-MPC approach has limitations, including high output current harmonics and the repeated selection of the same optimal vector over consecutive control cycles [6]. These issues lead to an unfixed switching frequency, complicating the design of filters for the inverter.

To reduce current harmonics, a multi-vector model predictive control (MV-MPC) strategy has been developed. By applying multiple voltage vectors within each control cycle, MV-MPC effectively reduces current harmonics and stabilizes the switching frequency. For example, a dual-vector model predictive control (DVMPC) strategy with optimized duty cycles was proposed in [7], selecting from six possible vectors to determine the optimal duty cycle. Similarly, a three-vector torque control approach introduced in [8] achieves precise control of torque and magnetic flux. However, utilizing multiple vectors increases the switching frequency, resulting in significant losses in high-power applications. While DVMPC strikes a balance between improved control and reduced switching frequency [9], its performance deteriorates over longer cycles. To address this, [10] refines DVMPC for current total harmonic distortion (THD) minimization by selecting optimal voltage vectors and duty cycles through area minimization, although this approach is limited to single-phase converters.

Furthermore, model mismatches caused by system parameter variations can significantly affect predicted values and degrade FCS-MPC performance [11]. To address these mismatches, recent research has focused on incorporating disturbance observers and model parameter estimation techniques into FCS-MPC for power electronics applications [12,13,14]. An emerging alternative is model-free predictive control (MFPC), which mitigates mismatches by eliminating the need for precise system modeling and instead deriving control strategies directly from input-output data. For instance, Ref. [15] employs MFPC with a look-up table to store and update current gradients for each vector within a control cycle. However, gradient updates may stagnate, leading to increased prediction errors. Moreover, as inverter voltage levels increase, the required number of gradient updates grows substantially, placing high demands on the computational capabilities of the controller.

MFPC based on the ultra-local model (ULM) has gained significant attention in recent years. For example, Ref. [16] introduces a fully parameter-free ULM-based MFPC approach that employs finite-time gradient descent to adaptively estimate the controller’s gain, eliminating dependency on initial values. Similarly, Ref. [17] proposes an MFPC method combining ULM with an extended state observer (ESO), leveraging only current and past current data to predict future values, thereby avoiding reliance on PMSM parameters. Additionally, Ref. [18] presents an adaptive MFPC for T-type inverters with LCL filters, utilizing a novel ULM to decouple the system into known linear and unknown nonlinear components. A sliding mode perturbation observer is then used to estimate the unknown nonlinear terms, ensuring robust control. However, these methods do not account for or analyze the influence of observer gain on overall controller performance.

To address these challenges, this article proposes an entire-time-domain current THD optimization strategy based on robust DVMPC (R-DVMPC). Within the DVMPC framework, a real-time virtual vector representing the real-time current is constructed, and the cost function is redefined using current THD as the optimization criterion. The sequences and duty cycles of the dual vectors are optimized to minimize THD. A novel ULM is introduced to enable rapid modeling of the T-type inverter, while an adaptive bandwidth-gaining ESO is employed to estimate the unknown components of the ULM. The ESO dynamically adjusts its bandwidth, increasing it under disturbances to enhance perturbation resistance and decreasing it under steady-state conditions to improve noise rejection, thereby ensuring robust control performance. The proposed strategy is validated on a 10 kW T-type inverter experimental platform. Both simulation and experimental results demonstrate the effectiveness of the method in significantly reducing current THD and mitigating the impact of parameter perturbations on the system.

2. Model Predictive Current Control for T-Type Three-Level Inverters

2.1. Mathematical Model Analysis

The T-type inverter, as introduced in [4], features the topology shown in Figure 1a. The input stage contains two equal capacitance capacitors $C_1$ and $C_2$, maintaining balanced upper/lower voltages expressed as $u_{C1}=u_{C2}=0.5u_{dc}$. Each phase employs four switching tubes, with 27 total switching states generated through combinatorial tube activation in the three-phase configuration. The corresponding space–voltage vector diagram appears in Figure 1b. The switching function is defined to explicitly characterize the output voltage–switching state relationship:

$$S_x=\left\{\begin{array}{cc}1\hfill & u_x=0.5u_{dc}\hfill \\ 0\hfill & u_x=0\hfill \\ -1\hfill & u_x=-0.5u_{dc}\hfill \end{array}\right.x=a,b,c$$

(1)

Assuming that the voltages across capacitors $C_1$ and $C_2$ in Figure 1a are equal, the dynamic model of the T-type inverter in the $\alpha \beta$ coordinate system can be expressed as

$$\frac{\mathrm{d}{i}_{\alpha \beta }}{\mathrm{d}t}=f(i_{\alpha \beta },u_{\alpha \beta })=-\frac{R}{L}{i}_{\alpha \beta }+\frac{1}{L}{u}_{\alpha \beta }$$

(2)

where L is the AC side inductance; R is the load resistance on the AC side.

A key challenge in controlling the T-type inverter is addressing the imbalance of the midpoint potential. To account for this, the dc-side capacitive current is also modeled:

$$i_{Cn}\left(t\right)=C_n\frac{\mathrm{d}{u}_{Cn}}{\mathrm{d}t}$$

(3)

where $n\in \{1,2\}$.

2.2. Model Predictive Current Control Analysis

FCS–MPC operates based on a discrete-time model. Given that the control period is sufficiently small, the current change can be approximated as linear within one control cycle. According to Equation (2), assuming the slope of the current change is $f\left(i_{\alpha \beta }\left(k\right),u_{\alpha \beta }\left(k\right)\right)$ during the first control cycle, the discrete model of the T-type inverter can be expressed as

$$i_{\alpha \beta }^p\left(k+1\right)=i_{\alpha \beta }\left(k\right)+T_s·f\left(i_{\alpha \beta }\left(k\right),u_{\alpha \beta }\left(k\right)\right)$$

(4)

where $i_{\alpha \beta }\left(k\right)$ is the current sampling value at the kth moment; $i_{\alpha \beta }^{\mathrm{p}}(k+1)$ represents the predicted output current at the k+1th moment; $u_{\alpha \beta }\left(k\right)$ denotes the control input during the kth control cycle; and $T_{\mathrm{s}}$ is the duration of the control period.

Substituting Equation (2) into Equation (4) yields the following:

$$i_{\alpha \beta }^p(k+1)=\left(1-\frac{T_{s}R}{L}\right)i_{\alpha \beta }\left(k\right)+\frac{T_s}{L}{u}_{\alpha \beta }\left(k\right)$$

(5)

After deriving the prediction model of the T-type inverter, the next step is to formulate an appropriate cost function to achieve the desired control objectives. For instance, in the context of current control, the cost function can be expressed as

$$J\left(v_i\right)=\left|\right|i_{\alpha \beta }^{\ast }(k+1)-i_{\alpha \beta }^p(k+1)|{|}_2^2$$

(6)

where $v_i$ represents the control set of the FCS-MPC, with $v_i\in {S_a,S_b,S_c}^3$, and $i_{\alpha \beta }^{\ast }(k+1)$ denotes the reference currents at the $k+1$th moment.

In addition, to ensure the midpoint voltage balance of the T-type inverter, a midpoint voltage balance term should be incorporated into the cost function.

$$J\left(v_i\right)=\left|\right|i_{\alpha \beta }^{\ast }(k+1)-i_{\alpha \beta }^p(k+1)|{|}_2^2+\lambda \Delta u_C{(k+1)}^2$$

(7)

where $\lambda$ is the weighting factor, allowing the equalization of current control and midpoint voltage balance by selecting an appropriate value for $\lambda$. $\Delta u_C{(k+1)}^2$ represents the midpoint potential difference at the $k+1$th moment.

The midpoint voltage difference can be obtained by discretizing Equation (3), assuming $C_1=C_2=C$. The discrete-time model of the capacitive voltage on the dc side is then

$$u_{C_n}(k+1)=u_{C_n}\left(k\right)+\frac{T_s}{C}{i}_{C_n}\left(k\right)$$

(8)

According to Equation (8), the midpoint voltage difference at time $k+1$ is obtained as

$$\begin{array}{cc}\hfill \Delta u_C(k+1)& =u_{C_1}(k+1)-u_{C_2}(k+1)\hfill \\ \hfill & =u_{C_1}\left(k\right)-u_{C_2}\left(k\right)+\frac{T_s}{C}\left[i_{C_1}\left(K\right)-i_{C_2}\left(K\right)\right]\hfill \end{array}$$

(9)

Considering the midpoint current $i_O=i_{C_1}-i_{C_2}$, Equation (9) is replaced by

$$\Delta u_C(k+1)=u_{C_1}\left(k\right)-u_{C_2}\left(k\right)+\frac{T_s}{C}{i}_O\left(k\right)$$

(10)

By substituting all 27 voltage vectors into Equation (7), the voltage vector that minimizes the cost function is identified as the optimal vector. This optimal vector is then determined by solving the constrained least squares problem in Equation (11).

$$\min J\left(v_i\right)$$

(11)

It is worth mentioning that FCS-MPC for T-type inverters faces the challenge of high computational complexity. To address this issue, the method proposed in [19] is applied, which enables rapid determination of the optimal control set through an algebraic approach. By limiting the number of control sets to three, this method significantly reduces the computational burden while enhancing system performance.

3. Entire-Time-Domain Current THD Optimization for DVMPC

DVMPC can effectively reduce current harmonics and maintain a fixed switching frequency; however, the sequences and duty cycle of DVMPC still require further optimization. To address this, this section proposes a DVMPC method for entire-time-domain current harmonic optimization. The method introduces a real-time virtual vector that is equivalent to the real-time current error and uses this as an evaluation index to optimize the sequences and duty cycle of voltage vectors, thereby minimizing the current THD.

3.1. Principles and Design of Conventional DVMPC

The conventional DVMPC obtains the optimal vector $v_{o1}$ and the suboptimal vector $v_{o2}$ by searching for the optimal vector using the cost function in Equation (6). The optimal output vector is then obtained by reasonably allocating the action times of $v_{o1}$ and $v_{o2}$. According to the analysis in [19], the current deviation is positively correlated with the distance between the reference vector and the output vector, with the optimal output vector being the one closest to the reference vector.

After obtaining the optimal and suboptimal vectors, the duty cycle of the two vectors needs to be calculated. Assume that the equivalent output vector resulting from the joint action of the optimal vector $v_{o1}$ and the suboptimal vector $v_{o2}$ during one control cycle is $v_o$. According to the principle of vector computation, when $v^{\ast }-v_o$ is perpendicular to $v_{o1}-v_{o2}$, the output vector $v_o$ is closest to the reference vector $v^{\ast }$. Assuming that the lengths of the three sides of the triangle formed by $v^{\ast }-v_{o1}-v_{o2}$ are x, y, and z, the duty cycle $d^{\ast }$ of the optimal vector $v_{o1}$ can be determined using the cosine rule of the triangle:

$$d^{\ast }=\frac{y^2+z^2-x^2}{2z^2}$$

(12)

The duty cycle of the suboptimal vector $v_{o2}$ is automatically assigned as $1-d^{\ast }$.

3.2. Determination of the Virtual Output Vector in the Entire-Time Domain

In order to measure the performance of FCS-MPC, the current THD is considered as an evaluation index. According to [10], when the reference signal is sinusoidal, the current THD can be defined as

$$\mathrm{THD}=\frac{I_{hrms}}{I_{brms}}=\frac{\sqrt{\frac{1}{T}{\int }_T{\left[i^{\ast }\left(t\right)-i\left(t\right)\right]}^{2}dt}}{I_{brms}}$$

(13)

where $I_{hrms}$ is the sum of the harmonic current RMS values; $I_{brms}$ is the RMS value of the fundamental current; $i\left(t\right)$ is the real-time output current; $i^{\ast }\left(t\right)$ is the real-time reference current; and T is the sampling period.

According to the definition of current THD, it is positively correlated with the area between $i^{\ast }\left(t\right)$ and $i\left(t\right)$. Therefore, the cost function can be redefined based on this area, enabling current THD minimization by selecting the voltage vector that minimizes the cost function. To accurately calculate the current THD, the control error must be evaluated in the entire time domain. Accordingly, the following three definitions are introduced:

Definition 1.

Define the reference vector $v^{\ast }$ as the virtual vector that makes the deviation of the current prediction part of the cost function zero. Let the coordinates of $v^{\ast }$ be $\left(u_{\alpha }^{\ast },u_{\beta }^{\ast }\right)$. According to the dead-beat prediction control, the current prediction value obtained by substituting into Equation (5) should be equal to the desired value, i.e.,

$$\left(1-\frac{T_{s}R}{L}\right)i_{\alpha \beta }\left(k\right)+\frac{T_s}{L}{v}^{\ast }=i_{\alpha \beta }^{\ast }(k+1)$$

(14)

Definition 2.

Define the reference zero vector $v_0^{\ast }$ as the virtual vector that keeps the output current constant. Let the coordinates of $v_0^{\ast }$ be $\left(u_{\alpha 0}^{\ast },u_{\beta 0}^{\ast }\right)$. The predicted value of the current obtained by substituting into Equation (5) should be equal to the current value, i.e.,

$$\left(1-\frac{T_{s}R}{L}\right)i_{\alpha \beta }\left(k\right)+\frac{T_s}{L}{v}_0^{\ast }=i_{\alpha \beta }^{\ast }\left(k\right)$$

(15)

Definition 3.

Assuming that the output vector is $v\left(u_{\alpha },u_{\beta }\right)$ in one control cycle, define the real-time current prediction value $i_{\alpha \beta }\left(t\right)$ as calculated by the following:

$$\left(1-\frac{tR}{L}\right)i_{\alpha \beta }\left(k\right)+\frac{t}{L}v=i_{\alpha {\beta }^{\ast }\left(t\right)}$$

(16)

Substituting Equation (16) into the cost function in Equation (6), the real-time current deviation is calculated. Combining Equations (14) and (15), the real-time current deviation is obtained as

$${\left|\left|i_{\alpha \beta }^{\ast }-i_{\alpha \beta }\left(t\right)\right|\right|}_2^2=\frac{T_s^2}{L^2}{\left|\left|v^{\ast }-\left[m·v+(1-m)v_0^{\ast }\right]\right|\right|}_2^2$$

(17)

where $m=t/T_s$, and $0\le m\le 1$.

Let the real-time virtual vector $v_t^{\ast }=m·v+(1-m)v_0^{\ast }$. According to Equation (17), the distance between $v_t^{\ast }$ and $v^{\ast }$ is positively correlated with the real-time current deviation, and the distance between $v_t^{\ast }$ and $v^{\ast }$ can be expressed as

$$h={\left|\left|v^{\ast }-v_t^{\ast }\right|\right|}_2$$

(18)

Thus, the cost function can be designed as an integral of h over a control period, which is equivalent to the magnitude of the current THD, i.e.,

$$J={\int }_0^{T_s}{h}^2\mathrm{d}t$$

(19)

By minimizing Equation (19), the optimal output vector can be selected, thereby minimizing the current THD of the model predictive control output. This results in a reduction of current ripple. Given the difficulty of performing the integral operation in the actual system, Equation (19) is transformed to obtain:

$$\begin{array}{cc}\hfill J& ={\int }_0^{T_s}\left|\left|\left(v^{\ast }-v_0^{\ast }\right)-m·\left(v-v_0^{\ast }\right)\right|\right|\mathrm{d}t\hfill \\ \hfill & =T_s\left[{\left(v^{\ast }-v_0^{\ast }\right)}^2-\left(v^{\ast }-v_0^{\ast }\right)\left(v-v_0^{\ast }\right)+\frac{1}{3}{\left(v-v_0^{\ast }\right)}^2\right]\hfill \end{array}$$

(20)

3.3. Optimization of Dual-Vector Sequences and Duty Cycle

3.3.1. Sequences Optimization

The purpose of DVMPC is to minimize the current tracking error after $v_{o1}$ and $v_{o2}$ act together for one cycle. Currently, the primary focus of research is on optimizing the duty cycle to achieve the optimal output, without considering the sequences of DVMPC. However, different sequences strategies can significantly affect the current THD.

As shown in Figure 2a, $v_j,j\in (1,2)$ represents the optimal and suboptimal vectors obtained through the cost function Equation (20). Although the output vectors use the same duty cycle and the final output values of left and right are the same at the moment $k+1$, it is evident that the different vector output sequences result in different current THDs at the end of the control cycle (due to the differing integrals of the current errors). The right is a better solution than the left because it achieves a smaller current THD. Therefore, it is necessary to evaluate the output sequence of the optimal dual vector after determining it.

The ideal sequence should allow the actual output vector to approach the reference vector as quickly as possible at the beginning of each control cycle. This means that the Euclidean distance between $\left(v^{\ast }-v_0^{\ast }\right)$ and $d·\left(v_j-v_0^{\ast }\right)$ should be minimized when the duty cycle d is small. Therefore, the angle between $\left(v^{\ast }-v_0^{\ast }\right)$ and $v_j$ should be small, and the modulus of $v_j$ should be large. This can be obtained by

$$\begin{array}{cc}\hfill \underset{d\to 0}{lim}{h}^2& =\underset{d\to 0}{lim}{\left|\left|\left(v^{\ast }-v_0^{\ast }\right)-d·\left(v_j-v_0^{\ast }\right)\right|\right|}_2^2\hfill \\ \hfill & =\underset{d\to 0}{lim}{r}_1^2-2r_1{r}_{2}d+r_2^2{d}^2\hfill \\ \hfill & =\underset{d\to 0}{lim}{r}_1^2-2r_1{v}_{j}d+2r_1{r}_{3}d+r_2^2{d}^2\hfill \end{array}$$

(21)

where $r_1=v^{\ast }-v_0^{\ast }$, $r_2=v_j-v_0^{\ast }$, $r_3=v_0^{\ast }$.

When d tends to 0, $r_2^2{d}^2$, $r_1^2$, and $r_1{r}_3$ can be omitted as constant values. Therefore, only $r_1{v}_j$ needs to be computed and used as a judgment condition. By substituting $v_1$ and $v_2$ into $r_1{v}_j$ as outputs respectively, the vector corresponding to the larger result is selected as the first in the sequence. To summarize, let the first output vector be $v_{o1}$, with $v_{o2}$ representing the other output vector. The relationship between $v_{o1}$, $v_{o2}$, $v_1$, and $v_2$ is shown in

Table 1. The sequences of optimal output voltage vector.

Condition	${\mathit{v}}_{\mathit{o}1}$	${\mathit{v}}_{\mathit{o}2}$
$\left(v^{\ast }-v_0^{\ast }\right)\left(v_1-v_2\right)>0$	$v_1$	$v_2$
$\left(v^{\ast }-v_0^{\ast }\right)\left(v_1-v_2\right)<0$	$v_2$	$v_1$

.

3.3.2. Duty Cycle Optimization

The DVMPC determines the duty cycle based on the distance between the two optimal output vectors and the reference vector, but this approach is evidently not optimal. As shown in Figure 2b, shifting the duty cycle d to the left results in a current THD that is significantly smaller than that in left, according to the definition of current THD. Therefore, to minimize current harmonics, the duty cycle in dual vector control must also be optimized.

As shown in Figure 3, $v_{o1}$ and $v_{o2}$ are the optimal and suboptimal vectors identified in Table 1, respectively. In the previous analysis, it can be derived that the expression for the real-time virtual vector, denoted as $v_t^{\ast }=m·v+(1-m)v_0^{\ast }$. This vector is then transformed into $v_t^{\ast }=v_0^{\ast }+m·(v-v_0^{\ast })$, indicating that $v_t^{\ast }$ moves along the direction of $v-v_0^{\ast }$, starting from $v_0^{\ast }$, and reaches v at $m=T_s$ within one control cycle. With double-vector control, $v_t^{\ast }$ first moves in the direction of $v_{o1}-v_0^{\ast }$ under the influence of $v_{o1}$, reaching $v_0^{\prime }=v_0^{\ast }+d{T}_s{v}_{o1}-v_0^{\ast }$ at $m=d{T}_s$. Subsequently, $v_t^{\ast }$ moves in the direction of $v_{o2}-v_0^{\prime }$ under the influence of $v_{o2}$ and finally reaches $v_o$ at $m=T_s$.

Thus, the cost function in Equation (19) becomes a segmented function.

$$J={\int }_0^{d·T_s}{h}_1^2\mathrm{d}t+{\int }_{d·T_s}^{T_s}{h}_2^2\mathrm{d}t$$

(22)

where $h_1^2={\left|\left|\left(v^{\ast }-v_0^{\ast }\right)-t·\left(v_{o1}-v_0^{\ast }\right)\right|\right|}_2^2$, $h_2^2={\left|\left|\left(v^{\ast }-v_0^{\ast }\right)-d·\left(v_{o1}-v_{o2}\right)-t·\left(v_{o2}-v_0^{\ast }\right)\right|\right|}_2^2$.

Solving Equation (22) yields the cost function J as a one-dimensional cubic function with respect to d as follows

$$J=\frac{1}{3}a{d}^3-\frac{1}{2}(a+b)d^2+bd+c$$

(23)

where $a=\left(v_{o1}-v_{o2}\right)\left(v_0^{\ast }+v_{o2}-2v_{o1}\right)$, $b=\left(v_{o1}-v_{o2}\right)\left(v_0^{\ast }+v_{o2}-2v^{\ast }\right)$, $c={\left(v^{\ast }-v_0^{\ast }\right)}^2-\left(v^{\ast }-v_0^{\ast }\right)\left(v_{o2}-v_0^{\ast }\right)+\frac{1}{3}{\left(v_{o2}-v_0^{\ast }\right)}^2$.

The problem of solving the minimum of Equation (23) can be transformed into a problem of finding the extreme value of its derivative by taking the derivative of it. After derivation:

$$\frac{\partial J}{\partial d}=a{d}^2-(a+b)d+b=(ad-b)(d-1)=0$$

(24)

Through Equation (24), the extreme points of J can be obtained as $d_1=b/a$ and $d_2=1$. Therefore, the duty cycle is calculated as follows:

$$d_{min\left(J\right)}\in \left\{0,\frac{b}{a},1\right\}$$

(25)

4. Robust MPC Using Ultra-Local Model and an Improved Extended State Observer

Although the time-domain current harmonic optimization strategy in Section 3 enhances DVMPC performance, an accurate prediction model is essential, as its accuracy directly impacts control performance. In practical scenarios, component parameters, such as those of inductors, may vary due to temperature fluctuations and other conditions. To address parameter mismatch, this section employs the ultra-local model (ULM) for rapid system modeling and uses the ESO with adaptive bandwidth gain to estimate nonlinear unknowns. This approach enables model-free predictive current control and enhances system robustness.

4.1. Novel Ultra-Local Model

For a nonlinear Single Input Single Output (SISO) system, the ultra-local model can be represented as

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{y}^{\left(\mathrm{v}\right)}=\alpha u+f\left(x\right)\hfill \\ y=cx\hfill \end{array}\right.\end{array}$$

(26)

where u and y are the input and output variables of the system, respectively; x is the state variable; $y^{\left(v\right)}$ denotes the v-th derivative of y; $f\left(x\right)$ represents the unknown perturbation; and $\alpha$ is the system’s input gain.

In traditional ULM theory, $f\left(x\right)$ contains both a linear unknown term and a nonlinear unknown term related to the state of the system. Estimating these two terms simultaneously often leads to inaccurate results. To address this issue, Ref. [20] proposes a novel ULM that further decomposes $f\left(x\right)$, namely:

$$f\left(x\right)=\beta x+F$$

(27)

where $\beta$ is the control system state gain, and F is the unknown nonlinear part that satisfies Lebesgue measurability and Lipschitz boundedness.

$$\begin{array}{c}\hfill \left\{\begin{array}{c}\frac{\mathrm{d}{i}_{\alpha \beta }}{\mathrm{d}t}=\alpha u_{\alpha \beta }+\beta i_{\alpha \beta }+F\hfill \\ F=\frac{\Delta R·L-R·\Delta L}{L(L+\Delta L)}{i}_{\alpha \beta }-\frac{\Delta L}{L(L+\Delta L)}{u}_{\alpha \beta }\hfill \\ \hfill & \frac{\mathrm{d}F}{\mathrm{d}t}=G\hfill \end{array}\right.\end{array}$$

(28)

where $\alpha =1/L,\beta =-R/L$.

For T-type inverter system, it is common to set the order of the ULM to be first order, i.e., $v=1$. Let the state variable $x=i_{\alpha \beta }$ be used to realize the current control of the inverter, while also considering the inductive perturbation $\Delta L$ and the resistive perturbation $\Delta R$. The new ULM for a T-type inverter can be expressed as Equation (28).

4.2. Enhanced Design of the Extended State Observer

The core idea of the extended state observer (ESO) is to treat the unmodeled dynamics and external perturbations of the system as an “extended state” and design an observer to estimate this state. By choosing the output current and perturbation $z={\left[i_{\alpha \beta }F\right]}^T$ as the state variables, the state-space equation of the ULM-based T-type inverter is given by

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}\hfill \dot{z}& =Az+Bu+E\dot{F}\hfill \\ \hfill y& =Cz+Du\hfill \end{array}\right.\end{array}$$

(29)

where $A=\left[\begin{array}{cc}\beta & 1\\ 0& 0\end{array}\right]$, $B=\left[\begin{array}{c}\alpha \\ 1\end{array}\right]$, $E=\left[\begin{array}{c}0\\ 1\end{array}\right]$, $C=\left[10\right]$, $d=\left[0\right]$.

Based on Equation (29), the second-order ESO is constructed as the follows:

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}\hfill \dot{\widehat{z}}& =A\widehat{z}+Bu+L(y-\widehat{y})\hfill \\ \hfill \widehat{y}& =C\widehat{z}+Du\hfill \end{array}\right.\end{array}$$

(30)

where $\widehat{z}={\left[{\widehat{i}}_{\alpha \beta }\widehat{F}\right]}^T$ is the estimated value of output current and disturbance.

Substituting Equation (29) into Equation (30) yields the following:

$$\begin{array}{c}\hfill \left\{\begin{array}{c}\dot{\widehat{z}}=[A-LC]\widehat{z}+[B-LDL]u_c\hfill \\ {\widehat{y}}_c=\widehat{z}\hfill \end{array}\right.\end{array}$$

(31)

where $u_c={\left[uy\right]}^T$, $L={\left[l_1{l}_2\right]}^T$ are the observer gains.

The observer gain $l_1$, $l_2$ can be obtained from the characteristic root $\sigma$ of the system matrix $[A-LC]$ of the state observer, i.e.,

$$\left|\sigma I-(A-LC)\right|=\left|\begin{array}{cc}\sigma -\beta +l_1& -1\\ l_2& \sigma \end{array}\right|={\sigma }^2+(l_1-\beta )\sigma +l_2$$

(32)

To stabilize the system, the characteristic roots of the system matrix $[A-LC]$ must have negative real parts. Here, $\sigma =-{\omega }_0$, where ${\omega }_0$ is the ESO bandwidth. Thus, the gain coefficients $l_1$ and $l_2$ are

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}\hfill l_1& =2{\omega }_0+\beta \hfill \\ \hfill l_2& ={\omega }_0^2\hfill \end{array}\right.\end{array}$$

(33)

According to the Routh criterion, the stabilization condition of Equation (31) is $l_1-\beta >0$, $l_2>0$, which means the stabilization condition is equal to ${\omega }_0>0$.

However, in practice, the control algorithm is executed periodically in the Interrupt Service Program (ISR) with limited execution frequency. Therefore, it is also necessary to determine the stabilization conditions of ESO in the discrete domain.

The discrete ESO is obtained by transforming Equation (31) into the discrete domain using the forward Euler discretization method:

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\epsilon }_1\left(k\right)={\widehat{z}}_1\left(k\right)-i\left(k\right)\hfill \\ {\widehat{z}}_1(k+1)={\widehat{z}}_1\left(k\right)+T_s\left[{\widehat{z}}_2\left(k\right)+\alpha u\left(k\right)+\beta {\widehat{z}}_1\left(k\right)\right]-T_s{l}_1{\epsilon }_1\left(k\right)\hfill \\ {\widehat{z}}_2(k+1)={\widehat{z}}_2\left(k\right)-T_s{l}_2{\epsilon }_1\left(k\right)\hfill \end{array}\right.\end{array}$$

(34)

Define ${\epsilon }_2\left(k\right)={\widehat{z}}_2\left(k\right)-F\left(k\right)$, and obtain Equation (35) by substituting ${\epsilon }_2\left(k\right)$ into Equation (34).

$$\left[\begin{array}{c}{\epsilon }_1\left[k+1\right]\\ {\epsilon }_2\left[k+1\right]\end{array}\right]=\left[\begin{array}{cc}1-T_s(l_1-\beta )& T_s\\ -T_s{l}_2& 1\end{array}\right]\left[\begin{array}{c}{\epsilon }_1\left[k\right]\\ {\epsilon }_2\left[k\right]\end{array}\right]+\left[\begin{array}{c}0\\ -T_{s}G\left[k\right]\end{array}\right]$$

(35)

Clearly, the stabilization condition for Equation (34) requires that all eigenvalues of the matrix $M=\left[\begin{array}{cc}1-T_s(l_1-\beta )& T_s\\ -T_s{l}_2& 1\end{array}\right]$ lie within the unit circle of the z-plane. The characteristic equation for Equation (34) is derived as follows:

$$z^2-(2-T_s(l_1-\beta ))z+\left(1-T_s{l}_1+T_s^2{l}_2\right)=0$$

(36)

Based on the Jury criterion, the stabilization conditions are derived as follows:

$$\left\{\begin{array}{c}\mid 2-T_s(l_1-\beta )\mid <2-T_s(l_1-\beta )+T_s^2{l}_2\hfill \\ \left|1-T_s(l_1-\beta )+T_s^2{l}_2\right|<1\hfill \end{array}\right.$$

(37)

Substituting $l_1$ and $l_2$ into Equation (37) yields the stabilization condition: $0<{\omega }_0<2/T_s$.

To further illustrate the effect of bandwidth gain on the controller, the bandwidth of the ESO will be investigated in detail in this paper by analyzing the frequency response characteristics of the closed-loop transfer function of F. According to Equation (32), the transfer function of the ESO is obtained as

$$\frac{\widehat{F}\left(s\right)}{F\left(s\right)}=\frac{l_2}{s^2+(l_1-\beta )s+l_2}$$

(38)

From the transfer function of the ESO, it can be seen that the ESO is a typical second-order low-pass system, and its Bode diagram is shown in Figure 4.

As can be seen from the bode plot, the input signal is attenuated in the high-frequency region. Therefore, when the ESO is observing high-frequency disturbances, its limited bandwidth leads to large observation errors. However, expanding the bandwidth of the ESO introduces more noise and causes interference. It is analyzed that higher bandwidth can improve the ESO’s convergence speed, tracking accuracy, and disturbance immunity, but its noise suppression performance will worsen [12]. In practical applications, the design of the bandwidth should be reasonable to strike a balance between speed and noise immunity. To this end, the adaptive bandwidth extended state observer (AESO) is proposed to balance disturbance rejection and noise suppression, i.e.,

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\omega }_0={\omega }_{0\min }+k·\left|e\right|\hfill \\ {\omega }_{0\min }<{\omega }_0<{\omega }_{0\max }\hfill \end{array}\right.\end{array}$$

(39)

where $e={\widehat{i}}_{\alpha \beta }-i_{\alpha \beta }$ is the current observation error; ${\omega }_{0\min }$ and ${\omega }_{0\max }$ are the minimum and maximum bandwidths, respectively; and k is the slope of the bandwidth gain with respect to the error.

According to [12], ${\omega }_0>\left|G\right|·|\dot{e}|$ can be obtained from the stability analysis of the closed-loop system, but it is difficult to obtain the value of $\left|G\right|·|\dot{e}|$ in the actual system. Therefore, ${\omega }_{0\min }$ is determined through experimental trial and error. A small margin should be left in the actual system to ensure stability, so ${\omega }_{0\max }$ is set to $1.5/T_s$. k is a large positive integer. When the system is disturbed, the adaptive bandwidth gain is increased to enhance the system’s perturbation resistance. In the steady state, the adaptive bandwidth gain is decreased to improve noise suppression.

4.3. Robust Predictive Control Based on a Novel Ultra-Local Model

According to the designed novel ULM, the input of the system is obtained as

$$u\left(k\right)=\frac{i_{\alpha \beta }(k+1)-(1+\beta T_s)i_{\alpha \beta }\left(k\right)}{\alpha T_s}-\frac{\widehat{F}\left(k\right)}{\alpha }$$

(40)

where $\widehat{F}\left(k\right)$ is obtained from Equation (29).

Assuming the current reaches the reference value $i^{\ast }(k+1)$ at moment $k+1$, the reference voltage can be calculated as

$$u^{\ast }\left(k\right)=\frac{i_{\alpha \beta }^{\ast }(k+1)-(1+\beta T_s)i_{\alpha \beta }\left(k\right)}{\alpha T_s}-\frac{\widehat{F}\left(k\right)}{\alpha }$$

(41)

After obtaining $u^{\ast }\left(k\right)$, it is used as the reference signal for the entire-time-domain DVMPC, which helps in selecting the optimal vector and computing the timing and duty cycle of the dual vectors. This process ultimately enables robust model predictive control that is independent of system parameters. The block diagram of the ultra-local model based on AESO is shown in Figure 5.

5. Simulation and Experimental Verification

To verify the feasibility and effectiveness of the proposed strategy, the entire-time-domain optimization and ULM-based robust DVMPC are analyzed experimentally using Matlab R2018b/Simulink simulations and a physical platform. These methods are compared with conventional approaches, and the experimental parameters are summarized in

Table 2. Experimental parameters.

Parameters	Values
DC voltage/V	200
AC load/$\mathsf{\Omega }$	10
AC inductance/mH	5
DC BUS capacitance/µF	480
Control frequency/kHz	10

.

5.1. Simulation Verification

5.1.1. Current Harmonic Optimization Verification

To verify the effectiveness of the current harmonic optimization strategy proposed in this article, the voltage and current waveforms of FCS-MPC, DVMPC, and the entire-time-domain DVMPC are compared. The results are shown in Figure 6. Compared to FCS-MPC, DVMPC exhibits more frequent line voltage switching and reduced current distortion, indicating that DVMPC prevents the repeated use of the same optimal vector across multiple control cycles. This helps stabilize the switching frequency and improves current harmonics to some extent. Additionally, the entire-time-domain DVMPC enhances both steady-state and dynamic performance when compared to conventional DVMPC.

In order to compare the effect of the proposed strategy on current harmonic optimization more intuitively, the a-phase current is subjected to fast Fourier transform (FFT) analysis, and the results are shown in Figure 7. After the adoption of DVMPC, the current THD is reduced from 4.89% to 2.52%, while the entire-time-domain DVMPC proposed in this article reduces about 30% on the basis of the conventional DVMPC, and the steady-state performance is significantly improved.

To comprehensively evaluate the advantages of the proposed the entire-time-domain DVMPC, the current THD values of the three control strategies are compared across different output currents, with the results shown in Figure 8. As seen in the figure, the current THD values for all three strategies decrease as the output current increases. However, the control performance of DVMPC is significantly better than that of single-vector MPC. By employing the entire-time-domain optimization, the current THD is further reduced in all cases compared to the conventional DVMPC.

5.1.2. Robustness Analysis

In order to verify the effectiveness of the proposed AESO, the estimates of the current and the perturbation are compared without parameter mismatch, and the results are shown in Figure 9. From the figure, it can be seen that the AESO is able to track the current accurately when the reference current is varied, and the error between is limited to 0.3 A, which indicates that the designed AESO has a good performance.

Considering a sudden mismatch in the inductor parameters, the control effectiveness of the proposed robust MPC is compared with the entire-time-domain DVMPC, and the results are shown in Figure 10. When the inductor mismatch occurs, the control performance of the entire-time-domain DVMPC decreases sharply and the current THD increases to 3.95%, while the robust MPC still has a good control effect under the parameter mismatch by estimating the perturbation generated by the parameter mismatch through the AESO, which verifies that the algorithm proposed in this paper has a good robustness in addition to improving the current control effect.

5.2. Experimental Verification

An experimental platform for the T-inverter is developed using a DSP (TMS320F28335) and a Field Programmable Gate Array (FPGA) (XC6SLX9). The DSP is responsible for deploying the control algorithms, while the FPGA handles pulse-width modulation (PWM) waveform generation and sensor data acquisition. The entire system includes hardware, software, real-time display software on the host computer, and the corresponding communication equipment. The complete inverter experimental platform is realized through the main circuit, control system, and host computer interactive software, as shown in Figure 11.

To comprehensively validate the feasibility of the proposed method, a comparative analysis was conducted against the classical vector modulation algorithm (SVPWM), conventional single-vector MPC, and DVMPC. The comparison metrics included algorithm execution time, dynamic response time, current THD, and steady-state error. The results of this analysis are summarized in

Table 3. Steady-state and dynamic performance of different algorithms.

Algorithms	Computational Time in One Cycle (µs)	Dynamic Response Time (ms)	Current THD (%)	Steady-State Error (A)
SVPWM	49.6	2.135	3.87	0.62
FCS-MPC	80.7	0.763	6.99	1.02
DVMPC	44.5	0.579	4.02	0.71
R-DVMPC	54.1	0.475	3.65	0.54

.

From the results presented in Table 3, it is evident that SVPWM exhibits a relatively small steady-state error and low computational demand. However, the inherent viscous effect of SVPWM leads to a significantly slower dynamic response, which is nearly three times slower than that of MPC. In contrast, while conventional MPC requires substantial computational resources, its computational complexity can be effectively reduced to a level comparable to SVPWM. Moreover, MPC demonstrates significant improvements in dynamic response, steady-state error, and current THD. These results underscore the effectiveness of the algorithm proposed in this article.

5.2.1. Steady-State Experimental Analysis

The line voltage, three-phase current, and A-phase current THD under different strategies are demonstrated in Figure 12. From Figure 12, it can be seen that all three control strategies can achieve the stable control of the system, and the current THD value of the entire-time-domain DVMPC is the smallest among the three strategies, which also demonstrates that the strategy proposed in this article has a considerable effect on the optimization of current harmonics.

Figure 13 compares the steady-state current errors of three strategies. The proposed entire-time-domain DVMPC achieves the smallest error of 0.54 A, while single-vector control has the largest error of 1.02 A with significant fluctuations. Although DVMPC reduces these fluctuations, some deviation remains. By optimizing the timing and duty cycle of dual vectors, the proposed strategy further aligns the output and reference currents, as shown in the enlarged view of Figure 13c. These results confirm its effectiveness in minimizing current harmonics.

5.2.2. Dynamic Experimental Analysis

Figure 14 illustrates the dynamic response performance of the three control strategies. Following a sudden change in the reference current, all three strategies are able to reach the new steady state within a short period. However, the dynamic response time for the single-vector control is the longest among the three strategies. By employing DVMPC, the response time is reduced, and the entire-time-domain DVMPC further shortens the response time and enhances the control performance compared to the conventional DVMPC.

5.2.3. Robustness Analysis

When the system parameters change, the control performance is affected. As shown in Figure 15a, when the inductance perturbation is −50%, the output current of the entire-time-domain DVMPC experiences significant distortion, and current harmonics increase substantially. However, by adopting the robust MPC, which observes and compensates for system perturbations, current distortion is suppressed. Similarly, when the inductance perturbation is +50%, the entire-time-domain DVMPC also experiences increased current distortion, while the robust MPC continues to deliver superior control performance.

Figure 15b shows the effect of resistance mismatch on current amplitude. A 50% resistance increase raises the amplitude to 10A, while a −50% decrease lowers it to 6 A. With the robust MPC, the amplitude stabilizes at 8A, confirming the strategy’s effectiveness against resistance mismatch.

6. Conclusions

In this article, a robust dual-vector model predictive control strategy is proposed for optimizing current harmonics across the entire time domain to enhance the output current quality of T-type inverters and reduce switching losses. The following conclusions are drawn from theoretical analysis and experimental verification:

(1)

Compared to single-vector FCS-MPC, DVMPC effectively reduces current ripple and THD while maintaining a fixed switching frequency. Moreover, it achieves lower switching losses than the three-vector FCS-MPC, making it more suitable for high-power applications.

(2)

Compared to conventional DVMPC, the entire-time-domain DVMPC introduced in this paper optimizes inverter output current harmonics by incorporating simple vector timing and duty cycle optimization. This approach enhances system control performance with only a minimal increase in computational effort.

(3)

A novel ultra-local model of the T-type inverter is developed, with an ESO utilized to accurately estimate perturbations. By adaptively processing the ESO bandwidth, the system improves tracking performance and effectively mitigates the impact of parameter mismatches, demonstrating superior robustness compared to conventional MPC approaches.