method = forward, backward, bilinear, prewarp, matched, zoh, or foh

Specifies the conversion method.

frequency = realcons

Specifies the warping or critical frequency in rad/s. This is used only if method = prewarp, otherwise it is ignored.

The ToContinuous command converts a discrete-time system, sys, to a continuous-time system.

The method option specifies the discrete-time to continuous-time conversion method. The following methods are supported: forward rectangle rule (forward), backward rectangle rule (backward), bilinear rule (bilinear), bilinear with prewarping (prewarp), matched poles and zeros (matched), zero-order hold (zoh), and first-order hold or non-causal triangle-hold (foh).

The bilinear rule is also known as the Tustin or Trapezoid rule.

Not all inverse-discretization methods can be applied to all system types. The following table describes the usage. An x indicates that the method can be used by the system type. A D indicates that the method is the default for the system type.

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

$\mathrm{sys}≔\mathrm{TransferFunction}\left(\left[10\,-5\right]\,\left[1\,-0.8\right]\,\mathrm{discrete}\,\mathrm{sampletime}=0.5\right)\:$$\mathrm{PrintSystem}\left(\mathrm{sys}\right)$

$\left[\begin{array}{l}\mathbf{Transfer\; Function}\\ \mathrm{discrete;\; sampletime\; =\; .5}\\ \mathrm{1\; output(s);\; 1\; input(s)}\\ \mathrm{inputvariable}\=\left[\mathrm{u1}\left(z\right)\right]\\ \mathrm{outputvariable}\=\left[\mathrm{y1}\left(z\right)\right]\\ {\mathrm{tf}}_{1,1}\=\frac{10.z-5.}{z-0.8000000000}\end{array}\right.$

$\mathrm{sys\_s1}≔\mathrm{ToContinuous}\left(\mathrm{sys}\,\mathrm{method}=\mathrm{forward}\right)\:$$\mathrm{PrintSystem}\left(\mathrm{sys\_s1}\right)$

$\left[\begin{array}{l}\mathbf{Transfer\; Function}\\ \mathrm{continuous}\\ \mathrm{1\; output(s);\; 1\; input(s)}\\ \mathrm{inputvariable}\=\left[\mathrm{u1}\left(s\right)\right]\\ \mathrm{outputvariable}\=\left[\mathrm{y1}\left(s\right)\right]\\ {\mathrm{tf}}_{1,1}\=\frac{10.\left(s+1.\right)}{s+0.4000000000}\end{array}\right.$

$\mathrm{sys\_s2}≔\mathrm{ToContinuous}\left(\mathrm{sys}\,\mathrm{method}=\mathrm{backward}\right)\:$$\mathrm{PrintSystem}\left(\mathrm{sys\_s2}\right)$

$\left[\begin{array}{l}\mathbf{Transfer\; Function}\\ \mathrm{continuous}\\ \mathrm{1\; output(s);\; 1\; input(s)}\\ \mathrm{inputvariable}\=\left[\mathrm{u1}\left(s\right)\right]\\ \mathrm{outputvariable}\=\left[\mathrm{y1}\left(s\right)\right]\\ {\mathrm{tf}}_{1,1}\=\frac{6.250000000\left(s+2.\right)}{s+0.5000000000}\end{array}\right.$

$\mathrm{sys\_s3}≔\mathrm{ToContinuous}\left(\mathrm{sys}\,\mathrm{method}=\mathrm{bilinear}\right)\:$$\mathrm{PrintSystem}\left(\mathrm{sys\_s3}\right)$

$\left[\begin{array}{l}\mathbf{Transfer\; Function}\\ \mathrm{continuous}\\ \mathrm{1\; output(s);\; 1\; input(s)}\\ \mathrm{inputvariable}\=\left[\mathrm{u1}\left(s\right)\right]\\ \mathrm{outputvariable}\=\left[\mathrm{y1}\left(s\right)\right]\\ {\mathrm{tf}}_{1,1}\=\frac{8.333333333\left(s+1.333333333\right)}{s+0.4444444444}\end{array}\right.$

$\mathrm{sys\_s4}≔\mathrm{ToContinuous}\left(\mathrm{sys}\,\mathrm{method}=\mathrm{prewarp}\,\mathrm{frequency}=0.3\right)\:$$\mathrm{PrintSystem}\left(\mathrm{sys\_s4}\right)$

$\left[\begin{array}{l}\mathbf{Transfer\; Function}\\ \mathrm{continuous}\\ \mathrm{1\; output(s);\; 1\; input(s)}\\ \mathrm{inputvariable}\=\left[\mathrm{u1}\left(s\right)\right]\\ \mathrm{outputvariable}\=\left[\mathrm{y1}\left(s\right)\right]\\ {\mathrm{tf}}_{1,1}\=\frac{8.333333333\left(s+1.330832395\right)}{s+0.4436107985}\end{array}\right.$

$\mathrm{sys\_s5}≔\mathrm{ToContinuous}\left(\mathrm{sys}\,\mathrm{method}=\mathrm{zoh}\right)\:$$\mathrm{PrintSystem}\left(\mathrm{sys\_s5}\right)$

$\left[\begin{array}{l}\mathbf{Transfer\; Function}\\ \mathrm{continuous}\\ \mathrm{1\; output(s);\; 1\; input(s)}\\ \mathrm{inputvariable}\=\left[\mathrm{u1}\left(s\right)\right]\\ \mathrm{outputvariable}\=\left[\mathrm{y1}\left(s\right)\right]\\ {\mathrm{tf}}_{1,1}\=\frac{10.\left(s+1.115717756\right)}{s+0.4462871026}\end{array}\right.$

$\mathrm{sys\_s6}≔\mathrm{ToContinuous}\left(\mathrm{sys}\,\mathrm{method}=\mathrm{foh}\right)\:$$\mathrm{PrintSystem}\left(\mathrm{sys\_s6}\right)$

$\left[\begin{array}{l}\mathbf{Transfer\; Function}\\ \mathrm{continuous}\\ \mathrm{1\; output(s);\; 1\; input(s)}\\ \mathrm{inputvariable}\=\left[\mathrm{u1}\left(s\right)\right]\\ \mathrm{outputvariable}\=\left[\mathrm{y1}\left(s\right)\right]\\ {\mathrm{tf}}_{1,1}\=\frac{8.264233654\left(s+1.350055919\right)}{s+0.4462871026}\end{array}\right.$

$\mathrm{p0}≔\mathrm{MagnitudePlot}\left(\mathrm{sys}\,\mathrm{range}=0.1..10\,\mathrm{color}=\mathrm{red}\,\mathrm{thickness}=2\,\mathrm{legend}=''discrete''\right)\:$

$\mathrm{p1}≔\mathrm{MagnitudePlot}\left(\mathrm{sys\_s1}\,\mathrm{range}=0.1..10\,\mathrm{color}=\mathrm{blue}\,\mathrm{legend}=''forward''\right)\:$

$\mathrm{p2}≔\mathrm{MagnitudePlot}\left(\mathrm{sys\_s2}\,\mathrm{range}=0.1..10\,\mathrm{color}=\mathrm{green}\,\mathrm{legend}=''backward''\right)\:$

$\mathrm{p3}≔\mathrm{MagnitudePlot}\left(\mathrm{sys\_s3}\,\mathrm{range}=0.1..10\,\mathrm{color}=\mathrm{black}\,\mathrm{legend}=''bilinear''\right)\:$

$\mathrm{p4}≔\mathrm{MagnitudePlot}\left(\mathrm{sys\_s4}\,\mathrm{range}=0.1..10\,\mathrm{color}=\mathrm{grey}\,\mathrm{legend}=''prewarp''\right)\:$

$\mathrm{p5}≔\mathrm{MagnitudePlot}\left(\mathrm{sys\_s5}\,\mathrm{range}=0.1..10\,\mathrm{color}=\mathrm{cyan}\,\mathrm{legend}=''zoh''\right)\:$

$\mathrm{p6}≔\mathrm{MagnitudePlot}\left(\mathrm{sys\_s6}\,\mathrm{range}=0.1..10\,\mathrm{color}=\mathrm{magenta}\,\mathrm{legend}=''foh''\right)\:$

$\mathrm{plots}\left[\mathrm{display}\right]\left(\mathrm{p0}\,\mathrm{p1}\,\mathrm{p2}\,\mathrm{p3}\,\mathrm{p4}\,\mathrm{p5}\,\mathrm{p6}\right)$

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

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