checkstability = truefalse

The NormH2 command computes the H2 norm of a linear system sys. Both continuous-time and discrete-time systems, and both single-input single-output (SISO) and multiple-input multiple-output (MIMO) systems are supported.

For both continuous and discrete-time systems, the H2 norm is finite if the LTI system is asymptotically stable. It follows that for unstable systems, the H2 norm is infinite.

A deterministic interpretation of the H2 norm is that it measures the energy of the impulse response of the LTI system.

A stochastic interpretation of the H2 norm is that it measures the energy of the output response to unit white Gaussian noise inputs. A white noise process $w\left(t\right)$ has an expected or mean value $𝔼\left(w\left(t\right)\right)=0$ and covariance matrix $𝔼\left(w\left(t\right)·{w\left(t+\mathrm{\tau }\right)}^T\right)=𝕀·\delta \left(\mathrm{\tau }\right)$, where $𝕀$ is the Identity Matrix and $\mathrm{\delta }$ is the Dirac delta function. It follows that the H2 norm is equivalent to: ${\mathrm{\Vert H\Vert }}_2=\sqrt{\mathrm{Trace}\left(\mathrm{Covariance}\left('\mathrm{sys}'\,𝕀\right)\right)}$ from the interpretation above and DynamicSystems[Covariance].

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

$\mathrm{sys1}≔\mathrm{TransferFunction}\left(\frac{10\left(2z+1\right)}{10z^2+2z+5}\,\mathrm{discrete}\,\mathrm{sampletime}=0.1\right)\:$

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

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

$\mathrm{h2norm1}≔\mathrm{NormH2}\left(\mathrm{sys1}\right)$

$\mathrm{h2norm1}≔2.46238673166698$

$\mathrm{sys2}≔\mathrm{StateSpace}\left(⟨⟨-5\,3⟩|⟨3\,-4⟩⟩\,⟨⟨2\,3⟩|⟨1\,1⟩⟩\,⟨⟨1\,-2⟩|⟨\frac{1}{2}\,1⟩⟩\,⟨⟨0\,0⟩|⟨0\,0⟩⟩\right)\:$

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

$\left[\begin{array}{l}\mathbf{State\; Space}\\ \mathrm{continuous}\\ \mathrm{2\; output(s);\; 2\; input(s);\; 2\; state(s)}\\ \mathrm{inputvariable}\=\left[\mathrm{u1}\left(t\right)\,\mathrm{u2}\left(t\right)\right]\\ \mathrm{outputvariable}\=\left[\mathrm{y1}\left(t\right)\,\mathrm{y2}\left(t\right)\right]\\ \mathrm{statevariable}\=\left[\mathrm{x1}\left(t\right)\,\mathrm{x2}\left(t\right)\right]\\ \mathrm{a}\=\left[\begin{array}{cc}\mathrm{-5}& 3\\ 3& \mathrm{-4}\end{array}\right]\\ \mathrm{b}\=\left[\begin{array}{cc}2& 1\\ 3& 1\end{array}\right]\\ \mathrm{c}\=\left[\begin{array}{cc}1& \frac{1}{2}\\ \mathrm{-2}& 1\end{array}\right]\\ \mathrm{d}\=\left[\begin{array}{cc}0& 0\\ 0& 0\end{array}\right]\end{array}\right.$

$\mathrm{h2norm2}≔\mathrm{NormH2}\left(\mathrm{sys2}\right)$

$\mathrm{h2norm2}≔2.52637601270590$

$\mathrm{sys3}≔\mathrm{Coefficients}\left(\left[1\,-2.841\,2.875\,-1.004\right]\,\left[1\,-2.417\,2.003\,-0.5488\right]\,\mathrm{discrete}\,\mathrm{sampletime}=0.1\right)\:$

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

$\left[\begin{array}{l}\mathbf{Coefficients}\\ \mathrm{discrete;\; sampletime\; =\; .1}\\ \mathrm{1\; output(s);\; 1\; input(s)}\\ \mathrm{inputvariable}\=\left[\mathrm{u1}\left(z\right)\right]\\ \mathrm{outputvariable}\=\left[\mathrm{y1}\left(z\right)\right]\\ {\mathrm{num}}_{1,1}\=\left[1\,\mathrm{-2.841}\,2.875\,\mathrm{-1.004}\right]\\ {\mathrm{den}}_{1,1}\=\left[1\,\mathrm{-2.417}\,2.003\,\mathrm{-0.5488}\right]\end{array}\right.$

$\mathrm{h2norm3}≔\mathrm{NormH2}\left(\mathrm{sys3}\right)$

$\mathrm{h2norm3}≔1.24382062647607$

$\mathrm{sys4}≔\mathrm{DiffEquation}\left(\mathrm{diff}\left(\mathrm{diff}\left(x\left(t\right)\,t\right)\,t\right)=-10x\left(t\right)-\mathrm{diff}\left(x\left(t\right)\,t\right)+w\left(t\right)\,\left[w\left(t\right)\right]\,\left[x\left(t\right)\right]\right)\:$

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

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

$\mathrm{h2norm4}≔\mathrm{NormH2}\left(\mathrm{sys4}\right)$

$\mathrm{h2norm4}≔0.223606797749979$

$\mathrm{sys5}≔\mathrm{StateSpace}\left(⟨⟨-5\,3⟩|⟨3\,-4⟩⟩\,⟨⟨2\,3⟩|⟨1\,1⟩⟩\,⟨⟨1\,-2⟩|⟨\frac{1}{2}\,1⟩⟩\,⟨⟨2\,1⟩|⟨3\,7⟩⟩\right)\:$

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

$\left[\begin{array}{l}\mathbf{State\; Space}\\ \mathrm{continuous}\\ \mathrm{2\; output(s);\; 2\; input(s);\; 2\; state(s)}\\ \mathrm{inputvariable}\=\left[\mathrm{u1}\left(t\right)\,\mathrm{u2}\left(t\right)\right]\\ \mathrm{outputvariable}\=\left[\mathrm{y1}\left(t\right)\,\mathrm{y2}\left(t\right)\right]\\ \mathrm{statevariable}\=\left[\mathrm{x1}\left(t\right)\,\mathrm{x2}\left(t\right)\right]\\ \mathrm{a}\=\left[\begin{array}{cc}\mathrm{-5}& 3\\ 3& \mathrm{-4}\end{array}\right]\\ \mathrm{b}\=\left[\begin{array}{cc}2& 1\\ 3& 1\end{array}\right]\\ \mathrm{c}\=\left[\begin{array}{cc}1& \frac{1}{2}\\ \mathrm{-2}& 1\end{array}\right]\\ \mathrm{d}\=\left[\begin{array}{cc}2& 3\\ 1& 7\end{array}\right]\end{array}\right.$

$\mathrm{h2norm5}≔\mathrm{NormH2}\left(\mathrm{sys5}\right)$

Error, (in DynamicSystems:-NormH2) H2 norm is infinite for continuous 'sys' with D<>0 (system is not strictly causal).

$\mathrm{sys6}≔\mathrm{TransferFunction}\left(\frac{4s+3}{5s^4+7s^3+4s^2+3s+1}\right)\&colon;$

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

$\left[\begin{array}{l}\mathbf{Transfer\; Function}\\ \mathrm{continuous}\\ \mathrm{1\; output(s);\; 1\; input(s)}\\ \mathrm{inputvariable}\&equals;\left[\mathrm{u1}\left(s\right)\right]\\ \mathrm{outputvariable}\&equals;\left[\mathrm{y1}\left(s\right)\right]\\ {\mathrm{tf}}_{1,1}\&equals;\frac{4s+3}{5s^4+7s^3+4s^2+3s+1}\end{array}\right.$

$\mathrm{h2norm6}≔\mathrm{NormH2}\left(\mathrm{sys6}\right)$

Error, (in DynamicSystems:-NormH2) H2 norm is infinite for unstable systems. Unstable eigenvalues of 'sys': .0324596324047337-.6550790709001*I, .0324596324047337+.6550790709001*I

The DynamicSystems[NormH2] command was introduced in Maple 18.

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