Observer Pole Placement for Desired Error Dynamics

Observer Design as a "Dual" Problem

Before we discuss where to place the poles, it's helpful to see the strong parallel between controller design and observer design.

• Controller Design (Regulator Problem): We chose a feedback gain $K$ to place the poles of the closed-loop system, $\lambda (A-BK)$, in desired locations to shape the system's response. This was possible if the pair $(A,B)$ was controllable.
• Observer Design (Estimator Problem): We need to choose an observer gain $L$ to place the poles of the error dynamics, $\lambda (A-LC)$, in desired locations to shape the error's convergence. This is possible if the pair $(A,C)$ is observable.

These two problems are so similar they are called dual problems in control theory. The mathematics are nearly identical. Notice the structure:

• Controller: $A\to A-BK$
• Observer: $A\to A-LC$

In fact, the eigenvalues of $(A-LC)$ are the same as the eigenvalues of its transpose, $(A-LC{)}^T=A^T-C^T{L}^T$. This looks exactly like a regulator problem for a system with state matrix $A^T$, input matrix $C^T$, and feedback gain $L^T$. We'll use this insight in the next lesson to calculate $L$. For now, this duality reminds us that all our intuition about pole placement for controllers can be repurposed for designing observers.

The Guiding Principle: Faster Than the Controller

The whole point of the observer is to provide an accurate state estimate for our feedback controller, $u=-K\hat{x}$. If the estimate $\hat{x}$ converges to the true state $x$ very slowly, our controller will spend a long time operating on bad information, leading to poor performance.

The estimation error must converge to zero much faster than the system's own dynamics unfold. This leads to the most important rule of thumb for observer design.

As the notes explain, the core idea is to make the observer's dynamics significantly faster than the controller's dynamics. A common practice is to make the observer poles correspond to time constants that are 2 to 6 times smaller than the dominant time constant of the closed-loop controller. In the s-plane, this means placing the real parts of the observer poles 2 to 6 times further to the left than the real parts of the dominant controller poles.

For example, if your controller design resulted in dominant closed-loop poles at $-2\pm 3j$, a good starting point for your observer poles might be at $-10$ and $-12$, or a complex pair with a real part of $-10$, such as $-10\pm 5j$.

The Practical Trade-Off: Speed vs. Noise Sensitivity

Making the observer as fast as possible might seem like the obvious goal. However, there is a critical trade-off that is especially relevant given your background in signal processing. In the real world, sensor measurements are never perfect; they are always corrupted by noise.

Let's see how this affects our observer.

As the reading explains, a more realistic model includes process noise $w$ (disturbances, model uncertainty) and measurement noise $\nu$. The error dynamics then become:
$\dot{\tilde{x}}=(A-LC)\tilde{x}+w-L\nu$
This equation clearly shows the trade-off:

• Choosing fast poles requires a large gain matrix $L$. A large $L$ makes the term $(A-LC)$ highly stable, so the error $\tilde{x}$ from initial conditions or process noise $w$ dies out quickly. However, a large $L$ also amplifies the measurement noise $\nu$, injecting it into your state estimate. This makes your state estimate $\hat{x}$ noisy, which in turn makes your control signal $u=-K\hat{x}$ noisy and jittery.

• Choosing slow poles results in a small gain matrix $L$. This minimizes the effect of measurement noise $\nu$ (you get a smoother estimate), but the estimation error converges slowly. The observer is trusting the system model more and the noisy measurements less.

This creates a fundamental design conflict: fast estimation vs. low noise sensitivity. The "2 to 6 times faster" rule is a starting point, but the final choice depends on how noisy your sensors are. For your rotary inverted pendulum, if the angle sensors are very precise, you can afford a faster observer. If they are noisy, you may need to slow the observer down to get smooth control action.

An Example of Pole Selection

Let's look at a concrete numerical example of this design choice in action.

In this example:

• Regulator (Controller) Poles: The desired time constant is ${\tau }_c=0.25$ sec. The poles are placed at $-4\pm 4j$. The speed of the system response is primarily dictated by the real part of these poles, $\text{Re}(s)=-4$.
• Estimator (Observer) Poles: The designer wants the estimator to be faster, choosing a time constant of ${\tau }_e=0.1$ sec. They place both observer poles at $-10$.

Notice that $-10$ is 2.5 times "faster" than $-4$, fitting perfectly within our rule of thumb. This ensures the estimation error decays with a time constant of 0.1 seconds, which is significantly quicker than the system's response time constant of 0.25 seconds.

Conclusion

Today we've moved from the theory of why observers work to the practical art of how to design them for good performance.

Key Takeaways:

• The primary goal is to make the observer dynamics faster than the closed-loop controller dynamics. A common rule of thumb is to place the observer poles so they are 2 to 6 times faster than the dominant controller poles.
• "Faster" means having a more negative real part in the s-plane, corresponding to a smaller time constant.
• There is a critical trade-off between estimation speed and noise sensitivity. Very fast poles require a high-gain observer, which amplifies sensor noise and can lead to poor performance.
• The final choice of observer poles is an engineering judgment call, balancing the need for a quick, accurate estimate against the reality of noisy measurements.

Preview of the Next Lesson:
We now know the principles for selecting where to place our observer poles. The next logical step is to calculate the observer gain matrix, $L$, that will place the poles in these desired locations. In the next lesson, we will cover the techniques for computing $L$, leveraging the concept of duality and methods you've already seen, like Ackermann's formula.