## 4 Boundedness and Stability
### Set of Bounded Trajectories  
The hyperbolic line is stability preserving and satisfies the following deviation bound:  
\[\left\|G(\rho)-G_{1}\right\|_{\infty}\leq f(G_{1},G_{2})=\max\{g_{ 1}(G_{1},G_{2}),g_{2}(G_{1},G_{2})\}\] \[\left\|G(\rho)-G_{2}\right\|_{\infty}\leq f(G_{1},G_{2})=\max\{g_ {1}(G_{1},G_{2}),g_{2}(G_{1},G_{2})\} \tag{21}\]  
where  
\[g_{1}(G_{1},G_{2})=\sum_{i=1}^{n}\Biggl{(}\max\{|r_{i1}|,|r_{i2}| \}\max\biggl{\{}\left|\frac{1}{1-|p_{i1}|}\right|,\left|\frac{1}{1-|p_{i2}|} \right|\biggr{\}}+\left|\frac{r_{i1}}{\frac{p_{i2}}{|p_{i1}|}-p_{i1}}\right| \Biggr{)}\] \[g_{2}(G_{1},G_{2})=\sum_{i=1}^{n}\Biggl{(}\max\{|r_{i1}|,|r_{i2}| \}\max\biggl{\{}\left|\frac{1}{1-|p_{i1}|}\right|,\left|\frac{1}{1-|p_{i2}|} \right|\biggr{\}}+\left|\frac{r_{i2}}{\frac{p_{i2}}{|p_{i2}|}-p_{i2}}\right| \Biggr{)}. \tag{22}\]  
The detailed derivation can be found in [20]. In order to create a self-contained derivation, the first steps of the original derivation in [20] are repeated. The rest of the section is devoted to the construction of a set that contains all the trajectories that satisfy Equation (21). As assumed the systems \(G(\rho)\) and \(G_{1}\) have a partial fraction decomposition:  
\[\left\|G(\rho)-G_{1}\right\|_{\infty}=\left\|\sum_{i=1}^{n}\left(P_{i}(\rho)- P_{i1}\right)\right\|_{\infty} \tag{23}\]  
where  
\[P_{i}(\rho)=\frac{r_{i}(\rho)}{z-p_{i}(\rho)},\ \ P_{i1}=\frac{r_{i1}}{z-p_{i1}}.\]  
Exploiting the subadditivity of the \(H_{\infty}\) norm from (23):  
\[\left\|G(\rho)-G_{1}\right\|_{\infty}\leq\sum_{i=1}^{n}\left\|P_{i}(\rho)-P_{ i1}\right\|_{\infty} \tag{24}\]In the following, an upper bound is given on \(\left\|P_{i}(\rho)-P_{i1}\right\|_{\infty}\). \[\left\|P_{i}(\rho)-P_{i1}\right\|_{\infty}=\left\|\frac{r_{i}(\rho )}{z-p_{i}(\rho)}-\frac{r_{i1}}{z-p_{i1}}\right\|_{\infty}\leq\] \[\left|r_{i}(\rho)\right|\left\|\frac{1}{z-p_{i}(\rho)}\right\|_{ \infty}+\left\|\frac{r_{i1}}{z-p_{i1}}\right\|_{\infty}. \tag{25}\]  
Compute an upper bound term-by-term in (25). The last term can be calculated exactly since it is a first-order system:  
\[\left\|\frac{r_{i1}}{z-p_{i1}}\right\|_{\infty}=\max_{z=e^{k}}\left|\frac{r_{i 1}}{z-p_{i1}}\right|=\left|\frac{r_{i1}}{\left|\frac{p_{i1}}{\left|p_{i1} \right|}-p_{i1}\right|} \tag{26}\]  
The residual \(r_{i}(\rho)\) in the first term is interpolated linearly:  
\[\left\|G(\rho)-G_{2}\right\|_{\infty}\leq f(G_{1},G_{2})=\max\{g_ {1}(G_{1},G_{2}),g_{2}(G_{1},G_{2})\} \tag{21}\]0  
The second term in (25) is also a first-order system like the last term, but it depends on \(\rho\). As the first step, one can write:  
\[\left\|G(\rho)-G_{2}\right\|_{\infty}\leq f(G_{1},G_{2})=\max\{g_ {1}(G_{1},G_{2}),g_{2}(G_{1},G_{2})\} \tag{21}\]1 \[\left\|G(\rho)-G_{2}\right\|_{\infty}\leq f(G_{1},G_{2})=\max\{g_ {1}(G_{1},G_{2}),g_{2}(G_{1},G_{2})\} \tag{21}\]2  
The expression  
\[\left\|G(\rho)-G_{2}\right\|_{\infty}\leq f(G_{1},G_{2})=\max\{g_ {1}(G_{1},G_{2}),g_{2}(G_{1},G_{2})\} \tag{21}\]3  
is less than or equal to one because it represents the pseudo-hyperbolic distance between the complex numbers \(p_{i1}\) and \(\rho w_{i}\) in the unit disk \(\mathbb{D}\). Pseudo-hyperbolic distance is always less than or equal to one [25]. It is equal to one when one of the points is on the unit circle, which cannot happen since we assumed stable systems. In order to construct an upper bound, the maximum of the expression  
\[\left\|G(\rho)-G_{2}\right\|_{\infty}\leq f(G_{1},G_{2})=\max\{g_ {1}(G_{1},G_{2}),g_{2}(G_{1},G_{2})\} \tag{21}\]4  
is needed. Since we discussed that the inner part \(\left|\frac{p_{i1}-\rho w_{i}}{1-\rho w_{i}\overline{p_{i1}}}\right|\) is less than one, we can move the maximum operation inside the expression:  
\[\left\|G(\rho)-G_{2}\right\|_{\infty}\leq f(G_{1},G_{2})=\max\{g_ {1}(G_{1},G_{2}),g_{2}(G_{1},G_{2})\} \tag{21}\]5  
A geometric approach is used to find the maximum of \(\left|\frac{p_{i1}-\rho w_{i}}{1-\rho w_{i}\overline{p_{i1}}}\right|\) (see Figure 4).