Guaranteed cost control of mobile sensor networks with Markov switching topologies

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Guaranteed cost control of mobile sensor networks with Markov switching topologies Yuan Zhao, Ge Guo n, Lei Ding School of Information Sciences and Technology, Dalian Maritime University, Linghai Road 1#, Dalian 116026, China

art ic l e i nf o

a b s t r a c t

Article history: Received 18 October 2014 Received in revised form 29 March 2015 Accepted 26 May 2015 This paper was recommended for publication by Y. Chen

This paper investigates the consensus seeking problem of mobile sensor networks (MSNs) with random switching topologies. The network communication topologies are composed of a set of directed graphs (or digraph) with a spanning tree. The switching of topologies is governed by a Markov chain. The consensus seeking problem is addressed by introducing a global topology-aware linear quadratic (LQ) cost as the performance measure. By state transformation, the consensus problem is transformed to the stabilization of a Markovian jump system with guaranteed cost. A sufficient condition for global meansquare consensus is derived in the context of stochastic stability analysis of Markovian jump systems. A computational algorithm is given to synchronously calculate both the sub-optimal consensus controller gains and the sub-minimum upper bound of the cost. The effectiveness of the proposed design method is illustrated by three numerical examples. & 2015 ISA. Published by Elsevier Ltd. All rights reserved.

Keywords: Mobile sensor networks Consensus Markov switching topologies Mean-square stability Guaranteed cost control

1. Introduction In the past decade, wireless sensor networks have received a great deal of research attention due to their diverse applications in industrial automation, health monitoring, environment and climate monitoring, intruder detection, etc., [1]. In a dangerous or hostile environment, sensors cannot be manually deployed and fixed. It is necessary to deploy sensors mounted on mobile platforms such as unmanned vehicles, mobile robots, and spacecraft or man-made satellites. These sensors can collaborate among themselves to set up a sensing/actuating network, which is called a mobile sensor network (MSN). A typical MSN consists of hundreds or thousands of mobile sensor nodes distributed over a spatial region. Each sensor node has some level of capability for sensing, communication, signal processing and movement. The tendency that MSNs operated in a distributed manner will make use of small low power mobile devices may play revolutionary impact on many civil and military applications in exploration and monitoring. Due to the limitation of resource, MSNs have limited costs for communication, computation and motion sub-capabilities. As a result, power-aware algorithms have recently been the subjects of extensive research [2–6] regarding various key issues such as localization, deployment, environment estimation and coverage control, rendezvous and consensus. For example, energy-efficient

n

Corresponding author. E-mail address: [email protected] (G. Guo).

localization algorithms were proposed to reposition sensors in desired locations in order to recover or enhance network coverage or to maximize the covered area in [7,8] and [9]. In [8–10], the power-constrained deployment and coverage control issues were addressed by modeling energy consumption by the total traveling distance of the sensors. In [11] and [12], the vehicle speed management and the optimization problem of the number of agents for adequate coverage were addressed. In [13], a new algorithm for the maximum distance which an agent could travel by a dynamically changing energy radius was presented to solve the distributed deployment problem. An energy aware protocol which can prevent the agents from depleting their energy in achieving rendezvous was proposed in [14]. Consensus seeking, which means a group of mobile sensors achieve agreement upon a common state (i.e., position, velocity and direction), is another interesting problem in cooperative control of MSNs. There have been many papers studying consensus problems with cost optimization. To mention a few, an optimal consensus control method was proposed in [15] to minimize energy cost of sensors deployed in intelligent buildings for resource allocation. In [16], by introducing the cost functions to weigh both the consensus regulation performance and the control effort, an LQR consensus method was derived for multivehicle systems with single integrator dynamics. In [17], an optimal consensus seeking problem was studied in a network of general linear multi-agents. In [18], a two-step sub-optimal consensus control algorithm guaranteeing minimum energy cost for mobility and communication sub-tasks were derived.

http://dx.doi.org/10.1016/j.isatra.2015.05.013 0019-0578/& 2015 ISA. Published by Elsevier Ltd. All rights reserved.

Please cite this article as: Zhao Y, et al. Guaranteed cost control of mobile sensor networks with Markov switching topologies. ISA Transactions (2015), http://dx.doi.org/10.1016/j.isatra.2015.05.013i

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However, most of the above researches assume a static communication topology of MSNs. In practice, MSNs may have a dynamic network topology caused by link failure, packet dropout or environmental disturbances. [19] proposed a theoretical framework to study the consensus problem of multi-agent systems with switching topology. Based on nonnegative matrix theory, [20] and [21] investigated consensus control of multi-agent systems under dynamic topology. [22] considered a tradeoff between system performance and control effort of multi-agent systems with switching topologies. In some cases, due to random network conditions or environmental factors (e.g., sea wave, the wind and weather condition, [23]), an MSN may experience a randomly switching communication topology. Recently, increasing research attention has been paid to multi-agent systems with randomly switching network topologies, especially those with Markov switching topologies [24–27]. For example, [28] studied the almost sure convergence to consensus for agent network with Markovian switching topologies. By using the pth moment exponential stability theory and M-matrix approach, [29] considered the average consensus for the wireless sensor networks with Markovian switching topology and stochastic noise. In these results, it is required that Markov chains are ergodic, which implies that the multi-agent systems experience switching topologies in infinite time horizon. In other words, the systems cannot stay in a certain topology. In many practical applications, it is however more reasonable that systems may go though from switching topologies to a certain fixed topology. An example can be found when the systems pass from unsteady environment to a settled one. In fact, the control cost of a mobile sensor network depends on the communication and mobility behaviors of the sensors as well as the network topology. Therefore, for MSNs with Markov switching topologies, it is of great importance how to delicately involve the network topology factor into the control cost in setting up a low cost consensus control protocol. However, there are few results available on guaranteed cost control for consensus of multi-agent systems with Markov switching topologies. In this paper, we aim to investigate the problem of guaranteed cost consensus seeking of MSNs with Markov switching topologies. We consider a collection of mobile sensors whose dynamics is described by a discrete-time state space equation. The communications topologies are assumed to be a set of directed graphs with a spanning tree. The switching of network topology is modeled as a Markov chain. A topology-dependent consensus protocol without local feedback is proposed, where the subtle structural dynamics of the switching topology is involved. A global LQ cost function depending on the control input and the state errors of neighboring sensors is introduced. Then, using graph theory and model transformation, the consensus problem with guaranteed cost is transformed to the problem of guaranteed cost stabilization of a reduced order Markov jumping system. A sufficient condition which guarantees global exponential consensus of the MSN in the mean square sense is derived based on stochastic Lyapunov functional method. A computational algorithm by which the consensus controller gains and a minimum upper bound of the cost can be calculated is given. The effectiveness of the consensus control method is illustrated by three numerical examples. The remainder of this paper is organized as follows. Section 2 gives some preliminaries of graph theory and the problem formulation. Section 3 contains the main results on the sufficient condition of consensus and controller design for MSNs with Markov switching topology. Numerical examples are given in Section 4, which is followed by the conclusion in Section 5. Notations: Rn denotes n dimensional Euclidean space, Rn
m represents the family of n
m dimensional real matrices. In is the identity matrix of dimension n. For a given vector or a matrix

X, X Τ and j j X j j denotes its transpose and its Euclidean norm. ρðXÞ means the eigenvalue of matrix X. For a square nonsingular matrix X, X  1 denotes its inverse matrix. And diag{…} stands for a blockdiagonal matrix. For symmetric matrices P and Q, P 4Q (respectively, P ZQ ) means that matrix P–Q is positive define (positive semi-definite). The sign  represents matrix Kronecker product. 1 denotes a column vector whose entries equal to one. Similar notation is adopted for 0. The symmetric elements of a symmetric matrix are demoted by n. E(y) and Pro(y) are the mathematical expectation and probability of stochastic variable y. N þ stands for non-negative integers.

2. Preliminaries and problem formulation 2.1. Preliminaries of graph theory We use a directed graph (digraph) Gðυ; ε; ΛÞ to model the interactions among sensors, where υ A fυ1 ; ⋯; υN g is the set of N nodes, ε D υ
υ is the set of edges, Λ ¼ ½aij  is the adjacency matrix with its elements associated with the edges, i.e., if υi ; υj ; A ε, aij 4 0, otherwise ðυi ; υj Þ2 = ε, aij ¼0. In the paper we will consider graphs without selfedge, i.e., aii ¼ 0. Each edge ðυi ; υj Þ A ε implies that node υi can receive information from node υj . A sequence of edges ðυi ; υk Þ, ðυk ; υl Þ, … , ðυm ; υj Þ is called a directed path from node υj to node υi . A digraph is said to have a spanning tree, if there is a root (which has only children but no parent) such that there is a directed path from the root to any other nodes in the graph. The set of neighbors of node υi is denoted by N i ¼ ðυj A υ : ðυi ; υj Þ A εÞ. Define the in-degree of node υi as P di ¼ N j ¼ 1 aij and in-degree matrix Δ ¼ diagfd1 ; ⋯dN g. The Laplacian matrix of the directed graph G is defined as L ¼ Δ  Λ. Accordingly, P o define the out-degree of node υi as di ¼ N j ¼ 1 aji and the out-degree o o o matrix Δ ¼ diagfd1 ; ⋯; dN g. The graph column Laplacian matrix of o T the directed graph G is defined as Lo ¼ Δ  Λ . An important property of L is that all of its row sums are zero, thus 1 is an eigenvector of L associated with eigenvalue zero. Zero is a simple eigenvalue of L if and only if the directed graph has a spanning tree, and the other eigenvalues are with positive real parts. 2.2. Markov switching topology Consider a mobile sensor network with N identical sensors. At every instant k, the interconnection of these sensors can be considered as a directed graph with a spanning tree. The communication topology is switching but not fixed due to a certain random event. Assume that the topology is switching within a     given set of graphs G θðkÞ A GðkÞ, GðkÞ ¼ G1 ; G2 ; ⋯; Gq , where þ fθðkÞ; k A N g is the switching signal. Here, θðkÞ A S ¼ f1; ⋯; qgis assumed to be a Markov chain taking values in a finite set. Its transition probability is given as  Profθðk þ 1Þ ¼ vθðkÞ ¼ lg ¼ π lv ; with ProðθðkÞ ¼ lÞ ¼ π l ðkÞ wherel; v A S, π l ðkÞ is the transition probability of Gl at time k with initial probability Proðθð0Þ ¼ lÞ ¼ π 0l , and π lv is the single step transition probability from mode l to mode v, which satisfies q P π lv ¼ 1. The adjacency matrix ΛðθðkÞÞ and Laplacian matrix v¼1       of graph G θðkÞ are defined as Λ θðkÞ A Λ1 ; Λ2 ; ⋯Λq and     L θðkÞ A L1 ; L2 ; ⋯Lq , respectively. Denote the whole topology modal probability distribution by matrix Π ðkÞ ¼ ½π 1 ðkÞ; ⋯; π q ðkÞT , with initial probability distribution Π 0 ¼ ½π 01 ; ⋯; π 0q T . Let π ¼ ½π lv q
q be the transition probability matrix. Then we have Π ðk þ 1Þ ¼ π T Π ðkÞ.

Please cite this article as: Zhao Y, et al. Guaranteed cost control of mobile sensor networks with Markov switching topologies. ISA Transactions (2015), http://dx.doi.org/10.1016/j.isatra.2015.05.013i

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Remark 1. Since θðkÞ A S ¼ f1; ⋯; qg is a finite state Markov chain, stationary distribution always exists. In such a Markov chain, there exists at least one positive recurrent closed set. As each topology has a spanning tree, it is not necessary for the Markov chain to be ergodic, which is required in related papers such as [24,26] and [27]. 2.3. Problem formulation The dynamics of sensor i in the MSN is described by: xi ðk þ 1Þ ¼ Axi ðkÞ þ Bui ðkÞ; i ¼ 1; …; N; m

ð1Þ

n

where ui ðkÞ A R and xi ðkÞ A R are the control input and the state of sensor i, with xi ð0Þ being the initial state, A A Rn
n and B A Rn
m are constant matrices. To make our study nontrivial, we assume matrix A is not Schur stable, i.e., all eigenvalues of A lie outside of the open unit disk. It is obvious that if A is Schur, the MSN will converge to zero and consensus can be achieved under zero consensus gain. As in many other studies [30], it is assumed that a number of mobile base stations are involved in the system to detect the topology information of the MSN and broadcast the knowledge about the topology to the sensors. In other words, this is a practical way that the sensors can know the present topology of the MSN at any time. Therefore, we use the following mode dependent control protocol for sensor i. X ui ðkÞ ¼ KðθðkÞÞ aij ðθðkÞÞðxi ðkÞ  xj ðkÞÞ; θðkÞ A S; ð2Þ j A Nj

where KðθðkÞÞ is the controller gain matrix to be determined. Furthermore, the control protocols should be able to guarantee a limited overall cost, which is defined below: 0 1 1 X N X X  T   T @ J¼ E aij ðθ ðkÞÞ xi ðkÞ  xj ðkÞ Q xi ðkÞ  xj ðkÞ þ ui ðkÞRui ðkÞA k¼0i¼1

j A N i ðkÞ

ð3Þ where Q A Rn
n Z 0, R A Rm
m 4 0 are constant matrices. Before stating our objective of this paper, we first give the following definition. Definition 1. The MSN is said to achieve mean square consensus (MS-consensus) under protocol (2) with Markov switching topologies in set GðkÞ, if for any finite xi ð0Þ, θðkÞ A S ¼ f1; ⋯; qg, the following holds true for any i, j ¼1, …, N lim E½‖xi ðkÞ  xj ðkÞ‖2  ¼ 0:

k-1

ð4Þ

Thus, our objective is to design controllers in the form of (2) such that the collection of sensors reach mean square consensus with J r J~ , where J~ is a given cost constraint. Remark 2. Note that the cost defined in (4) covers both mobility energy cost and communication energy cost. To be more specific, let the mobile sensors have mass mi and dynamics x_ i ðtÞ ¼ ui ðtÞ, where xi(t) and ui(t) denote the position and the velocity of the sensors. The mobility energy of sensor i amounts to EM ðtÞ ¼ mi u2i ðtÞ=2, and the wireless transmission energy is 2 EC ðtÞ ¼ lbi dij ðtÞ, where dij ðtÞ ¼ j xi ðtÞ  xj ðtÞj is the distance between sensors i and j [5], l denotes the bits of data transmitted, and bi is the channel constant. Clearly, the overall energy cost of agent i can P 2 2 be given by J i ¼ 1 k ¼ 0 mi ui ðkÞ=2 þlbi ðxi ðkÞ  xj ðkÞÞ , which is a special case of the cost defined in (4). Remark 3. Different from the one introduced in [18], the topology-dependent consensus protocol given in (2) has no local feedback part. Here, we aim to design controllers guaranteeing consensus of MSNs under Markov switching topologies with a

3

sub-minimum cost bound. In [18], the local feedback part is used in the consensus protocol to prevent the mobility energy from growing infinitely large as time goes to infinity. Remark 4. The consensus seeking method in [21] based on nonnegative matrix theory requires the joint graph of the communication topologies to have a spanning tree. However, the method is not applicable to the guaranteed cost consensus control problem with stochastically switching topologies. Besides the entirely different problem setup, the protocol studied in [21] is independent of the topology mode, while the protocol in this paper is mode-dependent. Furthermore, the method in [21] requires that matrix B is of full rank, while in our method B can be any matrix with compatible dimension. 3. Main results In this section, we will first derive a sufficient MS-consensus condition for MSN (1) with guaranteed cost. Based on this condition, we will give a consensus controller design algorithm. Let I k ¼ fxðt Þ; θðtÞ; t ¼ 0; 1; 2; ⋯kg be the admissible information set. Clearly, I k  I k þ 1  I 1 . Then, we can rewrite system (1) with protocol (2) at the network level in the following form Xðk þ 1Þ ¼ ðI N  A þ Lθ  BK θ ÞXðkÞ;

ð5Þ

where Laplacian matrix Lθ ¼ LðθðkÞÞ, and controller gain K θ ¼ KðθðkÞÞ. Also, the total energy cost in (3) is written as 1       X  ð6Þ E X T ðkÞ Lθ þ Loθ  Q þ LTθ Lθ  K Tθ RK θ XðkÞ ; J¼ XðkÞ ¼ ½xT1 ðkÞ; xT2 ðkÞ; ⋯xTN ðkÞT ,

k¼0

where column Laplacian matrix Loθ ¼ Lo ðθðkÞÞ. The following lemma will be useful for obtaining the main results, which is a minor extension of the result in [31]. Lemma 1. Given h i 1 T ¼ pffiffiNffi1N T o A RN
N ;

ð7Þ

where To is the orthogonal complement of 1N satisfying T To T o ¼ I N  1 , then for any Laplacian matrix L A RN
N of a directed graph, the similarity transformation of L , Lo , and LTL are 2 3 2 3 p1ffiffiffi1T LT o p1ffiffiffi1T Lo T o 0 0 N N N N T T o 5; T L T ¼ 4 5; T LT ¼ 4 0N  1 T To LT o 0N  1 T To Lo T o " # 0 0TN  1 T T T L LT ¼ : ð8Þ 0N  1 T To LT LT o According to Lemma 1, a state transformation can be conducted as follows, X~ ðkÞ ¼ ðT  I n ÞT XðkÞ;

ð9Þ

Then, partitioning X~ ðkÞ A R into two parts, i.e.,X~ ðkÞ ¼ T T ½X~ 1 ðkÞ; X~ 2 ðkÞT , where X~ 1 ðkÞ A Rn is a vector consisting of the first n elements of X~ ðkÞ, the dynamics of system (5) can be written as nN

X~ 1 ðk þ 1Þ ¼ AX~ 1 ðkÞ þ Θθ  BK θ X~ 2 ðkÞ;

ð10Þ

X~ 2 ðk þ 1Þ ¼ ðI N  1  A þ Φθ  BK θ ÞX~ 2 ðkÞ;

ð11Þ

where Θθ ¼ p1ffiffiNffi1T Lθ T o , Φθ ¼ T o T Lθ T o . By [26], the MS-consensus of MSNs will be achieved if system (11) is mean square stable, which will be defined later. Also, we can write the energy cost (6) in the following form 1     X T o T J¼ EðX~ 2 ðkÞ Φθ þ Φθ  Q þ Φθ Φθ  K Tθ RK θ X~ 2 ðkÞÞ; k¼0

ð12Þ

Please cite this article as: Zhao Y, et al. Guaranteed cost control of mobile sensor networks with Markov switching topologies. ISA Transactions (2015), http://dx.doi.org/10.1016/j.isatra.2015.05.013i

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4

where Φθ ¼ T o T Loθ T o . Note that, by Lemma 1, we can easily prove that T is an orthogonal matrix. By matrix theory we know that the orthogonal transformation will not change the norm. Therefore, the energy cost J is not altered after the state transformation. Based on the above discussion, we can transform the consensus problem of system (5) into the stability problem of the reduced order system in (11) with cost function (12). Here we are interested in exponential mean square stability defined below. o

Definition 2. [32] System (11) is said to be exponentially mean square stable (MSS) with Markovian topologies set GðkÞ, if   k E ‖X~ 2 ðkÞ‖2 r βζ ‖X~ 2 ð0Þ‖22 ; k ¼ 0; 1; …; for any finite X~ 2 ð0Þ; β Z 1; 0 o ζ o 1: We are now in the position to establish a sufficient MSS condition for system (11) with a suboptimal guaranteed energy cost Jn. Theorem 1. MSN (1) with controllers (2) are mean square consensus under Markovian switching topologies set GðkÞ, if there exist matrices P l 4 0, P l A RðN  1Þn
ðN  1Þn , Q 40, R 40, l A S, such that the following condition holds: X Ψ Tl π lv P v Ψ l  P l þQ l þ Q ol þ ðI N  1  K l ÞT Rl ðIN  1 vAS

 K l Þ o 0; 8 l A S;

ð13Þ

where Ψ l ¼ I N  1  A þ Φl  BK l , Φl ¼ T o Ll T o , Ll ¼ LðθðkÞ ¼ lÞ, o o T Q l ¼ Φl  Q , Q ol ¼ Φl  Q , Φl ¼ T o T Lol T o , Rl ¼ ðΦl Φl Þ  R and T o is the orthogonal basis for the null space of 1. Moreover, the guaranteed cost upper bound X T J~ ¼ sup J ¼ X~ 2 ð0Þ π 0v P v X~ 2 ð0Þ T

S

vAS

(18), we can get EðVðkÞj I k  2 Þ ¼ EðEðV ðkÞj I k  1 Þj I k  2 Þ r ζ EðVðk  1Þj I k  2 Þ r ζ Vðk  2Þ: 2

ð19Þ By recursion like (17)–(19), we can have that T k k T EðVðkÞÞ ¼ EðX~ 2 ðkÞP θðkÞ X~ 2 ðkÞÞ r ζ V ð0Þ ¼ ζ X~ 2 ð0ÞP θð0Þ X~ 2 ðkÞ;

ð20Þ

from which it is straightforward that T k T EðX~ 2 ðkÞP θðkÞ X~ 2 ðkÞÞ r βζ X~ 2 ð0ÞX~ 2 ð0Þ;

ð21Þ

where β ¼ λmax ðP θð0Þ Þ=λmin ðP θð0Þ Þ 41, 0 o ζ o 1. Therefore, by Definition 2, the reduced system in (11) is exponentially mean square stable, namely, the MSNs under Markov switching topologies GðkÞ can achieve mean square consensus under controller (2) We then proceed to prove the guaranteed cost. By accumulating both sides of inequality (15) from k ¼0 to infinite, we can have J¼

1 X k¼0

  T T EðX~ 2 ðkÞF l X~ 2 ðkÞÞ r E Vð0; θð0ÞÞ ¼ EðX~ 2 ð0ÞP θð0Þ X~ 2 ð0ÞÞ; 8 l A S: ð22Þ

By (22), we can achieve the following performance bound X T π 0v P v X~ 2 ð0Þ; 8 l A S; J~ ¼ sup J ¼ X~ 2 ð0Þ S

ð23Þ

vAS

where X~ 2 ð0Þ ¼ ðT o  I n ÞT X 2 ð0Þ. This completes the proof. Remark 5. From Theorem 1, it  is easily seen that T Q l þ Q ol ¼ ðT To ðLl þ Lol ÞT o Þ  Q , and Q l þ Q ol ¼ ðT To ðLl þ Lol ÞT T o Þ  Q T . Since, by the definitions of the Laplacian and column Laplacian matrices of directed graphs, Q is a symmetric matrix, we know that  T Q l þ Q ol ¼ Q l þ Q ol , 8 l A S. Furthermore, as each topology has a spanning tree, zero is a simple eigenvalue of Ll and hence Φl ¼ T o T Ll T o has no zero eigenvalue. By the properties of Kronecker T product, ρðRl Þ ¼ ρðΦl Φl ÞρðRÞ. As R40, it is clear that Rl is reversible, 8 l A S.

Let θðkÞ ¼ l, θðk þ 1Þ ¼ v, l; v A S. Then, the backward difference of (14) is obtained as

Remark 6. This paper considers that each topology has a spanning tree. The results can be easily reduced to the case with connected undirected topologies. For a connected undirected graph, the formulations will be simplified since Lθ ¼ LTθ ¼ Loθ , 2 N i T T Lθ T ¼ diagð 0 Λθ Þ, Λθ ¼ diagð λθ ; ⋯; λθ Þ, λθ 4 0, i¼2, …, N, o Φl ¼ Λl , Ll ¼ Ll . In this case, the cost function in (6) will become 1       X E X T ðkÞ 2Lθ  Q þ LTθ Lθ  K Tθ RK θ XðkÞ ; J¼

ΔVðkÞ ¼ EðV ðk þ1; j I k Þ  VðkÞ

and system (11) will be reduced to

where X~ 2 ð0Þ ¼ ðT o  I n ÞT X 2 ð0Þ. Proof. We first prove the stability of system (11). Define the following stochastic Lyapunov functional: T Vðk; θðkÞÞ ¼ X~ 2 ðkÞP θðkÞ X~ 2 ðkÞ; θðkÞ A S ¼ f1; ⋯; qg:

T T ¼ EðX~ 2 ðk þ 1ÞP v X~ 2 ðk þ1Þj I k Þ  X~ 2 ðkÞP l X~ 2 ðkÞ ! X T T ¼ X~ ðkÞ Ψ π P v Ψ  P X~ 2 ðkÞ; 2

l

lv

l

l

ð14Þ

X~ 2 ðk þ 1Þ ¼ ðI N  1  A þ Λθ  BK θ ÞX~ 2 ðkÞ: ð15Þ

vAs

where Ψ l ¼ I N  1  A þ Φl  BK l . If (13) holds, we can have

ΔV r  X~ 2 ðkÞF l X~ 2 ðkÞ; T

whereF l ¼ Q l þ Q ol þ ðI N  1

ð16Þ T

 K l Þ Rl ðI N  1  K l Þ. Therefore, by (15) and (16), we can obtain that

 T λ ðF Þ EðV ðk þ1j I k Þ o VðkÞ  X~ 2 ðkÞF l X~ 2 ðkÞ r 1  min l VðkÞ ¼ ζ VðkÞ: λmax ðP l Þ ð17Þ

If inequality (13) holds, it is clearly seen that λmax ðF l Þ o λmax ðP l Þ, 0 o ζ o 1. To get the MSS condition in Definition 2, similarly, we can have the following equality EðV ðkÞj I k  1 Þ ¼ ζ Vðk  1Þ:

k¼0

ð18Þ

According to smooth characteristics of conditional mean [33], by

The presentation of communication cost in this case will be much less complex than our case. In what follows, we will give a controller design method. For this purpose, the following sufficient condition for the existence of the stochastic controller gain is derived based on stability criterion in Theorem 1. Theorem 2. For MSN (1) under Markovian switching topologies set GðkÞ, the protocols in (2) can drive the system to reach MS-consensus, if there exist matrices P l , M l A RðN  1Þn
ðN  1Þn , P l 4 0, M l 40, such that the following LMIs 2 3 T P l þ Q l þ Q lo ðI N  1  K l ÞT π Tl Ψ l 6 7 6 IN  1  K l ð24Þ  Rl 1 0 7 4 5 o 0: ~ πlΨ l 0 M h 1=2 i 1=2 1=2 T for all l A S, where π l ¼ π l1 π l2 ⋯ π lq , n o 1 ~ M M ⋯ M q with the constraint M l ¼ P l , l A S. M ¼ diag 1 2

hold

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Proof. Theorem 2 can be easily proved by using Theorem 1 and Schur complement lemma, so we omit it. As for the cost bound of the MSNs, we have obtained a feasible one in Theorem 1. Since the capacity of batteries for the MSNs is limited, a minimum cost bound is preferable. Thus, We will give the minimum guaranteed cost by minimizing the upper bound given in (23) over the feasibility set Ω ¼ ðP l ; M l ; K l Þ, l A S, determined by Theorem 2. Namely, we aim to find the infimum of J~ X T π 0v P v X~ 2 ð0Þ: ð25Þ J n ¼ inf J~ ¼ inf X~ 2 ð0Þ Ω

vAS

The minimization problem can be formulated as min δ T s: t: δ Z X~ 2 ð0Þ

ð26Þ X

π 0v P v X~ 2 ð0Þ

ð27Þ

vAS

which, by Schur complement, is equivalent to 2 3 T δ π T0 X~ 2 ð0Þ 5 4 Z0 ð28Þ ~ π 0 X~ 2 ð0Þ M h 1=2 i 1=2 1=2 T where π 0 ¼ π 01 π 02 ⋯ π 0q . The conditions in Theorem 2 contain some matrix inversion constraints which are equivalent to the rank constrained LMI " # " # Pl n Pl n Z 0; l A S; rank r ðN  1Þn ð29Þ I Ml I Ml Therefore, the above minimization problem can be reduced to min δ; s:t:ð24Þ; ð28Þ and ð29Þ:

ð30Þ

To solve the rank constrained LMI in (29), we can use the LMIRank solver [34], called the YALMIP interface and the underlying SeDuMi solver. However, LMIRank does not support objective functions, but only solves feasibility problems. To simultaneously determine the controller gain matrix in (5) and the sub-minimum quantity of energy cost in (7), we propose Algorithm 1.

topology, switching topology and switching-to-fixed topology. In other words, using Theorem 2, we can design mode-dependent consensus controllers for all these topologies, while the existing results are only applied to the cases with a fixed topology or a switching topology [24–27].

4. Numerical examples In this section, we present three numerical examples to illustrate the effectiveness of the proposed consensus protocol. Example 1. Consider a team of three identical sensors, whose dynamics can be described in the form of (1) with the following parameters [35] ( 2 r i ðk þ 1Þ ¼ r i ðkÞ þ hvi ðkÞ þ 12h ui ðkÞ ð31Þ vi ðk þ1Þ ¼ vi ðkÞ þ hui ðkÞ where ri(k), vi(k) and ui(k) are respectively the position, velocity and control input of sensor i at time t ¼kh, h is the sample interval.

1 0:6 When h¼0.6, we have for the system in (1) A ¼ , 0 1

0:18 B¼ . Choose the initial states as x1 ð0Þ ¼ ½ 4  3 T , 0:6 x2 ð0Þ ¼ ½  3:6 5 T , and x3 ð0Þ ¼ ½ 1:7 2 T . In the forthcoming simulation, we will use Q ¼ 2I 2 and R ¼ 0:5. All the possible information transmission relationships among sensors are given as a group of three directed graphs with each one a spanning tree (shown in Fig. 1). For simplicity, assuming that all the weights are equal to 1, we can have three Laplacian matrices of these graphs in Fig. 1 as follows 2 3 2 3 2 1 1 1 0 1 6 7 6 7 1 0 5; L2 ¼ 4  1 1 0 5; L1 ¼ 4  1 0 1 1 0 1 1 2

Algorithm 1. Step 1: Set the initial statexi ð0Þ (i¼1,…, N), weight matrices Q, R and the computational accuracy ε. Let a ¼ 0. Step 2: Find a feasible solution δ of δ by solving LMIs (24), (28) and the rank constrains LMI (29), set b ¼ δ. Step 3: Let c ¼ ða þ bÞ=2, δ ¼ c. Solving LMIs (24), (28) and the rank constrains LMI (29), if there exist the feasible solutions P l and K l (l A S), set b ¼ c. Otherwise, a ¼ c. Step 4: If j a  bj o ε, output δ,P l and K l . Otherwise, go to Step 3. Remark 7. In this paper, the ergodic characteristics (0 o π lv o 1;l; v A S) of the Markov chain is not required any longer since each topology is assumed to have a spanning tree. If π 0l ¼ π ll ¼ 1, the condition reduces to the case with a fixed topology Gl (namely, the sensors remain the initial topology all the time). If π 0l ¼ 0; π ll ¼ 1 (i.e., Gl is the absorbing state), the system will undergo from a switching topology to a fixed topology. This means that the method presented in this paper is more general than existing results, since it covers all cases of fixed

0 6 L3 ¼ 4  1 0

0 1 1

0

3

7 0 5: 1

The corresponding column Laplacian matrices are 2 3 2 3 1 1 0 1 1 0 6 7 o 6 7 o 2  1 5; L2 ¼ 4 0 1  1 5; L1 ¼ 4  1 1 0 1 1 0 1 2

1 6 Lo3 ¼ 4 0 0

1 1 0

3 0 7  1 5: 0

We assume that the communication topology of the MSN is switching according to a Markov chain with the following 4

Markov state

Ω

5

3 2 1 0

5

10

15

20

25

30

k

Fig. 1. Directed graph set GðkÞ ¼ fG1 ; G2 ; G3 g.

Fig. 2. Markov switching sequences of communication topologies in GðkÞ.

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6 30

30 20

position

position

20 10

10 sensor 1 sensor 2 sensor 3

0 0 -10

-10 0

5

10

15

20

25

0

5

10

15

20

25

30

k

30

k

Fig. 6. Position trajectories of MSNs.

Fig. 3. Position trajectories of MSNs. 6 4

velocity

6

velocity

4 2

2 0 -2

sensor1 sensor2 sensor3

-4

0

-6

-2

0

5

10

15

-6

20

25

30

k

-4 0

5

10

15

20

25

Fig. 7. Velocity trajectories of MSNs.

30

k 4

Markov state

Fig. 4. Velocity trajectories of MSNs.

4

Markov state

3

3 2 1

2

0

1

5

10

15

20

25

30

k

0

5

10

15

20

25

30

Fig. 8. Markov switching sequences of communication topologies in GðkÞ.

k Fig. 5. Markov switching sequences of communication topologies in GðkÞ.

30

transition probability matrix 0:7

0:2

0:1

7 π¼6 4 0:5 0:4 0:1 5; 0:3

0:5

20

3 ð32Þ

0:2

  and the initial probability distribution is Π 0 ¼ 0:5 0:4 0:1 . Using the presented design algorithm, we get the following controller gains K 1 ¼ ½  0:6422  0:8469 , K 2 ¼ ½ 0:9422  1:1842 , K 3 ¼ ½  1:7489  2:1834 . The mean energy cost in (4) is calculated as J¼220.6958, while the cost upper bound Jn is min δ ¼ 741.0021. The Markov switching sequences of topologies is shown in Fig. 2, where modes 1, 2 and 3 in the ordination denote topologies G1, G2, and G3, respectively. The simulation results of the three sensors with the obtained controllers are shown in Figs. 3 and 4. It can be clearly seen that the three sensors' states asymptotically reach agreement, which illustrates the effectiveness of the proposed method. A comparison with the controller design method in [31] is given to show the advantage of Algorithm 1 proposed in this paper. We consider the same systems (31) in Example 1. Assume that the communication topology set is the same in Fig. 1, and the transition probability matrix of Markov chain is same as (32). According to the controller design algorithm proposed in [31], we get the controller gains K c1 ¼ ½  0:1272 0:5001 , K c2 ¼ ½  0:124 0:4878 , K c3 ¼ ½  0:1088  0:4281 . The mean

position

2

10 sensor 1 sensor 2 sensor 3

0 -10

0

5

10

15

20

25

30

k Fig. 9. Position trajectories of MSNs.

energy cost defined in (4) can be calculated as J c ¼ 438:5833, while the energy upper bound J nc ¼ 2805:4. The simulation results of the three sensors with the controllers are shown in Figs. 5–7. Obviously, by comparisons with Example 1, we can have J o J c , J n o J nc , which show that our controller design method will guarantee the MSNs to achieve consensus under less cost consumption with Markov switching topologies. Example 2. The aim of this example is to show that the control method applies to MSNs with the switching to fixed topology. We consider the same systems as in Example 1. Assume that the communication topology set is the same as in Fig. 1 but is

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6

Acknowledgments

velocity

4

The authors acknowledge the financial support of the Natural Science Foundation Of China under Grants 61273107 and 61174060, and the Dalian Leading Talents, Dalian, China, the Fundamental Research Funds for Central Universities under Grant 3132013334, China.

2 0 -2

sensor 1 sensor 2 sensor 3

-4 -6

7

0

5

10

15

k

20

25

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References

Fig. 10. Velocity trajectories of MSNs.

switching according to a Markov chain with the following transition probability matrix 2 3 0 0:8 0:2 6 π^ ¼ 4 0 0:5 0:5 7 ð33Þ 5: 0 0 1 ^ ¼  0:7 0:3 0 . Note that The initial probability distribution is Π 0 this Markov chain is no longer an ergodic one, where graph G1 is a transient state and the G3 is an absorbing state. The Markov switching sequences of the topologies are shown in Fig. 8. Using Theorem 2, we can get the controller gains as K^ 1 ¼ ½  0:7398 0:9074 , K^ 2 ¼ ½  1:0377  1:2909 , and K^ 3 ¼ ½  1:6759  2:1437 . The mean energy cost is calculated as ^J ¼ 219.3023, while the energy upper n bound ^J is min δ^ ¼ 730.8635. The simulation results of the three sensors are shown in Figs. 9 and 10. Obviously, the MSN can achieve consensus with guaranteed cost when the switching topology finally settles down to a fixed topology.

5. Conclusion We have studied the consensus seeking problem for MSNs with Markov switching topologies. A sufficient condition for achieving exponential mean square consensus with guaranteed cost has been obtained. Complementing this condition by rank constrained LMIs, we have derived a numerical algorithm to calculate the controller gains and the sub-minimum cost bound. The method and the results are obtained based on Markov jumping system theory and state transformation. This paper investigates the feasibility and means of achieving cooperative control of MSNs with a guaranteed control cost. We just gave an upper bound of the control cost but not the minimum cost that could be guaranteed. Clearly, there is a problem of further decreasing the cost to a minimum, especially for energy-critical applications. In addition, circuit energy consumption cannot be overlooked in WSNs (unlike cellular networks) compared to the actual communication power. Thus, usual energy optimization techniques that minimize communication energy may not be effective in the case of wireless sensor networks. Besides counting in circuit energy consumption, multiple-input multiple-output (MIMO) technique may be another promising solution to energy limited wireless sensor networks due to large spectral efficiency. However, direct application of MIMO techniques is not practical for two reasons, (1) it requires complex transceiver circuitry and signal processing, implying large power consumption at the circuit level; (2) physical implementation of multiple antennas at a small node may not be realistic. Instead, one can consider cooperative MIMO [36] to achieve MIMO capability in a network of single antenna (single-input/single-output, SISO) sensors, as it has been shown that in some cases cooperative MIMO based sensor networks may lead to better energy optimization and smaller end-toend delay.

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