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State Estimation over Sensor Networks with Correlated Wireless Fading Channels
Daniel E. Quevedo, Anders Ahlen, Karl H. Johansson
TL;DR
The paper addresses Kalman filtering when fading wireless links cause correlated packet losses in centralized sensor-network estimation. It introduces a network-state model with Markov or semi-Markov temporal behavior and derives sufficient conditions for exponential boundedness of the estimator covariance. The framework also covers power and bit-rate control and contains earlier one-sensor stability results as special cases.
Problem
Wireless fading can cause correlated packet drops in centralized estimation over time-varying sensor networks, creating a need to characterize their effect on Kalman-filter stability.
Method
The paper models radio-environment configurations with a network state process whose link-gain distributions may be spatially correlated and whose temporal evolution is Markov or semi-Markov.
Results
The authors derive sufficient conditions ensuring that the Kalman-filter estimator covariance is exponentially bounded under Markovian and semi-Markov network states.
Takeaways & Limitations
The stability framework generalizes prior packet-dropout results and remains applicable when sensor transmission power and bit-rate are controlled from network state and channel gains.
Abstract
from arXiv · showhide
Stochastic stability for centralized time-varying Kalman filtering over a wireles ssensor network with correlated fading channels is studied. On their route to the gateway, sensor packets, possibly aggregated with measurements from several nodes, may be dropped because of fading links. To study this situation, we introduce a network state process, which describes a finite set of configurations of the radio environment. The network state characterizes the channel gain distributions of the links, which are allowed to be correlated between each other. Temporal correlations of channel gains are modeled by allowing the network state process to form a (semi-)Markov chain. We establish sufficient conditions that ensure the Kalman filter to be exponentially bounded. In the one-sensor case, this new stability condition is shown to include previous results obtained in the literature as special cases. The results also hold when using power and bit-rate control policies, where the transmission power and bit-rate of each node are nonlinear mapping of the network state and channel gains.
I. INTRODUCTION
The paper studies centralized state estimation over wireless sensor networks where fading causes correlated packet dropouts, and develops a stochastic framework for analyzing Kalman-filter stability. It models network topology, fading, temporal channel variation, and measurement aggregation while accounting for practical power and bit-rate control.
- Motivation: Wireless channels experience time-varying fading and interference that can cause packet errors, degrading estimation and control stability.Moving equipment, vehicles, and people can change the propagation environment; power and bit-rate control can partially compensate for fading.
- Problem setting: The study considers centralized estimation of linear time-varying systems over a wireless sensor network with a tree topology and correlated packet dropouts.Network delays are neglected because in-network processing is assumed faster than the system dynamics.
- Fading model: A network state process represents finite radio-environment configurations and determines link channel-gain distributions, allowing spatial and temporal fading correlations.Temporal correlations are modeled with a Markov or semi-Markov network-state process.
- Stability analysis: The paper derives sufficient stochastic-stability conditions under which the estimator covariance trace is exponentially bounded.For Markovian states, the conditions involve system and network parameters; the framework also includes power and bit-rate control policies and extends prior special cases.
- Sensor network architecture: Sensors aggregate their current measurements with packets from incoming nodes and forward the resulting packet toward a single gateway along the directed tree.Each sensor has one parent, does not buffer old data, and communicates through a unique path to the gateway.
III. SENSOR NETWORK CONNECTIVITY MODEL
The connectivity model represents fading through a finite network-state process that determines correlated link-gain distributions and may evolve Markovianly. It also models packet successes, retransmissions, and state- and channel-dependent power and bit-rate control.
- Network fading model: Channel gains capture path-loss, shadow fading, and small-scale fading, with shadow fading allowing correlations across links and time.Small-scale fading is commonly modeled with uncorrelated gain distributions.
- Network fading model: The network state is a finite-valued process representing physical-environment configurations and temporal correlations in the fading environment.The model first uses a time-homogeneous Markov chain and later permits arbitrary holding times.
- Network fading model: Conditional on a network state, each link’s channel-gain distribution can vary by state and links may remain spatially correlated.The framework accommodates Rayleigh, Rician, and Nakagami distributions and does not require channel gains or dropouts themselves to be Markovian.
- Packet loss: Packet transmissions are modeled as perfect reception or complete loss, with success probabilities determined by channel gains, bit-rates, and transmit powers.Fast retransmissions can be represented by modifying the link dropout function.
- Packet loss: Power and bit-rate controls may be nonlinear functions of channel gains and network state, while conditional link successes are independent over time and across links.This control class includes fixed settings, saturated controllers, and power-allocation policies.
IV. STATE ESTIMATION OVER A SENSOR NETWORK TREE WITH PACKET DROP-OUTS
The gateway estimates the system state from time-varying subsets of sensor measurements produced by a tree network with packet losses. Under the no-delay assumption, the resulting estimator is a stochastic Kalman filter driven by a random observation matrix.
- Network architecture: Each sensor aggregates its current measurement with successfully received upstream packets, so the gateway receives a subset of the current sensor measurements.Packets are assumed error-detectable, and sensors do not buffer old data.
- Connectivity and observations: Sensor-to-gateway connectivity is represented by binary processes indicating whether each sensor’s path transmission succeeds.With no network delay, these connectivity outcomes determine which measurements reach the gateway.
- Connectivity and observations: Conditional connectivity probabilities are expressed through individual link success probabilities, but paths sharing links can produce dependent sensor-to-gateway outcomes.The factorization into separate success probabilities does not generally hold for sensors on overlapping paths.
- Kalman filtering: The gateway’s estimator is a Kalman filter for a time-varying stochastic observation matrix induced by the packet-drop pattern.The conditional state distribution remains Gaussian, and the mean and covariance follow Kalman recursions.
- Kalman filtering: The observation matrix has 2^M possible values, making both the Kalman recursion and estimator covariance process stochastic.Its distribution depends on current channel gains, bit-rates, and power levels.
V. STABILITY ANALYSIS FOR MARKOVIAN NETWORK STATES
The stability analysis studies exponential boundedness of the Kalman covariance under Markovian network states and packet losses. It derives a sufficient condition involving system growth, network-state transitions, and conditional full-rank observation probabilities.
- Stability criterion: Packet dropouts can prevent covariance convergence and may cause intermittent divergence when the underlying system is unstable.The analysis therefore uses exponential boundedness as its stochastic stability notion.
- Stability criterion: Exponential boundedness controls the expected covariance and implies covariance stability for every time index.The implication follows from positive semidefiniteness of the covariance matrix.
- Stability criterion: The rank process records whether the random observation matrix is full rank, and ν_i is the conditional probability that it is not full rank.The analysis conditions this probability on the preceding network state.
- Main result: Theorem 1 gives a sufficient condition under which the Kalman filter with Markovian channel states and control policies is exponentially bounded.The condition requires a bound involving system-matrix spectral norms, channel-state transition probabilities, and conditional probabilities of rank failure.
- Main result: The conditional rank probabilities are determined by link success probabilities and can therefore be influenced through power and bit-rate policy design.Example 3 evaluates these probabilities for a six-node tree requiring at least three received measurements for full rank.
VI. THE ONE-SENSOR CASE FOR LTI SYSTEMS
The one-sensor LTI specialization applies the general stability theorem to single-link systems and recovers several established dropout models as special cases.
- Stability condition: Theorem 1 can be directly applied to the one-sensor setup, yielding a sufficient condition for exponential boundedness when C1 is full rank.Corollary 1 states the resulting condition and conclusion for M = 1.
- Connections to prior models: The hidden Markov and Gilbert loss models can also be recovered through appropriate choices of the conditional success parameters.Conversely, the Corollary 1 model can be represented using an aggregated state process {(γ1, Ξ)}.
- Connections to prior models: With |B| = 1, the model becomes an i.i.d. dropout system with transmission success probability Pr{γ1(k) = 1} = φ1|1.This case yields a previously studied single-state formulation.
VII. NETWORK STATES WITH ARBITRARY HOLDING TIMES
The semi-Markov extension permits arbitrary holding-time distributions for network states, allowing the model to represent slowly changing radio environments and virtual transitions.
- Model motivation: The semi-Markov connectivity model imposes minimum holding times on network states, targeting radio environments that change slowly.The earlier Markov model allowed the network state to change at every instant.
- Holding times: The times between network-state switches are allowed to follow an arbitrary probability distribution.Transition instants are represented by an ordered set, and holding times are modeled explicitly.
- Semi-Markov assumption: Under Assumption 3, holding times and the next state transition depend only on the current state and are conditionally independent of the past.The joint transition law is qijψi(δ).
- State trajectory: The renewal process describes the network-state trajectory at all sampling times and permits virtual transitions where the state remains unchanged.The model generalizes the Markov model by assigning holding-time distributions and can capture slowly changing environments.
- Example: For the illustrated four-state robot trajectory, transition probabilities q11 = 0.8, q12 = 0.1, q13 = 0, and q14 = 0.1 describe movement from the home state.The example also specifies holding-time distributions for selected states, including ψ2(δ) = ψ3(δ) = 1/8 for δ ∈ {1, 2, . . . , 8}.
B. Stability Analysis
The semi-Markov stability analysis averages observability failures over bounded holding-time horizons and derives sufficient conditions for exponential boundedness of the Kalman filter.
- Stability analysis: The analysis introduces the transition matrix Φ(ℓ, k) to characterize system behavior under the semi-Markov model.This matrix supports the subsequent observability and stability condition.
- Observability over holding times: The conditional probability in (31) measures when the observability matrix O(kℓ + δ − 1, kℓ) is not full rank.It is conditioned on the preceding network state and the holding time.
- Stability condition: If holding times satisfy ∆ℓ ≤ σ and condition (32) holds for some ρ ∈ [0, 1), the Kalman filter is exponentially bounded.The theorem assumes Assumptions 2 and 3 and bounded support for the holding times.
- Relation to the Markov case: Condition (32) extends the Markov condition by averaging non-full-rank observation outcomes over finite horizons.The averaging incorporates holding-time probabilities and the transition dynamics over the semi-Markov interval.
- Relation to the Markov case: The Markovian network model is recovered when p_ij = q_ij and ∆ℓ = 1 for every ℓ, reducing condition (32) to condition (22).The resulting condition recovers the earlier theorem.
- Rank requirement: Unlike Theorem 1, Theorem 2 does not require C(k) to be full rank with non-zero probability; it requires observability over horizons of length σ when no dropouts occur.This changes the rank requirement from an instantaneous condition to a finite-horizon observability condition.
C. Example
A one-sensor LTI example applies the semi-Markov theorem to two connectivity configurations and incorporates power and bit-rate control laws.
- Example setup: The example considers an LTI plant with M = 1 sensor and two possible connectivity configurations, Ξ(k) ∈ B = {1, 2}.The configurations obey Assumption 3 and are illustrated by the semi-Markov connectivity model.
- Control policies: Power and bit-rate control laws follow form (8) in the two-configuration example.The example therefore evaluates the model while allowing radio-control policies.
- Specialization: For the one-sensor LTI case, C(k) = γ1(k)C1 and Φ(k + t, k) = A^t.These identities reduce the general semi-Markov expressions to the single-sensor time-invariant plant.
- Stability result: Theorem 2 establishes exponential boundedness when its sufficient condition holds for the example's system and network parameters.The example verifies the relevant observability structure before applying the theorem.
- Conclusion: The paper concludes that sufficient exponential-boundedness conditions apply to fading networks with spatially and temporally correlated channel gains.The framework also accommodates fading mitigated by power and bit-rate control, while reducing to earlier stability results in particular cases.
- Future work: Future work includes deriving necessary conditions, designing control and rerouting strategies, and extending the fading model to more general topologies.The authors also identify closed-loop networked control systems as an extension area.
A. Proof of Theorem 1
The proof uses a Markovian composite process and a stochastic Lyapunov argument to establish exponential boundedness of the Kalman-filter covariance under sufficient rank and transition-probability conditions.
- The composite process Z is Markovian, enabling stochastic stability analysis of the estimator covariance.
- When observations are full rank, a predictor yields bounded estimation error covariance because the system, process-noise, and measurement-noise sequences are bounded.
- When no measurements are successfully received, the covariance is bounded by the worst case in which every transmission fails.
- The resulting bound establishes exponential boundedness with constants α = ρV0 and β = ¯β/(1 −ρ), although the intermediate bound is explicitly not tight.
- The proof uses a nonnegative Lyapunov candidate V_k whose drift is bounded through the Markov transition structure.
B. Proof of Theorem 2
The semi-Markov proof analyzes the filter at embedded-chain transition times and extends the resulting exponential bound across bounded holding intervals to obtain boundedness at every time instant.
- The embedded network-state chain at the selected time instants is Markovian, even though the original semi-Markov state process is not.
- At embedded times with received measurements, bounded system matrices make the suboptimal predictor’s covariance bounded, and Kalman optimality preserves that bound.
- At embedded times without received measurements, the covariance is upper-bounded by the worst case in which all transmissions fail throughout the holding interval.
- Exponential boundedness first follows at the embedded instants k_ℓ through constants α1 and β1.
- Bounded holding times, with Δ_ℓ ≤ σ, extend the embedded-time estimate to intermediate instants using constants α2 and W3.
- The proof concludes with E{tr P(k|k −1)} ≤ α1α2ρ^k + α2β1 + W3 for every k ∈ N0.