Work with transfer functions, state space models, or differential equations

Linearize systems

Analyze the controllability, observability, phase and gain margin, and more

Generate control plots, including Bode, root-locus and Nyquist plots

Work symbolically or numerically

$$\begin{gathered} \mathrm{restart}\: \\ \mathrm{with}\left(\mathrm{DynamicSystems}\right)\: \end{gathered}$$

$$\begin{gathered} \mathrm{eq\_sym}≔ \\ \mathrm{L}\cdot \stackrel{\.}{\mathrm{i}}\left(\mathrm{t}\right)\+\mathrm{R}\⋅\mathrm{i}\left(\mathrm{t}\right)\=\mathrm{v}\left(\mathrm{t}\right)\−\mathrm{K}\⋅\stackrel{\.}{\mathrm{\theta }}\left(\mathrm{t}\right)\, \\ \mathrm{J}\cdot \stackrel{..}{\mathrm{\theta }}\left(\mathrm{t}\right)\+\mathrm{b}\⋅\stackrel{\.}{\mathrm{\θ}}\left(\mathrm{t}\right)\+\mathrm{Ks}\⋅\mathrm{\θ}\left(\mathrm{t}\right)\=\mathrm{K}\⋅\mathrm{i}\left(\mathrm{t}\right) \: \end{gathered}$$

$\mathrm{params}≔\left[\mathrm{J}\=0.1\, \mathrm{b}\=0.1\, \mathrm{K}\=0.01\, \mathrm{R}\=1\, \mathrm{L}\=0.5\, \mathrm{Ks} \= 1\right]\:$

$\mathrm{eq\_num}≔\mathrm{eval}\left(\mathrm{eq\_sym}\,\mathrm{params}\right)\:$

$\mathrm{sys\_num} ≔ \mathrm{StateSpace}\left(\left[\mathrm{eq\_num}\right]\, \mathrm{inputvariable}\=\left[\mathrm{v}\left(\mathrm{t}\right)\right]\, \mathrm{outputvariable}\=\left[\mathrm{theta}\left(\mathrm{t}\right)\,\mathrm{i}\left(\mathrm{t}\right)\right]\right)\:$

$\mathrm{ControllabilityMatrix}\left(\mathrm{sys\_num}\right)$

$\left[\begin{array}{ccc}2& \mathrm{-4}& \frac{1999}{250}\\ 0& 0& \frac{1}{5}\\ 0& \frac{1}{5}& -\frac{3}{5}\end{array}\right]$

$\mathrm{RootLocusPlot}\left(\mathrm{sys\_num}\right)$

$\mathrm{sys\_sym} ≔ \mathrm{StateSpace}\left(\left[\mathrm{eq\_sym}\right]\, \mathrm{inputvariable}\=\left[\mathrm{v}\left(\mathrm{t}\right)\right]\, \mathrm{outputvariable}\=\left[\mathrm{theta}\left(\mathrm{t}\right)\,\mathrm{i}\left(\mathrm{t}\right)\right]\right)\:$

$\mathrm{ControllabilityMatrix}\left(\mathrm{sys\_sym}\right)$

$\left[\begin{array}{ccc}\frac{1}{\mathrm{L}}& -\frac{\mathrm{R}}{{\mathrm{L}}^2}& \frac{{\mathrm{R}}^2}{{\mathrm{L}}^3}-\frac{{\mathrm{K}}^2}{{\mathrm{L}}^2\mathrm{J}}\\ 0& 0& \frac{\mathrm{K}}{\mathrm{J}\mathrm{L}}\\ 0& \frac{\mathrm{K}}{\mathrm{J}\mathrm{L}}& -\frac{\mathrm{K}\mathrm{R}}{\mathrm{J}{\mathrm{L}}^2}-\frac{\mathrm{b}\mathrm{K}}{{\mathrm{J}}^2\mathrm{L}}\end{array}\right]$