Function Reference
norm

Compute LTI model norms

Syntax

• ```norm(sys)
norm(sys,2)

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

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

• ```sqrt(trace(covar(sys,1)))
```

`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

• ```H = tf([1 -2.841 2.875 -1.004],[1 -2.417 2.003 -0.5488],0.1)
norm(H)
ans =
1.2438
```

Compute its infinity norm by typing

• ```[ninf,fpeak] = norm(H,inf)
ninf =
2.5488

fpeak =
3.0844
```

These values are confirmed by the Bode plot of .

• ```bode(H)
```

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

• ```20*log10(ninf)
```

MATLAB returns

• ```ans =
8.1268
```

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.

`bode``        `Bode plot
`freqresp``    `Frequency response computation
`sigma``       `Singular value plot