| name | admm-distributed-kalman-observer |
| description | ADMM-Based Distributed Kalman-like Observer methodology for multi-agent systems state estimation. Implements information-form Kalman filtering with exponential forgetting factor and partition-based ADMM correction. Provides sparsity-preserving prediction, distributed QP solving, and UGES stability guarantee via two-time-scale analysis. Use for: cooperative localization, distributed state estimation in multi-agent systems, sensor networks, robot swarm localization, distributed power network state estimation. |
ADMM-Based Distributed Kalman-like Observer
Distributed state estimation framework for multi-agent systems with local and relative measurements, using information-form Kalman filtering with ADMM-based distributed correction.
Source Paper
Title: ADMM-Based Distributed Kalman-like Observer with Applications to Cooperative Localization
Authors: Nicola De Carli, Nicola Bastianello, Dimos V. Dimarogonas
arXiv: 2604.21608v1 [eess.SY]
Date: April 23, 2026
Institution: KTH Royal Institute of Technology, Stockholm, Sweden
Core Problem
Distributed state estimation for multi-agent systems where:
- Global state dimension scales with network size N
- Agents have local measurements (depending only on own state)
- Agents have relative measurements (involving neighboring agents)
- Centralized approaches become intractable as N grows
Mathematical Framework
System Model
x_{i,k+1} = A_{i,k} x_{i,k} + B_{i,k} u_{i,k}, i ∈ V = {1,...,N}
Graphs:
- Sensing graph G_s = (V, E_s): directed, edge (i,j) means relative measurement
- Communication graph G_c = (V, E_c): undirected, bidirectional information exchange
Measurements:
- Local: y^ℓ_{i,k} = H^ℓ_{i,k} x_{i,k}
- Relative: y^r_{ij,k} = H^r_{ij,i,k} x_{i,k} + H^r_{ij,j,k} x_{j,k}
Information-Form Representation
S_{k|k-1} := P_{k|k-1}^{-1} (information matrix)
z_{k|k-1} := S_{k|k-1} x̂_{k|k-1} (information vector)
Key Innovation 1: Sparsity-Preserving Prediction
Standard prediction (destroys sparsity):
S_{k+1|k} = (A_k S_{k|k}^{-1} A_k^⊤ + Q_k)^{-1} → becomes dense
Proposed prediction (preserves sparsity):
S_{k+1|k} = γ A_k^{-⊤} S_{k|k} A_k^{-1}, 0 < γ < 1
Since A_k is block-diagonal, the sparsity pattern of S_{k|k} is preserved.
Key Innovation 2: Distributed ADMM Correction
Correction formulated as sparse SPD linear system:
S_{k|k} ξ_k = b_k, where ξ_k = (1/ε)(x̂_{k|k} - x̂_{k|k-1})
Solved via partition-based ADMM with local communication only.
Implementation
import numpy as np
from scipy import sparse
class ADMMDistributedKalmanObserver:
"""
Distributed Kalman-like observer using ADMM for correction.
Based on: De Carli et al., "ADMM-Based Distributed Kalman-like
Observer with Applications to Cooperative Localization", 2026.
"""
def __init__(self, num_agents, state_dim, gamma=0.9, epsilon=0.5, rho=1.0):
self.N = num_agents
self.d = state_dim
self.gamma = gamma
self.epsilon = epsilon
self.rho = rho
self.S = {}
self.z = {}
self.x_hat = {}
def initialize_agent(self, agent_id, x0, P0):
"""Initialize an agent with initial state and covariance"""
S0 = np.linalg.inv(P0)
self.S[agent_id] = {
'self': S0,
'neighbors': {}
}
self.z[agent_id] = S0 @ x0
self.x_hat[agent_id] = x0
def predict(self, agent_id, A, B, u, neighbors):
A_inv = np.linalg.inv(A)
S_self_old = .S[agent_id][]
.S[agent_id][] = .gamma * A_inv.T @ S_self_old @ A_inv
neighbor_id neighbors:
neighbor_id .S[agent_id][]:
S_coupling = .S[agent_id][][neighbor_id]
A_n = ._get_neighbor_dynamics(neighbor_id)
.S[agent_id][][neighbor_id] = (
.gamma * A_inv.T @ S_coupling @ np.linalg.inv(A_n)
)
.x_hat[agent_id] = A @ .x_hat[agent_id] + B @ u
.z[agent_id] = .S[agent_id][] @ .x_hat[agent_id]
():
innovation = y_local - H_local @ .x_hat[agent_id]
R_inv = np.linalg.inv(R_local)
b = H_local.T @ R_inv @ innovation
S_kk = .S[agent_id][]
xi = np.zeros(.d)
lambda_dual = np.zeros(.d)
iteration (max_iter):
xi_new = np.linalg.solve(
S_kk + .rho * np.eye(.d),
b + .rho * (xi - lambda_dual)
)
neighbor_xis = ._exchange_with_neighbors(agent_id, xi_new, neighbors)
xi_avg = np.mean([xi_new] + neighbor_xis, axis=)
lambda_dual = lambda_dual + xi_new - xi_avg
xi = xi_new
.x_hat[agent_id] = .x_hat[agent_id] + .epsilon * xi
.z[agent_id] = .S[agent_id][] @ .x_hat[agent_id]
.x_hat[agent_id]
():
():
[np.zeros(.d) _ neighbors]
Two-Time-Scale Stability Analysis
System Decomposition
The interconnected observer separates into:
-
Slow subsystem (estimation error dynamics):
e_{k+1} = (I - ε S_{k|k}^{-1} Y_k) A_k e_k + noise
- Uniformly exponentially stable under observability
-
Fast subsystem (ADMM dynamics):
ξ^{(t+1)} = ξ^{(t)} - α ∇f(ξ^{(t)})
- Exponentially stable for strongly convex QP
Stability Theorem
Under standard assumptions (uniform boundedness, complete uniform observability, uniform invertibility), the overall distributed observer achieves Uniform Global Exponential Stability (UGES).
Comparison with Alternatives
| Method | Communication | Computation | Stability | Key Limitation |
|---|
| Centralized KF | All-to-all | O(N³) | Optimal | Not scalable |
| Covariance Intersection | Local | O(d³) | Guaranteed | Conservative, underuses info |
| Consensus-based | Iterative | O(d³) | Limited | Requires consensus on global state |
| ADMM Observer | One-hop | O(d³) | UGES | Requires bounded ADMM iterations |
Parameters
| Parameter | Symbol | Range | Effect |
|---|
| Forgetting factor | γ | (0, 1) | Adaptation rate vs. stability |
| Correction gain | ε | (0, 1] | Observer correction magnitude |
| ADMM penalty | ρ | (0.1, 10) | Convergence speed |
| ADMM iterations | max_iter | 10-100 | Accuracy vs. computation |
Applications
- Cooperative Localization: Multi-robot teams with relative measurements
- Distributed Power Networks: Smart grid state estimation
- Sensor Networks: Environmental monitoring with local sensing
- UAV Swarms: GPS-denied formation flying
- Autonomous Platoons: Vehicle string stability
References
De Carli, N., Bastianello, N., & Dimarogonas, D. V. (2026). ADMM-Based Distributed Kalman-like Observer with Applications to Cooperative Localization. arXiv:2604.21608v1 [eess.SY].
