Observability in State Estimation

Why We Need State Estimation

Before we can define observability, we must first understand the problem it solves. We often have a system where we can only measure a subset of the state variables. These measurements are called the outputs, represented by the vector $\mathbf{y}$.

For your rotary inverted pendulum, the state vector $\mathbf{x}$ might include the arm's angle, the pendulum's angle, and their respective angular velocities. However, you might only have encoders to measure the angles, not the velocities. In this case, how can you implement a control law that depends on the velocities?

The answer is to build a state estimator, also known as an observer. An observer is a software algorithm that runs in parallel with the real system. It takes the same control input $\mathbf{u}$ that is sent to the physical system and uses the measured outputs $\mathbf{y}$ to compute an estimate of the full state vector, denoted as $\hat{\mathbf{x}}$.

The document below gives an excellent introduction to this idea.

The key idea is that we "fuse" our mathematical model of the system with real-world measurements to produce the best possible estimate of the internal state. This is a cornerstone of modern control and is deeply related to concepts from signal processing like filtering and data fusion.

Defining Observability

An observer seems like a perfect solution, but it only works if the system has a specific property: observability.

Let's start building our understanding with a conceptual video.

As the video explained, observability means that all critical states can be known from the system's outputs. Let's formalize this.

Formal Definition of Observability:
A system is said to be observable if, for any unknown initial state $\mathbf{x}(0)$, there exists a finite time $t_1>0$ such that the knowledge of the input $\mathbf{u}(t)$ and the output $\mathbf{y}(t)$ over the time interval $[0,t_1]$ is sufficient to uniquely determine the initial state $\mathbf{x}(0)$.

Let's unpack this:

• If we can figure out the state at the beginning ($\mathbf{x}(0)$), and we know all the inputs applied since then, we can simulate our model forward to find the state $\mathbf{x}(t)$ at any time.
• In short, observability means we can reconstruct the full state trajectory just by watching the inputs and outputs.
• A system that is not observable has "hidden" states or modes that don't show up in the output, no matter what happens.

The following resource provides this formal definition.

The Physical Interpretation of Observability

The formal definition is precise, but physical intuition is what makes it useful. Observability is fundamentally about the relationship between the system's internal dynamics (the $A$ matrix) and how those dynamics are reflected in the measurements (the $C$ matrix).

An Extreme Example: Driving Blind

What happens when a system is completely unobservable?

This example shows that if your output matrix $C$ is all zeros, you get no information about the state. The system is completely unobservable.

Inferring States vs. Measuring States

A crucial point is that observability does not require you to measure every state directly. It only requires that the behavior of unmeasured states has some effect on the states you do measure.

From your signal processing background, you know that differentiating a noisy signal is problematic. This highlights the practical trade-offs involved. While you can estimate velocity from a noisy position measurement, the estimate itself will be very noisy. Sometimes, adding a velocity sensor is the better engineering choice, even if the system is technically observable without it.

Sensor Placement Creates Blind Spots

A system can be made unobservable by poor sensor placement. If a sensor is located at a point that is unaffected by a certain internal motion (a "node"), then that motion will be invisible to the sensor, and the corresponding state will be unobservable.

Let's look at two excellent examples of this.

1. The Flexible Beam:

2. The Two-Tank System:

   This is a classic example that should resonate with your electrical engineering background. Imagine two connected water tanks. We can pump water between them ($u(t)$), and we can measure the difference in their water levels ($y=x_1-x_2$). The states are the individual water levels, $x_1$ and $x_2$.

   This figure, from the UC Berkeley resource, shows two tanks connected by a pump. The states are the individual water levels, but the measurement is only the difference between them.

   Now, ask yourself: if we start with levels $x_1=3$ and $x_2=1$, the output is $y=2$. If we start with $x_1=5$ and $x_2=3$, the output is also $y=2$. Just by looking at the output, we can't distinguish between these two initial states. The system is unobservable.

   Specifically, we can determine the difference mode ($x_1-x_2$), but we can never determine the common mode ($x_1+x_2$). If both tank levels increase by the exact same amount, the output doesn't change at all. This is directly analogous to differential and common-mode signals in an electrical circuit. This unobservable "common mode" is a hidden internal behavior.

   You can read the full mathematical breakdown of this example in the following text.

Conclusion

Today, we've explored the crucial concept of observability, the necessary counterpart to controllability.

Key Takeaways:

• State Estimation: When we can't measure all states, we use an observer to estimate them using a model and available sensor data.
• Observability: A system is observable if we can determine its entire internal state by observing its inputs and outputs over a period of time. It is the fundamental property that makes state estimation possible.
• Physical Interpretation: Observability is determined by the system's structure ($A$) and the sensor configuration ($C$). A poor choice of sensors can create "blind spots" (unobservable modes), making it impossible to know the full state of the system.
• Duality: Controllability is about whether inputs can affect every state. Observability is about whether every state affects the outputs.

Preview of the Next Lesson:

Just as we developed a mathematical test for controllability using the controllability matrix, there is a parallel, or "dual," test for observability. In our next lesson, we will introduce the observability matrix and learn how to use its rank to provide a definitive yes-or-no answer to the question, "Is this system observable?" This will be the next tool you'll apply to your inverted pendulum model.