## 1. Introduction

The convergence of sensing, computing, and communication in low cost, low power devices is enabling a revolution in the way we interact with the physical world. The technological advances in wireless communication make possible the integration of many devices allowing flexible, robust, and easily configurable systems of wireless sensor networks (WSNs). This chapter is devoted to the estimation problem in such networks.

Since sensor networks are usually large‐scale systems, centralization is difficult and costly due to large communication costs. Therefore, one must employ distributed or decentralized estimation techniques. Conventional decentralized estimation schemes involve all‐to‐all communication [1]. Distributed schemes seem to fit better. In this class of schemes, the system is divided into several smaller subsystems, each governed by a different agent, which may or may not share information with the rest. There exists a vast literature that study the distributed estimation for sensor networks in which the dynamics induced by the communication network (time‐varying delays and data losses mainly) are taken into account [2–10]. Millan et al. [6] have studied the distributed state estimation problem for a class of linear time‐invariant systems over sensor networks subject to network‐induced delays, which are assumed to have taken values in

One of the constraints is the network‐induced time delays, which can degrade the performance or even cause instability. Various methodologies have been proposed for modeling and stability analysis for network systems in the presence of network‐induced time delays and packet dropouts. The Markov chain can be effectively used to model the network‐induced time delays in sensor networks. In Ref. [11], the time delays of the networked control systems are modeled by using the Markov chains, and further an output feedback controller design method is proposed.

The rest of the chapter is organized as follows. In Section 2, we analyze the available delay information and formulate the observer design problem. In Section 3, the sufficient and necessary conditions to guarantee the stochastic stability are presented first and the equivalent LMI conditions with constraints are derived. Simulation examples are given to illustrate the effectiveness of the proposed method in Section 4.

*Notation*: Consider a network with

## 2. Problem formulation

Assume a sensor network intended to collectively estimate the state of a linear plant in a distributed way. Every observer computes a local estimation of the plant's states based on local measurements and the information received from neighboring nodes. Observers periodically collect some outputs of the plant and broadcast some information of their own estimation. The information is transmitted through the network, so network‐induced time delays and dropouts may occur.

In this work, the system to be observed is assumed to be an autonomous linear time‐invariant plant given by the following equations:

where

Besides the system's output

### 2.1. Delays modeled by Markov chains

The communication links between neighbors may be affected by delays and/or packet dropouts. The equivalent delay

The Markov chain is a discrete‐time stochastic process with Markov property. One way to model the delays is to use the finite state Markov chain as in Refs. [7–9]. The main advantages of the Markov model are that the dependencies between delays are taken into account since in real networks the current time delays are usually related to the previous delays [8]. In this note,

where

Remark 1: In the real network, the network‐induced delays are difficult to measure. Using the stochastic process to model the delays is more practical. For sensor networks, the communication link between different pairs of nodes is also different, so the data may experience different time delays. It is more reasonable to model the delays by different Markov chains.

### 2.2. Observation error system

The structure of the observers described in the following is inspired by that given in Ref. [6]. To estimate the state of the plant, every node is assumed to run an estimator of the plant's state as:

The observers’ dynamics are based on both local Luenberger‐like observers weighted with

The observation error of observer

Define

where

(8) |

Remark 2: The observation error system (Eq. (7)) depends on the delays

Definition 1 [7]: The system in Eq. (7) is stochastically stable if for every finite

## 3. Observers’ design

In this section, we first derive the sufficient and necessary conditions to guarantee the stochastic stability of system (Eq. (7)) with Definition 1. For ease of presentation, when the system's delays are

we denote

Theorem 1: Under the observer (Eqs. (4) and (5)), the observation error system (Eq. (7)) is stochastically stable if and only if there exists symmetric

(11) |

holds for all

Proof: *Sufficiency:* For the system Eq. (7), construct the Lyapunov function

Calculating the difference of

(13) |

Define

Then, Eq. (13) can be evaluated as

(14) |

Thus, if

where

Then we have

According to Definition 1, the observation error system Eq. (7) is stochastically stable.

*Necessity:* For necessity, we need to show that if the system Eq. (7) is stochastically stable, then there exists symmetric

(18) |

Define

Assuming that

(20) |

Since it is valid for any

From Eq. (20), we obtain

(22) |

The second term in Eq. (22) equals to

(23) |

Substituting Eq. (23) into Eq. (22) gives rise to

(24) |

Letting

As it is clearly seen from Eq. (11) that the matrix inequality to be solved in order to design the observers is nonlinear. To handle this, Proposition 1 gives the equivalent LMI conditions with nonconvex constraints. It can be solved by several existing iterative LMI algorithms. Product reduction algorithm in Ref. [10] is employed to solve the following conditions.

Proposition 1: There exist observers Eqs. (4) and (5) such that the observation error system Eq. (7) is stochastically stable if and only if there exists matrices

for all

(27) |

Proof: As we know

Since

## 4. Numerical example

Consider a plant whose dynamics is given by:

.Assume the network has two nodes, with two links, one is from node 1 to node 2, and the other is from node 2 to node 1. The matrices are given as follows:

The random delays are assumed to be

**Figure 1** shows part of the simulation run of the delay

By using Proposition 1, we design the observers with the following matrices:

The initial values of the plant and the observers are **Figure 2** represents the evolution of the plant's states (solid lines) and the estimated states (dashed lines) for observer 2. It is observed that the estimation of the observers converge to the plant's state.

## 5. Conclusion

This chapter addresses the problem of distributed estimation considering random network‐induced delays and packet dropouts. The delays are modeled by Markov chains. The observers are based on local Luenberger‐like observers and consensus terms to weight the information received from neighboring nodes. Then the resulting observation error system is a special discrete‐time jump linear system. The sufficient and necessary conditions for the stochastic stability of the observation error system are derived in the form of a set of LMIs with nonconvex constraints. Simulation examples verify its effectiveness.