The DynamicSystems package allows for the creation of six different types of System Objects.

A DynamicSystems object is a data container structure that encapsulates all of the information about a system.

The following example shows how to create a Differential Equation System Object. The differential equation describing the DC motor's electrical system is defined by $\mathrm{eqElectrical}$.

$\mathrm{eqElectrical}≔\left[L\cdot \stackrel{\.}{i}\left(t\right)\+R\cdot i\left(t\right)\=V\left(t\right)-K\cdot \stackrel{\.}{\mathrm{\θ}}\left(t\right)\right]$

Note: Maple provides two distinct methods to write derivatives with respect to $t$. The first method produces $\frac{\ⅆ}{\ⅆt}$ and can be obtained using the $\frac{\ⅆ}{\ⅆx}f$ operator from the Expressions palette, while the second method makes use of Maple's dot notation for time derivatives. To get the dot notation, first type the variable and then press Ctrl + Shift + '' to raise the cursor above the variable. Now, type a period (.) above the variable to indicate the derivative. 

Example: Time derivative of $y$ in 2-D Math:

1.

2. Press Ctrl + Shift + " :

3. Type (.):

4. Press Ctrl + = :

  For more details and shortcuts for using 2-D Math notation to enter derivatives, see 2-D Math Shortcut Keys and How Do I Enter an ODE?.

Now that the equation is defined, you can use the DiffEquation command to create a DynamicSystems System Object which can then be accessed by other commands in the DynamicSystems package.

$\mathrm{sysElectrical}≔\mathrm{DiffEquation}\left(\mathrm{eqElectrical}\, \mathrm{inputvariable}\=\left[V\left(t\right)\right]\,\mathrm{outputvariable}\=\left[\mathrm{\θ}\left(t\right)\right]\right)$

The System Object created in (9) provides us with information about the system. For instance, $\mathrm{sysElectrical}$ has 1 input and 1 output. This information becomes very useful when dealing with large scale systems.

Once a System Object is created you can use the PrintSystem command to print the contents of the System Object.

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

An alternative method for creating a DynamicSystems Object is through the context-sensitive menu. Following is an illustration of the use of the context menu to create a System Object of the DC motor's mechanical system.

The differential equation describing the DC motor's mechanical system is defined by $\mathrm{eqMechanical}$.. To create a Differential System Object from this equation, right-click on the equation and select Differential Equation from the DynamicSystems>System Creation menu. A dialog appears asking you to define a name for the System Object. For this example, use the name sysMechanical. Then, you are asked to specify the input and outputs of the system. In this case the input of the system is $i\left(t\right)$ and the output is $\mathrm{theta}\left(t\right)$.

$\mathrm{eqMechanical}≔\left[J\cdot \frac{{\ⅆ}^2}{\ⅆ t^2}\mathrm{\θ}\left(t\right)\+b\cdot \frac{\ⅆ}{\ⅆ t}\mathrm{\θ}\left(t\right)\=K\cdot i\left(t\right)\right]$

$\stackrel{\text{create differential equation}}{\to }$

By combining equations in (8) and (11) you can create a Differential Equation System Object describing the entire DC motor model in terms of the voltage, $V\left(t\right)$, and angular position, $\mathrm{\θ}\left(t\right)$.

$\mathrm{eqOverall}≔\left[\mathrm{eqElectrical}\left[\right]\, \mathrm{eqMechanical}\left[\right]\right]$

Note:  The syntax equation[ ] is used to extract the equations from the list.  For instance, $\mathrm{eqElectrical}\left[\right]$ = $L\left(\frac{\ⅆ}{\ⅆt}i\left(t\right)\right)\+Ri\left(t\right)\=V\left(t\right)-K\left(\frac{\ⅆ}{\ⅆt}\mathrm{\θ}\left(t\right)\right)$.  The two equations are then combined in a new list.

$$\begin{gathered} \mathrm{sysOverall} ≔ \mathrm{DiffEquation}\left(\mathrm{eqOverall}\, \mathrm{inputvariable}\=\left[V\left(t\right)\right]\, \mathrm{outputvariable}\=\left[\mathrm{\θ}\left(t\right)\right]\right)\: \\ \mathrm{PrintSystem}\left(\mathrm{sysOverall}\right) \end{gathered}$$

A Transfer Function System Object can be obtained directly from the Differential Equation System Object $\mathrm{sysOverall}$ by applying the TransferFunction command to the System Object.

$$\begin{gathered} \mathrm{sysOverallTF}≔\mathrm{TransferFunction}\left(\mathrm{sysOverall}\right)\: \\ \mathrm{PrintSystem}\left(\mathrm{sysOverallTF}\right) \end{gathered}$$

The equation for the Transfer Function Model can be extracted from the Transfer Function System Object by accessing the exports of the Transfer Function System Object. This is done by appending $\:-\mathrm{tf}$ to the end of the System Object variable.

$\mathrm{exports}\left(\mathrm{sysOverallTF}\right)$

$\mathrm{sysOverallTF}:-\mathrm{tf}$

Information about the exports of each System Object type can be found in the help page SystemObject.  For general information about accessing exports using ':-', see colondash.

Similarly, a Zero Pole Gain System Object can be obtained by applying the ZeroPoleGain command to $\mathrm{sysOverall}$.

$$\begin{gathered} \mathrm{sysOverallZPK}≔\mathrm{ZeroPoleGain}\left(\mathrm{sysOverall}\right)\: \\ \mathrm{PrintSystem}\left(\mathrm{sysOverallZPK}\right) \end{gathered}$$

The zero, pole and gain matrices are exports of the System Object that can be extracted from the Zero Pole Gain model by appending $:-z\, :-p$ and $:-k$ to the end of the System Object variable.

$\mathrm{sysOverallZPK}:-z$

$\mathrm{sysOverallZPK}:-p$

$\mathrm{sysOverallZPK}:-k$

In the last section, you saw how easy it is to convert between different DynamicSystems System Object Representations. The DynamicSystems package, through the DynamicSystems context menus, allows you to convert between different DynamicSystems representations without first having to create a System Object.

For instance, the differential equation defined in $\mathrm{eqOverall}$ can be quickly converted into a Transfer Function or to a sequence of State Space or Zero Pole Gain Matrices. This is done by right-clicking on the differential equation defined in $\mathrm{eqOverall}$ and selecting from the choices found in the DynamicSystems>Conversions menu.

$\mathrm{eqOverall}$

$\stackrel{\text{convert to transfer function}}{\to }$

$\stackrel{\text{convert to zero pole gain}}{\to }$