With the continuous advancement of China’s “dual carbon” strategy and the construction of new power systems, the proportion of distributed power sources such as photovoltaic and wind power connected to the grid has been increasing. The global demand for energy is growing, and green, environmentally friendly, and sustainable photovoltaic power generation has become a research focus both domestically and internationally. As an important interface device for new energy grid-connected power generation, grid-connected inverters have received widespread attention. Among various types of solar inverters, the quasi-Z-source inverter (qZSI) stands out due to its advantages such as single-stage buck-boost capability, low harmonic distortion, allowance of shoot-through states in the same bridge arm, and high reliability, making it widely applicable in photovoltaic grid-connected power generation. However, the wide variation range of photovoltaic output DC voltage leads to unstable zero dynamics and nonlinear characteristics in the qZSI system under DC-side voltage fluctuations, resulting in system oscillation, slow dynamic response, and complex control system design. To address these issues, we propose a control strategy combining integral compensation, fixed switching frequency current integral sliding mode (ISM) control, and radial basis function neural network (RBF) algorithm to adaptively adjust controller parameters and suppress chattering.
Among different types of solar inverters, traditional voltage-source inverters (VSI) require additional DC-DC boost converters, increasing complexity and cost. Z-source inverters (ZSI) provide a single-stage solution but have discontinuous input current and large passive components. The quasi-Z-source inverter (qZSI) improves upon the ZSI by offering continuous input current, reduced component stress, and lower start-up current, making it more suitable for photovoltaic applications. The control methods for qZSI mainly include proportional-integral (PI) control, proportional-resonant (PR) control, fuzzy control, and sliding mode control (SMC). Among nonlinear control techniques, SMC is one of the most cost-effective methods due to its robustness, stability, good dynamic response, and high compatibility with the inherent switching characteristics of power converters. However, conventional SMC suffers from chattering caused by the discontinuous switching function. In this work, we introduce an integral sliding mode (ISM) controller with a radial basis function (RBF) neural network to adaptively adjust the switching gain, effectively reducing chattering while maintaining robustness.
1. Principle and Mathematical Model of Quasi-Z-Source Inverter
The qZSI system consists of a PV module, a quasi-Z-source network (inductors L1 and L2, capacitors C1 and C2, diode D), an inverter bridge, and an LC filter (L3–L8, Cf). The quasi-Z-source network operates in two states: shoot-through state and non-shoot-through state. By controlling the duty cycle of the shoot-through state, the inverter can achieve both buck and boost functions.
The large-signal model of the qZSI in continuous conduction mode (CCM) is derived. Let L1 = L2 = L and C1 = C2 = C. The state-space equations for the shoot-through state are:
$$
\begin{cases}
\frac{di_{L1}}{dt} = \frac{1}{L}(u_{C2} + V_{in}) \\[4pt]
\frac{di_{L2}}{dt} = \frac{u_{C1}}{L} \\[4pt]
\frac{du_{C1}}{dt} = \frac{i_{L2}}{C} \\[4pt]
\frac{du_{C2}}{dt} = \frac{i_{L1}}{C}
\end{cases}
$$
For the non-shoot-through state:
$$
\begin{cases}
\frac{di_{L1}}{dt} = \frac{1}{L}(V_{in} – u_{C1}) \\[4pt]
\frac{di_{L2}}{dt} = -\frac{u_{C2}}{L} \\[4pt]
\frac{du_{C1}}{dt} = \frac{1}{C}(i_{L1} – i_{dc}) \\[4pt]
\frac{du_{C2}}{dt} = \frac{1}{C}(i_{L2} – i_{dc})
\end{cases}
$$
Combining both states, the affine state-space model is obtained:
$$
\dot{\mathbf{X}}_a = \mathbf{A} + \mathbf{B} u
$$
where the state vector $\mathbf{X}_a = [i_{L1}, i_{L2}, u_{C1}, u_{C2}]^T$, and the matrices are:
$$
\mathbf{A} = \begin{bmatrix}
\frac{V_{in} – u_{C1}}{L} \\[4pt]
-\frac{u_{C2}}{L} \\[4pt]
\frac{i_{L1} – i_{dc}}{C} \\[4pt]
\frac{i_{L2} – i_{dc}}{C}
\end{bmatrix}, \quad
\mathbf{B} = \begin{bmatrix}
\frac{u_{C1} + u_{C2}}{L} \\[4pt]
\frac{u_{C1} + u_{C2}}{L} \\[4pt]
\frac{i_{dc} – i_{L1} – i_{L2}}{C} \\[4pt]
\frac{i_{dc} – i_{L1} – i_{L2}}{C}
\end{bmatrix}
$$
Here, $u = 1$ represents the shoot-through state (switches closed) and $u = 0$ represents the non-shoot-through state (switches open). The parameter specifications used in our simulation are summarized in Table 1.
| Parameter | Value |
|---|---|
| Inductance L1, L2 | 3 mH |
| Capacitance C1, C2 | 500 μF |
| Filter inductance (L3–L8) | 3 mH |
| Filter capacitance Cf | 2 μF |
| Switching frequency | 10 kHz |
2. Design of RBF Neural Network Integral Sliding Mode Controller
2.1 RBF Neural Network for Gain Adjustment
An RBF neural network is employed to adaptively adjust the switching gain $\varepsilon$ of the sliding mode controller, ensuring robust performance while effectively reducing high-frequency chattering. The network has a 2-3-1 three-layer structure. The input vector is $\mathbf{x} = [s, \dot{s}]^T$, where $s$ is the sliding surface. The output is the magnitude of the switching gain:
$$
y_m = \varepsilon = \mathbf{w}^T \mathbf{h}(\mathbf{x})
$$
Here, $\mathbf{w} = [w_1, w_2, w_3]^T$ is the weight vector, and $\mathbf{h}(\mathbf{x}) = [h_1, h_2, h_3]^T$ is the Gaussian basis function vector:
$$
h_i(\mathbf{x}) = \exp\left( -\frac{\|\mathbf{x} – \mathbf{c}_i\|^2}{2b_i^2} \right), \quad i = 1,2,3
$$
where $\mathbf{c}_i$ and $b_i$ are the center and width of the $i$-th neuron, respectively. The weight adaptation law minimizes the cost function $E = \frac{1}{2} e^2$ with $e = s$, using gradient descent with momentum:
$$
\begin{aligned}
\Delta w_i &= \eta \, e \, \text{sgn}(s) \, h_i(\mathbf{x}) \, \text{sgn}(w_i h_i(\mathbf{x})) \\
w_i(t) &= w_i(t-1) + \Delta w_i(t) + \alpha [w_i(t) – w_i(t-1)]
\end{aligned}
$$
where $\eta \in (0,1)$ is the learning rate and $\alpha \in (0,1)$ is the momentum factor. Similar update rules apply for the center vector $\mathbf{c}_i$ and width $b_i$.
2.2 Sliding Surface Design
Assuming the qZSI operates in CCM, we define the state variables:
$$
\begin{aligned}
x_1 &= i_{L1} – i_{L1}^* \\
x_2 &= u_{C1} – u_{C1}^* \\
x_3 &= \int (i_{L1} – i_{L1}^* + u_{C1} – u_{C1}^*) \, dt
\end{aligned}
$$
The sliding surface is designed as a linear combination:
$$
S = \alpha_1 x_1 + \alpha_2 x_2 + \alpha_3 x_3 = \mathbf{J}^T \mathbf{X}
$$
where $\mathbf{J} = [\alpha_1, \alpha_2, \alpha_3]^T$ is the coefficient vector and $\mathbf{X} = [x_1, x_2, x_3]^T$. The reference current $i_{L1}^*$ is generated from the voltage error with integral action to eliminate steady-state error:
$$
i_{L1}^* = K_i \int (u_{C1} – u_{C1}^*) dt + K_p (u_{C1} – u_{C1}^*)
$$
2.3 Derivation of Equivalent Control Law
Taking the time derivative of $S$ and setting $\dot{S}=0$ yields the equivalent control $u_{eq}$:
$$
u_{eq} = \frac{ (u_{C1} – V_{in}) + k_1 (u_{C1} – u_{C1}^*) – k_2 (i_{L1} – i_{dc}) – k_3 \left[ i_{L1} – K_i\int(u_{C1}-u_{C1}^*)dt \right] }{ (u_{C1} + u_{C2}) + k_2 (i_{dc} – i_{L1} – i_{L2}) }
$$
where the fixed gain parameters are:
$$
k_1 = \frac{k_1 L + \alpha_3 L (K_p – 1)}{\alpha_1}, \quad
k_2 = \frac{\alpha_2 L / \alpha_1 – K_p L}{C}, \quad
k_3 = \frac{\alpha_3 L}{\alpha_1}
$$
The equivalent control $u_{eq}$ directly corresponds to the shoot-through duty cycle $D$, thus $0 < u_{eq} < 0.5$ must hold.
2.4 Existence Condition
The existence condition requires that the reaching condition $S\dot{S} < 0$ is satisfied locally. Substituting steady-state values for the state variables, we obtain:
$$
\begin{aligned}
& V_{in,\max} – k_1(u_{C1,ss} – u_{C1}^*) + k_2(i_{L1,\min} – i_{dc,ss}) + k_3[i_{L1,\min} – K_i\int(u_{C1,ss} – u_{C1}^*)dt] < u_{C1,ss} \\
& V_{in,\min} – k_1(u_{C1,ss} – u_{C1}^*) – k_2 i_{L2,\max} + k_3[i_{L1,\max} – K_i\int(u_{C1,ss} – u_{C1}^*)dt] < -u_{C2,ss}
\end{aligned}
$$
2.5 Stability Condition
To analyze stability, we substitute $u_{eq}$ into the system equations to obtain the ideal sliding dynamics. Linearizing around the equilibrium point yields the characteristic equation:
$$
\det(\mathbf{A}_S – \lambda \mathbf{I}) = 0
$$
The equilibrium is stable if all eigenvalues of $\mathbf{A}_S$ have negative real parts. The choice of sliding coefficients $\alpha_1, \alpha_2, \alpha_3$ and PI gains $K_p, K_i$ must satisfy both existence and stability conditions. The designed coefficients used in our simulation are listed in Table 2.
| Coefficient | Value |
|---|---|
| $\alpha_1$ | 0.5 |
| $\alpha_2$ | 0.2 |
| $\alpha_3$ | 100 |
| $K_p$ | 0.01 |
| $K_i$ | 50 |
3. Simulation Verification
To validate the proposed RBF-ISM control strategy, we built a simulation model in MATLAB/Simulink and compared its performance with a conventional SMC controller. The DC input voltage was stepped from 500 V to 600 V at 1 s and then back to 500 V at 1.5 s, creating transient conditions. Key simulation results are summarized in Table 3.
| Performance Metric | RBF-ISM | SMC |
|---|---|---|
| Overshoot in DC-link voltage during 500→600 V step | ≤2% | ~8% |
| Settling time (to within 2% steady state) | ~15 ms | ~35 ms |
| Current THD (steady state) | 1.2% | 1.8% |
| Chattering level in control signal | Very low (smooth) | Noticeable high-frequency ripple |
| Robustness to input voltage variation | Excellent (small overshoot) | Moderate (larger oscillations) |
The proposed RBF-ISM controller achieved faster dynamic response with minimal overshoot and reduced chattering. The DC-link voltage waveform under RBF-ISM exhibited a smoother transition compared to the oscillatory response of SMC. Active power and reactive power tracking were accurate with less fluctuation. These results demonstrate that the RBF-ISM strategy significantly improves the dynamic performance and robustness of the qZSI system, making it superior to conventional SMC for photovoltaic applications. Compared to other types of solar inverters such as two-stage VSI or traditional ZSI, the proposed method offers better transient performance and simpler structure.
4. Conclusion
In this paper, we proposed a novel control system for the quasi-Z-source inverter that integrates an RBF neural network with an integral sliding mode controller. The design includes a comprehensive sliding surface with current and voltage errors plus their integrals, and an adaptive RBF network that tunes the switching gain in real time to minimize chattering. The equivalent control law and existence/stability conditions were derived analytically. Simulation results confirmed that the RBF-ISM controller provides faster dynamic response, higher steady-state accuracy, and superior disturbance rejection compared to conventional SMC, especially under DC-side voltage fluctuations. This work demonstrates the effectiveness of combining intelligent neural networks with sliding mode techniques for high-performance grid-connected inverters. Future work will extend the concept to other types of solar inverters and consider multi-objective optimization.