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

Consider an interconnection of two subsystems $${G}_{1}$$ and $${G}_{2}$$ in series. The interconnected system $$H$$ is given by the mapping from input $$u$$ to output $${y}_{2}$$.

In contrast with parallel and feedback interconnections, passivity of the subsystems $${G}_{1}$$ and $${G}_{2}$$ does not guarantee passivity for the interconnected system $$H$$. Take for example

$${G}_{1}(s)=\frac{5{s}^{2}+3s+1}{{s}^{2}+2s+1},\phantom{\rule{0.2777777777777778em}{0ex}}\phantom{\rule{0.2777777777777778em}{0ex}}\phantom{\rule{0.2777777777777778em}{0ex}}{G}_{2}(s)=\frac{{s}^{2}+s+5s+0.1}{{s}^{3}+2{s}^{2}+3s+4}.$$

Both systems are passive as confirmed by

G1 = tf([5 3 1],[1,2,1]); isPassive(G1)

`ans = `*logical*
1

G2 = tf([1,1,5,.1],[1,2,3,4]); isPassive(G2)

`ans = `*logical*
1

However the series interconnection of $${G}_{1}$$ and $${G}_{2}$$ is not passive:

H = G2*G1; isPassive(H)

`ans = `*logical*
0

This is confirmed by verifying that the Nyquist plot of $${G}_{2}{G}_{1}$$ is not positive real.

nyquist(H)

While the series interconnection of passive systems is not passive in general, there is a relationship between the passivity indices of $${G}_{1}$$ and $${G}_{2}$$ and the passivity indices of $$H={G}_{2}{G}_{1}$$. 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 series interconnection $$H$$ satisfy

$$\nu \ge -\frac{0.125}{{\rho}_{1}{\rho}_{2}},\phantom{\rule{0.2777777777777778em}{0ex}}\phantom{\rule{0.2777777777777778em}{0ex}}\phantom{\rule{0.2777777777777778em}{0ex}}\rho \ge -\frac{0.125}{{\nu}_{1}{\nu}_{2}}.$$

In other words, the shortage of passivity at the inputs or outputs of $$H$$ is no worse than the right-hand-side expressions. For details, see the paper by Arcak, M. and Sontag, E.D., "Diagonal stability of a class of cyclic systems and its connection with the secant criterion," *Automatica*, Vol 42, No. 9, 2006, pp. 1531-1537. Verify these lower bounds for the example above.

% Output passivity index for G1 rho1 = getPassiveIndex(G1,'output'); % Output passivity index for G2 rho2 = getPassiveIndex(G2,'output'); % Input passivity index for H=G2*G1 nu = getPassiveIndex(H,'input')

nu = -1.2886

```
% Lower bound
-0.125/(rho1*rho2)
```

ans = -2.4194

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

% Input passivity index for G1 nu1 = getPassiveIndex(G1,'input'); % Input passivity index for G2 nu2 = getPassiveIndex(G2,'input'); % Output passivity index for H=G2*G1 rho = getPassiveIndex(H,'output')

rho = -0.6966

```
% Lower bound
-0.125/(nu1*nu2)
```

ans = -6.0000