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| R2012a Documentation → Control System Toolbox | |
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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 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.
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 |
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. |
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
sqrt(trace(covar(sys,1)))
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).
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 =
1.2438To compute the H∞ infinity norm, enter:
[ninf,fpeak] = norm(H,inf)
This command returns the result:
ninf =
2.5488
fpeak =
3.0844You 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
norm uses the same algorithm as covar for the H2 norm. For the H∞ norm, norm uses the algorithm of [1]. In both cases, norm first converts sys to a state space model.
[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.
Learn more about resources for designing, testing, and implementing control systems.
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