# Frequency Response of a MIMO System

This example shows how to examine the frequency response of a multi-input, multi-output (MIMO) system in two ways: by computing the frequency response, and by computing the singular values.

Calculate the frequency response of a MIMO model and examine the size of the output.

H = rss(2,2,2); H.InputName = 'Control'; H.OutputName = 'Temperature'; [mag,phase,w] = bode(H); size(mag)

`ans = `*1×3*
2 2 89

The first and second dimension of the data array `mag`

are the number of outputs and inputs of `H`

. The third dimension is the number of points in the frequency vector `w`

. (The `bode`

command determines this number automatically if you do not supply a frequency vector.) Thus, `mag(i,j,:)`

is the frequency response from the `j`

th input of `H`

to the `i`

th output, in absolute units. The phase data array `phase`

takes the same form as `mag`

.

Plot the frequency response of each input/output pair in `H`

.

bode(H)

`bode`

plots the magnitude and the phase of the frequency response of each input/output pair in `H`

. (Because `rss`

generates a random state-space model, you might see different responses from those pictured.) The first column of plots shows the response from the first input, `Control(1)`

, to each output. The second column shows the response from the second input, `Control(2)`

, to each output.

Plot the singular values of `H`

as a function of frequency.

sigma(H)

`sigma`

plots the singular values of the MIMO system `H`

as a function of frequency. The maximum singular value at a particular frequency is the maximum gain of the system over all linear combinations of inputs at that frequency. Singular values can provide a better indication of the overall response, stability, and conditioning of a MIMO system than a channel-by-channel Bode plot.

Calculate the singular values of `H`

between 0.1 and 10 rad/s.

[sv,w] = sigma(H,{0.1,10});

When you call `sigma`

with output arguments, the command returns the singular values in the data array `sv`

. The cell array input `{0.1,10}`

tells `sigma`

to calculate the singular values at a grid of frequencies between 0.1 and 10 rad/s. `sigma`

returns these frequencies in the vector `w`

. Each row of `sv`

contains the singular values of `H`

at the frequencies of `w`

.

## See Also

`bode`

| `bodeplot`

| `sigma`

| `sigmaplot`