# 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  
Recall that the output of the converter is given by  
\[y(t)=C_{0}x(t)+D_{0}(x(t))\lambda(t)=\sum_{i=1}^{M}\lambda_{i}(t)(C_{0}+C_{i}) x(t)\]  
where the notations are the ones introduced in Section 2. In Section 3, we see that a reference \(x_{ref}(t)\) is admissible if it is a solution of the relaxed converter model recalled below  
\[\dot{x}(t)=A_{0}x(t)+B_{0}(x(t))\lambda(t)=\sum_{i=1}^{M}\lambda_{i}(t)(A_{0}+ A_{i})x(t)+\sum_{i=1}^{M}\lambda_{i}(t)(B_{0}+B_{i})v_{in}\]  
for \(\lambda(t)=\lambda_{ref}(t)\in\Lambda_{R}\). The output reference signal is known because it is the signal that the output must track. We suppose that it is a periodic signal written as  
\[y_{ref}(t)=C_{0}x_{ref}(t)+D_{0}(x_{ref}(t))\lambda_{ref}(t) \tag{23}\]  
The problem is to deduce the state reference signal from the knowledge of the output reference signal \(y_{ref}(t)\). In general, the maximal frequency contained in the spectrum of the reference \(y_{ref}\) is significantly lower than the bandwidth of the converter. Then, the output reference can be seen as a succession of equilibrium points, and, if the output reference \(y_{ref}(t)\) is admissible, there exists a function \(\lambda_{app}\) such that \(\lambda_{app}(t)\in\Lambda_{R}\) and  
\[y_{ref}(t)=\sum_{i=1}^{M}\lambda_{app_{i}}(t)(C_{0}+C_{i})x_{app}(t) \tag{24}\]  
where, for all \(t\), \(x_{app}(t)\) is the solution of the following equation:  
\[A_{0}x_{app}(t)+B_{0}(x_{app}(t))\lambda_{app}(t)=0 \tag{25}\]  
Now, supposing that the conditions of Lemma 1 are satisfied, from the previous equation, we can deduce that  
\[x_{app}(t)=\left[\sum_{i=1}^{M}\lambda_{app_{i}}(t)(A_{0}+A_{i})\right]^{-1} \left[\sum_{i=1}^{M}\lambda_{app_{i}}(t)(B_{0}+B_{i})\right] \tag{26}\]  
the invertibility being guaranteed by the conditions of Lemma 1. Replacing in (24), we obtain  
\[y_{ref}(t)=\left[\sum_{i=1}^{M}\lambda_{app_{i}}(t)(C_{0}+C_{i})\right]\left[ \sum_{i=1}^{M}\lambda_{app_{i}}(t)(A_{0}+A_{i})\right]^{-1}\left[\sum_{i=1}^{ M}\lambda_{app_{i}}(t)(B_{0}+B_{i})\right]v_{in}=G(\lambda_{app}(t))v_{in} \tag{27}\]  
\(G(\lambda_{app})\) is simply the gain of the converter for \(\lambda=\lambda_{app}\). In general, if \(y_{ref}(t)\) is periodic of period \(T\), \(\lambda_{app}(t)\) is also periodic of period \(T\). We suppose that this is the case. We can use \(\lambda_{app}(t)\) to generate an approximated reference \(x_{a}(t)\) as the solution of the following differential system:  
\[\left\{ \begin{array}{rcll}\hat{x}_{a}(t)&=&A_{0}x_{a}(t)+B_{0}(x_{a}(t)) \lambda_{app}(t)&=&\sum_{i=1}^{M}\lambda_{app_{i}}(t)(A_{0}+A_{i})x_{a}(t)+ \sum_{i=1}^{M}\lambda_{app_{i}}(t)(B_{0}+B_{i})v_{in}\\ &&=&\mathcal{A}_{\lambda_{app}}x_{a}(t)+\mathcal{B}_{\lambda_{app}}v_{in}\\ y_{a}(t)&=&C_{0}x(t)+D_{0}(x_{a}(t))\lambda_{app}(t)&=&\sum_{i=1}^{M}\lambda_{ app_{i}}(t)(C_{0}+C_{i})x_{a}(t)\\ &&=&\mathcal{C}_{\lambda_{app}}x_{a}(t),\quad x_{a}(0)=x_{app}(0)\end{array} \right. \tag{28}\]  
Summarizing, the approximated state reference signal is determined as follows:  
1. From the given periodic output reference \(y_{ref}(t)\), and \(G(\lambda_{app})v_{in}\), we can compute \(\lambda_{app}(t)\);
2. From \(\lambda_{app}(t)\) obtained in step i), we can compute \(x_{a}(t)\), integrating the differential system (28), because \(\lambda_{app}(t)\in\Lambda_{R}\), \(x_{a}(t)\) is an admissible reference. The computation can be performed off-line and, because of the periodicity of the signals, it can be restricted to a period and stored in memory. This last aspect is important for implementation issues. When using the approximated reference, it is important to quantify the impact on the tracking error. To evaluate this, note that, combining (25) and (28), we have  
\[\left\{ \begin{array}{rcl}\hat{x}_{a}(t)&=&\mathcal{A}_{\lambda_{app}}(x_{a }(t)-x_{app}(t))\\ \Delta y(t)&=&y_{a}(t)-y_{ref}(t)=\mathcal{C}_{\lambda_{app}}(x_{a}(t)-x_{ app}(t)),\quad x_{a}(0)=x_{app}(0)\end{array} \right.