Full-Order State Observer: State-Space Equations

The Structure of a Luenberger Observer

At its heart, a state observer is a clever copy of the system it's observing. It runs a simulation of the plant in real-time and uses the plant's actual output to correct its simulation, ensuring the estimate stays on track.

Let's start with our standard LTI system model:
$\dot{x}=Ax+Bu$
$y=Cx$

The observer's job is to generate an estimate $\hat{x}$ that converges to $x$. The core idea is to model the dynamics of $\hat{x}$ using what we know about the plant, $A$ and $B$, and then add a correction term based on the measurement, $y$.

The equation for the state estimate $\hat{x}$ is constructed as follows:

$\dot{\hat{x}}=(\text{Model of plant dynamics})+(\text{Correction term})$

1. Model of Plant Dynamics: We build a copy of the plant's dynamics, but using our current estimate $\hat{x}$: $A\hat{x}+Bu$.
2. Correction Term: We compare the plant's actual measured output, $y$, with the estimated output from our observer, $\hat{y}=C\hat{x}$. The difference, $y-\hat{y}$, is the output estimation error. We feed this error back, weighted by our observer gain matrix $L$, to "nudge" the estimate in the right direction. The correction term is $L(y-\hat{y})$.

Combining these gives the fundamental equation of the Luenberger Observer:

$\dot{\hat{x}}=A\hat{x}+Bu+L(y-C\hat{x})$

The following video provides an excellent walkthrough of how this structure is developed.

The Observer in Standard State-Space Form

The equation above is intuitive, but to analyze and implement the observer, we need to rearrange it into the standard state-space form, $\dot{z}=A_{sys}z+B_{sys}{u}_{sys}$. Let's do that by gathering the terms involving $\hat{x}$ and the terms involving the observer's inputs, which are $u$ and $y$.

Starting with:
$\dot{\hat{x}}=A\hat{x}+Bu+L(y-C\hat{x})$

We can distribute the gain $L$ and regroup:
$\dot{\hat{x}}=A\hat{x}+Bu+Ly-LC\hat{x}$
$\dot{\hat{x}}=(A-LC)\hat{x}+Bu+Ly$

Finally, we can write the input part in matrix form:
$\dot{\hat{x}}=(A-LC)\hat{x}+\left[\begin{array}{cc}B& L\end{array}\right]\left[\begin{array}{c}u\\ y\end{array}\right]$

This is the state-space representation of the full-order observer. Let's identify its components:

• State Vector: The state of the observer is our estimated state vector, $\hat{x}$.
• System Matrix: $A_{obs}=A-LC$. The eigenvalues of this matrix are the observer poles that we carefully placed in the previous lessons to control the speed of convergence.
• Input Vector: The observer is a dynamic system that requires two inputs: the original system's control input $u$ and its measured output $y$.
• Input Matrix: $B_{obs}=\left[\begin{array}{cc}B& L\end{array}\right]$.
• Output: The primary output of the observer is the state estimate itself, $\hat{x}$. So, the output equation is simply ${\hat{x}}_{out}=I\hat{x}$.

The following reading from the University of Michigan provides a concise summary and a clear block diagram of this structure.

Why Does This Work? The Error Dynamics

We've constructed the observer, but let's quickly re-verify why it works. The goal is for the estimation error, defined as $e=x-\hat{x}$, to go to zero. Let's find the dynamics of this error.

$\dot{e}=\dot{x}-\dot{\hat{x}}$

Substitute the expressions for $\dot{x}$ and $\dot{\hat{x}}$:
$\dot{e}=(Ax+Bu)-(A\hat{x}+Bu+L(y-C\hat{x}))$

The $Bu$ terms cancel out. Now substitute $y=Cx$:
$\dot{e}=Ax-A\hat{x}-L(Cx-C\hat{x})$

Factor out the common terms:
$\dot{e}=A(x-\hat{x})-LC(x-\hat{x})$

Since $e=x-\hat{x}$, we arrive at the error dynamics:
$\dot{e}=(A-LC)e$

This is a beautiful result. It shows that the estimation error follows its own simple, unforced linear dynamics. The evolution of the error does not depend on the system's input $u$. If we choose $L$ such that the matrix $(A-LC)$ is stable (i.e., all its eigenvalues are in the left-half plane), then the error $e(t)$ will exponentially decay to zero, meaning $\hat{x}(t)$ will converge to $x(t)$.

This confirms that our pole placement procedure from the previous lesson was precisely for this purpose: to guarantee that the error dynamics are stable and fast.

The video from Steve Brunton also provides a great derivation of these error dynamics.

Conclusion

In this lesson, we have formally constructed the state-space model for a full-order Luenberger observer. You can now take the system matrices ($A,B,C$) and a calculated observer gain $L$ and write the complete set of differential equations that define the observer as a standalone dynamic system.

Key Takeaways:

• An observer is a copy of the plant model that uses measurement feedback to correct its state estimate.
• The fundamental observer equation is $\dot{\hat{x}}=A\hat{x}+Bu+L(y-C\hat{x})$.
• In standard state-space form, the observer is described by $\dot{\hat{x}}=(A-LC)\hat{x}+\left[\begin{array}{cc}B& L\end{array}\right]\left[\begin{array}{c}u\\ y\end{array}\right]$.
• The dynamics of the estimation error $e=x-\hat{x}$ are governed by $\dot{e}=(A-LC)e$, which is why we design $L$ to make $(A-LC)$ stable.

Preview of the Next Lesson:
We have now designed a controller (gain $K$) and an observer (gain $L$) in separate efforts. The final step is to integrate them. In our next lesson, we will connect the observer's output $\hat{x}$ to the controller by implementing the control law $u=-K\hat{x}$. We will then formulate the dynamic equations for the complete observer-based controller system and introduce the famous Separation Principle, which justifies our approach of designing the controller and observer independently.