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norm

Norm of linear model

Syntax

n = norm(sys)
n = norm(sys,2)
n = norm(sys,inf)
[n,fpeak] = norm(sys,inf)
[...] = norm(sys,inf,tol)

Description

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.

Input Arguments

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

Output Arguments

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.

Examples

This example uses norm to compute the H2 and H norms of a discrete-time linear system.

Consider the discrete-time transfer function

H(z)=z32.841z2+2.875z1.004z32.417z2+2.003z0.5488

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:

20*log10(ninf)

This command produces the following result:

ans =
    8.1268

More About

collapse all

H2 norm

The H2 norm of a stable continuous-time system with transfer function H(s), is given by:

H2=12πTrace[H(jω)HH(jω)] dω.

For a discrete-time system with transfer function H(z), the H2 norm is given by:

H2=12πππTrace[H(ejω)HH(ejω)]dω.

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:

H22=limEt{y(t)Ty(t)},       E(w(t)w(τ)T)=δ(tτ)I.

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)))

H-infinity norm

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:

H(s)=maxω|H(jω)|                   (SISO)H(s)=maxωσmax(H(jω))        (MIMO)

where σmax(· ) denotes the largest singular value of a matrix.

For a discrete-time system H(z):

H(z)=maxθ[0,π]|H(ejθ)|                   (SISO)H(z)=maxθ[0,π]σmax(H(ejθ))(MIMO)

The H norm is infinite if sys has poles on the imaginary axis (in continuous time), or on the unit circle (in discrete time).

Algorithms

norm first converts sys to a state space model.

norm uses the same algorithm as covar for the H2 norm. For the H norm, norm uses the algorithm of [1]. norm computes the H norm (peak gain) using the SLICOT library. For more information about the SLICOT library, see http://slicot.org.

References

[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.

See Also

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Introduced before R2006a

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