output = outcovar or statecovar or list of these names.

checkstability = truefalse

The Covariance command computes the steady-state output and state covariance of a linear system sys driven by white Gaussian noise inputs. In state-space form, this system is represented as:

$\stackrel{.}{x}=\mathrm{Ax}+\mathrm{Bw}$

$y=\mathrm{Cx}+\mathrm{Dw}$

where: ${\mathrm{A\; \∈\; \ℝ}}^{\mathrm{n\×n}}$, ${\mathrm{B\; \∈\; \ℝ}}^{\mathrm{n\×m}}$, ${\mathrm{C\; \∈\; \ℝ}}^{\mathrm{p\×n}}$, and ${\mathrm{D\; \∈\; \ℝ}}^{\mathrm{p\×m}}$, and $n$, $m$, and $p$ are the number of states, inputs and outputs of the linear system respectively.

The input disturbances $w$ are zero-mean white Gaussian noise inputs with spectral densities or intensities ${\mathrm{Rw\; \∈\; \ℝ}}^{\mathrm{m\×m}}$. In other words, the expected value of $w$ is $𝔼\left(w\left(t\right)\right)=0$ and its covariance matrix is $𝔼\left(w\left(t\right)·\left(w^T\right)\left(t+\mathrm{\tau }\right)\right)=\mathrm{Rw}·\delta \left(\mathrm{\tau }\right)$ where $\mathrm{\delta }$ is the Dirac delta function. In the scalar case, $\mathrm{Rw}={\mathrm{\sigma }}_w^2$ or variance of the random process.

The steady-state covariance matrix of the state $x$ is ${\mathrm{Px\; \∈\; \ℝ}}^{\mathrm{n\×n}}$. The state covariance matrix computed value is returned when sys is in the state-space form.

The steady-state covariance matrix of the output $y$ is ${\mathrm{Py\; \∈\; \ℝ}}^{\mathrm{p\×p}}$. The output covariance Matrix is returned for any sys type.

The covariance matrices are computed by solving the following equations:

The Lyapunov equation approach to finding the steady-state covariance matrices is based on the assumption that the system is stable. For unstable systems, $\mathrm{Px}$ and $\mathrm{Py}$ are infinite.

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

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

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

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

$\mathrm{Rw}≔\mathrm{Matrix}\left(\left[\left[5\right]\right]\right)$

$\mathrm{Rw}≔\left[\begin{array}{c}5\end{array}\right]$

$\mathrm{Py}≔\mathrm{Covariance}\left(\mathrm{sys1}\,\mathrm{Rw}\right)$

$\mathrm{Py}≔\left[\begin{array}{c}30.3167420814479\end{array}\right]$

$\mathrm{Px}≔\mathrm{Covariance}\left(\mathrm{sys1}\,\mathrm{Rw}\,\mathrm{output}=\mathrm{statecovar}\right)$

$\mathrm{Px}≔\left[\begin{array}{}\end{array}\right]$

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

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

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

$\mathrm{Rws}≔\mathrm{Matrix}\left(\left[\left[2\right]\right]\right)$

$\mathrm{Rws}≔\left[\begin{array}{c}2\end{array}\right]$

$\mathrm{Py}≔\mathrm{Covariance}\left(\mathrm{sys2}\,\mathrm{Rws}\,\mathrm{output}=\mathrm{outcovar}\right)$

$\mathrm{Py}≔\left[\begin{array}{cc}9.00000000000000& \mathrm{-4.00000000000000}\\ \mathrm{-4.00000000000000}& 1.81818181818182\end{array}\right]$

$\mathrm{outlist}≔\mathrm{Covariance}\left(\mathrm{sys2}\,\mathrm{Rws}\,\mathrm{output}=\left[\mathrm{outcovar}\,\mathrm{statecovar}\right]\right)$

$\mathrm{outlist}≔\left[\left[\begin{array}{cc}9.00000000000000& \mathrm{-4.00000000000000}\\ \mathrm{-4.00000000000000}& 1.81818181818182\end{array}\right]\,\left[\begin{array}{cc}3.36363636363637& 4.27272727272728\\ 4.27272727272728& 5.45454545454546\end{array}\right]\right]$

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

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