Function Reference    

Compute LTI model norms



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

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


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:

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.


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

MATLAB returns


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


[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