## 4 Approximate Prediction of Fast-Scale Instability and Adaptive Slope Compensation
### Fast-Scale Stability Analysis  
The stability analysis at the fast scale can be performed by linearizing (7) together with (16) resulting in the following small-signal model  
\[\partial_{n+1}\approx\lambda(\theta)\partial_{n} \tag{17}\]  
where \(\lambda\) is the total derivative of \(\sigma_{n+1}\) with respect to \(\sigma_{n}\) that can be expressed as follows  
\[\lambda(\theta):=\frac{\mathrm{d}\sigma_{n+1}}{\mathrm{d}\sigma_{n}}=\frac{ \partial\sigma_{n+1}}{\partial\sigma_{n}}+\frac{\partial\sigma_{n+1}}{ \partial d_{n}}\frac{\partial d_{n}}{\partial\sigma_{n}}=\frac{R_{s}m_{0}( \theta)+m_{r}}{R_{s}m_{1}(\theta)+m_{r}} \tag{18}\]  
The solution of (17) starting from an initial condition \(\sigma_{0}\) can be expressed as follows  
\[\partial_{n}\approx\lambda^{n}(\theta)\partial_{0} \tag{19}\]  
Based on the previous equation, the current loop stability at the fast scale can be ensured if \(|\lambda(\theta)|<1\) since in this case a small initial disturbance will vanish as \(n\) increases. In quasi-steady-state operation one has that \(\sigma_{n+1}\approx\sigma_{n}\), \(d_{n}=D(\theta)\) and according to (7) one has \(m_{1}D(\theta)\approx-m_{0}(1-D(\theta))\). Consequently, the expression of \(\lambda(\theta)\) in (18) becomes as follows  
\[\lambda(\theta)=\frac{R_{s}m_{0}(\theta)+m_{r}}{-\frac{R_{s}m_{0}(\theta)(1- D(\theta))}{D(\theta)}+m_{r}} \tag{20}\]  
Expression (20) gives how the eigenvalue \(\lambda\) evolves when the phase angle varies in the full range \((0,2\pi)\). However, for determining the boundary of the fast-scale instability, only the behavior of \(\lambda(\theta)\) within the half range \((0,\pi)\) can be considered. Indeed, from Figure 9 it can be observed that the fast-scale instability emerges first within some switching cycles during the first half cycle of the sinewave signal eventually in the neighborhood to the quarter of the cycle where the signal \(D(\theta)\) is maximum for \(\theta\in(0,\pi)\) during which the quasi-static duty cycle \(D\in(0.5,1)\). Therefore, the analytical study that will be presented below will be restricted to this range of phase angle \(\theta\). The eigenvalue \(\lambda(\theta)\) is plotted in Figure 10 in terms of the phase angle \(\theta\) for different values of compensating signal amplitude \(V_{M}\) or equivalently its slope \(m_{r}\). It can be observed that the eigenvalue \(\lambda(\theta)\) has its minimum value at the phase angle \(\pi/2\). If the minimum value of the eigenvalue is larger than \(-1\) the system will not exhibit fast-scale instability. This is the case for instance for \(V_{M}=4\) V and for \(V_{M}=5\) V. On the other hand, for \(V_{M}=3\) V fast-scale instability will be exhibited for the phase angle \(\theta\in(1.17,1.96)\) rad because within this interval of \(\theta\) one has \(\lambda(\theta)<-1\). The limits of this interval are indicated by two dashed lines in Figure 10.  
Figure 10: The eigenvalue \(\lambda(\theta)\) as the phase angle is varied within \((0,\pi)\) for different values of \(V_{M}\). At the boundary of fast-scale instability, one has \(\lambda_{\min}=-1\). By imposing this condition in (20), the following equation must hold  
\[m_{r}=\frac{R_{s}m_{0}(\theta)}{2}\frac{1-2D(\theta)}{D(\theta)} \tag{21}\]  
An upper bound of the left-hand side of (21) is \(-R_{s}m_{0}(\theta)/2=R_{s}v_{01}(\theta)/(2L)\) which is obtained for \(D=1\) or equivalently for \(\theta=\pi/2\). Therefore, the slope \(m_{r}=R_{s}v_{01}(\theta)/(2L)\) i.e., equal to one half the absolute value of the falling slope of the control signal \(\sigma\) will guarantee stability within the full range of \(\theta\in(0,2\pi)\). Therefore, to avoid the fast-scale instability for the full range of \(\theta\), an adaptive compensation scheme can be used where the amplitude of the compensating signal \(v_{r}\) is computed for each switching cycle according to the sampled voltage \(v_{01}(nT)\), \(n\in\mathbb{N}\), the inductance value \(L\) and sensing resistance \(R_{s}\).