Control System Toolbox™

norm - Compute LTI model norm

Syntax

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

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)

This command produces the following result.

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.

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.