## Appendix A Proof of Lemma 4  
Proof. : _Sufficiency._ Let us consider the discrete model of (1) including a zero order hold and a sampling period \(T=\frac{\tau}{n}\) with \(n\in\mathbb{N}\) and the output injection strategy described by Figure 11 (or its equivalent in Figure 11). Then,  
\[\frac{Y(z)}{U_{1}(z)}=\frac{\alpha_{1}\alpha_{2}b}{ab_{1}z^{n}(z-\mathrm{e}^{ aT})\big{(}\mathrm{e}^{b_{1}T}z-1\big{)}+\alpha_{1}b\big{(}b_{1}g_{1}\big{(} \mathrm{e}^{b_{1}T}z-1\big{)}+\alpha_{2}g_{2}\big{)}}G_{s}(z), \tag{11}\]  
with \(\alpha_{1}=\mathrm{e}^{aT}-1\) and \(\alpha_{2}=\mathrm{e}^{b_{1}T}-1\). Under the assumption that \(G_{s}(s)\) has stable poles (previously assumed) and, as a consequence, also \(G_{\bar{s}}(z)\), the stability of (11) only depends on the transfer function \(Y(z)/U_{2}(z)\). In this way, the characteristic equation of \(Y(z)/U_{2}(z)\) is given by  
\[\alpha_{1}\Big{(}g_{1}z-g_{1}\mathrm{e}^{-b_{1}T}+\alpha_{2}g_{2}\Big{)}+z^{ n}\Big{(}z-\mathrm{e}^{aT}\Big{)}\Big{(}z-\mathrm{e}^{-b_{1}T}\Big{)}=0. \tag{12}\]  
This lemma can be proved by showing that all the roots of Equation (12) are inside the unit circle if and only if \(\tau<\frac{1}{a}-\frac{1}{b_{1}}+\sqrt{\frac{1}{a^{2}}+\frac{1}{b_{1}^{2}}}\) when \(T\to 0\), or equivalently when \(n\to\infty\)The general idea of the proof is presented below. Later, a formal analysis using a root locus approach is presented. Let us take \(n=5\), for simplicity and without loss of generality, in the analysis. Then, in the following steps, consider the root locus diagram generated by the characteristic Equation (A2). Step 1. Assume \(g_{1}>0\) and \(g_{2}=0\). When \(g_{1}\) varies from zero towards positive values, \(n-1\) poles located at the origin move towards infinity: one pole moves in the direction of a breaking point located at the real axis, and the pole located at \(z=e^{aT}\) also moves from right to left on the real axis in the direction of to the breaking point, and for some values of \(g_{1}>0\), the trajectory enters the stability region (unit circle). These trajectories are depicted in Figure A3. In Figure A3, the gray points in the trajectories indicate the location of the roots for a specific value of \(g_{1}>0\), denoted by \(\bar{g}_{1}\). Notice that the pole on the right real axis (\(z=e^{-b_{1}T}\)) does not move since \(g_{1}\) has no influence on this pole. Step 2. With \(g_{1}=\bar{g}_{1}\) and \(g_{2}>0\) variable. This step is explained by using Figure A4 when \(g_{2}\) varies from zero towards positive values, the pole located at \(z=e^{-b_{1}T}\) is involved in the locus of the roots, which moves from left to the right on the real axis, causing a breaking point with the relocated pole (originally the pole outside the unit circle) in step one. Note that under these circumstances, a breaking point inside the unit circle is obtained. Gray points in the trajectories in Figure A4 indicate the location of the roots for a specific value of \(g_{2}>0\), denoted by \(\bar{g}_{2}\). The facts depicted in Steps 1 and 2 suggest that it is possible to obtain \(g_{1}=\bar{g}_{1}\) and \(g_{2}=\bar{g}_{2}\) such that all the roots of the locus are inside the unit circle and therefore the discrete injection scheme shown in Figure A2 is stable. Now, considering the previous developments, the breakpoint derived from the aforementioned analysis is calculated. Consider Step 2, where a breaking point is located in the real axis between the pole located at \(z=e^{-b_{1}T}\) and the relocated pole that was moved into the unit circle in Step 1. Then, to achieve closed-loop stability, the breaking point should be inside the unit circle. Then, the location of the breaking point is calculated. This can be done by solving for \(g_{2}\) from Equation (A1), and then its derivative is  
\[\frac{dg_{2}}{dz}=b_{1}\Big{(}e^{b_{1}T}z-1\Big{)}\big{(}az^{n}+anz^{n-1}\Big{(} z-e^{aT}\Big{)}\Big{)}+ab_{1}e^{b_{1}T}z^{n}\Big{(}z-e^{aT}\Big{)}+\alpha_{1}bb_{1 }e^{b_{1}T}g_{1}.\] (A3)  
The breakpoint of interest is obtained by calculating the roots of Equation (A3).