# Sensorless Control for DC-DC Boost Converter via Generalized Parameter Estimation-Based Observer
## 3 Sensorless Controller Design
### GPEBO Design  
The state \(x_{2}\) is measured. To pose the observer design problem, we define a matrix \(C=[0\,1]^{T}\) such that the vector  
\[y_{m}=Cx, \tag{15}\]  
is available for measurement.  
To streamline the presentation of the result, we define the matrix  
\[\Lambda(u):=(\mathcal{J}u-\mathcal{R})Q, \tag{16}\]  
and make the following interval excitation assumption on a scalar signal \(\Delta\), which is defined below.  
**Assumption 1**.: _Fix a small constant \(\mu\in(0,1)\) and a positive number \(\gamma\). There is a time \(t_{c}\in(0,\infty)\) such that_  
\[\int_{0}^{t_{c}}\Delta^{2}(\tau)d\tau\geq-\frac{1}{\gamma}\ln(1-\mu). \tag{17}\]  
A GPEBO is designed to estimate the unmeasured state \(x_{1}\). The main idea of this technique is to transform the estimation problem of the state to observe the parameter.  
**Proposition 2**.: _Consider the system (5) with the measurable output \(y_{m}\) (15). Define the GPEBO via_  
\[\dot{\xi} =\Lambda(u)\xi+\zeta \tag{18}\] \[\dot{\Phi} =\Lambda(u)\Phi,\ \Phi(0)=I_{n}\] (19) \[\dot{Y} =-\lambda Y+\lambda\Phi^{\top}C^{\top}(C\xi-y_{m})\] (20) \[\dot{\Omega} =-\lambda\Omega+\lambda\Phi^{\top}C^{\top}C\Phi\] (21) \[\dot{\omega} =-\gamma\Delta^{2}\omega,\ \omega(0)=1\] (22) \[\dot{\theta} =\gamma\Delta(\mathcal{Y}-\Delta\hat{\theta}), \tag{23}\]  
_with \(\lambda>0,\ \gamma>0\), free tuning parameters and_  
\[\mathcal{Y}=adj\{\Omega\}Y,\ \Delta=\det\{\Omega\}, \tag{24}\]  
_where \(adj\{\cdot\}\) is the adjunct matrix. The state estimate_  
\[\hat{x}=\xi+\Phi\hat{\theta}_{\text{FCT}} \tag{25}\]  
_where we introduced the vector_  
\[\hat{\theta}_{\text{FCT}}:=\frac{1}{1-\omega_{c}}\big{[}\hat{\theta}-\omega_{ c}\hat{\theta}(0)\big{]} \tag{26}\]  
_and \(\omega_{c}\) is defined via the clipping function_  
\[\omega_{c}=\begin{cases}\omega&\text{if}\ \ \omega\leq 1-\mu,\\ 1-\mu&\text{if}\ \ \omega>1-\mu.\end{cases}\]  
_ensures that, for all initial conditions \((\xi(0),Y(0),\Omega(0),\hat{\theta}(0))\in\mathbb{R}^{2}\times\mathbb{R}^{2 }\times\mathbb{R}^{2\times 2}\times\mathbb{R}^{2}\), we have that_  
\[\hat{x}(t)=x(t),\ \ \forall t>t_{c} \tag{27}\]  
_with all signals bounded provided \(\Delta\) verifies the above assumption._  
Proof.: Define the signal \(e:=x-\xi\). From Equations (5) and (18), one obtains the linear time-varying (LTV) system  
\[\dot{e}=A(t)e,\]  
where we have defined \(A(t):=\Lambda(u(t))\). The state transition matrix of this LTV system satisfies Equation (19) and, consequently,  
\[e:=\Phi\theta,\]  
To prove the boundedness of \(\Phi\), consider the quadratic function \(U(\Phi)=\frac{1}{2}\text{tr}\{\Phi^{\top}L\Phi\}\) whose derivative, along the trajectories of Equation (19), is  
\[\dot{U} =\sum_{i=1}^{n}\Phi_{i}^{\top}L\Lambda(u)\Phi_{i}\] \[=\sum_{i=1}^{n}\Phi_{i}^{\top}LRL\Phi_{i}\leq-cU, \tag{38}\]  
where \(\text{tr}\{\cdot\}\), denotes the trace, \(\Phi_{i}\in\mathbb{R}^{2}\) is the \(i\)-th column of the matrix \(\Phi\) and \(c>0\). Now, from Equation (21), we can see that boundedness of \(\Phi\) implies boundedness of \(\Omega\). Finally, the fact that \(Y\) is bounded follows from the fact that \(Y=\Omega\theta\). This completes the proof.