Objetivos específicos

To summarize, we have shown that making each link error-free in a wireless network is sub-optimal. Thus, a multihop approach, in which every relay node decodes the received message, is not necessarily the correct approach for all wireless networks. We have proposed a scheme for network operation that is of use in practical networks and in which operations performed by a node are restricted to decoding and forwarding – both of which are common operations performed in a network setting. We have suggested an algorithm that finds the optimum policy without exhaustive search over an exponential number of policies and also proposed a method to converge to the correct policy without having a central decision-making agent.

The algorithm of Section 4.7 can find the maximum rate and optimum policy for any Gaussian or wireless erasure network. In addition, the bounds presented in Section 4.11 give us some idea of what sort of optimal rates to expect. However, we still do not know what sort of policies are optimal in what ranges of erasure probabilities or SNR. The examples of Section 4.3 suggest that when the links are poor (high erasure probabilities or low SNR), it is better to decode. It would be interesting to know if this is true for general networks and what thresholds exist below which a certain operation is always preferred.

Also, Corollary 4.2 tells us that the algorithm returns the largest decoding set. Since decoding is the more costly of the two operations considered here, an algorithm that finds the smallest decoding set such that the maximum rate is obtained is of interest.

Finally, we note that in this chapter we considered only two types of operations. However, it is possible to imagine a larger set of operations and the optimal choice

of operation from among these. Finding practical schemes that improve upon the present algorithm is an interesting avenue for future work.

Chapter 5

Estimation over Wireless Erasure

Networks

5.1

Introduction

Recent advances in Micro-Electro-Mechanical Systems (MEMS) technology have pro- vided us with cheap, low power, customizable sensors capable of sensing, signal pro- cessing, and communication in wireless media (for university and industrial prototype of these sensors, see [64, 65]). These advances have given rise to an increasing number of applications for networks of sensors in different aspects of our life. As mentioned earlier in Chapter 1, examples of these applications appear in environmental monitor- ing, industrial, transportation, and home systems automation, control of distributed embedded systems (such as robots or UAVs), and even medical services [66, 67].

One important feature of these applications is that not in all of them the main objective is high-data rate communication between components of the network. Dif- ferent tasks such as distributed computation, detection, and control can be the main purpose for deploying these networks.

Given the increasing use of wireless sensor networks for different tasks other than data communication, a theoretical framework for analysis of the ultimate performance and the optimal schemes of operation for each of these tasks is required.

networked control systems in which components communicate over wireless links or communication networks that may also be used for transmitting other unrelated data (see, e.g., [68, 69] and the references therein). The estimation and control performance in such systems is severely affected by the properties of the communication channels. Communication links introduce many potentially detrimental phenomena, such as quantization error, random delays, data loss, and data corruption to name a few, that lead to performance degradation or even stability loss. As emergent applications in distributed control mature, these issues have gained a lot of focus from the community. In the previous chapters of this thesis, we looked at the performance of different classes of wireless networks for different network problems. In these problems the main objective is maximizing the reliable rate ofcommunication between the nodes of the network. In this chapter, we look at another task, namely control and estimation, over these networks. We are interested in the problem of estimation and control of a dynamical process across the wireless erasure network model introduced in Section 2.3. We consider dynamical process evolving in time that is being observed by a sensor. The sensor needs to transmit the data over a network to a sink (destination) node, which can either be an estimator or a controller. However, the links in the network stochastically erase packets.

Prior work in this area has focused on studying the effect of packet erasures by a single link in an estimation or control problem. Assuming certain statistical models for the packet erasure process, stability and control performance of such systems were analyzed in [70]-[73]. To counteract the degradation in performance, some approaches have been proposed in the literature [74]-[79]. In particular, in [76], a sub-optimal estimator and regulator is proposed that minimizes a quadratic cost. This approach was later extended by [77, 78]. [79] also considered the related problem of optimal estimation across a packet erasure link that erases packets in an independent and identically distributed (i.i.d.) fashion, and obtained bounds on the expected error

covariance.

Most of the above designs aimed at designing a packet-loss compensator. The compensator accepts those packets that the link successfully transmits and comes up with an estimate for the time steps when data is lost. If the estimator is used inside a control loop, the estimate is then used by the controller. A more general approach is to design both an encoder and a decoder for the communication link to counteract the effect of stochastic packet erasures. This was considered for the case of a single communication link in [80] and [81]. It was demonstrated that using encoders and decoders can improve both the stability margin and the performance of the system.

In this chapter, we consider the design of encoders and decoders for wireless era- sure network model introduced in Chapter 2. The optimal transmission strategy over general networks for the purpose of estimation and control is largely an open prob- lem. In [82], Tatikonda studied some issues related to quantization rates required for stability when data was being transmitted over a network of digital memoryless channels. Also relevant is the work of Robinson and Kumar [83], who considered the problem of optimal placement of the controller when the sensor and the actuator are connected via a series of communication links. They ignore the issue of delays over paths of different lengths (consisting of different number of links), and under a

Long Packet Assumption come up with the optimal controller structure. There are

two main reasons why the problem of encoding data for transmission is much more complicated in the case of transmission over a network:

• If the intermediate nodes are allowed to process data, it introduces an element of memory. The network is not equivalent to an erasure channel with probability of successful transmission as the reliability of the network.

• There are potentially many paths from the source to any node that offer data with varying amounts of delay.

We begin by proving a separation principle that allows us to separate the control problem into one of designing a state-feedback optimal controller, and another of transmitting information across unreliable links. This also allows us to identify the information that needs to be made available to the controller for achieving optimal performance. We then propose a simple recursive algorithm that ensures that this information is available to the controller. Even though the algorithm requires a constant amount of memory, transmission, and processing at any node, it is optimal for any packet erasure pattern and has many additional desirable properties that we illustrate. The analysis of the algorithm identifies a property of the network called the max-cut probability that is relevant for the purpose of stability of the control loop. We also provide a framework to analyze the performance of our algorithm. The main contributions of this chapter are as follows:

(a) We identify the optimal information processing strategy that should be followed by the nodes of the network to allow the sink to calculate the optimal estimate at every time step. This algorithm is optimal for any packet erasure process, yet requires a constant amount of memory, processing, and transmission by any node per time step. Due to a separation principle, the algorithm also solves the optimal control problem.

(b) We analyze the stability of the expected error covariance for this strategy when the packet erasure events are independent from one time step to the next and across channels. For any other scheme (e.g., transmitting measurements without any processing), our conditions are necessary for stability. For channels with correlated erasures, we show how to extend this analysis.

(c) We calculate the performance for our algorithm for channels that erase packets independently. We provide a mathematical framework for evaluating the per- formance for a general network and provide expressions for networks containing

links in series and parallel. We also provide lower and upper bounds for the performance over general networks. For any other strategy, these provide lower bounds for achievable performance.

Our results can also be used for synthesis of networks to improve estimation per- formance. We consider a simple example in which the optimal number of relay nodes to be placed is identified for estimation performance. We also consider optimal rout- ing of data in unicast networks.1 _{Simulation results are provided to illustrate the}

results.

The chapter is organized as follows. In the next section, we set up the problem and state the various assumptions. Then, we state a separation principle that allows us to focus on the optimal estimation problem. In Section 5.4 we identify a recursive yet optimal processing and transmission algorithm. We then specialize to the case of packet erasure events occurring in a memoryless fashion and independently across different links. We first do a stability analysis of the algorithm in Section 5.5 to obtain conditions on the packet erasure probabilities under which the estimate error at the sink retains a bounded covariance. Following that, in Section 5.6 we analyze the performance of the algorithm. We derive an expression for general networks and evaluate it explicitly for specific classes of networks. We also provide bounds for general networks. We then illustrate the results using some examples. Finally, we consider some extensions of the analysis by considering correlated erasures and using the results already derived for optimal routing in unicast networks and for network synthesis.

Figure 5.1: The set-up of the control across communication networks problem. For most of the discussion in the chapter, we will ignore the network between the controller and the actuator. See, however, Section 5.4.2.