Compute passivity index of linear system

`getPassiveIndex`

computes various
measures of the excess or shortage of passivity for a given system.

A linear system *G*(*s*) is *passive* if
all its I/O trajectories (*u*(*t*),*y*(*t*)) satisfy:

$${\int}_{0}^{T}y{\left(t\right)}^{T}u\left(t\right)dt}>0,$$

for all *T* > 0.
Equivalently, a system is passive if its frequency response is positive
real, such that for all *ω* >
0,

$$G\left(j\omega \right)+G{\left(j\omega \right)}^{H}>0$$

(or the discrete-time equivalent).

`R = getPassiveIndex(G)`

`nu = getPassiveIndex(G,'input')`

`rho = getPassiveIndex(G,'output')`

`tau = getPassiveIndex(G,'io')`

`DX = getPassiveIndex(G,dQ)`

`index = getPassiveIndex(___,tol)`

`index = getPassiveIndex(___,tol,fband)`

`[index,FI] = getPassiveIndex(___)`

```
[index,FI,Qout,dQout]
= getPassiveIndex(___)
```

computes
the relative passivity index. `R`

= getPassiveIndex(`G`

)`G`

is passive when `R`

is
less than one. `R`

measures the relative excess
(`R`

< 1) or shortage (`R`

>
1) of passivity.

For more information about the notion of passivity indices, see About Passivity and Passivity Indices.

computes
the input passivity index. The system is `nu`

= getPassiveIndex(`G`

,'input')*input strictly
passive* when `nu`

> 0. `nu`

is
also called the input feedforward passivity (IFP) index. The value
of `nu`

is the minimum feedforward action such
that the resulting system is passive.

For more information about the notion of passivity indices, see About Passivity and Passivity Indices.

computes
the output passivity index. The system is `rho`

= getPassiveIndex(`G`

,'output')*output strictly
passive* when `rho`

> 0. `rho`

is
also called the output feedback passivity (OFP) index. The value of `rho`

is
the minimum feedback action such that the resulting system is passive.

For more information about the notion of passivity indices, see About Passivity and Passivity Indices.

computes
the combined I/O passivity index. The system is `tau`

= getPassiveIndex(`G`

,'io')*very strictly
passive* when `tau`

> 0.

`[index,`

also
returns the frequency at which the returned index value is achieved.`FI`

] = getPassiveIndex(___)

`getSectorCrossover`

| `getSectorIndex`

| `isPassive`

| `nyquist`

| `passiveplot`

| `sectorplot`