Guaranteed convergence control for consensus of mobile sensor networks with dynamical topologies

Abstract

Mobile sensors need to reach consensus in many applications such as unmanned vehicles, multi-agent systems and environmental monitoring. In these scenarios, it is very important to reduce the power consumption as well as make sensors reach consensus as fast as possible. In this article, we propose a novel guaranteed convergence control algorithm to switch topologies of mobile sensors so that we can reduce power consumption in the sensor network and make the mobile sensors reach consensus with guaranteed convergence as well. The topology graphs over the mobile sensors are unconnected, directed, dynamical and switched periodically at any moment, while the joint graphs of dynamical topologies are connected in a bounded time period. The guaranteed convergence rate of consensus is derived on a method based on the variable decomposition. Some illustrative examples are provided to demonstrate the validity and effectiveness of the results.

Keywords

Consensus network topology switching control mobile sensor networks guaranteed convergence

Introduction

Mobile sensor networks have attracted much research attention in control theory field during recent years, which has been found in many applications such as industrial automation, health monitoring, environment and climate monitoring.^1,2 In these sceneries, deployment, rendezvous and consensus usually are needed in certain tasks of mobile sensor networks. Consensus seeking is a very interesting problem in control of mobile sensor networks.³ Consensus refers to the group behaviour that all of the agents asymptotically reach a certain common agreement through a local distributed protocol,⁴ which plays a very useful role in information fusion, deployment and optimization of mobile sensors.^5,6

The network topology of sensors plays a very important role in achieving consensus. The system’s performance and stability are affected by the communication networks. Sometimes, the network is deterministic, and sometimes it is time-dependent. Some network topology structures for consensus are investigated, in which the topology is time-varying or stochastic.^7–11 On the other hand, convergence speed is also an important problem, which determines how fast consensus can be achieved. The maximum convergence speed has been conducted by designing weights associated with the edges of topology graph.^12–14 The convergence rate of networked systems is derived.¹⁵

The current studies on consensus over network topology are as follows: (1) consensus protocol design for various mobile sensor systems;^5,6 (2) conditions for consensus over deterministic or dynamical topology;^16–20 and (3) network topology design to optimize the system’s performance.^10,11,21

Most of the above researches consider switching communication topologies. However, the energy limitation is not concerned in these researches. Due to limitation of batteries, mobile sensors have limited energy consumption in communication, computation and data processing. So it is very important for mobile sensor systems to lower the power consumption in communication topology networks of mobile sensors. To the best of our knowledge, the research, which considers both of dynamical topologies for saving energy and fast convergence speed for consensus, has not been considered yet.

In this article, we aim to investigate the control algorithm for consensus of mobile sensors with dynamical topologies and guaranteed convergence rate. In order to reduce the power in communication topologies, we make as less communication edges as possible to reduce the cost as well as to obtain the fastest convergence speed. On one hand, some directed and unconnected topologies may be chosen to make less communication links among agents at the same time. On the other hand, different topologies may be jointly connected to be a spanning tree in a finite time. Here, we consider the case that the network topology is directed and periodically connected. This is very practical in engineering application.

The remainder of this article is organized as follows. In section ‘Graph theory’, some basic notations of graph theory are introduced. Section ‘Convergence rate for periodically connected topologies’ describes the convergence rate of mobile sensor systems over switching topologies. An algorithm for calculating and maximizing the convergence speed is given in section ‘Guaranteed convergence switching control’. In section ‘Simulation examples’, some illustrative examples and simulations are provided. Finally, concluding remarks are made in section ‘Conclusion’.

Graph theory

Let $G = (V, E, A)$ be a weighted graph of order n with the set of vertexes ${v_{1}, v_{2}, \dots, v_{n}}$ , set of edges $E \subseteq V \times V$ and a weighted adjacency matrix $A = [w_{ij}]$ with non-negative adjacency elements $w_{ij}$ . The agent indices belong to a finite index set $I = {1, 2, \dots, n}$ . A directed path of G is denoted by $e_{ij} = (v_{i}, v_{j})$ which indicates that there is a path from agent j to i. The adjacency elements associated with the edges of the graph are positive, that is, $e_{ij} \in E \Leftrightarrow w_{ij} > 0$ ; otherwise, $w_{ij} = 0$ . Moreover, assume that there are no self-cycles, that is, $w_{ii} = 0$ for all $i \in I$ . The set of neighbours of agent $v_{i}$ is denoted by $N_{i} = {v_{j} \in V : (v_{i}, v_{j}) \in E}$ . The in-degree of agents are, respectively, defined as follows

\deg_{in} (v_{i}) = \sum_{j = 1}^{n} w_{ij}

(1)

Definition 1. Periodically connected.

Let $σ (t)$ be a switching signal satisfying $G_{σ (t)}$ directed graph. Here, $G_{σ (t)} \in {G_{0}, G_{1}, \dots, G_{m}}$ and m is bounded positive integer indicating the number of switching graphs in the finite time interval T. If the union of the graphs ${G_{0}, G_{1}, \dots, G_{m}}$ is a directed spanning tree in the periodical time interval T, then the graph is periodically connected.

Convergence rate for periodically connected topologies

In this section, we aim to derive the convergence rate of system (4). First, we define the dynamics of multi-agent systems and provide a different description for system (4) using the method of variable decomposition. The variable decomposition method for consensus was first presented.²² Second, we derive the convergence rate of system (4).

Dynamics of mobile sensor systems

For simplicity, we consider switching n sensors with dynamics of single integrators

{\overset{\cdot}{x}}_{i} = u_{i}, i \in {1, 2, \dots, n}

(2)

The consensus protocol is designed as follows

u_{i} = - \sum_{j \in N_{i}} w_{ij} (x_{i} (t) - x_{j} (t))

(3)

Here, $x_{i}$ denotes the state of sensor $v_{i}$ , which might represent physical variables including altitude, position, temperature, speed, voltage and so on. $u_{i}$ is the control input. The coefficient $w_{ij}$ is non-negative element of adjacency matrix associated with the graph G over mobile sensors defined in equation (1). $N_{i}$ is the set of neighbours of sensor $v_{i}$ defined in section ‘Graph theory’. Considering equations (2) and (3), the dynamics of mobile sensor systems can be expressed in a compact form as follows

\overset{\cdot}{x} = - L_{σ (t)} x

(4)

Here, L is the Laplacian of graph G, $σ (t)$ is the switching signals from one topology graph to another.

Model description

Let $α$ be a normalized vector in the space S spanned by vector $1_{n \times 1}$ , which represents the common value of the states of sensors. Then, we can get

L_{σ (t)} α = 0

Let $α_{⊥} \in R^{n \times (n - 1)}$ be the orthonormal complement of $α$ , $α_{⊥}^{T}$ be the transpose of $α_{⊥}$ . Then the following equations hold

α_{⊥}^{T} α_{⊥} = I_{n - 1}, α_{⊥}^{T} α = 0, α_{⊥} α_{⊥}^{T} + α α^{T} = I_{n}

(5)

and

\begin{matrix} α_{⊥}^{T} L_{σ (t)} = α_{⊥}^{T} L_{σ (t)} (α_{⊥} α_{⊥}^{T} + α α^{T}) \\ = α_{⊥}^{T} L_{σ (t)} α_{⊥} α_{⊥}^{T} \end{matrix}

(6)

Applying the property $L_{σ (t)} α = 0$ , we can decompose the state variable as follows

x = (\begin{matrix} α_{⊥} & α \end{matrix}) (\begin{matrix} θ \\ δ \end{matrix})

and system (4) can be rewritten as follows

\overset{\cdot}{x} = (\begin{matrix} α_{⊥} & α \end{matrix}) (\begin{matrix} \overset{\cdot}{θ} \\ \overset{\cdot}{δ} \end{matrix}) = - L_{σ (t)} (\begin{matrix} α_{⊥} & α \end{matrix}) (\begin{matrix} θ \\ δ \end{matrix})

Then, considering equation (5), we have

\begin{matrix} (\begin{matrix} \overset{\cdot}{θ} \\ \overset{\cdot}{δ} \end{matrix}) = (\begin{matrix} α_{⊥} & α \end{matrix})^{- 1} L_{σ (t)} (\begin{matrix} α_{⊥} & α \end{matrix}) (\begin{matrix} θ \\ δ \end{matrix}) \\ = - (\begin{matrix} α_{⊥}^{T} \\ α^{T} \end{matrix}) L_{σ (t)} (\begin{matrix} α_{⊥} & α \end{matrix}) (\begin{matrix} θ \\ δ \end{matrix}) \\ = - (\begin{matrix} α_{⊥}^{T} L_{σ (t)} α_{⊥} & α_{⊥}^{T} L_{σ (t)} α \\ α^{T} L_{σ (t)} α_{⊥} & α^{T} L_{σ (t)} α \end{matrix}) (\begin{matrix} θ \\ δ \end{matrix}) \end{matrix}

Noting that $L_{σ (t)} α = 0$

(\begin{matrix} \overset{\cdot}{θ} \\ \overset{\cdot}{δ} \end{matrix}) = - (\begin{matrix} α_{⊥}^{T} L_{σ (t)} α_{⊥} & 0 \\ α^{T} L_{σ (t)} α_{⊥} & 0 \end{matrix}) (\begin{matrix} θ \\ δ \end{matrix})

Here, $θ$ and $δ$ represent non-consensus and consensus components of mobile sensor systems, respectively. If the non-consensus parts of systems asymptotically stable, then the states of system tend to consensus. We will show that as follows. Consider the non-consensus components

\overset{\cdot}{θ} = α_{⊥}^{T} (- L_{σ (t)}) α_{⊥} θ

(7)

We note that

δ (t) = δ (0) + \int_{0}^{t} α^{T} L_{σ (t)} α_{⊥} e^{α_{⊥}^{T} L_{σ (t)} α_{⊥} τ} θ (0) d τ

lim_{t \to \infty} θ (t) = 0

then we have

x (t) = α_{⊥} θ (t) + α δ (t) \to α (δ (0) + \int_{0}^{\infty} α^{T} L_{σ (t)} α_{⊥} e^{α_{⊥}^{T} L_{σ (t)} α_{⊥} τ} θ (0) d τ)

Since $(δ (0) + \int_{0}^{\infty} α^{T} L_{σ (t)} α_{⊥} e^{α_{⊥}^{T} L_{σ (t)} α_{⊥} τ} θ (0) d τ)$ is bounded. Then, $x (t)$ will converge to a point in the space S. Thus, the convergence rate of system (4) to consensus is equivalent to convergence rate of equation (7) to zero.

Convergence rate

First, we give a description of a periodical connecting. Given a bounded time T, in this interval switching signal, $σ (t)$ , switching among m topologies which are jointly connected with Laplacian matrices $L_{1}, L_{2}, \dots, L_{m}$ , respectively. The dwell time of each topology is denoted as $λ_{i} T, i = 1, 2, \dots, m$ . Here $λ_{i}$ is the weight coefficients and satisfies

\sum_{j = 1}^{m} λ_{i} = 1, 1 \geq λ_{i} \geq 0

Figure 1 shows the switching diagram in each period. In Figure 1, the union of the m graphs is periodical connected. There are m switching graphs in the kth period $[(k - 1) T, kT]$ . How to switch the m graphs is determined by switching control law.

Figure 1.

Switching diagram.

Note that to derive the convergence rate for consensus is equivalent to the convergence rate of equation (7). According to Figure 1 and equation (7), we get

θ (t) = Φ (t, k T) \cdot Φ (k T, (k - 1) T) \dots Φ (2 T, T) \cdot Φ (T, 0) \cdot θ (0)

Here, $Φ (\cdot)$ is the state transition matrix of system (6), T is the switching period. Using equation (5) we have

\begin{array}{l} e^{- λ_{i} (α_{⊥}^{T} L_{i} α_{⊥}) T} \\ = I_{n - 1} + (- λ_{i} α_{⊥}^{T} L_{i} α_{⊥} T) + \frac{(- λ_{i} α_{⊥}^{T} L_{i} α_{⊥} T) (- λ_{i} α_{⊥}^{T} L_{i} α_{⊥} T)}{2!} + \dots \\ = I_{n - 1} + (- λ_{i} α_{⊥}^{T} L_{i} α_{⊥} T) + \frac{- λ_{i}^{2} (α_{⊥}^{T} L_{i} α_{⊥} α_{⊥}^{T} L_{i} α_{⊥}) T^{2}}{2!} + \dots \\ = I_{n - 1} + (- λ_{i} α_{⊥}^{T} L_{i} α_{⊥} T) + \frac{- λ_{i}^{2} (α_{⊥}^{T} L_{i} L_{i} α_{⊥}) T^{2}}{2!} + \dots \\ = I_{n - 1} + (- λ_{i}) (α_{⊥}^{T} L_{i} α_{⊥}) (T) + \frac{- λ_{i}^{2} (α_{⊥}^{T} L_{i}^{2} α_{⊥}) T^{2}}{2!} + \dots \\ = α_{⊥}^{T} α_{⊥} + α_{⊥}^{T} (- λ_{i}) L_{i} T α_{⊥} + \frac{α_{⊥}^{T} (- λ_{i}^{2}) L_{i}^{2} T^{2} α_{⊥}}{2!} + \dots \\ = α_{⊥}^{T} (I_{N - 1} + (- λ_{i}) L_{i} T + \frac{(- λ_{i}^{2}) L_{i}^{2} T^{2}}{2!}) α_{⊥} + \dots \\ = α_{⊥}^{T} e^{- λ_{i} L_{i} T} α_{⊥} \end{array}

Then

\begin{array}{l} Φ (k T, (k - 1) T) = e^{- λ_{1} (α_{⊥}^{T} L_{1} α_{⊥}) T} \cdot e^{- λ_{2} (α_{⊥}^{T} L_{2} α_{⊥}) T} \cdot e^{- λ_{m - 1} (α_{⊥}^{T} L_{m - 1} α_{⊥}) T} e^{- λ_{m} (α_{⊥}^{T} L_{n} α_{⊥}) T} \\ = e^{- α_{⊥}^{T} (λ_{1} L_{1} + λ_{2} L_{2} + \dots + λ_{m} L_{m}) T α_{⊥}} \\ = α_{⊥}^{T} e^{- (λ_{1} L_{1} + λ_{2} L_{2} + \dots + λ_{m} L_{m}) T} α_{⊥} \end{array}

Let $L (λ, t) = (λ_{1} L_{1} + λ_{2} L_{2} + \dots + λ_{m} L_{m})$ ,

Then

\begin{matrix} Φ (kT, (k - 1) T) = α_{⊥}^{T} e^{- L (λ, t) T} α_{⊥} \end{matrix}

Noting that $Φ (kT, (k - 1) T)$ is a function depending on the coefficients $λ_{i}, i = 1, 2, \dots, m$ and switching period T. Define the supremum of $∥ Φ (kT, (k - 1) T) ∥$ by

M = \sup_{λ_{i}, T} ∥ Φ (kT, (k - 1) T) ∥

Then

θ (t) \leq Φ (t, kT) M^{k} θ (0)

Utilizing $∥ Φ (t, kT) ∥ \leq 1$ , we have $∥ θ (t) ∥ \leq M^{k} ∥ θ (0) ∥$ and the convergence rate of system (6) as follows

\begin{matrix} μ = lim_{t \to \infty} \frac{1}{t} \ln \frac{∥ θ (t) ∥}{∥ θ (0) ∥} \\ = lim_{k \to \infty} \frac{1}{kT + (t - KT)} \ln \frac{∥ θ (t) ∥}{∥ θ (0) ∥} \\ \leq lim_{k \to \infty} \frac{k}{kT + (t - KT)} lnM \\ = lim_{k \to \infty} \frac{lnM}{T + ((t - kT) / k)} \\ = \frac{lnM}{T} \end{matrix}

Thus, the convergence rate of system (4) for consensus is derived as follows

μ \leq \frac{1}{T} \ln (M)

(8)

According to the above analysis, $μ$ is not greater than M. Moreover, $μ$ should be negative if the sensors can reach consensus. Thus, M should be as small as possible so as to have a fast convergence speed as possible. That is, if we make M as small as possible, we can get a convergence speed as fast as possible. In the next section, we discuss how to acquire the smallest value of M, which can be denoted as $M_{\min}$ .

Guaranteed convergence switching control

Theorem 1

Given the dynamics of a group of sensors as defined in equation (4) and the network topology graphs over the sensors are periodically connected and dynamically switched under the switching signal $σ (t)$ . System (4) reaches consensus with a guaranteed convergence rate at least $(1 / T) \ln M_{\min}$ if $λ_{i}$ satisfy the following conditions

\begin{matrix} minimize M, \\ subject to λ_{1} + λ_{2} + \dots + λ_{m} = 1 \\ 1 \geq λ_{i} \geq 0, i = 1, 2, \dots, m \end{matrix}

(9)

Proof

The state transition matrix of system (7) is as follows

Φ (kT, (k - 1) T) = α_{⊥}^{T} e^{- (λ_{1} L_{1} + λ_{2} L_{2} + \dots + λ_{n} L_{n}) T} α_{⊥}

and the convergence rate is as follows

μ \leq \frac{\ln (M)}{T} = \frac{\ln (\sup_{λ_{i}, T} ∥ Φ (kT, (k - 1) T) ∥)}{T}

When the topology graphs over the sensors are periodically connected, which means the joint graph of $G_{0}, G_{1}, G_{2}, \dots, G_{m}$ is a directed spanning tree in a bounded time interval T, we have $M \leq 1$ . Then, if we can get the smallest value of M, that is, $M_{\min}$ , then $(1 / T) \ln M_{\min}$ will be a negative value and its absolute value is the largest one. The larger the absolute value of $μ$ is, the faster the convergence speed the system (7) is. If there exits $λ_{i}$ such that M goes to minimum $M_{\min}$ , system (6) reaches consensus with a convergence rate at least $\ln (M_{\min}) / T$ . We note that the convergence rate of system (7) is equivalent to the convergence rate of system (4). Thus, Theorem 1 is proved. Then, we give Algorithm 4 with its computation complexity no greater than $O (n^{m})$ to obtain the guaranteed convergence value of $μ$ and $λ_{i}$ .

Simulation examples

In this section, we give some examples to illustrate the method of obtaining optimal switching parameters. For simplicity, four sensors are considered. In these examples, the switching signal $σ (t)$ is periodic with the cycle time T. Two cases are considered according to the number of $λ_{i}$ , that is, $m = 2$ and $m = 3$ . In both cases, let the switching period $T = 3$ .

One-freedom case ( $m = 2$ )

In this case, the communication topology graph over four agents is switching between graphs in the collection of ${G (L_{1}), G (L_{2})}$ . The switching graphs are shown in Figure 2. According to Figure 2, the Laplacian matrices are as follows

L_{1} = (\begin{matrix} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ - 1 & 0 & 1 & 0 \\ - 0.5 & 0 & - 0.5 & 1 \end{matrix}), L_{2} = (\begin{matrix} 1 & - 1 & 0 & 0 \\ - 1 & 1 & 0 & 0 \\ 0 & - 1 & 1 & 0 \\ 0 & 0 & 0 & 0 \end{matrix})

Figure 2.

Graphs: (a) $G (L_{1})$ and (b) $G (L_{2})$ .

Then, applying Theorem 1 and Algorithm 1, we can obtain the curve shown in Figure 3. It can be seen that when the switching parameters $λ_{1} = 0.644, λ_{2} = 1 - 0.644 = 0.356$ , the convergence rate for consensus of sensors can reach its value $μ \leq - 0.499$ . The dwell times for $G (L_{1})$ and $G (L_{2})$ are $λ_{1} T = 0.644 \times 3 = 1.932 s$ and $λ_{2} T = 0.356 \times 3 = 1.068 s$ , respectively. Let the initial values of the states be $x (0) = [1 2 3 4]^{T}$ . In Figure 4, the states response curves for different $λ_{1}$ are shown.

Algorithm 1. The algorithm to obtain value of $μ$ and $λ_{i}$
1. Give the initial values to some vectors and matrices, such as $T, α, α_{⊥}, L_{1}, L_{2}, \dots, L_{m}$ ;
2. for $λ_{1} = 0 : step (1) : 1$ do
3. for $λ_{2} = 0 : step (2) : 1$ do
4. …
5. for $λ_{i} = 0 : step (i) : 1$ do
6. for $λ_{m - 1} = 0 : step (n - 1) : 1$ do
7. if $λ_{m} = 1 - \sum_{j = 1}^{m - 1} λ_{i} \geq 0$ do
8. Calculate M;
9. Store $M_{\min}$ and $μ$ ;
10. Save the values of $λ_{i}$ ;
11. else
12. exit current for-loop;
13. end if
14. end for
15. end for
16. end for
17. end for
18. end for
19. Obtain the guaranteed value of $μ$ and corresponding $λ_{i}$ .

Figure 3.

Convergence rate for consensus of four sensors (m = 2).

Figure 4.

States response under different $λ_{1}$ : (a) λ₁ = 0.1, (b) λ₁ = 0.2, (c) λ₁ = 0.3, (d) λ₁ = 0.4, (e) λ₁ = 0.5, (f) λ₁ = 0.644, (g) λ₁ = 0.7, (h) λ₁ = 0.8, (i) λ₁ = 0.9 and (j) λ₁ = 0.99. $- μ$ under different $λ_{1}$ : the subfigures from (a) to (j) clear show that when the control parameter $λ_{1}$ varies from 0.1 to 1, the convergence speed of sensors gradually increases and then decreases with the optimal value at $λ_{1} = 0.644$ .

Two-freedom case ( $m = 3$ )

When the number of switching topologies among the sensors changes from two to three, the number of switching control parameters becomes two. There are three parameters $λ_{1}, λ_{2}, λ_{3}$ need to be calculated to obtain the optimal convergence rate. At the same time, it has to meet another constraint condition that $λ_{1} + λ_{2} + λ_{3} = 1$ . We only need to solve two of them, that is why we call this two-freedom case. In this scenario, we consider the following three Laplacian matrices corresponding to the graphs shown in Figure 5

L_{a} = (\begin{matrix} 1 & - 1 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ - 1 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 \end{matrix}), L_{b} = (\begin{matrix} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & - 1 & 1 & 0 \\ 0 & 0 & - 0.5 & 0.5 \end{matrix}), L_{c} = (\begin{matrix} 0 & 0 & 0 & 0 \\ - 1 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ - 0.5 & 0 & 0 & 0.5 \end{matrix})

Figure 5.

Graphs: (a) $G (L_{a})$ , (b) $G (L_{b})$ and $G (L_{c})$ .

In every switching period $T = 3 s$ , the dwell times for the communication graph $G (L_{a}), G (L_{b}), and G (L_{c})$ are $λ_{1} T, λ_{2} T, and λ_{3} T s$ , respectively. The switching control law $σ (t)$ is as follows

σ (t) = {\begin{matrix} a, (k - 1) T \leq t < (k - 1) T + λ_{1} T \\ b, (k - 1) T + λ_{1} T \leq t < (k - 1) T + λ_{1} T + λ_{2} T \\ c, (k - 1) T + λ_{1} T + λ_{2} T \leq t < kT \end{matrix}

Here, $k = 1, 2, \dots$ . According to Theorem 1, the optimal parameters combination of $λ_{1}, λ_{2}$ and $λ_{3}$ can be calculated. Figure 6 shows the convergence rate for different $λ_{1}, λ_{2}$ . It can be seen from the figure that when $λ_{1} = 0.231, λ_{2} = 0.183, λ_{3} = 1 - λ_{1} - λ_{2} = 0.586$ , the convergence rate of consensus reaches its value at $μ \leq - 0.3397$ . Table 1 shows the different convergence rates of sensors under different $λ_{i}$ . The states responses are shown in Figure 7 with several values of $λ_{i}$ .

Figure 6.

$- μ$ under different $λ_{1}$ and $λ_{2}$ .

Table 1.

$- μ$ under different $λ_{i}$ .

Case	$λ_{1}$	$λ_{2}$	$λ_{3}$	$μ$
1	0.054	0.102	0.844	−0.1056
2	0.661	0.209	0.13	−0.1047
3	0.052	0.230	0.718	−0.2
4	0.512	0.186	0.302	−0.2007
5	0.093	0.455	0.452	−0.2995
6	0.372	0.034	0.594	−0.3071
7	0.333	0.334	0.333	−0.2788
8	0.231	0.183	0.586	−0.3397*

The best value from calculation.

Figure 7.

States response under different $λ_{1}$ and $λ_{2}$ : the subfigures from (a) to (h) show that when the control parameters $λ_{1}$ and $λ_{2}$ vary from 0 to 1, the convergence rate changes with the optimal value at $λ_{1} = 0.231$ and $λ_{2} = 0.183$ : (a) λ₁ = 0.054, λ₂ = 0.102; (b) λ₁ = 0.661, λ₂ = 0.209l (c) λ₁ = 0.052, λ₂ = 0.230; (d) λ₁ = 0.512, λ₂ = 0.186; (e) λ₁ = 0.093, λ₂ = 0.455; (f) λ₁ = 0.372, λ₂ = 0.034; (g) λ₁ = 0.333, λ₂ = 0.334; and (h) λ₁ = 0.231, λ₂ = 0.183.

Effects of the switching period T on convergence rate $μ$

In both Case 1 and Case 2, the switching period T is fixed to be $T = 3 s$ . But how does the convergence rate $μ$ vary with T ? In Figure 8, the subfigures labelled (a) and (b) are corresponding to Case 1 and Case 2, respectively. It shows that the convergence rate of the system increases as the switching period T increases. This is very reasonable. In fact, increasing T is equivalent to increasing the weights of the communication graphs, which enhances the interaction among the sensors.

Figure 8.

$- μ$ varies with $λ_{i}$ : (a) Case 1 and (b) Case 2.

Conclusion

In this article, the problem of designing the switching control law for switching topologies over mobile sensor nodes has been addressed. In order to derive the consensus convergence rate of the sensors described as single-integrator nodes, a novel guaranteed convergence control method based on state variables decomposition is proposed. The system’s states are decomposed into consensus parts and non-consensus parts. When the non-consensus parts converge to zero point, the states of the whole system converge to the space spanned by $1_{n \times 1}$ vector. An algorithm is proposed to determine the switching control parameters so as to guarantee certain convergence rate. Two examples are provided to illustrate the validity and effectiveness of results.

Footnotes

Academic Editor: Jose Pelegri-Sebastia

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by the National Natural Science Foundation of China under Grant No. 61203032.

References

Djenouri

Merabtine

Mekahlia

. Fast distributed multi-hop relative time synchronization protocol and estimators for wireless sensor networks. Ad Hoc Netw 2013; 11: 2329–2344.

Yick

Mukherjee

Ghosal

Wireless sensor network survey. Comput Netw 2008; 52(12): 2292–2330.

Zhao

Guo

Wei

. Consensus control of mobile sensor networks with Markov switching topology under a given energy budget. In: Proceedings of the 34th Chinese control conference, Hangzhou, 28–30 July 2015, pp.6906–6911. New York: IEEE.

Cao

Ren

. An overview of recent progress in the study of distributed multi-agent coordination. IEEE T Ind Inform 2013; 9(1): 427–438.

Yang

Shi

Sensor selection schemes for consensus based distributed estimation over energy constrained wireless sensor networks. Neurocomputing 2012; 87(5): 132–137.

Guo

Zhao

Yang

Cooperation of multiple mobile sensors with minimum energy cost for mobility and communication. Inform Sciences 2014; 254: 69–82.

Tahbaz-Salehi

Jadbabaie

A necessary and sufficient condition for consensus over random networks. IEEE T Automat Contr 2008; 53(3): 791–795.

Proskurnikov

AV.

Average consensus in networks with nonlinearly delayed couplings and switching topology. Automatica 2013; 49: 2928–2932.

CW.

Synchronization and convergence of linear dynamics in random directed networks. IEEE T Automat Contr 2006; 51(7): 1207–1210.

10.

You

Xie

Consensus condition for linear multi-agent system over randomly switching topologies. Automatica 2013; 49(10): 3125–3132.

11.

Zhang

Tian

YP.

Consentability and protocol design of multi-agent systems with stochastic switching topology. Automatica 2009; 45(5): 1195–1201.

12.

Carli

Chiuso

Schenato

. Optimal synchronization for network of noisy double integrators. IEEE T Automat Contr 2011; 56(5): 1146–1152.

13.

Kim

Bisection algorithm of increasing algebraic connectivity by adding an edge. IEEE T Automat Contr 2010; 55(5): 170–174.

14.

Kim

Mesbahi

On maximizing the second smallest eigenvalue of a state-dependent graph Laplacian. IEEE T Automat Contr 2006; 51(1): 116–120.

15.

Wei

. On synchronizability of nonlinear networked systems with the switching topology and with the unit innercoupling matrix. In: Proceedings of the 30th Chinese control conference, Yantai, China, 22–24 July 2011, pp.916–921. New York: IEEE.

16.

Duan

Chen

. Consensus of multiagent systems and synchronization of complex networks: a unified viewpoint. IEEE T Circuits Syst 2010; 57(1): 213–224.

17.

Feng

Wen

Distributed consensus tracking for multi-agent systems under two types of attacks. Int J Robust Nonlin 2016; 26(5): 896–918.

18.

Xiao

Boyd

Fast linear iterations for distributed averaging. Syst Cont Lett 2004; 53(1): 65–78.

19.

Zheng

Wang

Consensus of switched multiagent systems. IEEE T Circuits-II 2016; 63(3): 314–318.

20.

Zheng

Wang

Finite-time consensus of heterogeneous multi-agent systems with and without velocity measurements. Syst Cont Lett 2012; 61(8): 871–878.

21.

Liu

Zhang

Liu

. Distributed stochastic consensus of multi-agent systems with noisy and delayed measurements. IET Control Theory A 2013; 7(10): 1359–1369.

22.

De Castro

Paganini

. Convex synthesis of controllers for consensus. In: Proceedings of 2004 American control conference, Boston, MA, 30 June–2 July 2004, pp.4933–4938. New York: IEEE.

Guaranteed convergence control for consensus of mobile sensor networks with dynamical topologies

Abstract

Keywords

Introduction

Graph theory

Definition 1. Periodically connected.

Convergence rate for periodically connected topologies

Dynamics of mobile sensor systems

Model description

Convergence rate

Guaranteed convergence switching control

Theorem 1

Proof

Simulation examples

One-freedom case ( m = 2 )

Two-freedom case ( m = 3 )

Effects of the switching period T on convergence rate μ

Conclusion

Footnotes

Declaration of conflicting interests

Funding

References

One-freedom case ( $m = 2$ )

Two-freedom case ( $m = 3$ )

Effects of the switching period T on convergence rate $μ$