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## Parallel Interconnection of Passive Systems

This example illustrates the properties of a parallel interconnection of passive systems.

### Parallel Interconnection of Passive Systems

Consider an interconnection of two subsystems ${G}_{1}$ and ${G}_{2}$ in parallel. The interconnected system $H$ maps the input $r$ to the output $y$.

If both systems ${G}_{1}$ and ${G}_{2}$ are passive, then the interconnected system $H$ is guaranteed to be passive. Take for example

`${G}_{1}\left(s\right)=\frac{0.1s+1}{s+2};\phantom{\rule{1em}{0ex}}{G}_{2}\left(s\right)=\frac{{s}^{2}+2s+1}{{s}^{2}+3s+10}$`

Both systems are passive.

```G1 = tf([0.1,1],[1,2]); isPassive(G1)```
```ans = logical 1 ```
```G2 = tf([1,2,1],[1,3,10]); isPassive(G2)```
```ans = logical 1 ```

We can therefore expect their parallel interconnection $H$ to be passive, as confirmed by

```H = parallel(G1,G2); isPassive(H)```
```ans = logical 1 ```

### Passivity Indices for Parallel Interconnection

There is a relationship between the passivity indices of ${G}_{1}$ and ${G}_{2}$ and the passivity indices of the interconnected system $H$. Let ${\nu }_{1}$ and ${\nu }_{2}$ denote the input passivity indices for ${G}_{1}$ and ${G}_{2}$, and let ${\rho }_{1}$ and ${\rho }_{2}$ denote the output passivity indices. If all these indices are nonnegative, then the input passivity index $\nu$ and the output passivity index $\rho$ for the parallel interconnection $H$ satisfy

`$\nu \ge {\nu }_{1}+{\nu }_{2},\phantom{\rule{0.2777777777777778em}{0ex}}\phantom{\rule{0.2777777777777778em}{0ex}}\phantom{\rule{0.2777777777777778em}{0ex}}\rho \ge \frac{{\rho }_{1}{\rho }_{2}}{{\rho }_{1}+{\rho }_{2}}.$`

In other words, we can infer some minimum level of input and output passivity for the parallel connection $H$ from the input and output passivity indices of ${G}_{1}$ and ${G}_{2}$. For details, see the paper by Yu, H., "Passivity and dissipativity as design and analysis tools for networked control systems," Chapter 2, PhD Thesis, University of Notre Dame, 2012. Verify the lower bound for the input passivity index $\nu$.

```% Input passivity index for G1 nu1 = getPassiveIndex(G1,'input'); % Input passivity index for G2 nu2 = getPassiveIndex(G2,'input'); % Input passivity index for H nu = getPassiveIndex(H,'input')```
```nu = 0.3777 ```
```% Lower bound nu1+nu2```
```ans = 0.1474 ```

Similarly, verify the lower bound for the output passivity index of $H$.

```% Output passivity index for G1 rho1 = getPassiveIndex(G1,'output'); % Output passivity index for G2 rho2 = getPassiveIndex(G2,'output'); % Output passivity index for H rho = getPassiveIndex(H,'output')```
```rho = 0.6443 ```
```% Lower bound rho1*rho2/(rho1+rho2)```
```ans = 0.2098 ```