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On the Controllability and Observability of Networked Dynamic Systems

Tong Zhou

arXiv:1401.4335v1eess.SYcs.SY

TL;DR

The paper develops conditions for analyzing controllability and observability in networked LTI systems and for determining when distributed prediction matches lumped Kalman filtering. Its conditions use subsystem transmission zeros and network connections; the estimator equivalence requires a block-diagonal steady-state Kalman update gain.

  • Problem

    Distributed predictors generally have lower accuracy than lumped estimators, although some coupled systems show matching steady-state accuracy, motivating conditions for when this occurs.

    Source: §IV · p. 12, A. A brief summary of the distributed… · p. 15

  • Method

    The paper derives necessary and sufficient controllability and observability conditions based essentially on subsystem transmission zeros and the connection matrix.

    Source: §I · pp. 2–3

  • Results

    The distributed predictor matches the lumped Kalman filter’s steady-state estimation accuracy only under necessary and sufficient system-matrix conditions that require a block-diagonal steady-state Kalman update gain.

    Source: §I · p. 3, B. Conditions on the equivalence · p. 18, APPENDIX: PROOF OF THE TECHNICAL RESULTS · p. 28

  • Takeaways & Limitations

    The conditions offer a basis for analyzing and synthesizing large-scale networked systems and characterizing when distributed prediction matches lumped Kalman accuracy.

    Source: §I · p. 3

    The stated analysis concerns LTI systems with time-independent connections.

    Source: §I · p. 2

Abstract

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Some necessary and sufficient conditions are obtained for the controllability and observability of a networked system with linear time invariant (LTI) dynamics. The topology of this system is fixed but arbitrary, and every subsystem is permitted to have different dynamic input-output relations. These conditions essentially depend only on transmission zeros of every subsystem and the connection matrix among subsystems, which makes them attractive in the analysis and synthesis of a large scale networked system. As an application, these conditions are utilized to characterize systems whose steady state estimation accuracy with the distributed predictor developed in (Zhou, 2013) is equal to that of the lumped Kalman filter. Some necessary and sufficient conditions on system matrices are derived for this equivalence. It has been made clear that to guarantee this equivalence, the steady state update gain matrix of the Kalman filter must be block diagonal.

I. INTRODUCTION

The paper derives necessary and sufficient controllability and observability conditions for networked LTI systems with heterogeneous subsystems and fixed connections. It also applies these conditions to characterize when a distributed predictor matches the lumped Kalman filter’s steady-state accuracy.

I. INTRODUCTION

  • Standard controllability and observability tests can be computationally prohibitive for systems with many states, motivating more efficient verification criteria.These properties are connected to outcomes such as optimal control, stabilization, and state-estimation convergence,,,,.Source: §I · pp. 1–2
  • Earlier approaches impose practical restrictions: assumes equal interaction strengths, while requires known zero locations and evaluates controllability probabilistically.Prior work also developed graph-based tests and driver-node identification tools.Source: §I · p. 2
  • The proposed necessary and sufficient conditions depend essentially on subsystem transmission zeros and the connection matrix, supporting analysis and synthesis of large-scale networked systems.The model permits different subsystem dynamics and arbitrary fixed connections.Source: §I · pp. 2–3
  • The paper derives necessary and sufficient system-matrix conditions for the distributed predictor in to match the lumped Kalman filter’s steady-state estimation accuracy.The Kalman filter’s steady-state update gain must be block diagonal for this equivalence.Source: §I · p. 3

II. SYSTEM DESCRIPTION AND SOME PRELIMINARIES

The system model represents a network of LTI subsystems with explicit interconnections and permits heterogeneous subsystem dynamics. The preliminaries introduce PBH-based tests and transmission-zero characterization for the subsequent analysis.

II. SYSTEM DESCRIPTION AND SOME PRELIMINARIES

  • Each subsystem has its own dynamics and internal inputs formed from other subsystems’ internal outputs, with time-independent connections represented by Φ.The model distinguishes internal input/output signals from external inputs and outputs.Source: §II · p. 4
  • The connection matrix is structured to encode which outputs connect to which inputs; influence strengths are instead included in subsystem parameter matrices to reduce verification complexity.Each row of Φ has one nonzero element equal to one, and Φ is generally low-dimensional relative to the full state.Source: §II · pp. 4–5
  • Compared with an earlier spatially connected plant model,,, this representation allows heterogeneous subsystem dynamics and arbitrary connections, including power networks and multi-agent systems.It describes subsystem connections more explicitly than a conventional plant state-space representation.Source: §II · p. 5
  • The analysis assumes time-invariant subsystem dynamics and connections; time-varying systems, delays, and packet loss are left for future investigation.The authors describe this setting as a first step because verification for time-varying systems is mathematically difficult and delays or packet loss introduce additional issues.Source: §II · p. 5
  • The preliminaries invoke the PBH test for controllability and observability and state that a transmission zero occurs when a nonzero input vector maps to zero output.For a full-column-normal-rank transfer function matrix G, λ0 is a transmission zero exactly when some nonzero z0 satisfies G(λ0)z0 = 0.Source: §II · pp. 5–6

III. CONTROLLABILITY AND OBSERVABILITY OF A NETWORKED SYSTEM

The section develops necessary and sufficient tests for observability and controllability of a well-posed networked LTI system, without requiring a state-space model of the whole network. The tests use subsystem models and the connection matrix, with observability tied to transmission zeros.

III. CONTROLLABILITY AND OBSERVABILITY OF A NETWORKED SYSTEM

  • The analysis assumes well-posed systems, meaning that I − ASSΦ is invertible.The paper treats well-posedness as an essential design property.Source: §III · p. 6
  • Observability holds exactly when the matrix-valued polynomial M(λ) has full column rank for every complex λ.Rank loss is equivalent to a transmission zero of G(λ).Source: §III · p. 7
  • Subsystems whose internal-input effects are not visible in their own outputs need signals transmitted to other subsystems to make those effects estimable.The theorem’s conditions therefore connect local output limitations to signals available elsewhere in the network.Source: §III · p. 9
  • The observability tests use subsystem parameter matrices and the connection matrix, not a state-space model of the whole network.The transfer-function matrix’s block-diagonal structure, apart from the connection matrix, supports computationally efficient procedures.Source: §III · pp. 7–9
  • Controllability likewise has a necessary and sufficient test and a corresponding verification procedure based on subsystem matrices and the connection matrix.The controllability test follows through the duality between controllability and observability.Source: §III · pp. 11–12

IV. CONDITIONS ON THE EQUIVALENCE OF THE CDOSSP AND THE KALMAN FILTER

Distributed estimation can be attractive when subsystems are numerous or spatially separated, but its accuracy is generally lower than lumped estimation. This section asks when the CDOSSP predictor matches the lumped Kalman filter’s accuracy.

IV. CONDITIONS ON THE EQUIVALENCE OF THE CDOSSP AND THE KALMAN FILTER

  • Distributed estimation is valued for numerous or spatially separated subsystems, although its accuracy is generally lower than lumped estimation.The distributed one-step predictor is generally globally optimal under the unbiasedness restriction and mean square error criterion, but equal steady-state accuracy is not guaranteed for every network.Source: §IV · p. 12
  • The section investigates when the distributed algorithm’s state-prediction accuracy equals that of the lumped Kalman filter.The investigation uses results from Section III.Source: §IV · p. 12
  • The paper calls the distributed one-step state predictor CDOSSP, short for Coordinated Distributed One-step State Predictor..Source: §IV · p. 12

A. A brief summary of the distributed prediction algorithm and some preliminaries

The distributed predictor mirrors the plant’s subsystem structure and uses local output measurements, with matrix dimensions tied to individual subsystems. The section also sets out conditions and assumptions used to analyze predictor performance and Kalman-filter convergence.

A. A brief summary of the distributed prediction algorithm and some preliminaries

  • The predictor analysis assumes stationary external inputs and time-invariant subsystem parameters and connections; its model also excludes direct paths from external inputs to internal and external outputs.The paper says these exclusions avoid complicated expressions; it notes that disturbance-measurement correlations are commonly decoupled by transformations.Source: A. A brief summary of the distributed… · p. 13
  • Each predictor subsystem mirrors the plant, and state estimation uses only that subsystem’s local output measurements.The method is suited to distributed estimation; its recursively computed matrices depend only on the dimensions of each subsystem.Source: A. A brief summary of the distributed… · p. 13
  • Under unbiasedness and error-variance minimization, the cited predictor result assumes white, uncorrelated inputs and measurement errors and a full-row-rank disturbance matrix for each subsystem.Under these conditions, the optimal observer gains can be computed recursively, with matrix dimensions depending only on subsystem dimensions.Source: A. A brief summary of the distributed… · pp. 13–14
  • The paper gives necessary and sufficient system-matrix conditions for the distributed predictor to match the Kalman filter’s steady-state estimation accuracy.The stated full-row-rank restriction on disturbance matrices is used to avoid cumbersome expressions, not as a claimed essential condition.Source: A. A brief summary of the distributed… · p. 15
  • The Kalman-filter convergence lemma assumes white Gaussian stationary disturbances and measurement errors, plus specified controllability and observability conditions.It guarantees convergence of the error covariance to a unique stabilizing DARE solution and of the update gain to a constant matrix.Source: A. A brief summary of the distributed… · p. 15

B. Conditions on the equivalence

The section derives conditions for matching the distributed predictor’s steady-state estimation accuracy to the Kalman filter’s, and relates Kalman-filter convergence to transmission-zero conditions. The equivalence requires a block-diagonal steady-state Kalman update gain, while applying the conditions to large systems can be difficult.

B. Conditions on the equivalence

  • If the plant and subsystem transmission-zero conditions stated in Corollary 2 hold, the networked system’s Kalman filter converges.The result concerns transmission zeros outside the unit disk for one transfer-function matrix and zeros on the unit circle for the other.Source: B. Conditions on the equivalence · p. 16
  • Under the stated time-invariance and positive-definite steady-state covariance assumptions, the predictor matches Kalman-filter steady-state accuracy if and only if the condition in Equation (24) holds for every subsystem.The paper derives this characterization from covariance comparisons; it also gives a preceding necessary-and-sufficient condition for a one-step covariance match when the covariances initially agree.Source: B. Conditions on the equivalence · pp. 17–18
  • A block-diagonal steady-state Kalman update gain is necessary to guarantee equal steady-state estimation accuracy for the distributed predictor and the Kalman filter.Source: B. Conditions on the equivalence · p. 18
  • The equivalence conditions are difficult to apply to large systems because they require whole-network matrices and the Kalman filter’s steady-state covariance, which may be hard to obtain.The paper also notes that its convergence proof for the distributed predictor initializes it with the Kalman filter’s steady-state covariance.Source: B. Conditions on the equivalence · p. 18

V. CONCLUDING REMARKS

The paper derives PBH-based controllability and observability conditions for networked LTI systems with arbitrary fixed connections and differing subsystem transfer-function matrices. It also characterizes when the distributed predictor matches the Kalman filter’s steady-state accuracy, requiring a block-diagonal steady-state Kalman update gain.

V. CONCLUDING REMARKS

  • The PBH-based controllability and observability conditions depend on the connection matrix and subsystem parameter matrices, supporting analysis of large networked systems.The subsystems may have different transfer-function matrices, and the connections may be arbitrary.Source: §V · pp. 18–19
  • The distributed predictor has the Kalman filter’s steady-state estimation accuracy only under characterized conditions that require a block-diagonal steady-state Kalman update gain.Source: §V · p. 19
  • The paper identifies random communication delays, missing data, and more computationally attractive equivalence conditions as open issues.Source: §V · p. 19

APPENDIX: PROOF OF THE TECHNICAL RESULTS

The appendix proves technical results on observability characterizations and estimator equivalence, including steady-state covariance equivalence between the Kalman filter and CDOSSP and its extension to other initial covariances via a temporal-variable shift. It also establishes that equal steady-state estimation accuracy requires a block-diagonal Kalman-filter update gain.

Proof of Theorem 1

  • Observability of Σ is equivalent to the MVP M(λ) having full column rank for every complex λ.The proof establishes both directions by deriving a contradiction between a nonzero unobservable eigenvector and a rank-deficient MVP.Source: APPENDIX: PROOF OF THE TECHNICAL RESULTS · p. 20

Proof of Lemma 3

  • The MVP M(λ) loses full column rank at some complex λ exactly when the transfer-function matrix G(λ) has a transmission zero.The proof constructs a nonzero null vector in each direction between rank deficiency and a transmission zero.Source: APPENDIX: PROOF OF THE TECHNICAL RESULTS · p. 21

Proof of Corollary 2

  • An eigenvector violating observability with |λ| ≥1 is equivalent to a vector in some Y[k] satisfying the corresponding ΦG[2](λ[k]) relation.The proof also gives the analogous equivalence for a left eigenvector with |λ| = 1 violating controllability, using the conjugate-vector relation.Source: APPENDIX: PROOF OF THE TECHNICAL RESULTS · p. 24

Proof of Theorem 4

  • Equal steady-state estimation accuracy requires the system-matrix condition in Equation (a.52), which implies a block-diagonal steady-state Kalman update gain.The proof states that the condition is equivalent to the displayed cross-block matrix constraint and derives block diagonality of the steady-state gain.Source: APPENDIX: PROOF OF THE TECHNICAL RESULTS · p. 28

APPENDIX: PROOF OF THE TECHNICAL RESULTS

  • For linear plants with normal external disturbances,,,, matching estimation accuracy requires equal estimator covariances at every time; if the Kalman filter converges, CDOSSP also converges and their steady-state prediction-error covariances match.The convergence conclusion is stated for this initial condition.Source: APPENDIX: PROOF OF THE TECHNICAL RESULTS · p. 29
  • The same result holds for another Kalman-filter initial covariance by shifting CDOSSP time as t_c = t_k + δt and taking δt → ∞.The proof says this follows from Theorem 3 by similar arguments; details are omitted.Source: APPENDIX: PROOF OF THE TECHNICAL RESULTS · p. 29
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