# FDI related utilities

Evaluation of the`fdhinfminus`

`H∞-`

index of the transfer function matrix of a descriptor system model.Evaluation of the maximum of column norm of the transfer function matrix of a descriptor system model.`fdhinfmax`

Computation of the weak or strong structure matrix of a descriptor system model.`fditspec_`

Computation of the strong structure matrix of a descriptor system model.`fdisspec_`

Computation of the column-gains sensitivity condition of the transfer function matrix of a descriptor system model.`fdiscond_`

`FaultDetectionTools.fdhinfminus`

— Function` fdhinfminus(sys,freq) -> (β, ind, fr)`

Compute for a stable descriptor system `sys = (A-λE,B,C,D)`

the `H∞-`

index `β`

of its transfer function matrix `G(λ)`

. If `freq = missing`

(default), then `β`

is the minimum `H∞-norm`

of the columns of `G`

, `ind`

is the index of the minimum-norm column and `fr`

is the frequency where the minimum `H∞-norm`

of the columns is achieved. If `freq`

is a real value or a real vector of frequency values, then `β`

is the minimum of the 2-norms of the columns of the frequency responses of `G`

evaluated for all values contained in `freq`

, `ind`

is the index of column for which the minimum is achieved and `fr`

is the corresponding frequency.

`FaultDetectionTools.fdhinfmax`

— Function` fdhinfmax(sys,freq) -> (γ, ind, fr)`

Compute for a descriptor system `sys = (A-λE,B,C,D)`

, `γ`

- the maximum norm of the columns of its transfer function matrix `G(λ)`

. If `freq = missing`

(default), then `γ`

is the maximum `H∞-norm`

of the columns of `G`

, `ind`

is the index of the maximum-norm column and `fr`

is the frequency where the maximum `H∞-norm`

of the columns is achieved. If `freq`

is a real value or a real vector of frequency values, then `γ`

is the maximum of the 2-norms of the columns of the frequency responses of `G`

evaluated for all values contained in `freq`

, `ind`

is the index of column for which the maximum is achieved and `fr`

is the corresponding frequency.

`FaultDetectionTools.fditspec_`

— Function```
S = fditspec_(sysrf::DescriptorStateSpace; FDfreq = missing, block = false, poleshift = false,
FDtol, FDStol, atol = 0, atol1 = atol, atol2 = atol, rtol, fast = true)
```

Compute the weak or strong binary structure matrix `S`

of the transfer function matrix of a linear time-invariant system `sysrf`

(typically representing the transfer channel from the fault inputs to residuals). `sysrf`

has a descriptor system realization of the form `sysrf = (Af-lambda*Ef,Bf,Cf,Df)`

with a `q x mf`

transfer function matrix `Rf(λ)`

. For the description of keyword parameters see the documentation of `fditspec`

.

`FaultDetectionTools.fdisspec_`

— Function```
fdisspec_(sysrf::DescriptorStateSpace, freq; block = false, stabilize = false, FDGainTol = 0.01,
atol, atol1, atol2, atol3, rtol, fast = true) -> (S, gains)
```

Compute the strong binary structure matrix `S`

of the transfer function matrix of a linear time-invariant system `sysrf`

(typically representing the transfer channel from the fault inputs to residuals). `sysrf`

has a descriptor system realization of the form `sysrf = (Af-lambda*Ef,Bf,Cf,Df)`

with a `q x mf`

transfer function matrix `Rf(λ)`

. For the description of keyword parameters see the documentation of `fdisspec`

.

`FaultDetectionTools.fdiscond_`

— Function` fdiscond_(sysrf::DescriptorStateSpace, freq) -> (scond, β, γ)`

Compute for a stable descriptor system `sysrf = (A-λE,B,C,D)`

with the transfer function matrix `Rf(λ)`

, `β`

- the H∞- index of `Rf(λ)`

, `γ`

- the maximum of the columns norms of `Rf(λ)`

and `scond`

- the column-gains sensitivity condition evaluated as `scond := β/γ`

. If `freq`

is a vector of real frequency values, then `β`

and `γ`

are evaluated over the frequencies contained in `freq`

.