# 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  
The schemes proposed in this article use the estimation method in [15] that estimates the parameter \(T(t)\) of the observed (master) system (1) from its free response with PO (2). The observation is made starting from the initial moment \(t_{o}\) when \(y\) has the value \(y_{o}\). \[T(t)\cdot\dot{y}(t)+y(t)=0,\;y(t_{o})=y_{o}\;. \tag{1}\]  
\[PO:\;\left\{ \begin{array}{l}z(t)=ln(y(t)),\\ \frac{d\hat{z}(t)}{dt}=\hat{c}(t),\;\;\hat{z}(0)=\hat{z}_{o},\\ \hat{c}(t)=z(t)-\hat{z}(t),\\ \hat{c}(t)=K_{p}\varepsilon(t)+K_{i}\int_{0}^{t}\hat{c}\left(\widetilde{t} \right)d\widetilde{t}+\hat{c}_{o}\;,\\ \hat{T}(t)=-\frac{1}{\hat{c}(t)}\;.\end{array} \right. \tag{2}\]  
PO (2) is a stable second-order tracking system of the variable \(z(t)=ln(y(t))\). It has the state variables \(\hat{z}\) and \(\hat{c}\). The variable \(\hat{c}\) provides through the opposite of its inverse the estimated \(\hat{T}\) of \(T\). The variable \(\varepsilon\) represents the tracking error, \(\hat{z}_{o}\) and \(\hat{c}_{o}\) are the initial values of the state variables, and \(K_{p}=2\cdot\omega_{0}\) and \(K_{i}=\omega_{0}^{2}\) are adjustable parameters of the PI block of the PO. They are adjusted so that the tracking process stabilizes as quickly as possible. For this purpose, \(\omega_{o}\) is chosen so that \(\tau\cdot\omega_{o}\geq 10\), \(\tau\) being the length of the time interval after which it is desired to install the tracking regime. The above principle is applied in Section 3 to the filter capacitors of two buck converters. For the first converter, the principle is used for a capacitor charge/discharge process, while for the second converter, the principle is only applied for processes of discharge, as in [15]. For the first scheme, the principle is applied to estimate the time constants \(T_{1}\) and \(T_{2}\) of a process with a variable structure of the form (3):  
\[T(t)\dot{v}(t)+v(t)=u(t),\;\;\text{with}\;\left\{ \begin{array}{l}T(t)=T_{1}, u(t)=0,t\in[t_{0},t_{1}),v(t_{0})=v_{0}\\ T(t)=T_{2},u(t)=K,\;t\in[t_{1},t_{2}]\end{array} \right.\;. \tag{3}\]  
The signals \(u(t)\) and \(v(t)\) are the input and output of the system (3). The modification of the structure occurs at the moment \(t_{1}\), simultaneously with the step variation, of amplitude \(K>0\), of the input signal. The initial state of the system is \(v_{0}\). Through a simple calculation, it results that the output signal has the expression (4), to which corresponds the curve in V from Figure 1. On the descending portion, the signal has the time constant \(T_{1}\), and on the ascending portion, the time constant \(T_{2}\). \[v(t)=\left\{ \begin{array}{l}v_{0}\cdot e^{-\frac{t-t_{0}}{t_{1}}},\quad t\in [t_{0},t_{1})\\ v_{0}\cdot e^{-\frac{t-t_{1}-t_{0}}{t_{1}}}+\left(K-v_{0}\cdot e^{-\frac{t_{1} -t_{0}}{t_{1}}}\right)\cdot\left(1-e^{-\frac{t-t_{1}}{t_{2}}}\right),\quad t \in[t_{1},t_{2}]\end{array} \right. \tag{4}\]  
Figure 1: V curve of Equation (4). To adapt the PO to an input signal of the form (4), we use the fact that this signal can be considered as the output of the exogenous system with variable structure (5). \[\left\{ \begin{array}{ll}T(t)\!:\!\dot{y}(t)+y(t)=0,&y(t_{0})=v_{o}, \quad\quad T(t)=\left\{\begin{array}{ll}T_{1},&t\in[t_{0},t_{1})\\ T_{2},&t\in[t_{1},t_{2}]\end{array}\right.\\ v(t)=\left\{\begin{array}{ll}y(t),&t\in[t_{0},t_{1})\\ K-y(t),&t\in[t_{1},t_{2}]\end{array}\right.\end{array} \right. \tag{5}\]  
If the signal \(v(t)\) of system (3) is measurable, then interpreting it also as the output of system (5) allows \(T(t)\) to be estimated using a PO. The statement is based on two observations: (i) if the signal \(v(t)\) is measurable, then the signal \(y(t)\) can be determined from it according to Equation (6) obtained from the system output Equation (5); (ii) given the signal \(y(t)\) the parameter \(T(t)\) can be estimated using a PO based on the state equation in (5). \[y(t)=\left\{