Distributed State Estimation of Discrete-Time LTI Systems via Jordan Canonical Representation

This paper proposes a distributed state estimation scheme for discrete-time LTI systems that utilizes the Jordan canonical form to combine local Luenberger observers with consensus-based strategies, thereby establishing necessary and sufficient conditions for asymptotic convergence while offering greater flexibility and less restrictive solvability conditions than previous work.

The Gist

Imagine a massive, complex machine (like a giant robot or a power grid) that is constantly moving and changing. This machine has thousands of internal parts (its "state"), but no single sensor can see the whole picture. Instead, we have a team of N sensors scattered around the machine.

Each sensor is like a blindfolded detective. They can only see a tiny, specific slice of the machine's activity. One might see the temperature, another the speed, and a third the pressure. Individually, none of them knows the full story. However, they can talk to their neighbors.

The goal of this paper is to teach these detectives how to share information so that, eventually, every single one of them knows the exact state of the entire machine, even though they only see a fraction of it.

Here is how the authors solved this puzzle, explained simply:

1. The Problem: The "Unseeable" Parts

In the past, trying to get these sensors to agree on the full picture was like trying to solve a jigsaw puzzle where some pieces are invisible.

• The Detectable Parts: Some parts of the machine are easy to see. If a sensor looks at a specific gear, it can figure out exactly how that gear is moving.
• The Undetectable Parts: Other parts are "hidden." No single sensor can see them directly. To figure these out, the sensors must rely on their neighbors.

2. The Old Way vs. The New Way

Previous methods (like the one in reference [2]) tried to force all sensors to use the same strict rule for how they talk to each other. Imagine a choir where everyone must sing at the exact same volume and pitch to harmonize. If the room is too big or the singers are too far apart, the harmony breaks, and the system fails.

This paper introduces a smarter, more flexible choir:
Instead of one rigid rule, the authors propose a strategy where each "hidden piece" of the machine gets its own custom rule for how the sensors talk.

3. The Secret Weapon: The "Jordan" Map

The authors use a mathematical trick called the Jordan Canonical Form. Think of this as a special organizing map for the machine.

Instead of looking at the machine as one giant, messy block, this map breaks the machine down into small, manageable "mini-blocks" (like individual gears or springs).

• Step 1: The map identifies which mini-blocks each sensor can see clearly.
• Step 2: It separates the "visible" blocks from the "hidden" blocks.

4. The Two-Step Strategy

Once the map is drawn, the sensors use a two-part strategy:

Part A: The Local Detective (Luenberger Observer)
For the parts of the machine a sensor can see, it acts like a super-smart local detective. It uses its own eyes and a quick calculation to figure out exactly what those specific parts are doing. This is fast and accurate.

Part B: The Team Huddle (Consensus Strategy)
For the parts the sensor cannot see, it joins a "team huddle."

• It asks its neighbors: "Hey, do you know what this hidden gear is doing?"
• If a neighbor knows (because they can see it), they share the info.
• If the neighbor doesn't know, they pass the question along.

The Big Innovation:
In the old method, the sensors used a single "volume knob" (coupling gain) for the whole team. If the volume was too low, they couldn't hear each other; too high, and they got confused.

In this new method, the authors realize that different hidden gears need different volumes.

• Some hidden gears are "shy" and need a loud, strong signal to be heard.
• Others are "loud" and need a gentle whisper.
• The new system allows the sensors to turn the volume knob individually for each mini-block. This makes it much easier to get everyone to agree without breaking the system.

5. The Result: A Perfect Picture

By using this flexible, customized approach, the paper proves that:

1. It always works (under certain logical conditions about the network).
2. It's easier to design because you don't have to force a "one-size-fits-all" rule.
3. It's more robust. Even if the network is messy or the machine is complex, the sensors can eventually reconstruct the entire state of the machine.

The Real-World Test

To prove it works, the authors simulated a group of 6 Pendubots (robotic arms with two joints).

• Each robot could only see its own joints.
• They had to figure out the angles of all 6 robots' joints.
• Using this new "custom volume" strategy, they successfully shared information until every robot knew the exact position of every other robot's joints.

Summary Analogy

Imagine a group of people trying to guess the shape of a giant elephant in a dark room.

• Old Method: Everyone shouts their guess at the same volume. If the room is echoey, no one hears the truth.
• New Method: The group realizes that the "trunk" is hard to hear, so they shout louder about the trunk. The "tail" is easy to hear, so they whisper about the tail. By tuning their voices to the specific difficulty of each body part, they quickly build a perfect mental image of the whole elephant.

This paper provides the mathematical "tuning guide" to make that happen for complex machines.

Technical Summary

Here is a detailed technical summary of the paper "Distributed State Estimation of Discrete-Time LTI Systems via Jordan Canonical Representation".

1. Problem Statement

The paper addresses the distributed state estimation problem for discrete-time Linear Time-Invariant (LTI) systems.

• System Model: A network of $N$ sensors observes a global system described by $x(t+1) = Ax(t) + Bu(t)$ and $y_i(t) = C_i x(t)$.
• Constraints:
  • Individual sensors only have access to partial measurements ($y_i$) and potentially partial control inputs.
  • Sensors cannot estimate the full state independently; they must rely on a communication network (directed graph $\mathcal{G}$) to exchange information with neighbors.
  • The goal is to design local observers such that the estimation error $e_i(t) = x(t) - \hat{x}_i(t)$ asymptotically converges to zero for all agents $i$.
• Challenge: Unlike continuous-time systems where high-gain consensus strategies often work, discrete-time systems are more sensitive to sampling and coupling gains. Previous discrete-time solutions often required restrictive assumptions on graph topology or specific gain structures.

2. Methodology

The proposed approach leverages the Jordan Canonical Form of the system matrix $A$ to decompose the estimation problem into manageable sub-problems based on the detectability properties of each agent.

A. System Decomposition and Permutation

The authors assume (without loss of generality) that $A$ is in Jordan form. They partition the state vector based on the observability of specific Jordan miniblocks ($A_{\ell, h_\ell}$) for each agent $i$:

1. Classification of Miniblocks: For each unstable eigenvalue $\lambda_\ell$ and its associated miniblocks, the authors categorize the indices $h_\ell$ into three sets for agent $i$:
   • $G_{i,1}^{\ell}$: Miniblocks where the output matrix block is zero (completely unobservable).
   • $G_{i,2}^{\ell}$: Miniblocks where the first column of the output block is zero, but the block is non-zero (partially observable).
   • $G_{i,3}^{\ell}$: Miniblocks where the first column is non-zero (completely observable/detectable).
2. State Permutation: A permutation matrix $Q_i$ (or $R_i$) is constructed to reorder the state vector into two main components:
   • Detectable Part ($z_{i,d}$): Contains states corresponding to stable eigenvalues and unstable eigenvalues that are fully or partially observable by agent $i$ (specifically the "observable" portions of $G_{i,2}^{\ell}$ and all of $G_{i,3}^{\ell}$).
   • Undetectable Part ($x_{i,u}$): Contains states corresponding to unstable eigenvalues that are completely unobservable by agent $i$ (indices in $G_{i,1}^{\ell}$ and the unobservable portions of $G_{i,2}^{\ell}$).

B. Hybrid Observer Design

The observer for each agent $i$ is split into two distinct mechanisms:

1. Local Luenberger Observer (for $z_{i,d}$):
   • Agent $i$ uses a standard Luenberger observer to estimate the detectable portion of the state.
   • The gain $L_{i,d}$ is chosen such that $(F_{i,d} - L_{i,d}H_{i,d})$ is Schur stable.
   • Crucial Refinement: Unlike previous works that might estimate the whole sub-vector, this method only retains the estimates of the entirely detectable Jordan miniblocks in the final local estimate.
2. Consensus-Based Strategy (for $x_{i,u}$):
   • For the undetectable components, agent $i$ relies on information from its neighbors.
   • The update law uses a consensus term weighted by the adjacency matrix of the communication graph.
   • Key Innovation: Instead of a single global coupling gain, the method assigns a specific scalar gain $k_{\ell, h_\ell}$ for each Jordan miniblock. This allows for tailored convergence rates for different modes of the system.

C. Error Dynamics Analysis

The estimation error dynamics are analyzed using the Kronecker product. The error for the undetectable part evolves according to:
$$e_{\ell, h_\ell}^u(t+1) = \left( (I - k_{\ell, h_\ell} L_{\ell, h_\ell}) \otimes A_{\ell, h_\ell} \right) e_{\ell, h_\ell}^u(t)$$
where $L_{\ell, h_\ell}$ is a reduced Laplacian matrix obtained by removing rows/columns corresponding to agents that can directly observe that specific miniblock.

3. Key Contributions

1. Jordan-Based Decomposition: The paper introduces a rigorous framework using Jordan canonical forms to separate the state into "locally estimable" and "consensus-requiring" components at the level of individual Jordan miniblocks.
2. Flexible Coupling Gains: Unlike the prior work [2] (which used a single global coupling gain), this approach allows distinct coupling gains ($k_{\ell, h_\ell}$) for each Jordan miniblock. This significantly relaxes the solvability conditions.
3. Simplified Solvability Conditions:
   • The paper derives necessary and sufficient conditions for the existence of a distributed observer.
   • The condition requires that for every unstable eigenvalue $\lambda_\ell$ and every miniblock index $h_\ell$, there exists a gain $k_{\ell, h_\ell}$ such that:
     $$|1 - k_{\ell, h_\ell} \mu| < \frac{1}{|\lambda_\ell|} \quad \forall \mu \in \sigma(L_{\ell, h_\ell})$$
   • This condition is less restrictive than previous results because it reduces the number of inequalities (one per miniblock rather than one per state entry) and allows for different gains.
4. Topology Independence: The method does not require modifying the graph topology (e.g., removing edges to form spanning trees) or edge weights, provided the joint detectability assumption holds and the specific reachability condition (directed spanning forest rooted at observable agents) is met for the reduced Laplacians.

4. Results

• Theoretical Proof: Theorem 4 establishes that the proposed observer guarantees asymptotic convergence if and only if the derived spectral condition on the reduced Laplacian matrices is satisfied.
• Numerical Example: A simulation involving 6 Pendubots (two-link underactuated robots) was conducted.
  • The system had complex eigenvalues and was transformed into real Jordan form.
  • The communication graph was a directed graph with specific connectivity.
  • The simulation demonstrated that all agents successfully estimated the joint angles of the entire formation, with errors converging to zero.
  • The example validated that the simplifying assumptions (real eigenvalues, pre-existing Jordan form) do not hinder the general applicability of the method.

5. Significance

• Advancement over State-of-the-Art: This work improves upon the framework in [2] by offering greater flexibility in gain selection and less restrictive solvability conditions. It simplifies the design process by focusing on entire Jordan miniblocks rather than individual state entries.
• Practical Applicability: By allowing different gains for different modes, the method is better suited for systems with widely varying eigenvalue magnitudes, where a single global gain might fail to stabilize the error dynamics.
• Foundation for Robustness: The authors note that this framework lays the groundwork for future extensions to systems with unknown inputs and disturbances, enhancing the robustness of distributed estimation in real-world industrial applications.

In summary, this paper provides a mathematically rigorous and practically flexible solution for distributed state estimation in discrete-time LTI systems, utilizing the structural properties of the Jordan form to decouple the estimation problem and optimize convergence through mode-specific consensus gains.