Observability Matrix & System Observability

The Observability Matrix and the Rank Test

Recall the standard LTI state-space model:

Observability is entirely determined by the relationship between the system dynamics (matrix $A$) and the sensor configuration (matrix $C$). The mathematical tool that captures this relationship is the observability matrix, denoted by $\mathcal{O}$.

For a system with $n$ states and $l$ outputs, the observability matrix is constructed as follows:

This is an $(l\cdot n)\times n$ matrix. Notice the structural similarity to the controllability matrix—this is an example of the principle of duality in control theory.

The condition for observability is straightforward and is known as the Kalman test for observability:

A system is observable if and only if its observability matrix $\mathcal{O}$ has full column rank, meaning $\text{rank}(\mathcal{O})=n$.

The following resource provides a concise definition of this test, directly in the context of a rotary pendulum project.

A Worked Example: Calculation and Rank Analysis

Let's see how this test is applied in practice with a simple 2-state system. The following video walks through the entire process step-by-step: building the $\mathcal{O}$ matrix and then calculating its determinant to check the rank.

The key takeaways from this example are:

• For a system with $n$ states and one output ($l=1$), the observability matrix $\mathcal{O}$ will be square ($n\times n$).
• For a square matrix, checking for full rank is equivalent to checking if its determinant is non-zero. If $\det (\mathcal{O})\ne 0$, the system is observable.
• If the rank of $\mathcal{O}$ is less than $n$, the rank's value indicates the number of observable states. The remaining $n-\text{rank}(\mathcal{O})$ states are unobservable.

Practical Application: The Inverted Pendulum

Now let's apply this test to the system you're interested in: the inverted pendulum. As we discussed last time, the choice of sensors is critical. We'll now see mathematically how different sensor choices (i.e., different $C$ matrices) affect observability.

The following video is an excellent demonstration in MATLAB. It shows how different $C$ matrices for the same pendulum system lead to different conclusions about observability.

This is a powerful result. The math confirms our physical intuition:

1. If you measure the cart's position x, you can infer its velocity $\dot{x}$ and, through the dynamic coupling, the pendulum's angle $\theta$ and angular velocity $\dot{\theta }$. The system is observable.
2. If you only measure the pendulum's angle $\theta$, you can never know the absolute position of the cart x. If the entire system translates left or right, the angle $\theta$ doesn't change, so this motion is invisible to your sensor. The x state is unobservable.

The CTMS page on the inverted pendulum reinforces this point and introduces the rank() command in MATLAB, which is more general than det() because it works for non-square observability matrices (which occur when you have multiple outputs).

Finally, let's look at the model for the specific rotary inverted pendulum system you are studying. The paper State feedback control of a rotary inverted pendulum provides the linearized model and confirms its properties.

Kinematics of the rotary inverted pendulum. The states are the arm angle $\theta$, pendulum angle $\alpha$, and their velocities.

The paper provides the full state-space model for this system and directly states the result of the observability test:

"The controllability and observability matrices are both full rank, and hence, the state-space model (24) is the minimal realization of the inverted pendulum system." - (Räsänen & Pyrhönen, 2019)

This confirms that with sensors for both the rotary arm angle ($\theta$) and the pendulum link angle ($\alpha$), the system is fully observable. You can find this statement in the resource LINK within Section 4, under Equation (24). This is great news, as it means you can proceed with designing an observer to estimate the angular velocities.

Conclusion

Today we've translated the abstract concept of observability into a concrete mathematical procedure. You now have a definitive test to determine if a system's internal state can be reconstructed from its outputs.

Key Takeaways:

• The observability matrix, $\mathcal{O}=[C^T(CA{)}^T\dots (C{A}^{n-1}{)}^T{]}^T$, encapsulates the relationship between system dynamics ($A$) and sensors ($C$).
• A system is observable if and only if the rank of its observability matrix equals the number of states, $n$.
• For practical analysis, software like MATLAB is used with commands like obsv to construct the matrix and rank or det to check for full rank.
• The choice of sensors (the $C$ matrix) is what determines observability. As seen with the pendulum, one sensor configuration may be observable while another is not.

Preview of the Next Lesson:

We've now seen that a system can have "unobservable states" when the observability matrix loses rank. But what does this mean in other representations? In our next lesson, we will explore the deep connection between observability and transfer functions. You will learn to relate uncontrollable or unobservable modes to pole-zero cancellations, discovering how these "hidden" internal dynamics manifest in the input-output behavior of the system. This will connect our state-space analysis back to the concepts you may have seen in classical, single-variable control.