## 4 Approximate Prediction of Fast-Scale Instability and Adaptive Slope Compensation
### Reduced-Order Discrete-Time Model for Predicting the Fast-Scale Instability in the DC-AC Inverter
According to (6), the phase-dependent raising slope \(m_{1}(\theta)\) (\(u=1\)) and the falling slope \(m_{0}(\theta)\) (\(u=0\)) of the signal \(\sigma\) are given by  
\[m_{1}(\theta) = R_{s}\frac{v_{o2}(\theta)}{L} \tag{8}\] \[m_{0}(\theta) = -R_{s}\frac{v_{o1}(\theta)}{L} \tag{9}\]  
where \(v_{o1}(\theta)\) and \(v_{o2}(\theta)\) can be expressed as follows [10; 35]  
\[v_{o1}(\theta) = \frac{V_{\mathrm{mpp}}}{1-D(\theta)} \tag{10}\] \[v_{o2}(\theta) = \frac{V_{\mathrm{mpp}}}{D(\theta)} \tag{11}\]  
\(D(\theta)\) is the time varying quasi-steady-state duty cycle that can be expressed as follows [10; 35]:  
\[D(\theta)=\frac{1}{2}-\frac{V_{\mathrm{mpp}}}{v_{g}(\theta)}+\text{sign}(v_{g }(\theta))\frac{1}{2}\sqrt{1+\frac{4V_{\mathrm{mpp}}^{2}}{v_{g}^{2}(\theta)}} \tag{12}\]  
It is worth noting that \(D(0^{-})=D(0^{+})=1/2\) and therefore \(D(\theta)\) is a continuous function of the phase angle \(\theta\). The quasi-steady-state duty cycle \(D(\theta)\) is plotted in Figure 9 together with the duty cycle obtained from numerical simulations corresponding to the results presented in Figures 2-4 for \(S=1000\) W/m\({}^{2}\) and in Figures 5-7 for \(S=200\) W/m\({}^{2}\) respectively. Notice that the plots corresponding to the numerical simulations and the one obtained from the theoretical expression (12) cannot be distinguished from each other when the system operates in the expected behavior without fast-scale instability. Notice also that when the system exhibits instability, bubbling phenomenon mentioned previously also occurs in the waveforms of the quasi-steady-state duty cycle. According to the switching decision corresponding to the control block diagram represented in Figure 1, the expression of the duty cycle \(d_{n}\) can be obtained by solving the following equation  
\[R_{s}(i_{1}(d_{n}T)-i_{2}(d_{n}T))+v_{r}(d_{n}T)-R_{s}i_{\text{ref}}(\theta)=0 \tag{13}\]  
By taking into account that \(\sigma=R_{s}(i_{1}-i_{2})\), (13) becomes as follows  
\[\frac{\mathrm{d}\sigma}{\mathrm{d}t}=\frac{R_{s}}{L}((v_{o1}(\theta)-v_{o2}( \theta))u-v_{o1}(\theta)) \tag{6}\]0  
Within the switching cycle \((nT,(n+1)T)\), \(n\in\mathbb{N}\), and taking into account that \(\sigma((n+d_{n})T)=\sigma_{n}+m_{1}d_{n}T=\) and \(v_{r}(d_{n}T)=m_{r}d_{n}T\) (See Figure 8), (14) can be expressed as follows  
\[\frac{\mathrm{d}\sigma}{\mathrm{d}t}=\frac{R_{s}}{L}((v_{o1}(\theta)-v_{o2}( \theta))u-v_{o1}(\theta)) \tag{6}\]1  
where, as stated previously, \(m_{r}=V_{M}/T\) is the slope of the ramp signal \(v_{r}\). The duty cycle \(d_{n}\) can be obtained by solving (15) while considering \(R_{s}i_{\text{ref}}(\theta)\) constant within a switching cycle. Solving for \(d_{n}\) one gets  
\[\frac{\mathrm{d}\sigma}{\mathrm{d}t}=\frac{R_{s}}{L}((v_{o1}(\theta)-v_{o2}( \theta))u-v_{o1}(\theta)) \tag{6}\]2  
The expressions (7) and (16) define the reduced-order model that can be used for performing the stability analysis of the DC-AC inverter at the fast-switching scale. Below this analysis will be performed and the stability criteria will be determined. \[\frac{\mathrm{d}\sigma}{\mathrm{d}t}=\frac{R_{s}}{L}((v_{o1}(\theta)-v_{o2}( \theta))u-v_{o1}(\theta)) \tag{6}\]3 The feedback signal \(\sigma\) and the control signal \(R_{s}\text{ref}(\theta)-v_{r}\). \[\frac{\mathrm{d}\sigma}{\mathrm{d}t}=\frac{R_{s}}{L}((v_{o1}(\theta)-v_{o2}( \theta))u-v_{o1}(\theta)) \tag{6}\]4 The time varying duty cycle calculated from (12) (\(D(t)\)) and obtained from numerical simulation (\(d(nT)\), \(n\in\mathbb{N}\)) for \(S=1000\) W/m\({}^{2}\) and \(S=200\) W/m\({}^{2}\).