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This example illustrates the properties of a 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

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

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

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