# Robust Optimal Tracking Control of a Full-Bridge DC-AC Converter
## 3 Proposed Control Strategy  
For ease of analysis, the second-order uncertain nonlinear error state system of the (3) can be reformulated as:  
\[\left\{ \begin{array}{l}\dot{e}_{1}=e_{2}\\ \dot{e}_{2}=f+\widetilde{w}+bu-\tilde{v}_{rac}\end{array} \right., \tag{4}\]  
where \(f=-\frac{1}{LC}e_{1}-\frac{1}{RC}e_{2}\), \(b=\frac{1}{LC}u\), \(u\) connotes the control input, and \(\widetilde{w}=-\frac{1}{LC}v_{rac}-\frac{1}{RC}\dot{v}_{rac}\) implies the internal parameter changes of the plant, the disturbances of external load, and the uncertainties of nonlinearity. The \(\widetilde{w}\) is expressed as the limitation of an inequality \(\|\widetilde{w}\|\leq\phi\), where \(\phi\) is positive constant. Aiming at the error state Equation (3), the sliding surface of the NFCTA can be designed as follows to ensure faster convergence without singularities and to track the reference sine wave more accurately:  
\[\sigma=e_{1}+g|e_{1}|^{m_{1}}sign(e_{1})+h|e_{2}|^{m_{2}}sign(e_{2}), \tag{5}\]  
where \(g\), \(h>0\), \(m_{1}>1\), and \(sign(\cdot)\) is approximated by \(\tanh(\cdot)\). \(1<m_{2}<2\). The (5) implies that the state is insensitive to changes in system parameters and external loading interferences during sliding mode action (sliding phase). Nevertheless, when the state trajectory is forced towards the sliding surface (reaching phase), the fast limited-time convergence and robustness is not always guaranteed; [PERSON] and [PERSON] first proposed a new concept of sliding-mode reaching law to solve this dilemma [76], and many researchers have improved classic sliding-mode reaching law for various applications [77; 78; 79; 80; 81]. Based on the principle in the [76] and [77], we reconstruct a power reaching law-based sliding mode as follows:  
\[\dot{\sigma}=-\gamma_{1}|\sigma|^{P_{1}}sign(\sigma)-\gamma_{2}|\sigma|^{P_{2 }}sign(\sigma)-\gamma_{3}|\sigma|^{P_{3}}\sigma, \tag{6}\]  
where \(\gamma_{1}\), \(\gamma_{2}\), \(\gamma_{3}>0\), \(0<p_{1}<1\), \(p_{2}>1\), \(p_{3}>0\), and \(sign(\cdot)\) is approximated by \(\tanh(\cdot)\)  
zone near the stagnant particles' location. Due to the stochastic nature of particles during initialization and evolution, updates are seemingly purposeless. Once \(X_{gbest}\) falls into a local extreme, then the entire swarm will show precocious convergence, reducing search performance. Owing to the non-repeatability of chaos, the CPSO can explore according to probability at a higher speed than random and traversal investigations. Therefore, a typical Logistic mapping adopted for chaos can be defined as:  
\[\chi(t+1)=\beta\cdot\chi(k)\cdot(1-\chi(t)), \tag{12}\]  
where the symbol \(\beta\) represents the control coefficient, which is usually set to 4 [58; 59; 60]. The speed update equation of the CPSO can be expressed as:  
\[V(t+1)=\Omega V(t)+\xi_{1}\cdot\chi(X_{gbest}(t)-X(t))\ +\xi_{2}\cdot\chi(X_{gbest}-X(t)), \tag{13}\]  
where \(\chi\) is a function based on the result of logistic mapping, and its value is amidst 0 and 1 [58; 59; 60]. Since this algorithm can achieve the ability to escape from the best solution in the local region, there is not much randomness and therefore, the solution changes within the present solution range. The \(\chi(t)\) can be set to be generated randomly amidst 0 and 0.1, thus allowing the particles to escape from the extreme value of the local region without too much randomness and difficulty in convergence. The process of the algorithm is represented in the following, and its Matlab code can refer to the literature [82]. Step 1: The initialization of position, velocity, and inertia weight is executed by the chaotic operation. The best individual position \(X_{gbest}\) as well as the best global position \(X_{gbest}\) are computed. An integrated absolute error (\(\text{IAE}=\int_{0}^{\infty}|\epsilon_{1}(\tau)|d\tau\)) is use as the objective function. Step 2: According to the (10) and (11), the particle's velocity and position can be updated.