# Robust Optimal Tracking Control of a Full-Bridge DC-AC Converter
## 2 Modeling Description of Full-Bridge DC-AC Converter  
Figure 1a depicts the full-bridge DC-AC converter, which is composed of three important units, namely the transistor switching elements, an inductor-capacitor low-pass filter, and a connected load; Figure 1b shows the gate control signals produced by the comparison of a reference sine waveshape and a high-frequency triangular carrier waveshape, which corresponds to DC-AC converter switches. Then, the DC bus voltage is defined as \(V_{dc}\), the \(R\) implies the resistive load, the output voltage, and current can be expressed as \(v_{o}\) and \(i_{o}\), respectively. Analyzing the voltage and current in Figure 1a using Kirchhoff's law and making the state variables \(x_{1}=v_{o}\) and \(x_{2}=\dot{v}_{o}\), then the state behavior of the full-bridge DC-AC converter can be expressed as:  
\[\left\{\begin{array}{l}x_{1}=\dot{x}_{2}\\ \dot{x}_{2}=-\frac{1}{LC}x_{1}-\frac{1}{RC}x_{2}+\frac{1}{LC}u\end{array} \right., \tag{1}\]  
where the control signal \(u\) equals \(d\cdot V_{dc}\), here \(d\) is duty ratio of switching limited amidst \(-1\) and \(1\).  
The design problem with the full-bridge DC-AC converter is a typical tracking control problem. Because the output voltage of the full-bridge DC-AC converter is a sinusoidal AC waveform, it must track the required reference AC waveform \(v_{\textit{r}ac}=\sqrt{2}\cdot V_{\textit{r}ms}\cdot\sin(\omega t)\)  
Figure 1: Common full-bridge DC-AC converter structure (**a**) Circuit diagram. (**b**) Switching pulse pattern.  
where \(V_{rms}\) denotes the root-mean-square value and \(\omega\) is the angular frequency. Therefore, based on the relationship amidst (1) and \(v_{rac}\), the error state variables can be defined as:  
\[\left\{\begin{array}{l}e_{1}=x_{1}-v_{rac}\\ e_{2}=x_{2}-\tilde{v}_{rac}\end{array}\right.. \tag{2}\]  
Then, using the (1) and (2), the error state equation for the full-bridge DC-AC converter can be written as follows:  
\[\left\{\begin{array}{l}\dot{e}_{1}=\dot{x}_{1}-\dot{v}_{rac}\ =e_{2}\\ \dot{e}_{2}=\ddot{x}_{1}-\ddot{v}_{rac}=\dot{x}_{2}-\ddot{v}_{rac}=-\frac{1}{ LC}e_{1}-\frac{1}{RC}e_{2}+\frac{1}{LC}u-\frac{1}{LC}v_{rac}-\frac{1}{RC} \dot{v}_{rac}-\ddot{v}_{rac}\end{array}\right.. \tag{3}\]  
The error dynamics of the (3) reveals that the control signal \(u\) needs to be carefully and well designed. In this way, the system states \(e_{1}\) and \(e_{2}\) will quickly reach and converge to the equilibrium point within a limited period of time. It is worth noting that in the path tracking control design, once the NFCTA establishes a fast limited-time closed-loop convergence, the system response exhibits strong insensitivity and eternal stability. The NFCTA is used in the control of full-bridge DC-AC converters not only to allow faster dynamic response, but also to afford more precise control of the steady state. In actual implementation, there is the possibility of drastic changes in the internal parameters of the plant, severe interference from external loads, and highly nonlinear uncertainties, thereby leading to system trembling or great steady-state errors. The CPSO has been considered as a useful methodology to overcome the problems of trembling and steady-state errors in practical non-linear system applications. Based on chaotic mapping, the CPSO makes implementation easier and will not converge untimely, so the system can avoid falling into the dilemma of local optimization. With the smart combination of NFCTA and CPSO, the controller parameters of the NFCTA can be optimized globally and produce a full-bridge DC-AC converter with higher performance.