Discrete Dynamics in Nature and Society

Volume 2015 (2015), Article ID 576341, 9 pages

http://dx.doi.org/10.1155/2015/576341

## Consensus Information Filtering for Large-Scale Systems with Application to Heat Conduction Process

^{1}School of Electronic and Control Engineering, Beijing University of Technology, Beijing 100124, China^{2}Key Laboratory of Computational Intelligence and Intelligent Systems, Beijing 100124, China^{3}School of Science, Communication University of China, Beijing 100024, China

Received 20 October 2015; Revised 9 November 2015; Accepted 17 November 2015

Academic Editor: Deqing Huang

Copyright © 2015 Liguo Zhang and Ying Lyu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

For large-scale distributed systems, the evolution of system dynamics is dominated by current states and boundary conditions simultaneously. This work describes a distributed consensus filtering for a class of large-scale distributed systems with unknown boundary conditions, which are monitored by a set of sensors. Because of the difference of spatial positions among the sensor network, only the single state variables or both the states and outside input jointly could be estimates with Kalman information filtering, respectively. On diffusion processing, we fuse the common state estimations of the local information filters using consensus averaging algorithms and algebraic graph theory. Stability and performance analysis is provided for this distributed filtering algorithm. Finally, we consider an application of distributed estimation to a heat conduction process. The performance of the proposed distributed algorithm is compared to the centralized Kalman filtering.

#### 1. Introduction

Large-scale systems have been studied extensively in the cutting-edge technology-based social sectors, ranging from ocean, atmospheric, and hydrological sciences [1, 2] and oil reservoir simulations [3] to intelligent transportation systems [4–6]. The large-scale systems are usually formulated with a set of partial differential equations to express their distributed, multiparameter coupled dynamical behaviors. In recent years, more attention has been given to cyber-physical systems, in which the physical plants usually are described with large-scale systems [1, 4, 7].

State estimate of large-scale systems is a challenging task, since the evolution of the large-scale systems usually codominated by the current system states and the outside boundary conditions. The troublesome problem is the boundary conditions are often unknown or inaccessible in advance in some applications. Two kinds of approaches have been developed in recent years to solve this problem. One method assumes that the boundary conditions are random walking variables and then estimates these variables as extended system states [8]. Another method treats these boundary conditions as outside system input and estimates system states and unknown input simultaneously based on the iterating two-step or multistep Kalman filtering [9, 10].

In this work, we present a distributed simultaneous state and input estimate method for a class of large-scale systems with unknown boundary conditions which are monitored by a sensor network. Based on the spatial positions of sensor nodes, the large-scale system is firstly decomposed into low-dimensional subsystems. If the sensor nodes are located at the boundary grids, information filtering for simultaneous state and input estimate is developed. Then consensus strategy fuses state estimations which are common among the local information filters using the consensus averaging algorithms [11].

The aims of this paper are (1) to decompose the large-scale distributed parameter system into spatial subsystems with reduced observation models; locations of the sensor group throughout the spatial domain significantly affect the outcome and quality of state estimation; we model sensors placed in different locations with different output operators, (2) to develop the information filtering algorithm for simultaneous state and boundary condition estimation for local sensor groups and design consensus estimator for every sensor so that the overall state can be estimated via local exchange of messages among neighboring nodes, and (3) to present a stability analysis distributed Kalman consensus filtering algorithm.

The subsequent sections are organized as follows. Some background on centralized and decentralized observation models is provided in Section 2. State-space partition of large-scale system is formulated in Section 3. In Section 4, we present the information filtering for simultaneous state and input estimate. Distributed Kalman consensus filters for large-scale systems with unknown boundary conditions are introduced in Section 5.1. Stability analysis is provided in Section 5.2. Simulation results and performance comparisons for a 1D heat conduction process are presented in Section 6. Conclusions and future work are discussed in Section 7.

#### 2. Problem Statement

##### 2.1. Spatially Distributed Processes

Consider the spatially distributed physical process represented by the following convection-diffusion equation:where , , and is the length of one spatial dimension. is the given outside environment, is the small diffusion coefficient, and is the convection coefficient.

Assume that the initial conditions of (1) are given by where is a given function. And the boundary conditions are imposed on both edges as where , are outside input functions.

When the initial and boundary conditions (2)-(3) are compatible, (1) is well posed in the sense that it has a unique solution .

##### 2.2. Spatiotemporal Discretization

The above PDEs could be solved numerically on a spatial uniformed mesh grid with the finite-difference discretization method. The central approximations of the first- and the second-order derivative are, respectively, where is the value of the random field at the th location of the spatial grid and is the length of the spatial grid.

Temporal discretization approximates the time derivative of PDEs (1) by using the forward Euler method. For those boundary grids whose outer grids are missing, the temporally discretized boundary conditions , are represented as discrete variable , where , is the discrete-time instant.

In general, the spatiotemporal discretization of the PDEs (1) is formulated by the following discrete-time state-space model: where the state vector is the collection of the sorted variables , by, for example, using the lexicographic ordering. Unknown input is the discretized boundary condition, and is the model approximation error assumed to be mutually uncorrelated, zero-mean, white random signals with known covariance matric . , are system state matrix and boundary input matrix, respectively.

*Remark 1. *For the system matrix of the discrete-time model (6), nonzero element means that the state representing the value of the spatial grid is affected directly by the outside boundary input ; otherwise, zero element of means that represents the inside grid.

##### 2.3. Distributed Observation Model

We assume that system (6) is monitored by a network of sensors, in which the boundary information is not accessible directly. Then the local observations at sensor at time are where is the local observation matrix, is the number of simultaneous observations made by sensor at time , and is the local observation noise assumed to be mutually uncorrelated, zero-mean, white random signals with known covariance matric .

*Remark 2. *The local measurement may be temperature or some other physical attribute at local sensor . The observation matrix is particular to the spatial position (corresponding to the nonzero element in ) of the local sensor , in the uniformed mesh grid.

We collect all observations in the network to get the global observation model. Let be the total number of observations at all sensors. Let the global observation vector, , the global observation matrix, , and the global observation noise vector, , be

Then the global observation model is given by Since the observation noises at the different sensors are independent, we can combine the local observation noise covariance matrices at each sensor into the global observation noise covariance matrix, , as

For large-scale systems with unknown boundary conditions, centralized and distributed filtering for simultaneous input and state estimation are discussed, respectively. No prior information about the unknown boundary input, , in the system model (7) is available or accessible in advance.

#### 3. Partitioned Large-Scale Systems

In this section, we partition the large-scale system into subsystems based on the different spatial positions of the local sensors. Each sensor has the capability to generate its own estimate using locally available measurements.

*Case 1. *The local sensor is placed at the boundary grids of the spatiotemporal discretization model (6), . is the corresponding index set.

In this case, , . We select submatrices containing the elements taken from the full-order matrix with indices belonging to , where the index set , , , and , such that where is an -dimensional vector composed of the components of selected by .

Then, the local subsystem models available to the sensor can be defined as. In order to estimate all boundary inputs , we further assume that .

*Case 2. *The local sensor is placed at the inside grids of the spatiotemporal discretization model (6); .

In this case, the elements of matrix corresponding to the inside grids are zeroes; that is, . Then, we define the local submatrices as and extract the local subsystem models as

*Example 3. *Consider the one-dimensional distributed parameter heat conduction model on a rod. Spatial discretization results in the grid shown in Figure 1. A number of sensors are located on the specified grid points and each sensor measures the specified temperature. Unknown boundary conditions on the end of the rod are , and the boundary system matrix isWhen the sensor is located at the grid point 1, the local observation matrix should be . Then, is obtained as a submatrix including the first column of the full-order boundary matrix ; that is, We define the input as .

When the sensor location is at the grid point 2, or the other inside grids, the local observation matrix should be . In this case, , and we define . The subsystem models are not including outside input.