Norm of linear model
n = norm(sys)
n = norm(sys,2)
n = norm(sys,inf)
[n,fpeak] = norm(sys,inf)
[...] = norm(sys,inf,tol)
Continuous- or discrete-time linear dynamic system model.
Positive real value setting the relative accuracy of the H∞ norm.
Frequency at which the peak gain of
This example uses
norm to compute the H2 and H∞ norms
of a discrete-time linear system.
Consider the discrete-time transfer function
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:
This command produces the following result:
ans = 8.1268
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:
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
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).
 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.