⚡ electrical engineering

Leader-following Consensus over Jointly Connected Switching Networks is Achievable for Exponentially Unstable Linear Systems

This paper demonstrates that leader-following exponential consensus and output-based distributed observer design are achievable for general linear multi-agent systems with exponentially unstable dynamics over jointly connected switching networks, thereby overcoming the previous restrictive requirement that system matrices must be marginally stable.

Original authors: Yuhan Chen, Tao Liu, Jie Huang

Published 2026-03-20

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CC BY 4.0

Original authors: Yuhan Chen, Tao Liu, Jie Huang

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

The Big Picture: The "Unstable Leader" Problem

Imagine a group of drones (the followers) trying to fly in formation with a lead drone (the leader).

In the past, engineers could only solve this problem if the lead drone was "calm." If the lead drone started wobbling or drifting slightly, the followers could adjust and stay in sync. However, if the lead drone was explosively unstable—meaning it was accelerating away, spinning out of control, or flying in a chaotic, diverging path—the old math said, "Game over. The followers can never catch up."

This was a huge limitation because many real-world machines (like rockets or fast-moving robots) are naturally unstable. They need constant correction to stay upright; left alone, they fall apart.

The Breakthrough:
This paper proves that even if the leader is flying into a chaotic, exponentially unstable storm, the followers can still perfectly track it, provided they talk to each other often enough and in the right pattern.

The Key Concepts (Translated)

1. The "Jointly Connected" Network (The Chatty Crowd)

Imagine the drones are in a large field. They can't all talk to everyone at once.

The Old Way: Everyone had to be in a perfect circle, talking to their neighbors constantly. If the network broke for even a second, the formation failed.
The New Way (Jointly Connected): The network is messy. Sometimes Drone A talks to Drone B, but not C. Five seconds later, Drone C talks to Drone D.
The Magic: As long as, over a period of time (say, every 10 seconds), the information eventually flows from the leader to every follower (even if it takes a relay race of messages), the group can stay synchronized. They don't need a perfect connection at every single second; they just need a "connected history."

2. The "Instability Limit" (The Speed of the Storm)

The paper introduces a new rule: How fast can the leader go crazy before the followers give up?

The authors found a mathematical "speed limit" for the leader's chaos.

The Variables:
- $\delta$ (Delta): Think of this as the "efficiency of the gossip chain." How well does the information about the leader's position spread through the group over time?
- $T_c$ (Time Constant): How long does it take for the network to complete one full cycle of connecting everyone?
The Rule: If the leader's instability (how fast it diverges) is slower than the group's ability to gossip and correct itself (determined by $\delta$ and $T_c$ ), the followers can catch up.
Analogy: Imagine the leader is running away from you at 10 mph. If your team can coordinate and run at 12 mph, you will catch up. But if the leader runs at 100 mph, no amount of teamwork will help. This paper calculates exactly what that "100 mph" limit is for any given team dynamic.

3. The "Dual Problem" (The Spy vs. The Leader)

The paper also solves a reverse problem: The Distributed Observer.

Scenario: The leader is a spy who is hiding their location. The followers can only see the leader's output (e.g., they see the smoke from the engine, but not the engine itself).
The Goal: The followers need to build a mental model to guess exactly where the spy is, even though the spy is moving chaotically.
The Result: Using the same math as the drone formation, they proved the followers can build a perfect "mental map" of the unstable spy's location, as long as the spy isn't moving too fast relative to how fast the followers can share their observations.

Why This Matters (The "So What?")

Before this paper, engineers had to design systems that were inherently stable (like a pendulum that naturally swings back to the center). They couldn't use systems that were naturally unstable (like a rocket that falls over if you don't steer it).

This paper removes that barrier.

It tells engineers: "You can use the most aggressive, high-performance, unstable machines you want. As long as your communication network is 'jointly connected' (information flows eventually), you can control them and keep them in sync."

Summary in One Sentence

This paper proves that a group of agents can perfectly track a leader that is flying into chaos, as long as the group communicates frequently enough to overcome the leader's speed of divergence.

1. Problem Statement

The paper addresses the leader-following exponential consensus problem for general linear multi-agent systems (MAS) operating over jointly connected switching networks.

System Dynamics: The system consists of one leader and $N$ followers. Both are modeled by linear time-invariant (LTI) systems. Crucially, the system matrix $A$ is allowed to be exponentially unstable (i.e., $\lambda_{\max}(A) > 0$ ).
Network Topology: The communication network is time-varying (switching). It does not need to be connected at every instant; instead, it satisfies the jointly connected condition, meaning the union of graphs over a finite time interval is connected.
The Challenge: Previous literature established that leader-following consensus over jointly connected networks was only solvable if the system matrix $A$ was marginally stable (eigenvalues on the imaginary axis or in the left half-plane). This restriction excluded common models like the double-integrator system (which has repeated eigenvalues at the origin) and any unstable systems. The paper aims to break this barrier by proving solvability for exponentially unstable systems.

2. Methodology

The authors employ a rigorous analytical approach combining switched system theory, graph theory, and duality principles.

A. System Formulation and Error Dynamics

The consensus error is defined as $\bar{x}_i(t) = x_i(t) - x_0(t)$ .
The closed-loop error dynamics form a linear switched system:
$\dot{\bar{x}}(t) = (I_N \otimes A - H_{\sigma(t)} \otimes BK) \bar{x}(t)$
where $H_{\sigma(t)}$ is the leader-follower matrix (a submatrix of the Laplacian involving the leader's connections).

B. Key Geometric Quantities

To handle the switching nature and instability, the authors define two critical quantities derived from the network topology:

$T_c$ (Dwell Time of Connectivity): The maximum time interval required for the union of switching graphs to become connected (from Assumption 3).
$\delta$ (Contraction Factor): A value strictly between 0 and 1, derived from the product of orthogonal projection matrices onto the kernel of the leader-follower matrix $H_{\sigma(t)}$ $H_{σ (t)}$ over a connectivity interval.
- Specifically, Lemma 1 proves that $\| \prod P_{\sigma(t_r)} \| \leq \delta < 1$ . This quantifies how much the "unobservable" or "uncontrolled" subspace shrinks over a connected interval.

C. Stability Analysis via Switched System Theory

The authors decompose the state space into two subspaces based on the eigenvectors of $H_{\sigma(t)}$ $H_{σ (t)}$ :
1. The kernel of $H_{\sigma(t)}$ (associated with zero eigenvalues).
2. The orthogonal complement (associated with non-zero eigenvalues).
They derive a closed-form solution for the error dynamics over switching intervals.
The Core Logic: The instability of $A$ causes state divergence, but the control input (via $K$ ) acts on the non-zero eigenvalue subspace to drive errors to zero. The projection matrices $P_{\sigma(t)}$ map the state back into the kernel during switching. The authors prove that if the "shrinking" effect of the network connectivity (quantified by $\delta$ ) is strong enough to counteract the "expansion" caused by the unstable $A$ (quantified by $\lambda_{\max}(A)$ ), consensus is achieved.

D. Duality for Observer Design

The paper leverages the duality principle between controllability and observability.
The consensus problem (stabilizing $A-BK$) is shown to be dual to the output-based distributed observer design problem (stabilizing $A-LC$).
By solving the consensus problem, they automatically establish the solvability of designing an observer that estimates the leader's state exponentially, even if the leader is exponentially unstable.

3. Key Contributions

Breaking the Marginal Stability Barrier: The paper demonstrates that leader-following consensus is achievable for exponentially unstable linear systems over jointly connected switching networks. This is a significant generalization over prior work which was limited to marginally stable systems.
Explicit Instability Bound: The authors derive a precise solvability condition:
$\lambda_{\max}(A) < -\frac{\ln \delta}{T_c}$
This inequality explicitly characterizes the maximum allowable degree of instability ( $\lambda_{\max}(A)$ ) based on the network's connectivity properties ( $\delta$ and $T_c$ ).
Novel Quantification of Network Connectivity: Unlike previous works that relied on infinite matrix products or specific graph properties, this paper introduces the use of orthogonal projection matrices onto the kernel of the leader-follower matrix to quantify the "synchronizability" of the switching graph.
Unified Observer Design: By exploiting duality, the paper provides a constructive method for designing distributed observers for exponentially unstable leaders, a problem previously unsolved under jointly connected switching conditions.

4. Main Results

Theorem 1 (Consensus): Under standard assumptions (controllability of $(A,B)$ , undirected follower subgraphs, and joint connectivity), the leader-following exponential consensus problem is solvable if the condition $\lambda_{\max}(A) < -\ln \delta / T_c$ holds. A distributed static state feedback control law $u_i = K \sum a_{ij}(x_j - x_i)$ is constructed using a weighted controllability Gramian.
Theorem 2 (Observer Design): Under the same conditions (plus observability of $(C,A)$ ), an exponentially convergent output-based distributed observer exists for the leader system if the same instability bound holds.
Numerical Validation: A simulation with 8 followers and an unstable leader (eigenvalues with positive real parts) confirms the theory. The followers successfully track the exponentially diverging leader trajectory, and the observers accurately estimate the leader's state.

5. Significance and Impact

Practical Relevance: Many real-world systems (e.g., UAVs, mobile robots, power grids) are modeled as double-integrators or unstable linear systems. The previous requirement for marginal stability made theoretical consensus results inapplicable to these common scenarios under switching topologies. This paper removes that limitation.
Theoretical Advancement: It resolves a long-standing open problem in multi-agent control. The explicit bound linking system instability to network connectivity parameters provides a new design guideline for engineers: if the system is highly unstable, the network must be "more connected" (smaller $T_c$ or larger $\delta$ ) to maintain consensus.
Duality Application: The successful application of duality to extend observer design to unstable systems over switching networks opens new avenues for distributed estimation in unstable environments.

In summary, this paper fundamentally advances the field of multi-agent systems by proving that network connectivity can compensate for system instability, provided the instability does not exceed a specific threshold determined by the switching graph's joint connectivity properties.