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A simple approach to distributed observer design for linear systems
Weixin Han, Harry L. Trentelman, Zhenhua Wang, Yi Shen
TL;DR
The note addresses distributed state estimation for continuous-time LTI systems when observers possess only partial outputs and communicate over a network. It presents a simple observer-design approach based on balanced-graph mirroring and LMI synthesis. Under global observability and strong graph connectivity, a distributed observer exists with any prescribed positive decay rate, while each observer retains the plant’s state dimension.
Problem
Distributed estimation asks how networked observers can estimate a plant state when no single observer has sufficient local measurements.
Method
The note designs local Luenberger observers using the mirror of a balanced communication graph and gain matrices obtained through LMIs.
Results
If (H, A) is observable and G is a strongly connected directed graph, a distributed observer exists with error decay rate at least any prescribed α > 0.
Takeaways & Limitations
The design provides a simple distributed observer whose local state dimension equals that of the observed plant.
Abstract
from arXiv · showhide
This note investigates the distributed estimation problem for continuous-time linear time-invariant (LTI) systems observed by a network of observers. Each observer in the network has access to only part of the output of the observed system, and communicates with its neighbors according to a given network graph. In this note we recover the known result that if the observed system is observable and the network graph is a strongly connected digraph, then a distributed observer exists. Moreover, the estimation error can be made to converge to zero at any a priori given decay rate. Our approach leads to a relatively straightforward proof of this result, using the mirror of the balanced graph associated with the original network graph. The numerical design of our distributed observer is reduced to solving linear matrix inequalities (LMI's). Each observer in the network has state dimension equal to that of the observed plant.
I. INTRODUCTION
Distributed state estimation addresses systems where no single observer can recover the full state from local measurements. This note develops a simple continuous-time LTI observer design using strongly connected communication and an auxiliary graph construction.
- Motivation: Distributed estimation is needed when each observer has only limited local measurements and must exchange information with neighboring observers.Classical single-observer methods therefore do not directly apply.
- Related work: Existing approaches include distributed Kalman-filter methods and observer-based methods, with some relying on consensus or state augmentation.The cited Kalman-filter methods use estimation updates together with data fusion based on average consensus.
- Related work: Scalar-gain approaches can be limited by tight coupling between network topology and plant dynamics.This coupling restricts the range of scalar-gain designs.
- Related work: Some prior state-augmentation observer constructions discuss only discrete-time systems.Other observer designs use local measurements for observable portions and consensus dynamics for unobservable portions.
- This note: The note studies continuous-time LTI systems whose observers access output portions, communicate over a strongly connected directed graph, and estimate the plant state asymptotically.Each local observer has the same order as the observed system.
- This note: The design decouples network topology from local gains through the mirror of a balanced auxiliary graph and reduces gain selection to LMIs.The procedure also makes the observer gain matrices more flexible and numerically feasible to compute.
A. Preliminaries
The preliminaries define the graph, Laplacian, matrix, and subspace notation used in the observer analysis. For a strongly connected digraph, balancing and mirroring produce a positive-semidefinite undirected-graph Laplacian.
- Graph notation: A directed graph specifies weighted information flow between nodes, with directed paths defining strong connectivity.Strong connectivity means a directed path exists between every pair of distinct nodes.
- Graph notation: The graph Laplacian is L = D − A, has eigenvector 1_N for eigenvalue zero, and has other eigenvalues in the open right-half plane when the graph is strongly connected.These spectral properties support the subsequent network analysis.
- Balanced and mirror graphs: For a strongly connected digraph, a positive diagonal matrix R is formed from a positive vector satisfying r^T L = 0, and RL is the Laplacian of a balanced digraph.The balancing adjusts the weights of the original graph.
- Balanced and mirror graphs: The matrix L_hat = RL + L^T R is the Laplacian of the undirected graph formed by adding each balanced-graph edge and its reverse.This graph is called the mirror of the balanced graph and is positive semidefinite.
B. Problem formulation
The problem is to design local Luenberger observers that collectively estimate a continuous-time LTI plant from partitioned outputs and neighbor communication. The formulation allows locally unobservable measurements while requiring global observability and strong graph connectivity.
- Problem formulation: The plant is a continuous-time LTI system with state x and output y, whose output is partitioned into measurements available at individual nodes.Node i receives only y_i = H_i x.
- Assumptions: The communication graph is assumed to be strongly connected, the aggregate pair (H, A) observable, and individual pairs (H_i, A) need not be observable or detectable.Thus, local observers cannot generally be designed using classical observer methods alone.
- Observer design: Each local observer maintains an n-dimensional estimate and combines local output injection with information exchange from neighboring observer estimates.The observer uses gain matrices L_i and M_i together with a coupling gain γ.
- Error analysis: The distributed observer achieves asymptotic omniscience exactly when the global error system is stable.This converts the estimation objective into a stability condition for the stacked observer errors.
- Observer design: Observability decomposition separates each local error into observable and unobservable components for analysis and synthesis.The unobservable subspace is represented through the kernel of the local observability matrix.
III. MAIN RESULTS
The paper develops a distributed-observer design for observable systems over strongly connected directed graphs, with convergence at any prescribed decay rate. The construction reduces gain selection to LMIs and establishes the result through graph and observability arguments.
- Observability argument: Observability of the local transformed pairs supports the construction by ensuring the relevant matrix has full row rank n.The rank argument explicitly invokes the standing observability assumption.
- LMI-based design: The observer gains Li and Mi are obtained when an associated LMI is feasible, and these solutions guarantee asymptotic omniscience with error decay rate at least α.The proof uses positive-definite matrices and a Lyapunov inequality for the error system.
- Convergence proof: The Lyapunov analysis establishes ˙V(e) < −2αV(e), implying convergence of the error system to zero with decay rate at least α.The decay guarantee follows directly from the derivative inequality.
- Main theorem: For any desired decay rate α > 0, an observable pair (H, A) and strongly connected directed graph admit a distributed observer whose error converges at least at rate α.This is the main existence result and guarantees asymptotic omniscience.
- Design procedure: A constructive procedure selects orthogonal transformations, coupling parameters, local observer gains, and Lyapunov solutions to implement the distributed observer.The procedure includes choosing γ sufficiently large and placing eigenvalues of Aio−LioHio to the left of −α.
7 Define
Under the standing assumptions, the proposed design parameters satisfy the required inequalities and yield a distributed observer with the prescribed convergence rate. The associated inequalities can be solved numerically as LMIs.
- Constructive guarantee: Choosing gi = 1 and sufficiently large γ makes the conditions of Lemma 5 hold, yielding an observer with omniscience and decay rate at least α.The construction relies on observability and strong connectivity.
- Feasibility: The design procedure provides one valid solution of inequality (22) under observability and strong connectivity assumptions.The paper states that the required solution always exists under these assumptions.
- Numerical implementation: Inequalities (8) and (22) are LMIs that can be solved numerically using MATLAB's LMI Toolbox or YALMIP.This makes the gain-design procedure computationally implementable.
- Undirected special case: For a connected undirected communication graph, the positive row vector satisfying rL = 0 can be normalized so that ri = 1 for every observer.The paper identifies this as a special case of the general directed-graph construction.
IV. CONCLUSIONS
The note presents a simple distributed-observer design for LTI systems over strongly connected directed graphs, with each local observer matching the plant’s state dimension. It recovers sufficient existence conditions and provides an LMI-based procedure for calculating observer gains.
- The proposed distributed observer exchanges information over a strongly connected directed graph.
- Each local Luenberger observer asymptotically estimates the dynamical system’s state while retaining the observed plant’s state dimension.
- The gain-matrix structure is analyzed using an observability decomposition of the local systems and an auxiliary undirected graph.
- A bank of linear matrix inequalities calculates the distributed-observer gain matrices, making the design numerically feasible.
- The approach recovers known sufficient conditions for distributed-observer existence and supplies a simple optional design algorithm.