Problem Setup and Motivation

The broader problem that one would like to solve can be cast as causal estimation of a random process{x_n} using a quantized version,{q_n}, of the associated measure- ment process {yn}. The encoding/quantization of {yn} into {qn} is determined by the information available at the encoder/observer at each time. We will limit our

attention to Gaussian state-space models, i.e., we consider the following system xn+1 =Fnxn+G1wn+G2un (2.1a)

yn=Hnxn+vn (2.1b)

where x_n ∈ R^d is the state, y_n ∈ R is the observation, and w_n ∈ R^p and v_n ∈ R are uncorrelated Gaussian white noises with zero means and covariances W and R, respectively. The initial state, x₀, of the system, is also a zero mean Gaussian with covarianceP0 and is uncorrelated with bothwnandvn. un is the control input, which is set to 0 whenever we consider open-loop estimation. For a given sequence of control inputs {un}, the minimum mean-squared error estimate of xn given y0:n, which we denote with ˆx^kf_n|n, can be computed recursively using the following Kalman filtering equations (e.g., [50])

ˆ

x^kf_n+1|n+1 = ˆx^kf_n+1|n+F_nP_n|n−1^kf H_n^T

H_nP_n|n−1^kf H_n^T +R−1

y_n−H_nxˆ^kf_n+1|n

(2.2a) ˆ

x^kf_n+1|n=F_nxˆ^kf_n|n+G₂u_n, xˆ^kf_0|−1 = 0 (2.2b) P_n+1|n^kf =FnP_n|n−1^kf F_n^T +W −FnP_n|n−1^kf H_n^T

HnP_n|n−1^kf H_n^T +R −1

HnP_n|n−1^kf F_n^T (2.2c) and P_0|−1^kf =P₀.

2.2.1 Motivation

In classical LQG control, the controller is colocated with the observer and hence, at each time n, has access to y0:n, i.e., all uncoded measurements up to time n.

The controller’s goal is then to determine the optimal control law u_n to minimize a given quadratic cost function. This problem is well understood. Increasingly many modern control systems employ multiple sensors and actuators that are not colocated.

Towards addressing this paradigm, there has been considerable amount of work on estimation and control under communication constraints, a representative sample

being [15, 66, 71, 86, 100, 114]. Here, the observer and the controller are separated by a communication channel. Hence the observer causally quantizes the measurements y_0:n to obtain q_n which is suitably encoded and communicated over the channel at time n.

Sensor networks provide a slightly different setting. A salient feature of [15, 66, 71, 86, 100, 114] is the presence of a single observer in the system that has access to all the uncoded measurements y_0:n. However, in sensor networks, each sensor acts as an observer. A time n, according to a given schedule, a particular sensor makes a measurement y_n, appropriately quantizes it to q_n and communicates it to the fusion center. Note that different sensors could use different measurement matrices. So, in general the measurement matrix H_n can vary with time. The fusion center uses the received quantized measurements q_0:n to estimate the state x_n. Figure 2.1 outlines the overall filtering paradigm¹. It is assumed that the sensors do not communicate between themselves. So, the quantized measurement q_n will be a function of the sensor’s own analog measurement y_n and potential feedback from the fusion center.

Unlike the classical case, there is no single entity in the network that has access to all the analog measurements y_0:n. Also, when a control input u_n is to be applied to the state-space process x_n, it is assumed that the fusion center determines u_n and applies the control input. So, we consider the setting where sensing takes place in a distributed manner but the controller is centralized.

In both cases above, the controller/fusion center needs to estimate the state using quantized measurements. Due to energy and bandwidth limitations, sensor networks provide a more compelling case for developing estimation algorithms using coarsely quantized measurements. Through most of the chapter, we focus only on estimation.

Except in Section 2.6, where we study the separation between estimation and control, the control input u_n in (2.1) is assumed to be absent, i.e., u_n= 0.

1Here, we assume that the sensor communicates with the fusion center using a discrete rate- limited noiseless channel.

+ +

+

S1 S2 . . . Sl

Fusion Center v_n

qn

Feedback xn+1=F xn+G1wn

Hnxn

yn=Hnxn+vn

Figure 2.1: WSN with a fusion center: The sensors act as data gathering devices.

Si denotes the i^th sensor and in the above figure, S` is making the n^th measurement using the measurement matrix H_n.

2.2.2 Quantized Innovations and the Gaussian Assumption

A popular quantization scheme proposed for sensor networks is ‘quantized innova- tions’. In this scheme, at each time n, the scheduled sensor makes the measure- ment yn and also receives feedback from the fusion center in the form of a predic- tion ˆyn|n−1 = Ey_n|q0:n−1. The sensor then quantizes its analog measurement y_n as qn = g(yn−yˆn|n−1) for some fixed finite quantizer g(·). Under the simplifying as- sumption that the prior x_n|q0:n−1 is Gaussian, filtering equations of the following form have been obtained for ˆxn|n,Exn|q0:n−1 in [80, 113].

ˆ

xn|n = ˆxn|n−1+L(qn) P_nH_n^T (H_nP_nH_n^T +R)^1/2 ˆ

xn+1|n=Fnxˆn|n

Pn|n=P_n−λ P_nH_n^TH_nP_n

H_nP_nH_n^T +R (2.3a)

Pn+1 ,Pn+1|n=FnPn|nF_n^T +G1W G^T₁ (2.3b) The value of λ and the mapping L(q_n) depend on the quantization scheme used and are detailed in [113]. In particular, if q_n = sign y_n−yˆ_n|n−1

, λ = _π² and L(q_n) =

q2

πq_n. Eqs. (2.3a) and (2.3b) constitute the MLQ-Riccati with parameter λ. The above filter is optimal if the conditional distribution, p(x_n|q0:n−1), is Gaussian, which we will prove is generally a bad approximation. [98, 99] provide examples where the error performance of the filters in [80, 113] do not track the MLQ-Riccati that they were predicted to, i.e., Eq. (2.3). In order to understand the problem better, we take a closer look at the conditional law of x_n|q_0:n in the following section. When {x_n} and {y_n} are jointly Gaussian, we will provide a novel stochastic characterization of x_n causally conditioned on the quantized measurement process {q_n}. This, in turn, allows us to identify the conditional density of x_n|q_0:n to be, what we refer to as, a generalized closed skew normal distribution. We also use it to propose a novel filtering technique for the above problem which reduces to an elegant particle filter when {x_n} and {y_n} have linear state-space structure and outperforms the filters proposed in [80, 113], while providing much needed theoretical insight into the problem. Although the present work is motivated by sensor network applications, the results obtained are quite general as will become evident.

A note about the subscripts in F_n andH_n: In order to reduce notational clutter, in the rest of the chapter, we will drop the subscripts and just write F and H. In other words, we will present all results for the ‘time invariant’ case. The corresponding time varying versions can be obtained by simply replacing F (H) with F_n (H_n) wherever needed. The only exception to this rule is Corollary 2.4 which is applicable only to the time invariant case.

2.3 A Stochastic Characterization of the Condi-