Distributed State Estimation for Linear Time-Varying Systems with Sensor Network Delays

Distributed KF (DKF) for wireless sensor networks, have been particularly studied, e.g., in [5, 1, Olfati2]. An approach is proposed in [5] to place $n$ sensor (equivalently $n$ nodes on the corresponding graph) and collect their measurement data, for which there exists a coordinator which aggregates all the information on the state as well as covariance matrix estimates. A consensus-based fusion of sensor data is proposed in [1]. A modification to the DKF was carried out in [5], wherein micro-filters were associated with each sensor in the node. Similarly, low-Pass and band-pass filters were used in [Olfati2]. The primary reason for the introduction of filters in [1] is to ensure that different measurement modes do not offer an impediment to sensor fusion. By introducing identical filters across all sensor nodes, the bandwidth would be restricted, hence, limiting the variations that can be caused by assuming different measurement models. The DKF algorithm in [1], [Olfati2] are proposed assuming sharing information from one node to its neighbours. Naturally, there is a requirement for the assumption of connectivity of the graph. The state vector and covariance matrix posterior estimates are updated based on a rule that involves the difference between the prior and posterior estimates of its neighbours. This entails a substantial computational burden for large and dense sensor networks. However, by doing so, one can avoid the disadvantages of packet loss. An extension towards the Continuous Time (CT) KF is also proposed along with stability guarantees. Another idea on consensus-based DKF [2] is considering a weighted network, wherein each node computes a weighted average of its neighbours and itself. This weighted graph is such that the weights between neighbours are updated towards achieving a minimum variance of the estimates.

In this paper, we consider the KF architecture that follows the lines of [5], wherein the posterior and prior estimates are obtained at each filter node upon aggregating along a measurement vector.

One of the major challenges in distributed sensor networks is the presence of delays [10] that occur due to packet losses, noise in the transmission channels, and bandwidth limitations, as well as random communication delays [8]. As far as missing packet data is concerned, a modification on the KF equations can be made by incorporating a random variable which indicates whether data arrives at a time instant or not [intermittentKF]. Along similar lines, in [8], a time-variant Bernoulli-distributed variable was defined, which would return a value of one if the event at a particular time was affected by the past state and zero if not. In [nagh], design of an observer-based controller is performed, assuming packet dropouts that may occur during transmission. A similar technique was used in [dong], wherein the authors consider controlling LTI networked systems with transmission delays. Upon constructing a suitable Lyapunov equation, an upper limit for the permissible transmission delays is derived, based on which, the feedback controller is designed. Since the model in the latter work is assumed to be LTI, it is possible to develop anticipatory estimates using delayed-differential equations. However, it is not trivial to extend this method to the Linear Time-Varying (LTV) systems, especially when the sensors are noisy and the processes have uncertainty.

A First-In First-Out (FIFO)-based queuing strategy was developed in [chan] wherein a network manager determines which data must be transmitted first, based on pre-set priorities. However, this is a serial communication technique and is not feasible for a large number of nodes. Moreover, there are additional queuing issues, e.g., when the total number of transmission delays from all the nodes exceed the sampling period.

Rather than using an augmented KF design to account for the presence/absence of estimates in each time instant, as studied in [intermittentKF, Chen], we propose to consider a subset of the total set of sensor nodes. In most realistic scenarios, it is not economically feasible to use all sensors for state estimation. In addition, considering more sensor nodes would make the estimates susceptible to the effect of delays.

Studies on subset selection exist in the literature, e.g., in [13], wherein a node selection scheme is proposed based on maximising a ”perception confidence value” (inverse of the trace of the posterior covariance matrix). Intuitively, these are the nodes that provide the minimum Mean-Squared Error (MSE).

In this work, we initially apply a greedy ”selection” algorithm by choosing appropriate thresholds on sensor characteristics, after which we observe whether similar performance can be achieved by choosing sensor nodes based on a stability criterion derived in IV-C.

The major contributions of this article regarding DKFs with the presence of delays are as follows:

1. 1.

   Improving the performance of estimates obtained by the DKF for LTV systems in the presence of delays. Current literature has focused extensively on LTI systems.

2. 2.

   Theoretical overview of stability criterion for DKFs on LTV system, extending the work [14] which considered LTI systems.

3. 3.

   Subset selection of sensor nodes based on the stability criterion and performance comparison with a greedy algorithm as a non-scalable benchmark.

The rest of this paper is structured as follows: Section III provides the mathematical overview of the DKF with delay considerations and the theoretical analysis of stability for DKF estimates applied on LTV systems. It also presents the intuition behind the two algorithms proposed in this paper. Section VI prsents the simulation results on a second-order LTV system, along with performance comparisons of the proposed algorithms. Section VIII presents a summary and concludes the paper.

We consider the state estimation problem for an LTV system being observed with a sensor network with $n$ sensors to measure the states of the system. Fig. 1 depicts the block diagram of the DKF. Distributed estimation is carried out at the ”filter nodes”, wherein the individual prior and posterior estimates are computed, after which they are aggregated at the ”state estimator” to produce state estimates of the LTV system. While sending information to the state estimator, each sensor is associated with a stochastic communication delay. Delays typically compromise the accuracy of state estimates, hence, the goal is to select the optimal number of sensors that produce accurate-enough estimates in the presence of delays. We consider a stochastic LTV system represented by the following discrete-time model

where $x(k)\!\in\!\mathbb{R}^{m}$ and $w(k)\!\sim\!\mathcal{N}(0_{m},Q)$ denote the states and the system’s exogenous disturbances, respectively. It is assumed that the additive disturbances are independent $\forall\ k\in\mathbb{Z}_{\geq 0}$. The states are assumed to be partially observable by individual sensors, with the partial measurements given by

where $z_{i}(k)\in\mathbb{R}^{p}$ is the measurement vector taken from the $i$-th sensor node at time-step $k$, $H_{i}\in\mathbb{R}^{p\times m}$, and $v_{i}\sim\mathcal{N}(0_{p},R_{i})$ is the measurement noise associated with the sensor $i$. The aggregate measurement, i.e., including measurements from all sensors, would then be expressed as

We now introduce the following notations: $\hat{x}(k|k),\hat{x}(k|k-1)$ being the posterior and prior estimates of the state, respectively. The conventional KF state update equation is given by:

where $L(k)$ is the time-varying Observer Gain (OG). We define the posteriori estimation error as $\epsilon(k|k)=\hat{x}(k|k)-x(k)$, and its associated posteriori covariance matrix by $\Sigma(k|k)=\mathbb{E}\{\epsilon(k|k)\epsilon(k|k)^{\top}\}$. By minimizing the trace of the covariance matrix, the recursive equations for the OG and the covariance matrix are obtained as follows

where $R=[R_{1},R_{2},\ldots,R_{n}]^{\top}$.

For the purpose of brevity, we represent the KF in a compact form [5] using the Information Filter (IF) framework, wherein we define $\mathcal{I}(k|k)=\Sigma(k|k)^{-1}$, and $\mathcal{I}(k|k-1)=\Sigma(k|k-1)^{-1}$ as the corresponding Information Matrices (IM) for the priori and posteriori state estimates, respectively. Using the IM, the expression (3) together with the OG in (2) can be written as

where, $Y(k)=H^{\top}R^{-1}H$. We further introduce variables $y,\hat{y}$ as Information Vectors (IV), such that $y(k)\!=\!H^{\top}R^{-1}z(k)$, and $\hat{y}(k|k)=\mathcal{I}(k|k)\hat{x}(k|k)$. We then have

The recursive expressions for the OG and IM, in the IF framework, can therefore be derived as

where, $M(k)=(A(k)^{-1})^{\top}\mathcal{I}(k|k)A(k)^{-1}$, $C(k)=M(k)(M(k)+Q^{-1})^{-1}$. In case $A(k)$ gets close to singularity, its inverse can be replaced by the pseudo-inverse [5].

At the state estimator, the difference $\mathcal{I}_{i}(k|k)-\mathcal{I}_{i}(k|k-1)$ is computed and the IM is then updated. Similarly, the corresponding IV is updated by computing $\hat{y}_{i}(k|k)-\hat{y}_{i}(k|k-1)$ $\forall i\in[n]$. Each filter node provides its corresponding prior and posterior IM, i.e., $\mathcal{I}_{i}(k|k)$ and $\mathcal{I}_{i}(k|k-1)$, and state estimates, i.e., $\hat{x}_{i}(k|k)$ and $\hat{x}_{i}(k|k-1)$. This information is forwarded to the system estimator that produces the following outputs for the overall set of state estimates:

In this section, we present the modifications required for the DKF in the presence of stochastic delays. Before proceeding, we assume that the open-loop stochastic LTV system in (1) is asymptotically stable in the mean-square sense. This requires $Q$ to be finite and the state transition matrix, $\Phi(k,0)\triangleq A(k-1)A(k-2)\dots A(0)\rightarrow 0$, $k\rightarrow{\infty}$ .

If there exists a delay $\tau:N-\tau>0$, where $N$ is some time index; $\hat{x}_{i}(k|k):k\in\{N-1,\ldots,N-\tau+1\}$ is unavailable from a particular filter node $i$, leading to

(for illustration, we assume that $\tau=3$):

We observe a recursive framework tracing backwards to the estimates at time instant $k-\tau$.

In a networked dynamical system with a large number of nodes, the classical tests for observability that include calculating the Grammian matrix would become easily complicated when scale grows. This motivates us to look into the concept of “structural observability” to identify the number of nodes to guarantee the system observability. Initially developed in [15] for structural controllability, the theory of duality enables us to extend this concept to structural observability. Next, we briefly outline the concept of structural observability, mainly relying on the results in [16].

The structured pair of matrices ($\bar{A},\bar{H}$) is said to be structurally observable if and only if a numerical realization $(A,H)$ is observable. Structurally observable pair of matrices is observable in a wide range of parameters except in a narrow range that renders the system unobservable. Clearly, structural observability is only a necessary condition for the system to be observable in Kalman’s sense and does not prove sufficiency. In [17] it is proven that the system cannot be structurally controllable, if in the associated state-control input graph (this graph is formed by taking the states and control inputs as the vertices, wherein a directed edge between the two states $x_{j}$ and $x_{i}$ exists if $a_{ij}$ is nonzero, and a directed edge between the control input $u_{j}$ and state $x_{i}$ exists if $b_{ij}$ is nonzero), one cannot reach all the states, starting from any of the control input nodes. By the theory of duality, we can extend the above result to define structural observability. The pair $(A,H)$ is said to be structurally observable if and only if, in the respective graph formed by $A^{\top}$ and $H^{\top}$, every state $x_{i}$ can be reached from the outputs $y_{j}$ and no dilation exists. This result also extends to linear time varying systems, as shown in [18].

A comprehensive stability analysis of DKF for LTI systems is presented in [14]. In this section, we extend that result by conducting a rigorous analysis of the stability for LTV systems using DKF and prove that the estimation error $x_{i}(k)-\hat{x}_{i}(k|k)$ remains bounded in the mean square sense. Before stating the main result, we state the following lemma.

Theorem 1 proves that the true error covariance resulting from the DKF Algorithm is upper bounded by the inverse of the corresponding IM (which is the inverse of the predicted error covariance). To prove the boundedness of the true error covariance, it would therefore be sufficient to show that individual IMs are lower bounded by a positive definite matrix. Having this, we now state the main stability result.

Although the authors prove stability for a consensus based DKF algorithm in [14], the stability result holds regardless of the number of consensus steps. Clearly, from Theorems 1 and 2, one can see that the estimation error is asymptotically bounded in mean square and the DKF algorithm generates estimates with bounded mean square errors for LTV systems. Theorems 1 and 2, in their present form can only be applied for systems in which $A(k)$ is invertible for all $k$. Extension to the cases in which $A(k)$ is not invertible should be investigated further.

Typically a higher number of sensor nodes results in more accurate state estimates in an ideal scenario, however, it is not always true in the presence of delays. Simulations show that (Fig. 2) in the presence of delays, the accuracy of estimates may decrease when the number of sensor nodes increases. To mitigate the effects of delay, we propose to select a subset of sensor nodes such that the compromising effect of delays is reduced by lowering the number of sensors used for estimation, while the selected sensors render a satisfactory estimation accuracy. We measure the influence of delays on the state estimates by determining the relative Maximum Deviations (MD), defined as follows:

We measure the influence of the measurement noise using the Mean Squared Error (MSE) between the ideal and observed estimates, $x$ and $\hat{x}$, respectively. We compute the MSE from the time where the state variables settle to $1\%$ of their final values. Intuitively, the influence of measurement noise can be best noticed (independent of the influence of delays) when the system settles down to its equilibrium point.

In Algorithm 1, we perform a greedy search for the best subset of nodes such that the influence of delays and measurement noise is reduced while maintaining a desired state estimate accuracy level. In each iteration, the subset selection algorithm includes all those sensor nodes with possessing delays and measurement noise variances beneath their corresponding thresholds. The search iterations start from the highest possible values for $R$ and $\tau$. In each iteration, all the possible sensor nodes with lower measurement noise variances and delays are selected, based on those the DKF is implemented. The values of $R_{m}$ and $\tau_{m}$ in Algorithm 1 correspond to the largest measurement noise covariance and delay, respectively.

We now propose a novel subset selection method to choose a subset of sensor nodes from the available nodes to ensure the estimation error covariance matrix remains bounded asymptotically, i.e., when $k\to\infty$. The advantage of this subset selection method is that, unlike the greedy algorithm, there is no need to be aware of the true state values, while the selected subset of nodes ensures the stability of the DKF in the presence of delays and noises in the collected measurements. The pseudo-code for subset selection based on stability is shown in Algorithm 2.

In Algorithm 2, $n$ is the total number of sensor nodes, $N$ is the total number of time samples, $\bar{k}\in[0,N]$ is the time instance from which the stability bound in Theorem 2 is satisfied, $\mathcal{T}=[\tau_{1}\quad\tau_{2}\quad\dots\quad\tau_{n}]^{\top}\in\mathbb{R}^{n}$ is a vector that contains the delays associated with the nodes, and $T_{s}\in\mathbb{R}^{+}$ is the sampling time of the filter. The essence of the algorithm is to calculate the value of the bounding matrix $\widetilde{\mathcal{I}}(k)$ and pick nodes $i$ such that the delayed IM associated with that node, i.e., ${I}_{i}(k-\frac{\tau_{i}}{T_{s}}|k-\frac{\tau_{i}}{T_{s}})$, is larger than the bound described in (16), $\forall k$. The algorithm includes only these nodes in the final calculation of estimates by the estimator.

Algorithm 2 always returns a set of nodes that satisfies the stability criterion in (16), $\forall k>\bar{k},k\leq N$ and thus the order of the evaluation of the nodes by the algorithm does not affect its output. The algorithm fails to return a non-empty set of nodes only when none of the nodes satisfies the stability criterion for the specified time horizon. This occurs when the delays are larger than the estimation horizon $N$.