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# 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

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Compute the and norms of a discrete-time linear system.

Create the following discrete-time transfer function with sample time 0.1 second.

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

Compute the norm of the transfer function.

`n = norm(H)`
```n = 1.2438 ```

`n` is the root-mean-square of the impulse response of `H`.

Compute the norm of the transfer function.

`[ninf,fpeak] = norm(H,inf)`
```ninf = 2.5715 ```
```fpeak = 3.0051 ```

`ninf` is the peak gain of the frequency response of `H`, and `fpeak` is the frequency at which the peak gain occurs.

Compute the peak gain in dB.

`ninf_dB = 20*log10(ninf)`
```ninf_dB = 8.2039 ```

Use the Bode plot of `H` to verfiy the computed values.

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

The plot confirms that the maximum gain is approximately 8 dB and the gain peaks at approximately 3 rad/sec.

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### 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.