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

Specifies the method used to discretize the system.

frequency = realcons

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

The ToDiscrete command converts a continuous system, sys, to a discrete system.

The optional parameter T is the sampling time. If not symbolic it must evaluate to a positive number. The default value is set by the sampletime option to DynamicSystems[SystemOptions].

The method option specifies the method of discretization. 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 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(\frac{2s^3+s^2+1}{s^3+2s^2+s-10}\right)\:$$\mathrm{PrintSystem}\left(\mathrm{sys}\right)$

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

$T≔0.1$

$T≔0.1$

$\mathrm{sys\_z1}≔\mathrm{ToDiscrete}\left(\mathrm{sys}\,T\,\mathrm{method}=\mathrm{forward}\right)\:$$\mathrm{PrintSystem}\left(\mathrm{sys\_z1}\right)$

$\left[\begin{array}{l}\mathbf{Transfer\; Function}\\ \mathrm{discrete;\; sampletime\; =\; .1}\\ \mathrm{1\; output(s);\; 1\; input(s)}\\ \mathrm{inputvariable}\=\left[\mathrm{u1}\left(z\right)\right]\\ \mathrm{outputvariable}\=\left[\mathrm{y1}\left(z\right)\right]\\ {\mathrm{tf}}_{1,1}\=\frac{2.\left(z^3-2.950000000z^2+2.900000000z-0.9495000000\right)}{z^3-2.800000000z^2+2.610000000z-0.8200000000}\end{array}\right.$

$\mathrm{sys\_z2}≔\mathrm{ToDiscrete}\left(\mathrm{sys}\,T\,\mathrm{method}=\mathrm{backward}\right)\:$$\mathrm{PrintSystem}\left(\mathrm{sys\_z2}\right)$

$\left[\begin{array}{l}\mathbf{Transfer\; Function}\\ \mathrm{discrete;\; sampletime\; =\; .1}\\ \mathrm{1\; output(s);\; 1\; input(s)}\\ \mathrm{inputvariable}\=\left[\mathrm{u1}\left(z\right)\right]\\ \mathrm{outputvariable}\=\left[\mathrm{y1}\left(z\right)\right]\\ {\mathrm{tf}}_{1,1}\=\frac{1.750833333\left(z^3-2.950975726z^2+2.903379343z-0.9519276535\right)}{z^3-2.841666667z^2+2.666666667z-0.8333333333}\end{array}\right.$

$\mathrm{sys\_z3}≔\mathrm{ToDiscrete}\left(\mathrm{sys}\,T\,\mathrm{method}=\mathrm{bilinear}\right)\:$$\mathrm{PrintSystem}\left(\mathrm{sys\_z3}\right)$

$\left[\begin{array}{l}\mathbf{Transfer\; Function}\\ \mathrm{discrete;\; sampletime\; =\; .1}\\ \mathrm{1\; output(s);\; 1\; input(s)}\\ \mathrm{inputvariable}\=\left[\mathrm{u1}\left(z\right)\right]\\ \mathrm{outputvariable}\=\left[\mathrm{y1}\left(z\right)\right]\\ {\mathrm{tf}}_{1,1}\=\frac{1.861634506\left(z^3-2.950856655z^2+2.902444973z-0.9511005426\right)}{z^3-2.816118048z^2+2.627695800z-0.8206583428}\end{array}\right.$

$\mathrm{sys\_z4}≔\mathrm{ToDiscrete}\left(\mathrm{sys}\,T\,\mathrm{method}=\mathrm{prewarp}\,\mathrm{frequency}=0.3\right)\:$$\mathrm{PrintSystem}\left(\mathrm{sys\_z4}\right)$

$\left[\begin{array}{l}\mathbf{Transfer\; Function}\\ \mathrm{discrete;\; sampletime\; =\; .1}\\ \mathrm{1\; output(s);\; 1\; input(s)}\\ \mathrm{inputvariable}\=\left[\mathrm{u1}\left(z\right)\right]\\ \mathrm{outputvariable}\=\left[\mathrm{y1}\left(z\right)\right]\\ {\mathrm{tf}}_{1,1}\=\frac{1.861625101\left(z^3-2.950853005z^2+2.902437835z-0.9510969468\right)}{z^3-2.816105866z^2+2.627669763z-0.8206464690}\end{array}\right.$

$\mathrm{sys\_z5}≔\mathrm{ToDiscrete}\left(\mathrm{sys}\,T\,\mathrm{method}=\mathrm{zoh}\right)\:$$\mathrm{PrintSystem}\left(\mathrm{sys\_z5}\right)$

$\left[\begin{array}{l}\mathbf{Transfer\; Function}\\ \mathrm{discrete;\; sampletime\; =\; .1}\\ \mathrm{1\; output(s);\; 1\; input(s)}\\ \mathrm{inputvariable}\=\left[\mathrm{u1}\left(z\right)\right]\\ \mathrm{outputvariable}\=\left[\mathrm{y1}\left(z\right)\right]\\ {\mathrm{tf}}_{1,1}\=\frac{2.\left(z^3-2.953151801z^2+2.902122691z-0.9485180932\right)}{z^3-2.814354067z^2+2.624028879z-0.8187307530}\end{array}\right.$

$\mathrm{sys\_z6}≔\mathrm{ToDiscrete}\left(\mathrm{sys}\,T\,\mathrm{method}=\mathrm{foh}\right)\:$$\mathrm{PrintSystem}\left(\mathrm{sys\_z6}\right)$

$\left[\begin{array}{l}\mathbf{Transfer\; Function}\\ \mathrm{discrete;\; sampletime\; =\; .1}\\ \mathrm{1\; output(s);\; 1\; input(s)}\\ \mathrm{inputvariable}\=\left[\mathrm{u1}\left(z\right)\right]\\ \mathrm{outputvariable}\=\left[\mathrm{y1}\left(z\right)\right]\\ {\mathrm{tf}}_{1,1}\=\frac{8.187307539\times {10}^{\mathrm{-10}}\left(z^4+2.268487496\times {10}^9{z}^3-6.692346287\times {10}^9{z}^2+6.580907699\times {10}^{9}z-2.155942816\times {10}^9\right)}{z^3-2.814354067z^2+2.624028879z-0.8187307530}\end{array}\right.$

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

$\mathrm{p1}≔\mathrm{MagnitudePlot}\left(\mathrm{sys\_z1}\,\mathrm{range}=0.1..10\,\mathrm{color}=\mathrm{blue}\,\mathrm{legend}=''forward''\right)\:$

$\mathrm{p2}≔\mathrm{MagnitudePlot}\left(\mathrm{sys\_z2}\,\mathrm{range}=0.1..10\,\mathrm{color}=\mathrm{green}\,\mathrm{legend}=''backward''\right)\:$

$\mathrm{p3}≔\mathrm{MagnitudePlot}\left(\mathrm{sys\_z3}\,\mathrm{range}=0.1..10\,\mathrm{color}=\mathrm{black}\,\mathrm{legend}=''bilinear''\right)\:$

$\mathrm{p4}≔\mathrm{MagnitudePlot}\left(\mathrm{sys\_z4}\,\mathrm{range}=0.1..10\,\mathrm{color}=\mathrm{grey}\,\mathrm{legend}=''prewarp''\right)\:$

$\mathrm{p5}≔\mathrm{MagnitudePlot}\left(\mathrm{sys\_z5}\,\mathrm{range}=0.1..10\,\mathrm{color}=\mathrm{cyan}\,\mathrm{legend}=''zoh''\right)\:$

$\mathrm{p6}≔\mathrm{MagnitudePlot}\left(\mathrm{sys\_z6}\,\mathrm{range}=0.1..10\,\mathrm{color}=\mathrm{magenta}\,\mathrm{legend}=''foh''\right)\:$

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

The DynamicSystems[ToDiscrete] command was updated in Maple 18.

The method option was updated in Maple 18.