Stability of FFLS-based Diffusion Adaptive Filter Under Cooperative Excitation Condition

Owing to the capability to process the collaborative data, wireless sensor networks (WSNs) have attracted increasing research attention in diverse areas, including consensus seeking [1][2], resource allocation [3][4], and formation control [5][6]. How to design the distributed adaptive estimation and filtering algorithms to cooperatively estimate unknown parameters has become one of the most important research topics. Compared with centralized estimation algorithms where a fusion center is needed to collect and process information measured by all sensors, the distributed ones can estimate or track an unknown parameter process of interest cooperatively by using local noisy measurements. Therefore, the distributed algorithms are easier to be implemented because of their robustness to network link failure, privacy protection, and reduction on communication and computation costs.

Based on classical estimation algorithms and typical distributed strategies such as the incremental, diffusion and consensus, a number of distributed adaptive estimation or filtering algorithms have been investigated (cf., [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19]), e.g., the consensus-based least mean squares (LMS), the diffusion Kalman filter (KF), the diffusion least squares (LS), the incremental LMS, the combination of diffusion and consensus stochastic gradient (SG), the diffusion forgetting factor least squares (FFLS). Performance analysis of the distributed algorithms is also studied under some information conditions. For deterministic signals or deterministic system matrices, Battistelli and Chisci in [7] provided the mean-square boundedness of the state estimation error of the distributed Kalman filter algorithm under a collectively observable condition. Chen et al. in [8] studied the convergence of distributed adaptive identification algorithm under a cooperative persistent excitation (PE) condition. Javed et al. in [9] presented stability analysis of the cooperative gradient algorithm for the deterministic regression vectors satisfying a cooperative PE condition. Note that the signals are often random since they are generated from dynamic systems affected by noises. For the random regression vector case, Barani et al. in [10] studied the convergence of distributed stochastic gradient descent algorithm with independent and identically distributed (i.i.d.) signals. Schizas et al. in [11] provided the stability analysis of a distributed LMS-type adaptive algorithm under the strictly stationary and ergodic regression vectors. Zhang et al. in [12] studied the mean square performance of a diffusion FFLS algorithm with independent input signals. Takahashi et al. in [13] established the performance analysis of the diffusion LMS algorithm for i.i.d. regression vectors. Lei and Chen in [14] established the convergence analysis of the distributed stochastic approximation algorithm with ergodic system signals. Mateos and Giannakis in [15] presented the stability and performance analysis of the distributed FFLS algorithm under the spatio-temporally white regression vectors condition.

We remark that most theoretical results mentioned in the above literature were established by requiring regression vectors to be either deterministic and satisfy PE conditions, or random but satisfy independency, stationarity and ergodicity conditions. In fact, the observed data are often random and hard to satisfy the above statistical assumptions, since they are generated by complex dynamic systems where feedback loops inevitably exist (cf., [20]). The main difficulty in performance analysis of distributed algorithms is to analyze the product of random matrices involved in estimation error equations. In order to relax the above stringent conditions on random regression vectors, some progress has been made on distributed adaptive estimation and filtering algorithms under undirected graphs. For estimating time-invariant parameters, the convergence analysis of distributed SG algorithm and distributed LS algorithm is provided in [21] and [22] under cooperative excitation conditions. For tracking a time-varying parameter, Xie and Guo in [16] and [23] proposed the weakest possible cooperative information conditions to guarantee the stability and performance of consensus-based and diffusion-based LMS algorithms. Compared with LMS algorithm, FFLS algorithm can generate more accurate estimates in the transient phase (see e.g.,[24]), and the stability analysis for the distributed FFLS algorithm is still lacking. In this paper, we focus on the design and stability analysis of distributed FFLS algorithm without relying on the independency, stationarity or ergodicity assumptions on regression vectors.

The information exchange between sensors is an important factor for the performance of distributed estimation algorithms, and previous studies often assume that the networks are undirected and time-invariant. In practice, they might not be bidirectional or time-invariant due to the heterogeneity of sensors and signal losses caused by the temporary deterioration in the communication link. One approach is to model the networks which randomly change over time as an i.i.d. process, see e.g., [25, 26]. However, the loss of connection usually occurs with correlations [27]. Another approach is to model the random switching process as a Markov chain whose states correspond to possible communication topologies, see [27, 28, 29, 30] among many others. Some studies on the distributed algorithms with deterministic or temporally independent measurement matrix under Markovian switching topologies are given in e.g.,[31, 32].

In this paper, we consider the distributed filtering problem over sensor networks where all sensors aim at collectively tracking an unknown randomly time-varying parameter vector. Based on the fact that recent observation data respond to the parameter changes faster than the early data, we introduce a forgetting factor into the local accumulative cost function formulated as a linear combination of local estimation errors between the observation signals and the prediction signals. By minimizing the local cost function, we propose the distributed FFLS algorithm based on the diffusion strategy over the fixed undirected graph. The stability analysis of the distributed FFLS algorithm is provided under a cooperative excitation condition. Moreover, we generalize the theoretical results to the case of Markovian switching directed sensor networks. The key difference from the fixed undirected graph case is that the adjacency matrix is an asymmetric random matrix. We employ the Markov chain theory to deal with the coupled relationship between random adjacency matrices and random regression vectors. The main contributions of this paper can be summarized as the following aspects:

• •

  In comparison with [16] and [21], the main difficulty is that the random matrices in the error equation of the diffusion FFLS algorithm are not symmetric and the adaptive gain is no longer a scalar. We establish the exponential stability of the homogeneous part of the estimation error equation and the bound of the tracking error by virtue of the specific structure of the proposed diffusion FFLS algorithm and stability theory of stochastic dynamic systems.

• •

  Different from the theoretical results of distributed FFLS algorithms in [12] and [15] where regression vectors are required to satisfy the independent or spatio-temporally uncorrelated assumptions, our theoretical analysis is obtained without relying on such stringent conditions, which makes it possible to be applied to the stochastic feedback systems.

• •

  The cooperative excitation condition introduced in this paper is a temporal and spatial union information condition on the random regression vectors, which can reveal the cooperative effect of multiple sensors in a certain sense, i.e., the whole sensor network can cooperatively finish the estimation task, even if any individual sensor cannot due to lack of necessary information.

The remainder of this paper is organized as follows. In Section 2, we give the problem formulation of this paper. Section 3 presents the distributed FFLS algorithm. The stability of the proposed algorithm under fixed undirected graph and Markovian switching directed graphs are given in Section 4 and Section 5, respectively. Finally, we conclude the paper with some remarks in Section 6.

Tracking a time-varying signal is a fundamental problem in system identification and signal processing. The well-known recursive least squares estimator with a constant forgetting factor $\alpha\in(0,1)$ is often used to track time-varying parameters, which is defined by

With some simple manipulations using the matrix inversion formula, we can obtain the following recursive FFLS algorithm (Algorithm 1) for an individual sensor.

However, due to the limited sensing ability of each sensor, it is often the case where the measurements obtained by each sensor can only reflect partial information of the unknown parameter. In such a case, if only local measurements of the sensor itself are utilized to perform the estimation task (see Algorithm 1), then at most part of the unknown parameter rather than the whole vector can be estimated. Thus, in this paper, we aim at designing a distributed adaptive estimation algorithm such that all sensors cooperatively track the unknown time-varying parameter $\bm{\theta}_{t}$ by using random regression vectors and the observation signals from its neighbors. To simplify the analysis, in this section, we use a fixed undirected graph $\mathcal{G}=(\mathcal{V},\mathcal{E},\mathcal{A})$ to model the communication topology of $n$ sensors.

We first introduce the following local cost function $\sigma_{t+1,i}(\bm{\beta})$ for each sensor $i$ at the time instant $t\geq 0$ recursively formulated as a linear combination of its neighbors’ local estimation error between the observation signal and the prediction signal,

with $\sigma_{0,i}(\bm{\beta})=0$. Set

Hence by (4), we have

which implies that

where $a^{(t+1-k)}_{ij}$ is the $i$-th row, $j$-th column entry of the matrix $\mathcal{A}^{t+1-k}$.

By minimizing the local cost function $\sigma_{t+1,i}(\bm{\beta})$ in (5), we obtain the distributed FFLS estimate $\bm{\hat{\theta}}_{t+1,i}$ of the unknown time-varying parameter for sensor $i$, i.e.,

Denote $\bm{P}_{t+1,i}=\left(\sum^{n}_{j=1}\sum^{t}_{k=0}\alpha^{t-k}a^{(t+1-k)}_{ij}\bm{\varphi}_{k,j}\bm{\varphi}^{T}_{k,j}\right)^{-1}.$ Then we write it into the following recursive form,

By (6), we similarly have

Note that in the above derivation, we assume that the matrix $\sum^{n}_{j=1}\sum^{t}_{k=0}\alpha^{t-k}a^{(t+1-k)}_{ij}\bm{\varphi}_{k,j}\bm{\varphi}^{T}_{k,j}$ is invertible which is usually not satisfied for small $t$. To solve this problem, we take the initial matrix $\bm{P}_{0,i}$ to be positive definite. Then (7) can be modified into the following equation,

Though, the estimate given by (8) has a slight difference with (6), which does not affect the analysis of the asymptotic properties of the estimates.

To design the distributed algorithm, we denote

By Lemma 2.1, we have $\bm{\bar{P}}_{t+1,i}=\frac{1}{\alpha}(\bm{P}_{t,i}-\frac{\bm{P}_{t,i}\bm{\varphi}_{t,i}\bm{\varphi}^{T}_{t,i}\bm{P}_{t,i}}{\alpha+\bm{\varphi}^{T}_{t,i}\bm{P}_{t,i}\bm{\varphi}_{t,i}})$. Hence,

Therefore, we get the following distributed FFLS algorithm of diffusion type, i.e., Algorithm 2.

Note that when $\mathcal{A}=\bm{I}_{n}$, the distributed FFLS algorithm will degenerate to the classical FFLS (i.e., Algorithm 1), and when $\alpha=1$, the distributed FFLS algorithm will degenerate to the distributed LS in [22] which is used to estimate the time-invariant parameter. The quantity $1-\alpha$ is usually referred to as the speed of adaption. Intuitively, when the parameter process $\{\bm{\theta}_{t}\}$ is slowly time-varying, the adaptation speed should also be slow (i.e., $\alpha$ is large). The purpose of this paper is to establish the stability of the above diffusion FFLS-based adaptive filter without independence or stationarity assumptions on random regression vector $\{\bm{\varphi}_{t,i}\}$.

In order to analyze the distributed FFLS algorithm, we need to derive the estimation error equation. Denote $\bm{\widetilde{\theta}}_{t,i}\triangleq\bm{\theta}_{t}-\bm{\hat{\theta}}_{t,i}$, then from (13) and (14), we have

By (1), (2), (11) and (12), we can obtain the following equation,

For convenience of analysis, we introduce the following set of notations,

Hence by (15) and (16), we have the following equation about estimation error,

From (17), we see that the properties of product of random matrices, i.e., $\prod_{t}\alpha\bm{P}_{t+1}\mathscr{A}\bm{P}^{-1}_{t}$, play important roles in stability analysis of the homogeneous part in error equation.

As we all know, the analysis of product of random matrices is generally a difficult mathematical problem if the random matrices do not satisfy the independency or stationarity assumptions. There is some work to study this problem, which focuses on either symmetric random matrix or scalar gain case. For example, [21] and [16] investigated the convergence of consensus-diffusion SG algorithm and the stability of consensus normalized LMS algorithm where the random matrices in error equations are symmetric. Note that the random matrices $\alpha\bm{P}_{t+1}\mathscr{A}\bm{P}^{-1}_{t}$ here are asymmetric. Although [23] studied the properties of the asymmetric random matrices in the LMS-based estimation error equation, the adaptive gain of distributed LMS algorithm in [23] is a scalar while the gain $\frac{\bm{P}_{t,i}}{\alpha+\bm{\varphi}^{T}_{t,i}\bm{P}_{t,i}\bm{\varphi}_{t,i}}$ in (11) of this paper is a random matrix. Hence the methods used in existing literature including [16, 21, 23] are no longer applicable to our case. One of the main purposes of this paper is to overcome the above difficulties by using both the specific structure of the diffusion FFLS and some results of FFLS on single sensor case (see [34]).

In this section, we will establish exponential stability for the homogeneous part of the error equation (17) and the tracking error bounds for the proposed distributed FFLS algorithm in Algorithm 2 without requiring statistical independence on the system signals. For this purpose, we need to introduce some definitions on the stability of random matrices (see [34]) and assumptions on the graph and random regression vectors.