文章目錄

• Matlab `hinfnorm()`
• • Syntax
  • Description
  • E.G.

《Essentials of Robust Control》

山東科技大學周克敏教授主講《魯棒控制》課程

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Matlab hinfnorm()

Syntax

ninf = hinfnorm(sys) ninf = hinfnorm(sys,tol) [ninf,fpeak] = hinfnorm(___)

Description

ninf = hinfnorm(sys) returns the H∞ norm in absolute units of the dynamic system model, sys.

• If sys is a stable SISO system, then the H∞ norm is the peak gain, the largest value of the frequency response magnitude.
• If sys is a stable MIMO system, then the H∞ norm is the largest singular value across frequencies.
• If sys is an unstable system, then the H∞ norm is defined as Inf.
• If sys is a model that has tunable or uncertain parameters, then hinfnorm evaluates the H∞ norm at the current or nominal value of sys.
• If is a model array, then hinfnorm returns an array of the same size as sys, where ninf(k) = hinfnorm(sys(:,:,k)) .

For stable systems, hinfnorm(sys) is the same as getPeakGain(sys).

ninf = hinfnorm(sys,tol) returns the H∞ norm of sys with relative accuracy tol.

[ninf,fpeak] = hinfnorm(___) also returns the frequency, fpeak, at which the peak gain or largest singular value occurs. You can use this syntax with any of the input arguments in previous syntaxes. If sys is unstable, then fpeak = Inf.

E.G.

Norm of MIMO System

Compute the H∞ norm of the following 2-input, 2-output dynamic system and the frequency at which the peak singular value occurs.

$$G(s)=\left[\begin{array}{cc}0& \frac{3s}{s^2+s+10}\\ \frac{s+1}{s+5}& \frac{2}{s+6}\end{array}\right]$$

G = [0 tf([3 0],[1 1 10]);tf([1 1],[1 5]),tf(2,[1 6])]; [ninf,fpeak] = hinfnorm(G)

ninf = 3.0150
fpeak = 3.1623

The H∞ norm of a MIMO system is its maximum singular value. Plot the singular values of G and compare the result from hinfnorm.

sigma(G), grid

The values ninf and fpeak are consistent with the singular value plot, which displays the values in dB.

https://www.mathworks.com/help/robust/ref/lti.hinfnorm.html

http://matrix.etseq.urv.es/manuals/matlab/toolbox/mutools/h2norm.html

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