The controllability matrix for the above system can be determined using the DynamicSystems[ControllabilityMatrix] command.

$\mathrm{CntrlMatrix}≔\mathrm{ControllabilityMatrix}\left(\mathrm{sys}\right)$

Rank of $\mathrm{CntrlMatrix}$:

$\mathrm{LinearAlgebra}\left[\mathrm{Rank}\right]\left(\mathrm{CntrlMatrix}\right)$

Determinant of $\mathrm{CntrlMatrix}$:

$\mathrm{LinearAlgebra}\left[\mathrm{Determinant}\right]\left(\mathrm{CntrlMatrix}\right)$

Even though the generic rank of the above matrix is 4 (i.e. matrix is generically full rank), we cannot say the system is controllable without verifying the conditions upon which the determinant of the controllability matrix becomes 0. For this system, the determinant becomes 0 when $\mathrm{l\_\_1}\=\mathrm{l\_\_2}$. From this we can conclude that the system is controllable if the lengths of the inverted pendulums differ from each other.

Assuming we have prior knowledge of the desired location of the closed-loop poles for our system, we can use the ControlDesign[StateFeedback][PolePlacement]  command to calculate the state feedback gain for a single-input system.

For this design, let us assume that the desired location of the closed-loop poles are:

$\mathrm{poles}\mathit{≔}\left[-3\, -\frac{5}{2}\+2\cdot \mathrm{I}\,-\frac{5}{2}-2\cdot \mathrm{I}\,-7\right]\:$

The state-feedback gain, $\mathrm{Kc}$, is then:

$\mathrm{Kc}\mathit{\,}\mathit{ }\mathrm{Kr}\mathit{≔}\mathrm{StateFeedback}\left[\mathrm{PolePlacement}\right]\left(\mathrm{sys}\mathit{\,}\mathrm{poles}\mathit{\,}\mathit{ }\mathit{\'}\mathrm{return\_Kr}\mathit{\'}\right)$

We can obtain the closed-loop state-space matrices using the ControlDesign[StateFeedbackClosedLoop] command. Then we can verify that the closed-loop system has its poles located at the desired pole locations.

$\mathrm{clsys} ≔ \mathrm{ControlDesign}\left[\mathrm{StateFeedbackClosedLoop}\right]\left(\mathrm{sys}\, \mathrm{Kc}\, \':-\mathrm{Kr}\' \=\mathrm{Kr}\right)\:$

$\mathrm{LinearAlgebra}\left[\mathrm{Eigenvalues}\right]\left(\mathrm{clsys}:-a\right)$

At this point, we can simulate the closed-loop system to verify if the controller that we designed is able to stabilize the inverted pendulums on the cart. Since the controller was developed symbolically we can perturb any number of the system parameters. Doing so, will give us a sense of the controller's robustness to parameter variations.

This section will examine the observability of the system under the following conditions: (1) $\mathrm{x\_\_1}$ is measured and $\mathrm{x\_\_2}$ is not, (2) $\mathrm{x\_\_2}$ is measured and $\mathrm{x\_\_1}$ is not, and (3) $\mathrm{x\_\_1}$ and $\mathrm{x\_\_2}$ are measured.  

3.1.1 -  $\mathrm{x\_\_1}$ is measured and $\mathrm{x\_\_2}$ is not

We get a subsystem using DynamicSystems[Subsystem] command where only $\mathrm{x\_\_1}$ is measurable: ${\mathrm{subsys}}_{x_1}≔\mathrm{DynamicSystems}\left[\mathrm{Subsystem}\right]\left(\mathrm{sys}\,\mathrm{all}\,\left\{1\right\}\right)\:$ The observability matrix can be determined by using the DynamicSystems[ObservabilityMatrix] command. $\mathrm{ObsvMatrix\_\_x\_\_1}≔\mathrm{DynamicSystems}\left[\mathrm{ObservabilityMatrix}\right]\left({\mathrm{subsys}}_{x_1}\right)$ Rank of $\mathrm{ObsvMatrix\_\_x\_\_1}$: $\mathrm{LinearAlgebra}\left[\mathrm{Rank}\right]\left(\mathrm{ObsvMatrix\_\_x\_\_1}\right)$ $4$ (4) Determinant of $\mathrm{ObsvMatrix\_\_x\_\_1}$: $\mathrm{LinearAlgebra}\left[\mathrm{Determinant}\right]\left(\mathrm{ObsvMatrix\_\_x\_\_1}\right)$ $-\frac{m^2{g}^2}{M^2{\mathrm{l\_\_1}}^2}$ (5) Since the observability matrix is calculated symbolically, knowing that the matrix is generically full rank does not provide us with enough information to say that the system is observable for all possible values of parameters. We must determine for what parameter values the determinant of the observability matrix becomes 0.  For this example, the system is observable for all values of the parameters.

3.1.2 -  $\mathrm{x\_\_2}$ is measured and $\mathrm{x\_\_1}$ is not

We get a subsystem using DynamicSystems[Subsystem] command where only $\mathrm{x\_\_2}$ is measurable: ${\mathrm{subsys}}_{x_2}≔\mathrm{DynamicSystems}\left[\mathrm{Subsystem}\right]\left(\mathrm{sys}\,\mathrm{all}\,\left\{2\right\}\right)\:$  The observability matrix can be determined by using the DynamicSystems[ObservabilityMatrix] command. $\mathrm{ObsvMatrix\_\_x\_\_2}≔\mathrm{DynamicSystems}\left[\mathrm{ObservabilityMatrix}\right]\left({\mathrm{subsys}}_{x_2}\right)$ Rank of $\mathrm{ObsvMatrix\_\_x\_\_2}$: $\mathrm{LinearAlgebra}\left[\mathrm{Rank}\right]\left(\mathrm{ObsvMatrix\_\_x\_\_2}\right)$ $4$ (6) Determinant of $\mathrm{ObsvMatrix\_\_x\_\_2}$: $\mathrm{LinearAlgebra}\left[\mathrm{Determinant}\right]\left(\mathrm{ObsvMatrix\_\_x\_\_2}\right)$ $-\frac{m^2{g}^2}{M^2{\mathrm{l\_\_2}}^2}$ (7) As in section 3.1.1, the system is observable for all parameter values$$. This means that the system is observable when either of the angles are measured.

3.1.3 -  $\mathrm{x\_\_1}$ and $\mathrm{x\_\_2}$ are measured

If the both states are measurable, the observability matrix of the system is: $\mathrm{ObsvMatrix\_\_x\_\_12}≔\mathrm{DynamicSystems}\left[\mathrm{ObservabilityMatrix}\right]\left(\mathrm{sys}\right)$ Rank of $\mathrm{ObsvMatrix\_\_x\_\_12}$: $\mathrm{LinearAlgebra}\left[\mathrm{Rank}\right]\left(\mathrm{ObsvMatrix\_\_x\_\_12}\right)$ $4$ (8) The observability matrix is full rank. Clearly, the system is observable when both angles are measured, as well.

In section 2.2, we showed how the ControlDesign toolbox could be used to design a state-feedback controller when both angles are measured. In this section, we will show how the ControlDesign toolbox can be used to design an observer-based control system when only one state, let us say $x_1$, is measured.

According to the separation principle, for linear time invariant systems, the state feedback and state observer can be designed independently. We select the desired poles for the observer error dynamic to be about 5-10 times further away from the $\mathrm{j\ω}$ axis than those of the state feedback gain design. This ensures that the state feedback poles are the dominant poles of the system.

For this example, the following values for the state-feedback poles and the observer poles were chosen. If you will recall, the state feedback poles that were chosen here are the same as those used in the state-feedback control design section.

$\mathrm{StateFeedbackPoles}≔\left[\mathrm{-3}\,-\frac{5}{2}+2\mathrm{I}\,-\frac{5}{2}-2\mathrm{I}\,-7\right]\:$

$\mathrm{ObserverPoles}≔\left[-25\,-27\,-26\+7\cdot \mathrm{I}\,-26-7\cdot \mathrm{I}\right]\:$

Using the ControlDesign[StateObserver][PolePlacement] and ControlDesign[StateFeedback][PolePlacement] commands the observer gain, $\mathrm{`L\_\_`}$,  and state feedback gain, $\mathrm{K\_\_c}$, to stabilize the inverted pendulum configuration on top of the cart are:

$L\mathit{≔}\mathrm{StateObserver}\left[\mathrm{PolePlacement}\right]\left({\mathrm{subsys}}_{x_1}\,\mathrm{ObserverPoles}\right)$

$\mathrm{Kc}\, \mathrm{Kr}≔\mathrm{StateFeedback}\left[\mathrm{PolePlacement}\right]\left({\mathrm{subsys}}_{x_1}\mathit{\,}\mathrm{StateFeedbackPoles}\mathit{\,}\mathit{ }\mathit{\'}\mathrm{return\_Kr}\mathit{\'}\right)$

Using the ControlDesign[ControllerObserver] command, the closed-loop system of the state-feedback controller and observer can be obtained. We can verify that closed-loop system poles match the desired pole locations.

$\mathrm{closedloop}≔\mathrm{ControlDesign}\left[\mathrm{ControllerObserver}\right]\left({\mathrm{subsys}}_{x_1}\,\mathrm{Kc}\,L\, \':-\mathrm{Kr}\'\=\mathrm{Kr}\, \'\mathrm{closedloop}\'\=\mathrm{true}\, \'\mathrm{stateerror}\' \=\mathrm{true}\right)\:$

      $\mathrm{eigcl}\mathit{≔}\mathrm{LinearAlgebra}\left[\mathrm{Eigenvalues}\right]\left(\mathrm{closedloop}:-a\right)$

We modify the state-space representation of the closed-loop system so that there are 8 outputs corresponding to all the states of the closed-loop system. The first four outputs represent the state outputs, while the last four outputs represent the observer error.

  $\mathrm{closedloop1}≔\mathrm{DynamicSystems}\left[\mathrm{StateSpace}\right]\left(\mathrm{closedloop}:-a\, \mathrm{closedloop}:-b\,\mathrm{LinearAlgebra}:-\mathrm{IdentityMatrix}\left(\mathrm{closedloop}:-\mathrm{statecount}\right)\, \mathrm{Matrix}\left(\mathrm{closedloop}:-\mathrm{statecount}\,\mathrm{closedloop}:-\mathrm{inputcount}\right)\right)\:$

As in the previous section, we can simulate the closed-loop system to verify if the observer-based controller that was designed can stabilize the two inverted pendulums on the cart system.