# A Unified Approach for the Control of Power Electronics Converters. Part II: Tracking
## 7 Approximated State Reference Signal from a Periodic Output Reference Signal
### Approximated State Reference Signal
\tag{29}\]  
The error \(\Delta y(t)\) between \(y_{a}(t)\) and \(y_{ref}(t)\) is the error induced by the approximated state reference signal. **Theorem 4**. : _Suppose that the conditions of Lemma 1 are satisfied. Consider an admissible output periodic reference signal of period \(T\), \(y_{ref}(t)\), and consider the associated approximated state _reference trajectory_ \(x_{a}(t)\) _obtained by integrating differential system (_28_). Then, the following LMI optimization problem_  
\[ \begin{array}{c}\min\limits_{\mu>0,S=S^{T}>0}\mu\\ \left[\begin{array}{cc}(A_{0}+A_{i})^{T}S+S(A_{0}+A_{i})&-S&(C_{0}+C_{i})^{T} \\ -S&-\mu I&0\\ (C_{0}+C_{i})&0&-I\end{array}\right]\leq 0,\ i=1,\ldots,M\\ \end{array}  \tag{30}\]  
_is feasible and, if \((\tilde{\mu},\tilde{S})\) denotes its optimal solution, the control_  
\[\lambda(t)=\arg\biggl{\{}\min\limits_{d\in\Lambda_{S}}e^{T}(t)S\big{[}A_{0}e( t)+B_{0}(x(t))d-B_{0}(x_{a}(t))\lambda_{app}(t)\big{]}\biggr{\}}=\arg\biggl{\{} \min\limits_{d\in\Lambda_{S}}e^{T}(t)\tilde{S}B_{0}(x(t))d\biggr{\}} \tag{31}\]  
_solves problem 1 for \(x_{ref}(t)=x_{a}(t)\). In addition, when \(t\to\infty\), it holds that_  
\[\int_{t}^{t+T}\|y_{a}(t)-y_{ref}(t)\|^{2}dt\leq\bar{\mu}\int_{t}^{t+T}\biggl{\|} \frac{d\lambda_{app}(t)}{dt}\biggr{\|}^{2}dt \tag{32}\]  
_where \(x_{app}(t)\) satisfies (25)._  
Proof. : If the conditions of Lemma 1 are satisfied, there exist positive definite symmetric matrices \(Q\) and \(P\) such that  
\[(A_{0}+A_{i})^{T}P+P(A_{0}+A_{i})+2Q<0,\ \ i=1,\ldots,M\]  
There exists \(\alpha>0\) such that \(2\alpha Q\geq(C_{0}+C_{i})^{T}(C_{0}+C_{i})\) for \(i=1,\ldots,M\) and we have  
\[(A_{0}+A_{i})^{T}\alpha P+\alpha P(A_{0}+A_{i})+(C_{0}+C_{i})^{T}(C_{0}+C_{i} )\leq(A_{0}+A_{i})^{T}\alpha P+\alpha P(A_{0}+A_{i})+2\alpha Q<0\]  
Denote \(S=\alpha P\). Then, there exists \(\mu>0\) sufficiently large such that for \(i=1,\ldots,M\)  
\[(A_{0}+A_{i})^{T}S+S(A_{0}+A_{i})+\mu^{-1}S^{2}+(C_{0}+C_{i})^{T}(C_{0}+C_{i} )<0\]  
which, by the Schur complement [49], can be transformed into the inequalities in (30). Then, Problem (30) has a solution denoted \((\bar{\mu},\tilde{S})\). From Lemma 1 and the admissibility of approximated reference \(x_{a}(t)\), the control (31) solves problem 1 for \(x_{ref}=x_{a}\). From Lemma 1, we can also conclude that (28) is an asymptotically stable T-periodic system. It results from Theorem 4.7, p. 101 in [52] or Theorem 1 in [53] that there exists a unique steady-state periodic solution \(x_{s}(t)\) to (28). This means that when \(t\to\infty\),  
\[x_{a}(t)\to x_{s}(t)\ \text{where}\ x_{s}(t+T)=x_{s}(t)\]  
Now, consider the function \(V(x_{a},x_{app})=(x_{a}-x_{app})^{T}S(x_{a}-x_{app})/2\). We have  
\[\tilde{V}=(x_{a}-x_{app})^{T}\tilde{S}\dot{x_{a}}+{x_{a}}^{T}\tilde{S}(x_{a}-x _{app})-(x_{a}-x_{app})^{T}\tilde{S}\dot{x}_{app}+\dot{x}_{app}^{T}\tilde{S}(x_ {a}-x_{app})\]  
From (29), \(\dot{x}_{a}=\mathcal{A}_{\lambda_{app}}(x_{a}-x_{app})\); then,  
\[\dot{V}=(x_{a}-x_{app})^{T}\Big{(}\mathcal{A}_{\lambda_{app}}^{T}\tilde{S}+ \tilde{S}\mathcal{A}_{\lambda_{app}}\Big{)}(x_{a}-x_{app})-2(x_{a}-x_{app})^{T} \tilde{S}\dot{x}_{app}\]  
Replace \(S\) with \(\tilde{S}\), \(\mu\) by \(\bar{\mu}\) in the inequalities (30). Multiplying each of them by \(\lambda_{app_{i}}(t)\), summing the obtained terms and taking into account that \(\lambda_{app}\in\Lambda_{R}\), we obtain  
\[\left[ \begin{array}{ccc}\mathcal{A}_{\lambda_{app}}^{T}\tilde{S}+\tilde{S} \mathcal{A}_{\lambda_{app}}&-\tilde{S}&\mathcal{C}_{\lambda_{app}}^{T}\\ -\tilde{S}&-\bar{\mu}I&0\\ \mathcal{C}_{\lambda_{app}}&0&-I\end{array} \right]\leq 0\]which, by the Schur complement, is also equivalent to  
\[\left[ \begin{array}{cc}\mathcal{A}_{A_{app}}^{T}\bar{S}+\bar{S}\mathcal{A}_{ \lambda_{app}}+\mathcal{C}_{\lambda_{app}}^{T}\mathcal{C}_{\lambda_{app}}&-\bar {S}\\ -\bar{S}&-\bar{\mu}I\end{array}