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

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

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

$${G}_{1}(s)=\frac{{s}^{2}+s+1}{{s}^{2}+s+4},\phantom{\rule{0.2777777777777778em}{0ex}}\phantom{\rule{0.2777777777777778em}{0ex}}\phantom{\rule{0.2777777777777778em}{0ex}}{G}_{2}(s)=\frac{s+2}{s+5}.$$

Both systems are passive as confirmed by

G1 = tf([1,1,1],[1,1,4]); isPassive(G1)

`ans = `*logical*
1

G2 = tf([1,2],[1,5]); isPassive(G2)

`ans = `*logical*
1

The interconnected system is therefore passive.

H = feedback(G1,G2); isPassive(H)

`ans = `*logical*
1

This is confirmed by verifying that the Nyquist plot of $$H$$ is positive real.

nyquist(H)

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 positive, then the input passivity index $$\nu $$ and the output passivity index $$\rho $$ for the feedback interconnection $$H$$ satisfy

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

In other words, we can infer some minimum level of input and output passivity for the closed-loop system $$H$$ from the input and output passivity indices of $${G}_{1}$$ and $${G}_{2}$$. For details, see the paper by Zhu, F. and Xia, M and Antsaklis, P.J., "Passivity analysis and passivation of feedback systems using passivity indices," *American Control Conference* , 2014, pp. 1833-1838. Verify the lower bound for the input passivity index $$\nu $$.

% Input passivity index for G1 nu1 = getPassiveIndex(G1,'input'); % Output passivity index for G2 rho2 = getPassiveIndex(G2,'output'); % Input passivity index for H nu = getPassiveIndex(H,'input')

nu = 0.1293

```
% Lower bound
nu1*rho2/(nu1+rho2)
```

ans = 7.1402e-11

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

% Output passivity index for G1 rho1 = getPassiveIndex(G1,'output'); % Input passivity index for G2 nu2 = getPassiveIndex(G2,'input'); % Output passivity index for H rho = getPassiveIndex(H,'output')

rho = 0.4441

```
% Lower bound
rho1+nu2
```

ans = 0.4000