# Sensorless Control for DC-DC Boost Converter via Generalized Parameter Estimation-Based Observer
## 3 Sensorless Controller Design
### PI-PBC Design  
First, the PI-PBC is proposed to stabilize the boost converter. Here, we assume that \(x_{1}\) is measured. **Proposition 1**. : _A PI-PBC is designed as_  
\[\left\{ \begin{aligned} \dot{x}_{c}&=\bar{y},\\ u&=-k_{p}\bar{y}-k_{i}x_{c}\end{aligned} \right. \tag{7}\]  
_with the passive output_  
\[\bar{y}=\bar{x}^{T}Qg_{\star}, \tag{8}\]  
_where \(g_{\star}=\mathcal{J}Qx_{\star}\). For all \(x_{\star}\in\mathcal{E}\), the equilibrium_  
\[(x,x_{c})=(x_{\star},-\frac{1}{k_{i}}u_{\star})\]  
_is globally exponentially stable with the following condition_  
\[\left\{ \begin{aligned} \varepsilon& <\max\biggl{\{}\frac{2}{|g_{\star}|\sqrt{k_{i}Q}},\frac{k_{p}}{k_{i}}\biggr{\}}, \\ \mathcal{R}+k_{p}|g_{\star}|^{2}&>\varepsilon \biggl{(}\frac{bb^{T}}{4|g_{\star}|^{2}|Q|^{2}}+k_{i}|g_{\star}|^{2}\biggr{)},\end{aligned} \right. \tag{9}\]  
_where \(\varepsilon\) is a free constant, \(|g_{\star}|^{2}:=g_{\star}g_{\star}^{T},|Q|^{2}:=QQ^{T}\)._  
Proof. : First, for convenience, \(g_{\star}=\mathcal{J}Qx_{\star}\) is defined. To obtain the error dynamics, subtracting Equation (6) from Equation (5) gives  
\[\left\{ \begin{aligned} \dot{\bar{x}}&=(\mathcal{J}u- \mathcal{R})Q\bar{x}+\bar{u}g_{\star}\\ \bar{y}&=\bar{x}^{T}Qg_{\star}.\end{aligned} \right. \tag{10}\]  
In addition, the PI-PBC (Equation (7)) can be rewritten as the error form  
\[\left\{ \begin{aligned} \dot{\bar{x}}_{c}&=\bar{x}^{T}g_{ \star},\\ \bar{u}&=-k_{p}\bar{x}^{T}g_{\star}-k_{i}\bar{x}_{c}. \end{aligned} \right. \tag{11}\]  
Defining the energy function  
\[W=\frac{1}{2}\bar{x}^{T}Q\bar{x}+\frac{k_{i}}{2}\bar{x}_{c}^{T}\bar{x}_{c},\]the closed-loop system combining Equation (10) with Equation (11) can be expressed as  
\[\left[ \begin{array}{c}\dot{\bar{x}}\\ \dot{\bar{x}}_{c}\end{array} \right]=\left[ \begin{array}{cc}\mathcal{J}u- \mathcal{R}_{a}&-\bar{s}_{*}^{T}\\ g_{*}&0\end{array} \right]\
abla W, \tag{12}\]  
where \(\mathcal{R}_{a}=\mathcal{R}+k_{p}g_{*}g_{*}^{T}\). Next, the exponential stability of the closed-loop system (12) should be proved. Towards this end, the following Lyapunov function is chosen. \[V=W+\varepsilon k_{i}\bar{x}_{c}g_{*}^{T}\bar{\pi},\]  
which can be also expressed as  
\[V=\frac{1}{2}\underbrace{\left[ \begin{array}{cc}Q\bar{x}^{T}&k_{i}\bar{x}_ {c}\\ \
abla^{T}W\end{array} \right]}_{\
abla^{T}W}\underbrace{\left[ \begin{array}{ cc}\frac{1}{Q}&\frac{\bar{\varepsilon}g_{*}}{2}\\ \frac{\varepsilon}{2}g_{*}^{T}&\frac{1}{\bar{x}_{i}}\end{array} \right]}_{M} \underbrace{\left[ \begin{array}{c}Q\bar{\varepsilon}\\ k_{i}\bar{x}_{c}\end{array} \right]}_{\
abla W}. \tag{13}\]  
Differentiating the function \(V\) along the trajectory (Equation (12)) with respect to time gives  
\[\dot{V} =-\bar{x}^{T}\mathcal{R}_{a}\bar{x}+\varepsilon k_{i}\bar{x}_{c}g _{*}^{T}[(Ju-R_{a})Q\bar{x}-g_{*}k_{i}x_{c}]+\varepsilon k_{i}\bar{x}^{T}g_{*} g_{*}^{T}\bar{x}\] \[=-\bar{x}^{T}\Big{[}\mathcal{R}+(k_{p}-\varepsilon k_{i})g_{*}g_{ *}^{T}\Big{]}\bar{x}-\varepsilon k_{i}^{2}g_{*}^{T}g_{*}\bar{x}_{c}^{T}\bar{x} _{c}+\varepsilon k_{i}g_{*}^{T}(Ju-R_{a})\bar{x}\bar{x}_{c}\] \[=-\big{[} \begin{array}{cc}Q\bar{x}^{T}&k_{i}\bar{x}_{c}\end{array}  \big{]}\underbrace{\left[ \begin{array}{cc}\frac{\mathcal{R}+(k_{p}- \varepsilon k_{i})g_{*}g_{*}^{T}}{QQ^{T}}&-\frac{\varepsilon b}{2Q}\\ \frac{b\bar{b}^{T}}{2Q}&\varepsilon g_{*}^{T}g_{*}\end{array} \right]}_{N} \left[ \begin{array}{c}Q\bar{x}\\ k_{i}\bar{x}_{c}\end{array} \right]\] \[=-\
abla^{T}WN\
abla W,\]  
where \(b^{T}=g_{*}^{T}(\mathcal{J}u-\mathcal{R}_{a})\). Hence, the condition \(M>0\) should be satisfied to ensure that the function \(V\) is a Lyapunov function. Moreover, the closed-loop system is exponentially stable as long as \(N>0\). The sufficient condition is constructed as follows. \[ \begin{cases}M>0\iff\varepsilon<\frac{2}{|g_{*}|\sqrt{k_{i}Q}},\\ N>0\iff\left\{\begin{array}{cc}\frac{k_{p}}{k_{i}}>\varepsilon\\ \mathcal{R}_{a}>\varepsilon\Big{(}\frac{bb^{T}}{4 QQ^{T}g_{*}g_{*}^{T}}+k_{i}g_ {*}g_{*}^{T}\Big{)}\end{array}\right.\end{cases}