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Norm of linear model
n = norm(sys)
n =
norm(sys,2)
n = norm(sys,inf)
[n,fpeak]
= norm(sys,inf)
[...] = norm(sys,inf,tol)
n = norm(sys) or n = norm(sys,2) return the H_{2} 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.
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 |
n |
H_{2} 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. |
This example uses norm to compute the H_{2} and H_{∞} norms of a discrete-time linear system.
Consider the discrete-time transfer function
with sample time 0.1 second.
To compute the H_{2} 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
[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.