Reduced-Order Observer Design

In modern control systems, from robotics to aerospace, knowing the complete internal state of a system is crucial for implementing sophisticated feedback laws. However, sensors often measure only a subset of state variables directly. A reduced-order observer is a specialized state estimator that reconstructs only the states not available from measurements, offering a leaner, more computationally efficient alternative to a full-state observer when some outputs are directly available. This approach is not just an academic exercise; it is a practical design choice that reduces software complexity and processor load in embedded real-time systems, allowing for faster sampling rates or the allocation of computational resources to other critical tasks.

Full-Order vs. Reduced-Order Observers

To appreciate the reduced-order observer, you must first understand its counterpart: the full-order observer. A full-order observer, such as the Luenberger observer, estimates every state variable in the system, regardless of whether some are directly measured. For a linear time-invariant system in state-space representation, $\dot{x}=Ax+Bu$ and $y=Cx$, a full-order observer has the form $\dot{\hat{x}}=A\hat{x}+Bu+L(y-C\hat{x})$. The observer's dimension is equal to the number of states, $n$.

A reduced-order observer fundamentally challenges this approach: why estimate what you already know? If the output $y(t)$ provides direct, noise-free measurements of some states, it is computationally wasteful to reconstruct them. The core idea is to partition the state vector into measured and unmeasured components, then build an observer whose order is only equal to the number of unmeasured states. This directly lowers the complexity of the online estimation algorithm, which is a significant advantage in digital implementation.

State Partitioning and the Design Framework

The design of a reduced-order observer begins with a coordinate transformation that explicitly separates measured and unmeasured states. Assume the output matrix $C$ has full row rank, meaning the measurements are independent. We can always find a transformation matrix $T$ such that the new state vector is partitioned as: $$\left[\begin{array}{c}y\\ z\end{array}\right]=Tx=\left[\begin{array}{c}C\\ T_2\end{array}\right]x$$ Here, $y$ is the measured output (a subset of the transformed states), and $z$ represents the remaining, unmeasured states we need to estimate. The goal is to construct an estimator for $z$ alone.

In these new coordinates, the system dynamics decompose. The derivatives become: $$\dot{y}=A_{11}y+A_{12}z+B_{1}u$$ $$\dot{z}=A_{21}y+A_{22}z+B_{2}u$$ You can see that the equation for $\dot{z}$ depends on the unknown $z$, the known $y$, and the known input $u$. The first equation, $\dot{y}$, can be seen as an "output equation" for the $z$-subsystem, where $\dot{y}-A_{11}y-B_{1}u=A_{12}z$. If $A_{12}$ has full column rank, the unmeasured states $z$ are detectable through this relationship.

Constructing the Lower-Dimensional Estimator

We now design an observer for the $z$-subsystem. We treat $y$ and $u$ as known inputs to this subsystem. The standard Luenberger observer structure for $\dot{z}=A_{22}z+(A_{21}y+B_{2}u)$ with "output" $A_{12}z$ is: $$\dot{\hat{z}}=A_{22}\hat{z}+A_{21}y+B_{2}u+L(\underset{A_{12}z}{\underbrace{(\dot{y}-A_{11}y-B_{1}u)}}-A_{12}\hat{z})$$ However, this formulation is problematic because it requires the derivative of the measurement, $\dot{y}$, which is often noisy and impractical to obtain. The key innovation is to define a new observer state variable that absorbs this derivative term. Define $w=\hat{z}-Ly$. Taking the time derivative and substituting the expressions above allows you to eliminate $\dot{y}$. After algebraic manipulation, you arrive at the implementable reduced-order observer:

$$\dot{w}=(A_{22}-L{A}_{12})w+(B_2-L{B}_1)u+(A_{21}-L{A}_{11}+(A_{22}-L{A}_{12})L)y$$ $$\hat{z}=w+Ly$$ $$\hat{x}=T^{-1}\left[\begin{array}{c}y\\ \hat{z}\end{array}\right]$$

The estimator dynamics are now of order $(n-\text{rank}(C))$, governed by the matrix $(A_{22}-L{A}_{12})$. The gain matrix $L$ is chosen to place the eigenvalues of this matrix in desired, stable locations to ensure the estimation error for $z$ decays rapidly to zero. The final full state estimate $\hat{x}$ is reconstructed by combining the direct measurement $y$ with the estimate $\hat{z}$.

A Worked Example: Second-Order System

Consider a simple system where position $x_1$ is measured, but velocity $x_2$ is not: $${\dot{x}}_1=x_2,{\dot{x}}_2=u,y=x_1$$ In state-space form: $A=\left[\begin{array}{cc}0& 1\\ 0& 0\end{array}\right]$, $B=\left[\begin{array}{c}0\\ 1\end{array}\right]$, $C=\left[\begin{array}{cc}1& 0\end{array}\right]$. Here, $y=x_1$ (measured) and $z=x_2$ (unmeasured). Following the partitioning, we have $A_{11}=0,A_{12}=1,A_{21}=0,A_{22}=0,B_1=0,B_2=1$.

The reduced-order observer for the single unmeasured state (velocity) is designed via the pole of $(A_{22}-L{A}_{12})=(0-L\ast 1)=-L$. To place this pole at, say, -10 for fast convergence, we set $L=10$.

The observer equations become: $$\dot{w}=(-10)w+(1)u+(0-0+(-10)\ast 0)y=-10w+u$$ $${\hat{x}}_2=w+10y$$ The final estimates are ${\hat{x}}_1=y$ (directly measured) and ${\hat{x}}_2=w+10y$. This single first-order differential equation for $w$ is simpler than a full second-order observer would be.

Common Pitfalls

1. Incorrect or Ill-Conditioned Partitioning: The transformation $T$ must be chosen so that the sub-matrix $A_{12}$ has full column rank. If it does not, the $z$-states may not be detectable from the available signals, making observer design impossible. Always check the observability/detectability of the pair $(A_{22},A_{12})$ after partitioning.
2. Ignoring Measurement Noise: The design assumes measurements are perfect. In practice, $y(t)$ is noisy. The reduced-order observer uses the measurement $y$ directly in the reconstruction $\hat{x}=T^{-1}[y;\hat{z}]$ and in the feedback $Ly$. High observer gains $L$ can amplify this noise, leading to poor estimates. A careful trade-off between convergence speed and noise immunity is required, often informed by Kalman filter theory.
3. Forgetting the Coordinate Transformation: The estimate $\hat{z}$ lives in the transformed coordinates. A frequent calculation error is failing to transform back to the original physical state coordinates via $\hat{x}=T^{-1}[y;\hat{z}]$ when implementing the control law $u=-K\hat{x}$.
4. Overlooking the Initial Condition: The observer state $w$ must be initialized. A poor initial guess for $w(0)$ leads to a transient estimation error, even if the observer dynamics are stable. The relationship $\hat{z}(0)=w(0)+Ly(0)$ should be used to set $w(0)$ consistent with your best initial guess for $z$.

Summary

• Reduced-order observers estimate only the subset of state variables not directly available from system measurements, leading to a lower-dimensional, computationally efficient estimator.
• The design process hinges on partitioning the state into measured and unmeasured components via a coordinate transformation and then constructing a Luenberger observer for the unmeasured subsystem alone.
• A key algebraic step eliminates the need for differentiating the measured output, resulting in a practically implementable set of dynamic equations for the observer.
• The primary advantage is reduced online computational burden, which is critical for high-sample-rate or resource-constrained embedded control systems.
• Successful implementation requires careful attention to the detectability of the unmeasured subsystem, the handling of measurement noise, and the correct transformation between coordinate systems.
• This approach exemplifies a fundamental engineering principle: build only what is necessary, optimizing design for both performance and practical implementation constraints.