Accelerating the pace of engineering and science

# norm

Norm of linear model

## Syntax

n = norm(sys)
n = norm(sys,2)
n = norm(sys,inf)
[n,fpeak] = norm(sys,inf)
[...] = norm(sys,inf,tol)

## Description

n = norm(sys) or n = norm(sys,2) return the H2 norm of the linear dynamic system model sys.

n = norm(sys,inf) returns the H norm of sys.

[n,fpeak] = norm(sys,inf) also returns the frequency fpeak at which the gain reaches its peak value.

[...] = norm(sys,inf,tol) sets the relative accuracy of the H norm to tol.

## Input Arguments

 sys Continuous- or discrete-time linear dynamic system model. sys can also be an array of linear models. tol Positive real value setting the relative accuracy of the H∞ norm. Default: 0.01

## Output Arguments

 n H2 norm or H∞ norm of the linear model sys. If sys is an array of linear models, n is an array of the same size as sys. In that case each entry of n is the norm of each entry of sys. fpeak Frequency at which the peak gain of sys occurs.

## Examples

This example uses norm to compute the H2 and H norms of a discrete-time linear system.

Consider the discrete-time transfer function

$H\left(z\right)=\frac{{z}^{3}-2.841{z}^{2}+2.875z-1.004}{{z}^{3}-2.417{z}^{2}+2.003z-0.5488}$

with sample time 0.1 second.

To compute the H2 norm of this transfer function, enter:

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

These commands return the result:

```ans =
1.2438```

To compute the H infinity norm, enter:

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

This command returns the result:

```ninf =
2.5488

fpeak =
3.0844```

You can use a Bode plot of H(z) to confirm these values.

```bode(H)
grid on;```

The gain indeed peaks at approximately 3 rad/sec. To find the peak gain in dB, enter:

```20*log10(ninf)
```

This command produces the following result:

```ans =
8.1268
```

expand all

### H2 norm

The H2 norm of a stable continuous-time system with transfer function H(s), is given by:

For a discrete-time system with transfer function H(z), the H2 norm is given by:

${‖H‖}_{2}=\sqrt{\frac{1}{2\pi }{\int }_{-\pi }^{\pi }\text{Trace}\left[H{\left({e}^{j\omega }\right)}^{H}H\left({e}^{j\omega }\right)\right]d\omega }.$

The H2 norm is equal to the root-mean-square of the impulse response of the system. The H2 norm measures the steady-state covariance (or power) of the output response y = Hw to unit white noise inputs w:

The H2 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 ω = ∞).

norm(sys) produces the same result as

```sqrt(trace(covar(sys,1)))
```

### H-infinity norm

The H norm (also called the L norm) of a SISO linear system is the peak gain of the frequency response. For a MIMO system, the H norm is the peak gain across all input/output channels. Thus, for a continuous-time system H(s), the H norm is given by:

where σmax(· ) denotes the largest singular value of a matrix.

For a discrete-time system H(z):

The H norm is infinite if sys has poles on the imaginary axis (in continuous time), or on the unit circle (in discrete time).

### Algorithms

norm first converts sys to a state space model.

norm uses the same algorithm as covar for the H2 norm. For the H norm, norm uses the algorithm of [1]. norm computes the H norm (peak gain) using the SLICOT library. For more information about the SLICOT library, see http://slicot.org.

## References

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