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A Distributed Observer for a Time-Invariant Linear System

L. Wang, A. S. Morse

arXiv:1609.05800v4eess.SY

TL;DR

The paper addresses distributed state estimation for a continuous-time, time-invariant linear system when agents have local measurements and exchange estimator information. It develops a distributed-observer construction using decentralized-control results and shows that, under nonzero local sensing, strong connectivity, and joint observability, the overall observer spectrum can be freely assigned.

  • Problem

    The problem is to construct distributed linear estimators whose local estimation errors converge to zero at a preassigned, arbitrarily fast rate without requiring each individual pair (C_i, A) to be observable or detectable.

  • Method

    The paper formulates observer design through estimator and error-dynamics equations and uses decentralized-control results to choose gains for strongly connected neighbor graphs.

  • Results

    For any symmetric set of mn + m − 1 complex numbers, the paper constructs a distributed observer whose overall spectrum equals that set and whose outputs estimate the state at the corresponding exponential rate.

  • Takeaways & Limitations

    Under nonzero C_i, strong connectivity, and joint observability, distributed observers can achieve arbitrary overall spectral placement without individual-agent observability assumptions.

Abstract

from arXiv · show

A time-invariant, linear, distributed observer is described for estimating the state of an $m>0$ channel, $n$-dimensional continuous-time linear system of the form $ \dot{x} = Ax,\ y_i = C_i x,\ i \in \{1,2,\cdots, m\}$. The state $x$ is simultaneously estimated by $m$ agents assuming each agent $i$ senses $y_i$ and receives the state $z_j$ of each of its neighbors' estimators. Neighbor relations are characterized by a constant directed graph $\mathbb{N}$ whose vertices correspond to agents and whose arcs depict neighbor relations. The overall distributed observer consists of $m$ linear estimators, one for each agent; $m-1$ of the estimators are of dimension $n$ and one estimator is of dimension $n+m-1$. Using results from classical decentralized control theory, it is shown that subject to the assumptions that (i) none of the $C_i$ are zero, (ii) the neighbor graph $\mathbb{N}$ is strongly connected, (iii) the system whose state is to be estimated is jointly observable, and nothing more, it is possible to freely assign the spectrum of the overall distributed observer.

I. INTRODUCTION

The paper studies distributed observers for continuous-time, time-invariant linear systems, generalizing centralized observer estimation across a network of agents.

  • A centralized observer estimates the state of a continuous-time, time-invariant linear system from its measured output.Existence requires observability of the matrix pair (C, A).
  • The paper extends this observer concept to a network of m agents.

A. The Problem

The problem is to construct distributed linear estimators whose agents recover the system state at an arbitrarily preassigned convergence rate using local measurements and neighbor information, without requiring individual observability.

  • Each agent senses its local output and can receive neighbors’ estimator states and measured outputs.Neighbor relations are represented by a directed graph, with each agent included among its own neighbors.
  • A distributed observer must make every agent’s estimation error converge to zero from arbitrary initializations.The required convergence rate is preassigned but can be arbitrarily fast.
  • The standing assumptions are nonzero C_i and joint observability of the aggregate system pair (C, A).
  • Unlike common formulations, the paper does not require each individual pair (C_i, A) to be observable or detectable.The authors identify this exclusion as distinguishing the posed problem from almost all distributed estimator problems in the literature.

B. Background

The paper builds on decentralized-control interpretations of distributed observer design and extends prior results to strongly connected continuous-time systems, including singular system matrices A.

  • Prior work linked stable distributed-observer design to stabilizing decentralized control.
  • Prior results showed that only one agent subsystem need exceed dimension n, with enlarged dimension no greater than n + m − 1.
  • This paper outlines a construction for strongly connected neighbor graphs that freely adjusts the observer’s spectrum.
  • The construction applies whether A is singular or nonsingular, enabling continuous-time observers where the prior construction required nonsingular A.

II. OBSERVER DESIGN EQUATIONS

The observer design equations encode consistency between each estimator, the plant dynamics, and the estimation-error dynamics; satisfying them with a stable error system yields distributed state estimation.

  • The design equations relate estimator matrices to the plant so that estimator outputs represent the plant state consistently.They are derived from the requirement that the observer remain correct when all estimates equal the true state.
  • The matrices V_i map the plant state into each estimator’s state coordinates.Their columns are defined using the unit vectors of the n-dimensional state space.
  • The resulting estimation errors satisfy a collective error-system equation determined by the estimator and interconnection matrices.
  • Equations (5) and (8) are called the observer design equations and apply to time-invariant continuous- and discrete-time state observers.
  • If the error system is exponentially stable, every estimator output asymptotically correctly estimates the plant state.The design problem is to choose the estimator and coupling matrices so the design equations hold and the error dynamics are stable.

III. CENTRALIZED OBSERVERS

The centralized observer framework defines linear estimators that reconstruct the system state from measured output while requiring stable estimation-error dynamics. It distinguishes full-state, minimal-state, and extended-state observers by estimator dimension and structure.

  • Observer framework: A centralized observer is a linear system driven by y=Cx whose output estimates x with exponentially fast convergence.The convergence rate can be preassigned and arbitrarily large when the observer dynamics are appropriately designed.
  • Observer framework: The observer design equations require I=MV+NC, VA=HV+KC, and a stable matrix H.The matrices H, K, M, N, and V must jointly satisfy the reconstruction equations.
  • Observer types: Full-state observers use an n-dimensional state estimate z1 and dynamics ˙z1=(A−KC)z1+Ky, with x1=z1.Stability is obtained by choosing K so that A−KC is stable, using spectrum assignment when appropriate.
  • Observer types: Minimal-state observers exploit partial measurements and have dimension at least dimension ker C.The lower bound follows from the requirement that V have enough linearly independent rows.
  • Observer types: Extended-state observers add a dynamic lower subsystem whose flexibility becomes useful in decentralized observer construction.This additional subsystem is not clearly beneficial for centralized observers alone but supports the later distributed design.

IV. DISTRIBUTED OBSERVERS

The distributed observer design converts local sensing and neighbor communication into a structured decentralized stabilization problem. Its matrices must satisfy observer identities while the aggregate error dynamics are made stable with controllable information exchange.

  • Design objective: The observer matrices are chosen to satisfy the observer design equations and make the aggregate system exponentially stable.Reducing neighbor-to-neighbor information transmission is a secondary design objective.
  • Design objective: Setting Vi=I simplifies the design equations, while matrices Mij must sum over neighbors to I.This choice makes each local observer output reconstruction depend directly on neighbor-coupling matrices.
  • Aggregate dynamics: The aggregate matrix H has block-diagonal terms −Ki and off-diagonal neighbor-coupling terms Hij.The zero pattern of H follows the directed neighbor graph.
  • Aggregate dynamics: Designing the distributed observer reduces to selecting feedback gains Fij that stabilize H and control its convergence rate.Such gain selection is generally impossible except under special graph and system conditions.

A. Strongly Connected Neighbor Graph N

For strongly connected neighbor graphs, the paper uses decentralized-control constructions to obtain controllability and observability, then assigns the distributed observer spectrum through an enlarged estimator.

  • Strongly Connected Neighbor Graph N: Strong connectivity guarantees gains Fij for which every pair (H,Bp) is controllable and every pair (Cpq,H) is observable.The controllability index of each pair (H,Bp) is m.
  • Strongly Connected Neighbor Graph N: The construction first selects Mij satisfying the reconstruction identity, then chooses Fij with the controllability and observability properties.A standard centralized construction is applied afterward to assign the desired spectrum.
  • Strongly Connected Neighbor Graph N: The enlarged estimator uses dimension n+m−1 for one agent while the remaining estimator dimensions remain n.The added subsystem matrices are selected to shape the spectrum of the resulting block matrix.
  • Strongly Connected Neighbor Graph N: Theorem 1 permits any symmetric set of mn+m−1 complex numbers as the spectrum of the overall observer matrix H.The theorem assumes joint observability, nonzero Ci, and a strongly connected neighbor graph.
  • Strongly Connected Neighbor Graph N: All agent outputs estimate x asymptotically at the decay rate of e^Ht, regardless of the initializations of the plant and estimators.Thus the assigned spectrum directly determines the observer’s convergence behavior.

B. Non-Strongly Connected Neighbor Graph N

When the neighbor graph is not strongly connected, construction proceeds through its strongly connected components. The necessary and sufficient condition is joint observability for every source component subsystem.

  • Non-Strongly Connected Neighbor Graph N: A source component receives no signal flow from any other component, so its associated subsystem must be jointly observable.For source components, this condition is also sufficient because each source graph is strongly connected.
  • Non-Strongly Connected Neighbor Graph N: A downstream component can use a source estimator state as a measurement of x with exponentially decaying additive noise.The noise is generated by the source estimator’s exponentially decaying estimation error.
  • Non-Strongly Connected Neighbor Graph N: Replacing a downstream subsystem’s readout with an augmented readout yields a jointly observable subsystem with exponentially decaying measurement noise.Strong connectivity within the downstream component then supports observer construction at the source error-convergence rate.
  • Non-Strongly Connected Neighbor Graph N: For graphs with multiple nonsource components, the same construction is applied sequentially across the corresponding component subsystems.This extends the source-based construction beyond a single downstream component.
  • Non-Strongly Connected Neighbor Graph N: For nonzero Ci, source component subsystems being jointly observable is necessary and sufficient for arbitrary preassigned convergence rates.The criterion applies to distributed observers for each component subsystem.

V. DECENTRALIZED CONTROL THEORY

The paper applies decentralized control theory to characterize when local feedback can make a networked system controllable and observable. The conditions require joint controllability and observability, plus completeness of every complementary subsystem.

  • Decentralized feedback: Decentralized feedback uses local laws u_i = F_i y_i to produce a closed-loop matrix H = A + Σ B_iF_iC_i.The feedback gains are constrained by each agent’s local input-output channel.
  • Decentralized feedback: For each agent pair, suitable gains must make (C_p, H, B_p) controllable and observable.These properties enable subsequent stabilization through standard centralized feedback techniques.
  • Completeness conditions: The existence conditions require the augmented system to be jointly controllable and jointly observable.Joint properties concern the combined channels rather than requiring every individual pair (C_i, A) to be observable.
  • Completeness conditions: Every complementary subsystem must be complete, meaning its transfer matrix is nonzero and its matrix pencil has rank at least n for every real and complex λ.Complementary subsystems are formed by selecting input channels from a subset and output channels from its complement.
  • Graph condition: The transfer-matrix nonzero condition can be established from connectivity of the graph associated with the decentralized system.That graph contains an arc from j to i when C_i(sI − A)^−1B_j is nonzero.

VI. ANALYSIS

The analysis proves the gain-selection proposition by reducing controllability and observability to graph and algebraic conditions. Strong connectivity and joint observability then imply that suitable gains exist generically for all required agent-channel pairs.

  • Complementary subsystems: Strong connectivity of the neighbor graph implies each complementary subsystem is complete.The proof establishes a nonzero transfer matrix and the required matrix-pencil rank condition.
  • Algebraic construction: A nonsingular transformation converts the controllability matrix built from H and B_p into one built from F and b_p.The construction uses Kronecker products and the product Δ = T_mT_{m−1} ··· T_1 of nonsingular triangular matrices.
  • Joint properties: The auxiliary system is jointly controllable because the input-channel vectors span the full space.The proof also establishes joint observability by using strong connectivity to equalize agent components and joint observability of (C, A) to force them to zero.
  • Gain selection: For each channel p, suitable gains make (F, b_p) controllable, and the same gains can be chosen to work for all p generically.The admissible gains exclude only a proper algebraic set.
  • Proposition 1: The proposition follows because generic gains make every (H, B_p) controllable and every (C_pq, H) observable.The result applies for all p and all q in N_p under strong connectivity.

VII. CONCLUDING REMARKS

The paper constructs distributed observers for a fixed neighbor graph that estimate the system state at a preassigned, arbitrarily fast convergence rate. It identifies observer dimension, communication, graph variation, and measured inputs as directions for further work.

  • Main conclusion: The constructed observer family achieves estimation at a preassigned but arbitrarily fast convergence rate for a given neighbor graph.The conclusion summarizes the main construction and convergence property.
  • Open problems: The required observer dimension and the amount of information transferred across the network form a trade-off for future study.The paper specifically links least-dimension observers to transmitting neighbors’ measured signals.
  • Open problems: Time-varying neighbor graphs remain an open problem because the resulting equations would be time-varying systems.The paper states that this case requires different mathematics.
  • Open problems: Observers for systems with measured agent inputs, ẋ = Ax + Σ B_i u_i, are also left for future work.Here u_i is an input signal measurable by agent i.
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