Function Reference |

**Syntax**

**Description**

```
norm
```

computes the or norm of a continuous- or discrete-time LTI model.

**H2 Norm**

The norm of a stable continuous system with transfer function , is the root-mean-square of its impulse response, or equivalently

- This norm measures the steady-state covariance (or power) of the output
response to unit white noise inputs .

**Infinity Norm**

The infinity norm is the peak gain of the frequency response, that is,

where denotes the largest singular value of a matrix.

The discrete-time counterpart is

**Usage**

`norm(sys)`

or `norm(sys,2)`

both return the norm of the TF, SS, or ZPK model `sys`

. This norm is infinite in the following cases:

`sys`

is unstable.`sys`

is continuous and has a nonzero feedthrough (that is, nonzero gain at the frequency ).

Note that `norm(sys)`

produces the same result as

`norm(sys,inf) `

computes the infinity norm of any type of LTI model `sys`

. This norm is infinite if `sys`

has poles on the imaginary axis in continuous time, or on the unit circle in discrete time.

`norm(sys,inf,tol) `

sets the desired relative accuracy on the computed infinity norm (the default value is `tol=1e-2`

).

`[ninf,fpeak] = norm(sys,inf) `

also returns the frequency `fpeak`

where the gain achieves its peak value.

**Example**

Consider the discrete-time transfer function

with sample time 0.1 second. Compute its norm by typing

Compute its infinity norm by typing

These values are confirmed by the Bode plot of .

The gain indeed peaks at approximately 3 rad/sec and its peak value in dB is found by typing

**Algorithm**

`norm`

uses the same algorithm as `covar`

for the norm, and the algorithm of [1] for the infinity norm. `sys`

is first converted to state space.

**See Also**

`bode`

Bode plot
` `

`freqresp`

Frequency response computation
` `

`sigma`

Singular value plot
` `

** References**

[1] Bruisma, N.A. and M. Steinbuch, "A Fast Algorithm to Compute the -Norm of a Transfer Function Matrix," *System Control Letters*, 14 (1990), pp. 287-293.

nichols | nyquist |