# DC-DC Buck Converters with Quasi-Online Estimation of Filter Capacitor Equivalent Parameters
## 2 Materials and Methods
### Analytical Support of the Estimation Methods Used in the Schemes Proposed in This Article
\begin{array}{ll}v(t),&t\in[t_{0},t_{1})\\ K-v(t),&t\in[t_{1},t_{2}]\end{array} \right.. \tag{6}\]  
The block diagram in Figure 2 is obtained by concatenating observations (i) and (ii). Block S contains the observed system (OS) with model (3) and the operations in (6) necessary to obtain \(y(t)\). The K and 0 blocks generate the constant values written inside. SW is a switch initially in position 2. At time \(t_{1}\) it switches to position 1. Figure 3 exemplifies the behavior of the system in Figure 2 when \(v(t)\) has Equation (7). The signal \(v(t)\) is represented in Figure 3a and the system response in Figure 3b. The results were obtained by simulation in the Matlab/Simulink environment. \[v(t)=\left\{ \begin{array}{ll}2.5\!\cdot\!e^{-\frac{t}{0.002}}, \quad\quad t\in[0,\ 0.008)\\ 2.5\!\cdot\!e^{-4}+\left(\underbrace{3}_{K}-2.5\!\cdot\!e^{-4} \right)\!\cdot\!\left(1-e^{-\frac{t-0.008}{0.001}}\right),\quad\ t\in[0.008,\ 0.014]\end{array} \right. \tag{7}\]  
Figure 3: The behavior of the system in Figure 2 when \(v(t)\) has the Equation (7): (**a**) input signal; (**b**) the estimated time constant \(\hat{T}(t)\). Figure 2: The connection of PO to the observed system OS in order to estimate the parameter \(T(t)\). For \(t\in[t_{0},t_{1})\) the switch SW is in position 2, and for \(t\in[t_{1},t_{2}]\) in position 1. From the perspective of Equation (4) and Figure 2, signal (7) is characterized by the parameters \(T_{1}=2\cdot 10^{-3}\) s, \(T_{2}=10^{-3}\) s, \(K=3\), \(t_{0}=0\) s, \(t_{1}=8\cdot 10^{-3}\) s, and PO (2) by the parameters \(\hat{z}_{o}=0,\hat{c}_{o}=0\), \(\tau=0.001\) s, \(\omega_{0}=50/\tau\), \(K_{p}=100,000\) s\({}^{-1}\), \(K_{i}=2\cdot 10^{9}\) s\({}^{-2}\). As expected, PO returns the values \(\hat{T}_{1}=2\cdot 10^{-3}\) s and \(\hat{T}_{2}=10^{-3}\) s with a short delay due to the transients, unavoidable regimes that occur at the start of the estimation and at the change of the monotony of signal \(v(t)\). Omitting the OS block, the structure in Figure 2 can be integrated into a new PO as in Figure 4a. This new structure, intended to operate both on the falling edge of the \(v(t)\) signal and on the rising edge, we call it a parameter observer on two edges (PO2). In the practical case, the processing of the signal \(v(t)\) is carried out in discrete time, so the connection of PO2 to the OS is carried out as in Figure 4b, PO2 operating according to Equation (8). The used notations have the following meanings: \(h\) is the sampling time with which \(v(t)\) is acquired and operates PO2, IA--Input Adapter, I--Integrator (modified model of OS), C--Comparator (provides tracking error \(\varepsilon\)), PI--Tracking Controller type PI, OA--Output Adapter. \[\text{PO2}:\ \left\{ \begin{array}{l}(IA):y[k]=\left\{\begin{array}{cc}v[k],&t \in[t_{c},t_{d})\\ K-v[k],&t\in[t_{d},t_{f}]\end{array}\right.,\ z[k]=\ln(y[k]),\\ (I):\ \hat{z}[k]=\hat{z}[k-1]+h\cdot\varepsilon[k],&\hat{z}[0]=\hat{z}_{o},\\ (C):\ \varepsilon[k]=z[k]-\hat{z}[k],\\ (PI):\ \hat{c}[k]=K_{p}(\varepsilon[k]-\varepsilon[k-1])+hK_{i}\varepsilon[k-1]+ \hat{c}[k-1],\ \hat{c}[0]=\hat{c}_{o},\\ (OA):\ \hat{T}[k]=-\frac{1}{\hat{c}[k]}.\end{array} \right. \tag{8}\]