output = norm or peakfreq or list of these names.

checkstability = truefalse

The NormHinf command computes the $H_{\mathrm{\infty }}$ norm of a linear system sys, with relative accuracy eps. Both continuous-time and discrete-time systems, and both single-input single-output (SISO) and multiple-input multiple-output (MIMO) systems are supported.

The $H_{\mathrm{\infty }}$ norm provides a measure of the worst-case system gain, i.e., the largest factor by which any sinusoidal input is magnified by the system. For instance, the $H_{\mathrm{\infty }}$ norm of the transfer function G from w (disturbance input) to y (output) provides a measure of the worst-case influence of the noise w on the output y of an LTI system.

For a SISO linear system, the $H_{\mathrm{\infty }}$ norm is the maximum gain of the frequency response of the system. In an analogous way, for a MIMO linear system, the $H_{\mathrm{\infty }}$ norm is the maximum gain across all inputs and outputs of the system.

The $H_{\mathrm{\infty }}$ norm of $G$ equals the peak value on the Bode magnitude plot of $G$. It also equals the distance from the origin to the farthest point on the Nyquist plot of $G$.

The $H_{\mathrm{\infty }}$ norm is finite if and only if the transfer function $G$ is proper (degree of denominator greater than or equal to degree of numerator) and has no poles on the imaginary axis (continuous-time) or on the unit circle (discrete-time).

$\mathrm{with}\left(\mathrm{DynamicSystems}\right)\:$

$\mathrm{sys1}≔\mathrm{TransferFunction}\left(\frac{100}{s+5}\right)\:$

$\mathrm{PrintSystem}\left(\mathrm{sys1}\right)$

$\left[\begin{array}{l}\mathbf{Transfer\; Function}\\ \mathrm{continuous}\\ \mathrm{1\; output(s);\; 1\; input(s)}\\ \mathrm{inputvariable}\=\left[\mathrm{u1}\left(s\right)\right]\\ \mathrm{outputvariable}\=\left[\mathrm{y1}\left(s\right)\right]\\ {\mathrm{tf}}_{1,1}\=\frac{100}{s+5}\end{array}\right.$

$\mathrm{hinfnorm1}≔\mathrm{NormHinf}\left(\mathrm{sys1}\,{10}^{-10}\right)$

$\mathrm{hinfnorm1}≔20.00000000$

$\mathrm{MagnitudePlot}\left(\mathrm{sys1}\,\mathrm{decibels}=\mathrm{false}\,\mathrm{range}=0.001..100\right)$

$\mathrm{mag}≔\mathrm{MagnitudePlot}\left(\mathrm{sys1}\,\mathrm{decibels}=\mathrm{false}\,\mathrm{range}=0.001..100\,\mathrm{output}=\mathrm{data}\right)\:$

$\mathrm{Hinfgraph}≔\max \left(\mathrm{mag}\left(1..-1\,2..2\right)\right)$

$\mathrm{Hinfgraph}≔19.9999996000000$

$\mathrm{sys2}≔\mathrm{DiffEquation}\left(\mathrm{diff}\left(\mathrm{diff}\left(x\left(t\right)\,t\right)\,t\right)=-10x\left(t\right)-\mathrm{diff}\left(x\left(t\right)\,t\right)+w\left(t\right)\,\left[w\left(t\right)\right]\,\left[x\left(t\right)\right]\right)\:$

$\mathrm{PrintSystem}\left(\mathrm{sys2}\right)$

$\left[\begin{array}{l}\mathbf{Diff.\; Equation}\\ \mathrm{continuous}\\ \mathrm{1\; output(s);\; 1\; input(s)}\\ \mathrm{inputvariable}\=\left[w\left(t\right)\right]\\ \mathrm{outputvariable}\=\left[x\left(t\right)\right]\\ \mathrm{de}\=\left[\frac{{\ⅆ}^2}{\ⅆt^2}x\left(t\right)=-10x\left(t\right)-\frac{\ⅆ}{\ⅆt}x\left(t\right)+w\left(t\right)\right]\end{array}\right.$

$\mathrm{hinfnorm2}≔\mathrm{NormHinf}\left(\mathrm{sys2}\,'\mathrm{output}'=\left[\mathrm{peakfreq}\,\mathrm{norm}\right]\right)$

$\mathrm{hinfnorm2}≔\left[3.08220698300548\,0.320256627866482\right]$

$\mathrm{MagnitudePlot}\left(\mathrm{sys2}\,\mathrm{decibels}=\mathrm{false}\right)$

$\mathrm{mag}≔\mathrm{MagnitudePlot}\left(\mathrm{sys2}\,\mathrm{decibels}=\mathrm{false}\,\mathrm{output}=\mathrm{data}\right)\:$

$\mathrm{member}\left(\max \left(\mathrm{mag}\left(1..-1\,2..2\right)\right)\,\mathrm{mag}\left(1..-1\,2..2\right)\,'p'\right)\:$$\mathrm{fHinf}≔\mathrm{mag}\left(p\right)$

$\mathrm{fHinf}≔3.079785057$

$\mathrm{Hinfgraph}≔\max \left(\mathrm{mag}\left(1..-1\,2..2\right)\right)$

$\mathrm{Hinfgraph}≔0.320252649756933$

$\mathrm{sys3}≔\mathrm{StateSpace}\left(⟨⟨0\,0\,-3⟩\left|⟨1\,0\,-4⟩\right|⟨0\,1\,-7⟩⟩\,⟨⟨0\,0\,1⟩⟩\,⟨⟨1⟩\left|⟨0⟩\right|⟨0⟩⟩\,\mathrm{Matrix}\left(1\,1\right)\right)\:$

$\mathrm{PrintSystem}\left(\mathrm{sys3}\right)$

$\left[\begin{array}{l}\mathbf{State\; Space}\\ \mathrm{continuous}\\ \mathrm{1\; output(s);\; 1\; input(s);\; 3\; state(s)}\\ \mathrm{inputvariable}\=\left[\mathrm{u1}\left(t\right)\right]\\ \mathrm{outputvariable}\=\left[\mathrm{y1}\left(t\right)\right]\\ \mathrm{statevariable}\=\left[\mathrm{x1}\left(t\right)\,\mathrm{x2}\left(t\right)\,\mathrm{x3}\left(t\right)\right]\\ \mathrm{a}\=\left[\begin{array}{ccc}0& 1& 0\\ 0& 0& 1\\ \mathrm{-3}& \mathrm{-4}& \mathrm{-7}\end{array}\right]\\ \mathrm{b}\=\left[\begin{array}{c}0\\ 0\\ 1\end{array}\right]\\ \mathrm{c}\=\left[\begin{array}{ccc}1& 0& 0\end{array}\right]\\ \mathrm{d}\=\left[\begin{array}{c}0\end{array}\right]\end{array}\right.$

$\mathrm{hinfnorm3}≔\mathrm{NormHinf}\left(\mathrm{sys3}\,'\mathrm{output}'=\left[\mathrm{norm}\,\mathrm{peakfreq}\right]\right)$

$\mathrm{hinfnorm3}≔\left[0.451322261502234\,0.559605319105211\right]$

$\mathrm{MagnitudePlot}\left(\mathrm{sys3}\,\mathrm{decibels}=\mathrm{false}\right)$

$\mathrm{mag}≔\mathrm{MagnitudePlot}\left(\mathrm{sys3}\,\mathrm{decibels}=\mathrm{false}\,\mathrm{output}=\mathrm{data}\right)\:$

$\mathrm{Hinfgraph}≔\max \left(\mathrm{mag}\left(1..-1\,2..2\right)\right)$

$\mathrm{Hinfgraph}≔0.451320291397442$

$\mathrm{member}\left(\mathrm{Hinfgraph}\,\mathrm{mag}\left(1..-1\,2..2\right)\,'p'\right)\:$$\mathrm{fHinf}≔\mathrm{mag}\left(p\right)$

$\mathrm{fHinf}≔0.5590478459$

$\mathrm{sys4}≔\mathrm{TransferFunction}\left(\mathrm{Matrix}\left(\left[\left[\frac{1}{s^3+s^2+5s+2}\right]\,\left[\frac{s}{s^3+s^2+5s+2}\right]\,\left[\frac{s^2}{s^3+s^2+5s+2}\right]\right]\right)\right)\:$

$\mathrm{PrintSystem}\left(\mathrm{sys4}\right)$

$\left[\begin{array}{l}\mathbf{Transfer\; Function}\\ \mathrm{continuous}\\ \mathrm{3\; output(s);\; 1\; input(s)}\\ \mathrm{inputvariable}\=\left[\mathrm{u1}\left(s\right)\right]\\ \mathrm{outputvariable}\=\left[\mathrm{y1}\left(s\right)\,\mathrm{y2}\left(s\right)\,\mathrm{y3}\left(s\right)\right]\\ {\mathrm{tf}}_{1,1}\=\frac{1}{s^3+s^2+5s+2}\\ {\mathrm{tf}}_{2,1}\=\frac{s}{s^3+s^2+5s+2}\\ {\mathrm{tf}}_{3,1}\=\frac{s^2}{s^3+s^2+5s+2}\end{array}\right.$

$\mathrm{hinfnorm4}≔\mathrm{NormHinf}\left(\mathrm{sys4}\,0.001\,'\mathrm{output}'=\left[\mathrm{norm}\,\mathrm{peakfreq}\right]\right)$

$\mathrm{hinfnorm4}≔\left[1.89966130541915\,2.180899209\right]$

$\mathrm{MagnitudePlot}\left(\mathrm{sys4}\,\mathrm{decibels}=\mathrm{false}\right)$

$\mathrm{mag}≔\mathrm{MagnitudePlot}\left(\mathrm{sys4}\,\mathrm{decibels}=\mathrm{false}\,\mathrm{output}=\mathrm{data}\right)\:$

$\mathrm{Hinfgraph}≔\max \left(\mathrm{mag}\left(1..-1\,2..2\right)\right)$

$\mathrm{Hinfgraph}≔1.69438112239631$

$\mathrm{member}\left(\mathrm{Hinfgraph}\,\mathrm{mag}\left[3\right]\left(1..-1\,2..2\right)\,'p'\right)\:$$\mathrm{fHinf}≔\mathrm{mag}\left[3\right]\left(p\right)$

$\mathrm{fHinf}≔2.184166359$

$\mathrm{sys5}≔\mathrm{TransferFunction}\left(\mathrm{Matrix}\left(\left[\left[\frac{1}{s^2+s+4}\,0\right]\,\left[0\,\frac{1}{s^2+s+4}\right]\right]\right)\right)\:$

$\mathrm{PrintSystem}\left(\mathrm{sys5}\right)$

$\left[\begin{array}{l}\mathbf{Transfer\; Function}\\ \mathrm{continuous}\\ \mathrm{2\; output(s);\; 2\; input(s)}\\ \mathrm{inputvariable}\=\left[\mathrm{u1}\left(s\right)\,\mathrm{u2}\left(s\right)\right]\\ \mathrm{outputvariable}\=\left[\mathrm{y1}\left(s\right)\,\mathrm{y2}\left(s\right)\right]\\ {\mathrm{tf}}_{1,1}\=\frac{1}{s^2+s+4}\\ {\mathrm{tf}}_{2,1}\=0\\ {\mathrm{tf}}_{1,2}\=0\\ {\mathrm{tf}}_{2,2}\=\frac{1}{s^2+s+4}\end{array}\right.$

$\mathrm{hinfnorm5}≔\mathrm{NormHinf}\left(\mathrm{sys5}\,'\mathrm{output}'=\left[\mathrm{norm}\,\mathrm{peakfreq}\right]\right)$

$\mathrm{hinfnorm5}≔\left[0.516398295858964\,1.87082283018653\right]$

$\mathrm{MagnitudePlot}\left(\mathrm{sys5}\,\mathrm{decibels}=\mathrm{false}\right)$

$\mathrm{mag}≔\mathrm{MagnitudePlot}\left(\mathrm{sys5}\,\mathrm{decibels}=\mathrm{false}\,\mathrm{output}=\mathrm{data}\right)\:$

$\mathrm{Hinfgraph}≔\max \left(\mathrm{mag}\left(1..-1\,2..2\right)\right)$

$\mathrm{Hinfgraph}≔0.516350854134402$

$\mathrm{member}\left(\mathrm{Hinfgraph}\,\mathrm{mag}\left[1\right]\left(1..-1\,2..2\right)\,'p'\right)\:$$\mathrm{fHinf}≔\mathrm{mag}\left[1\right]\left(p\right)$

$\mathrm{fHinf}≔1.863838004$

$\mathrm{sys6}≔\mathrm{StateSpace}\left(\mathrm{Matrix}\left(\left[\left[0\,1\right]\,\left[-25\,-0.1\right]\right]\right)\,\mathrm{Matrix}\left(\left[\left[0\right]\,\left[1\right]\right]\right)\,\mathrm{Matrix}\left(\left[\left[1\,0\right]\right]\right)\,\mathrm{Matrix}\left(\left[\left[0\right]\right]\right)\right)\:$

$\mathrm{PrintSystem}\left(\mathrm{sys6}\right)$

$\left[\begin{array}{l}\mathbf{State\; Space}\\ \mathrm{continuous}\\ \mathrm{1\; output(s);\; 1\; input(s);\; 2\; state(s)}\\ \mathrm{inputvariable}\=\left[\mathrm{u1}\left(t\right)\right]\\ \mathrm{outputvariable}\=\left[\mathrm{y1}\left(t\right)\right]\\ \mathrm{statevariable}\=\left[\mathrm{x1}\left(t\right)\,\mathrm{x2}\left(t\right)\right]\\ \mathrm{a}\=\left[\begin{array}{cc}0& 1\\ \mathrm{-25}& \mathrm{-0.1}\end{array}\right]\\ \mathrm{b}\=\left[\begin{array}{c}0\\ 1\end{array}\right]\\ \mathrm{c}\=\left[\begin{array}{cc}1& 0\end{array}\right]\\ \mathrm{d}\=\left[\begin{array}{c}0\end{array}\right]\end{array}\right.$

$\mathrm{hinfnorm6}≔\mathrm{NormHinf}\left(\mathrm{sys6}\,'\mathrm{output}'=\left[\mathrm{norm}\,\mathrm{peakfreq}\right]\right)$

$\mathrm{hinfnorm6}≔\left[2.00010200760040\,4.99949995099030\right]$

$\mathrm{MagnitudePlot}\left(\mathrm{sys6}\,\mathrm{decibels}=\mathrm{false}\right)$

$\mathrm{mag}≔\mathrm{MagnitudePlot}\left(\mathrm{sys6}\,\mathrm{decibels}=\mathrm{false}\,\mathrm{output}=\mathrm{data}\right)\:$

$\mathrm{Hinfgraph}≔\max \left(\mathrm{mag}\left(1..-1\,2..2\right)\right)$

$\mathrm{Hinfgraph}≔1.99995097863956$

$\mathrm{member}\left(\mathrm{Hinfgraph}\,\mathrm{mag}\left(1..-1\,2..2\right)\,'p'\right)\:$$\mathrm{fHinf}≔\mathrm{mag}\left(p\right)$

$\mathrm{fHinf}≔5.000110374$

$\mathrm{sys7}≔\mathrm{TransferFunction}\left(\frac{10\left(2z+1\right)}{10z^2+2z+5}\,\mathrm{discrete}\,\mathrm{sampletime}=0.1\right)\:$

$\mathrm{PrintSystem}\left(\mathrm{sys7}\right)$

$\left[\begin{array}{l}\mathbf{Transfer\; Function}\\ \mathrm{discrete;\; sampletime\; =\; .1}\\ \mathrm{1\; output(s);\; 1\; input(s)}\\ \mathrm{inputvariable}\=\left[\mathrm{u1}\left(z\right)\right]\\ \mathrm{outputvariable}\=\left[\mathrm{y1}\left(z\right)\right]\\ {\mathrm{tf}}_{1,1}\=\frac{20z+10}{10z^2+2z+5}\end{array}\right.$

$\mathrm{hinfnorm7}≔\mathrm{NormHinf}\left(\mathrm{sys7}\,{10}^{-8}\,'\mathrm{output}'=\left[\mathrm{norm}\,\mathrm{peakfreq}\right]\right)$

$\mathrm{hinfnorm7}≔\left[4.26497897082109\,17.0452791622670\right]$

$\mathrm{MagnitudePlot}\left(\mathrm{sys7}\,\mathrm{decibels}=\mathrm{false}\,\mathrm{range}=0.01..\frac{\mathrm{\pi }}{\mathrm{sys7}:-\mathrm{sampletime}}\right)$

$\mathrm{mag}≔\mathrm{MagnitudePlot}\left(\mathrm{sys7}\,\mathrm{decibels}=\mathrm{false}\,\mathrm{range}=0.01..\frac{\mathrm{\pi }}{\mathrm{sys7}:-\mathrm{sampletime}}\,\mathrm{output}=\mathrm{data}\right)\:$

$\mathrm{Hinfgraph}≔\max \left(\mathrm{mag}\left(1..-1\,2..2\right)\right)$

$\mathrm{Hinfgraph}≔4.289843575$

$\mathrm{member}\left(\mathrm{Hinfgraph}\,\mathrm{mag}\left(1..-1\,2..2\right)\,'p'\right)\:$$\mathrm{fHinf}≔\mathrm{mag}\left(p\right)$

$\mathrm{fHinf}≔16.66285136$

$\mathrm{sys8}≔\mathrm{TransferFunction}\left(\left[5\,-14.2\,14.4\,-5\right]\,\left[5\,-12.1\,10\,-2.7\right]\,\mathrm{discrete}\,\mathrm{sampletime}=0.5\right)\:$

$\mathrm{PrintSystem}\left(\mathrm{sys8}\right)$

$\left[\begin{array}{l}\mathbf{Transfer\; Function}\\ \mathrm{discrete;\; sampletime\; =\; .5}\\ \mathrm{1\; output(s);\; 1\; input(s)}\\ \mathrm{inputvariable}\=\left[\mathrm{u1}\left(z\right)\right]\\ \mathrm{outputvariable}\=\left[\mathrm{y1}\left(z\right)\right]\\ {\mathrm{tf}}_{1,1}\=\frac{5.z^3-14.20000000z^2+14.40000000z-5.}{5.z^3-12.10000000z^2+10.z-2.700000000}\end{array}\right.$

$\mathrm{hinfnorm8}≔\mathrm{NormHinf}\left(\mathrm{sys8}\,'\mathrm{output}'=\left[\mathrm{norm}\,\mathrm{peakfreq}\right]\right)$

$\mathrm{hinfnorm8}≔\left[4.63571003774221\,0.615651253991576\right]$

$\mathrm{MagnitudePlot}\left(\mathrm{sys8}\,\mathrm{decibels}=\mathrm{false}\,\mathrm{range}=0.01..\frac{\mathrm{\pi }}{\mathrm{sys8}:-\mathrm{sampletime}}\right)$

$\mathrm{mag}≔\mathrm{MagnitudePlot}\left(\mathrm{sys8}\,\mathrm{decibels}=\mathrm{false}\,\mathrm{range}=0.01..\frac{\mathrm{\pi }}{\mathrm{sys8}:-\mathrm{sampletime}}\,\mathrm{output}=\mathrm{data}\right)\:$

$\mathrm{Hinfgraph}≔\max \left(\mathrm{mag}\left(1..-1\,2..2\right)\right)$

$\mathrm{Hinfgraph}≔4.635634989$

$\mathrm{member}\left(\mathrm{Hinfgraph}\,\mathrm{mag}\left(1..-1\,2..2\right)\,'p'\right)\:$$\mathrm{fHinf}≔\mathrm{mag}\left(p\right)$

$\mathrm{fHinf}≔0.6152990634$

S. Boyd, V. Balakrishnan, P. Kabamba, On computing the $H_{\mathrm{\infty }}$ norm of a transfer matrix, 1988.

N. A. Bruinsma, M. Steinbuch, A fast algortihm to compute the $H_{\mathrm{\infty }}$-norm of a transfer function matrix, 1990.

The DynamicSystems[NormHinf] command was introduced in Maple 18.

For more information on Maple 18 changes, see Updates in Maple 18.