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Guaranteed cost synchronization for second-order wireless sensor networks with given cost budgets
Available online at www.sciencedirect.com
Journal of the Franklin Institute 356 (2019) 4061–4075 www.elsevier.com/locate/jfranklin
Guaranteed cost synchronization for second-order wireless sensor networks with given cost budgets Xinli Yin a,b, Bailong Yang b, Jianxiang Xi b, Xiaojun Yang a,∗ a The
School of Information Engineering, Guangdong University of Technology, Guangzhou 510006, PR China b Rocket Force University of Engineering, Xi’an 710025, PR China Received 16 June 2018; received in revised form 6 December 2018; accepted 6 January 2019 Available online 19 March 2019
Abstract The current paper addresses leader-following guaranteed cost synchronization with the cost budget given previously for the second-order wireless sensor networks. The published researches on guaranteed cost synchronization design criteria usually are based on the linear matrix inequality (LMI) techniques and cannot take the cost budget given previously into consideration. Firstly, the current paper proposes a guaranteed cost synchronization protocol, which can realize the tradeoff design between the battery power consumption and the synchronization regulation performance. Secondly, for the case without the given cost budget, sufficient conditions for leader-following guaranteed cost synchronization are presented and an upper bound of the cost function is shown. Thirdly, for the case that the cost budget is given previously, the criterion for leader-following guaranteed cost synchronization is proposed. Especially, the value ranges of control gains in these criteria are determined, which means that the existence of control gains in synchronization criteria can be guaranteed, but the LMI techniques can only determine the gain matrix and cannot give the value ranges of control gains. Moreover, these criteria are only associated with the minimum nonzero eigenvalue and the maximum eigenvalue, which can ensure the scalability of the wireless sensor networks. Finally, numerical simulations are given to illustrate theoretical results. © 2019 Published by Elsevier Ltd on behalf of The Franklin Institute.
∗ Corresponding author at: The School of Information Engineering, Guangdong University of Technology, Guangzhou 510006, PR China. E-mail address: [email protected] (X. Yang).
https://doi.org/10.1016/j.jfranklin.2019.01.048 0016-0032/© 2019 Published by Elsevier Ltd on behalf of The Franklin Institute.
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1. Introduction In the past decade, the synchronization of the wireless sensor networks has obtained considerable attention from researchers due to performance degradation caused by network congestion and increasingly higher requirement for the transmission speed and transmission performance [1–7]. As the information communication medium, the wireless sensor networks generally contain one base station (sink node) and many source nodes which can collect, process and transfer the information to the destination. In the process of the information transmission, source nodes send the collected information to the sink node directly or by other source nodes indirectly, which can be modeled as the type of leader-following structure. In this case, the one-many communication mode of the wireless sensor networks easily leads to the network congestion due to the various data transmission rates of bandwidth limited channels and the different queue length of limited buffer space of sensors in [8–12]. Hence, in many practical applications of wireless sensor networks, certain specific performance indexes of groups of nodes need to be synchronizable to suppress network congestion. Furthermore, the state of the leader in the wireless sensor networks is acted as the synchronization state. When the wireless sensor networks achieve synchronization, each sensor on certain specific performance indexes is identical. The resources of sensor nodes in the practical multi-node networks are usually limited such as battery power, memory, and process capabilities, so it is necessary and challenging to achieve the tradeoff design between the system energy consumption and the synchronization regulation performance. In certain published researches, different cost functions of multiple nodes were modeled as the optimization or suboptimization problem. In [13–18], the optimal estimated strategies of synchronization states for multi-node networks were proposed without taking the control energy consumption into consideration. Cao and Ren [18] studied the optimal synchronization problem based on the linear quadratic regulator, where the analysis approach was only applied to the special first-order multi-node networks. In [19,20], the designed suboptimal controllers can guarantee to provide a certain level of performance for the multi-node networks, where only the synchronization regulation performance was discussed. Guo et al. [21] investigated a minimum energy cooperation control problem for the multi-node networks to achieve synchronization, where the mobility energy cost and communication energy cost were considered. In [22], event-triggering containment control strategies for the multi-node networks were discussed, where the communication resources were mitigated. In [23], the synchronization problem with sampled data and packet losses for a class of multi-node systems was studied and some important and interesting results were proposed. In [13–24], these methods cannot achieve guaranteed cost synchronization with the given cost budget; that is, they cannot achieve the tradeoff design between the system energy consumption and the synchronization regulation performance with the cost budget given previously for the multi-node networks. When to explore the conditions of the guaranteed cost synchronization, the linear matrix inequality (LMI) technique is a powerful tool to solve the synchronization criteria for multinode networks. In [25], the minimization of the cost function subject to the synchronization constraint was solved through a set of LMIs, where the guaranteed cost synchronization was not taken into consideration. In [26], Sun proposed several sufficient conditions for average synchronization of the multi-node networks by using the LMI method, which cannot ensure the scalability of multi-node networks since the computational complexity greatly increases when the number of nodes increases. Wang et al. [28] discussed guaranteed cost synchronization for
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general second-order multi-node networks and proposed the synchronization criteria, where the computational complexity was decreased since those criteria are dependent on the minimum nonzero eigenvalue and the maximum eigenvalue of the Laplacian matrix. In [25–29], the synchronization design criteria for the multi-node networks are based on the LMI techniques, which cannot ensure the existence of the gain matrix and cannot determine the value ranges of control gains in synchronization criteria. In this case, based on the structure property of the second-order multi-node networks, it is meaningful to determine the value ranges of control gains in the synchronization design criteria, which can ensure the networks achieve leaderfollowing guaranteed cost synchronization with the cost budget given previously. Moreover, to the best of our knowledge, as the special type of multi-node networks, there are still many open problems to be further investigated for the wireless sensor networks. For the leader-following second-order wireless sensor networks, the current paper proposes a synchronization protocol to address guaranteed cost synchronization design problems, which can realize the tradeoff design between the battery power consumption and the synchronization regulation performance. For the case without giving the cost budget previously, the value ranges of control gains in the guaranteed cost synchronization criterion are determined by the inequality of the quadratic form and the function monotony. An upper bound of the sum of the system energy consumption term and the synchronization regulation term is obtained. For the case with the given cost budget, the relationship between the given cost budget and control gains is determined, which depends on the initial states, the minimum nonzero eigenvalue and the maximum eigenvalue and can ensure the scalability of the wireless sensor networks. The synchronization design criterion is proposed for the wireless sensor networks to achieve leader-following guaranteed cost synchronization. Moreover, the relationship between the given cost budget and control gains can ensure that the cost consumption of the wireless sensor networks is limited. Compared with the published results [28–30] about guaranteed cost synchronization for the multi-node networks, the current paper has the following contributions. Firstly, the value ranges of control gains are determined to achieve guaranteed cost synchronization, which can ensure the existence of control gains in the synchronization design criteria. The synchronization criteria in [28,29] are obtained by the gain matrix using LMI tools. In certain situations, the synchronization criteria may not obtain the feasible gain matrix. Secondly, the cost budget can be given previously, which can limit the energy consumption of the wireless sensor networks to extend the battery lifetime of sensors. For practical wireless sensor networks, each sensor usually has the limited energy, so the cost budget should be a finite value given previously. The research results in [28–30] determined different upper bounds of the guaranteed cost without considering the limited resources of the wireless sensor networks. The current paper takes the limited resources into account and gives the cost budget previously. The remainder of the current paper is organized as follows. In Section 2, some preliminaries and the problem description are presented, respectively. Section 3 gives a sufficient condition of second-order guaranteed cost synchronization for the leader-following wireless sensor networks without the given cost budget, obtains an upper bound of the guaranteed cost, and determines second-order guaranteed cost synchronization design criteria for the leaderfollowing wireless sensor networks with the given cost budget. Section 4 shows the numerical example to demonstrate theoretical results. Some concluding remarks are given in Section 5. Notations: Rn is the n-dimensional real column vector space and Rn×n is the set of n × n dimensional real matrices. In represents the n-dimensional identity matrix. PT = P < 0 and PT = P > 0 mean that the symmetric matrix P is negative definite and positive definite,
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respectively. The notation diag{d1 , d2 , . . . , dN } represents a diagonal matrix with diagonal elements d1 , d2 , . . . , dN . 2. Preliminaries and problem description In following section, the concepts and modeling interaction topology on graph theory are given firstly. Then, the problem description is presented. 2.1. Modeling interaction topology on graph theory The current paper models the interaction topology of the wireless sensor networks with N identical nodes by a graph G = (V (G ), E (G )), which is composed of a nonempty vertex set V (G ) = {v1 , v2 , . . . , vN } and the edge set E = {ei j = {(vi , v j )} ∈ V × V . The vertex vi represents sensor node i, the edge eij denotes the interaction channel from sensor node i to sensor node j, and the edge weight w ji of eij stands for the interaction strength from sensor node i to sensor node j. The index of the set of all neighbors of vertex v j is denoted by N between vertex vi1 and vertex vil is a sequence of edges j = i : (vi , v j ) ∈ E (G ) . A path v i 1 , v i 2 , v i 2 , v i 3 , . . . , vil−1 , vil . Define the Laplacian matrix of the graph G as L = l ji ∈ RN×N with l j j = i∈N j w ji and l ji = −w ji ( j = i ). A directed graph has a spanning tree if there exists a root node which has a directed path to any other nodes. It is assumed that the whole interaction topology of the wireless sensor networks has a spanning tree. The root node denotes the leader, where the leader does not obtain any state information from followers, only some follower nodes can obtain the state information from the leader node, and the local interaction topology among followers is undirected. More basic concepts and conclusions on graph theory can be found in [36]. 2.2. Problem description For the wireless sensor networks consisting of N identical second-order linear sensor nodes, one sets that sink node 1 is the leader and the other N − 1 nodes are followers. The dynamics of the jth sensor node is described by x˙ j (t ) = v j (t ), (1) v˙ j (t ) = u j (t ), where j = 1, 2, . . . , N, xj (t) ∈ R, v j (t ) ∈ R and uj (t) ∈ R are the queue length, the transmission rate and the control input, respectively. Since the leader does not receive any state information from other nodes, one can set that u1 (t) ≡ 0. For the given γ 1 , γ 2 ∈ R > 0 and η ∈ R > 0, a guaranteed cost synchronization protocol with a cost function is proposed as follows:
u j (t ) = w ji k1 xi (t ) − x j (t ) + k2 vi (t ) − v j (t ) , (2) ∞i∈N j Js = 0 (Ju (t ) + Jx (t ) )dt, where j = 2, 3, . . . , N, Nj denotes the neighbor set of node j , and Ju (t ) =
N j=1
ηu2j (t ),
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Jx (t ) =
N
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2 2 w ji γ1 xi (t ) − x j (t ) + γ2 vi (t ) − v j (t ) .
j=1 i∈N j
Ju (t) is constructed by the control inputs while Jx (t) is constructed by state errors. Furthermore, from the perspective of engineering, Ju (t) and Jx (t) represent the battery power consumption of wireless sensor nodes and the system synchronization performance on the queue length and the transmission rate, respectively. Positive parameters γ 1 , γ 2 and η can be chosen freely based on the demand for synchronization problems and all of them are independent of each other. Choosing different γ 1 , γ 2 and η mean different tradeoff design between Ju (t) and Jx (t). In this paper, our object is to determine control gains of synchronization protocols for the given upper bound of the linear quadratic index which means the given cost budget. One can set Js∗ > 0 as a given cost budget, then the definition of leader-following guaranteed cost synchronization with the given cost budget for the wireless sensor networks is proposed as follows. Definition 1. For any given Js∗ > 0, wireless sensor network (1) is said to be leaderfollowing by protocol (2) if there exist k1 and k2 such that guaranteed cost synchronizable limt→∞ x j (t ) − x1 (t ) = 0, limt→∞ v j (t ) − v1 (t ) = 0 ( j = 2, 3, . . . , N ) and Js ≤ Js∗ for any bounded disagreement initial states xj (0) and v j (0) ( j = 2, 3, . . . , N ). The main aim of this paper is to design the feasible value ranges of control gains k1 > 0 and k2 > 0 such that wireless sensor network (1) with the leader-following structure achieves guaranteed cost synchronization under the condition that the cost budget is previously given. Remark 1. Compared with protocols in [31–35] about the synchronization for networks, there are two critical features of protocol (2). The first one is that the relevant research results achieve guaranteed cost synchronization based on the LMI techniques, which need a feasp solver to obtain the feasible control gain matrix. In this case, the control gain matrix may not exist as shown in [28]. It should be pointed out that the feasp solver is a numerical algorithm, which cannot give the analytic solutions. This paper presents an approach to give the value ranges of k1 > 0 and k2 > 0 which can ensure the wireless sensor networks achieve leader-following guaranteed cost synchronization. The second one is that protocol (2) gives the cost budget previously. The main challenge is to determine the relationship between the cost budget given previously and an upper bound of the sum of battery power consumption and the synchronization regulation performance. It requires to design the specific k1 > 0 and k2 > 0 such that the existence of control gains can be ensured and the sum of battery power consumption and the synchronization regulation performance are less than the cost budget given previously. Moreover, our methods cannot deal with high-order multi-node networks. 3. Guaranteed cost synchronization for wireless sensor networks This section mainly proposes a sufficient condition for guaranteed cost synchronization of the second-order wireless sensor networks modeled as the leader-following structure without giving the cost budget previously, obtains an upper bound of the sum of the energy consumption term and the synchronization regulation term, and gives a sufficient condition to achieve the leader-following guaranteed cost synchronization with the cost budget given previously.
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Let Lff be the Laplacian matrix of the interaction topology among follower nodes, l f l = [w21 , w31 , . . . , wN1 ]T and fl represent the interaction weightsfrom the leader to followers. 0 0 . One can The whole Laplacian matrix L of the interaction topology is −l f l L f f + f l obtain that L f f + f l is symmetric and positive. Since the network interaction topology has a spanning tree and the local interaction topology among follower nodes is undirected, there exists an orthonormal matrix U˜ such that T ff fl ˜ ˜ U L + U = diag{λ2 , λ3 , . . . , λN } with 0 < λ2 ≤ λ3 ≤ . . . ≤ λN being eigenvalues of L f f + f l . Let x j (t ) = x j (t ) − x1 (t ), v j (t ) = v j (t ) − v1 (t )( j = 2, 3, . . . , N ), X (t ) = [x2 (t ), v2 (t ), x3 (t ), v3 (t ), . . . , xN (t ), vN (t )]T , and
T U˜ T [x2 (t ), x3 (t ), . . . , xN (t )]T = x˜2 (t ), x˜3 (t ), . . . , x˜N (t ) , T (3) U˜ T [v2 (t ), v3 (t ), · · · , vN (t )]T = v˜2 (t ), v˜3 (t ), · · · , v˜N (t ) . In this case, one can obtain by Eqs. (1)–(3) that ⎧ ⎨x˙ j (t ) = x˙ j (t ) − x˙1 (t ) = v j (t ) − v1 (t ) = v j (t ), w ji k1 xi (t ) − x j (t ) + k2 vi (t ) − v j (t ) , ⎩v˙ j (t ) = v˙ j (t ) − v˙1 (t ) =
(4)
i∈N j
⎧ T T ⎪ ⎨U˜ T [x˙2 (t ), x˙3 (t ), . . . , x˙N (t )] = v˜2 (t ), v˜3 (t ), . . . , v˜N (t ) , U˜ T [v˙2 (t ), v˙3 (t ), . . . , v˙N (t )]T = −k1 λ2 x˜2 (t ) − k2 λ2 v˜2 (t ), −k1 λ3 x˜3 (t ) − k2 λ3 v˜3 (t ), T ⎪ ⎩ . . . , −k1 λN x˜N (t ) − k2 λN v˜N (t ) . (5) Hence, one can obtain by Eqs. (3)–(5) that x˜˙ j (t ) = v˜ j (t ), v˙˜ j (t ) = −k1 λ j x˜ j (t ) − k2 λ j v˜ j (t ),
(6)
where j = 2, 3, . . . , N . Due to U˜ is an orthonormal matrix, one can obtain that x j (t ) = 0 and v j (t ) = 0 when x˜ j (t ) = 0 and v˜ j (t ) = 0 ( j = 2, 3, . . . , N ), which mean that the wireless sensor networks achieve synchronization. Theorem 1. Wireless sensor network (1) is leader-following guaranteed cost synchronizable by protocol (2) if there exist k1 and k2 such that k22 k22 k2 k2 γ1 2 γ1 − + − < k < − , 1 2 2 4η 16η ηλ2 2η 4η ηλ2 16ηγ1 1 1 2 γ2 max , 4 γ2 < k 2 < + − . 2 2 λ2 2ηλN ηλN 4 η λN Proof. Firstly, we give sufficient conditions such that limt→∞ x˜ j (t ) = 0 and limt→∞ v˜ j (t ) = 0 ( j = 2, 3, . . . , N ), which can obtain the leader-following guaranteed cost synchronization. One can adopt a Lyapunov function candidate as follows: V j (t ) = 0.5λ2j k22 + λ j k1 x˜2j (t ) + v˜2j (t ) + λ j k2 x˜ j (t )v˜ j (t ), (7)
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where j = 2, 3, . . . , N. Since k1 > 0, k2 > 0 and λj > 0 ( j = 2, 3, . . . , N ), one has 0.5λ2j k22 + 0.5λ2j k22 + λ j k1 0.5λ j k2 > 0. Hence, one can get that Vj (t) ≥ 0. Moreλ j k1 > 0 and det 0.5λ j k2 1 over, the time derivative of Vj (t) with Eq. (6) is V˙ j (t ) = −k1 k2 λ2j x˜2j (t ) − k2 λ j v˜2j (t ).
(8)
Due to k1 k2 λ2j > 0 and k2 λj > 0, one can get that V˙ j (t ) ≤ 0 and V˙ j (t ) = 0 if and only if x˜ j (t ) = 0 and v˜ j (t ) = 0. Hence, the wireless sensor networks can achieve synchronization. In the following, an upper bound of the guaranteed cost is determined. It can be obtained by Eq. (2) that 2 ∞ ff k1 k2 k1 T fl 2 X (t )dt , Ju = η X (t ) L + (9) k1 k2 k22 0 Jx = 2
∞
X T (t )
0
ff γ L + fl 1 0
0 γ2
X (t )dt .
(10)
Then 2 ff k1 fl 2 X (t ) L + k1 k2
k1 k2 k22
T
X T (t )
ff γ L + fl 1 0
0 γ2
N 2 X (t ) = λ2j k1 x˜ j (t ) + k2 v˜ j (t ) ,
(11)
j=2
N X (t ) = λ j γ1 x˜2j (t ) + γ2 v˜2j (t ) .
(12)
j=2
Hence, it can be obtained that ∞ Js = (Ju (t ) + Jx (t ) )dt 0 ⎞ ⎛ ∞ N N 2 ⎝η = λ2j k1 x˜ j (t ) + k2 v˜ j (t ) + 2 λ j γ1 x˜2j (t ) + γ2 v˜2j (t ) ⎠dt 0
=
j=2
N ∞ j=2
+
0
2 2 ηλ j k1 + 2λ j γ1 x˜2j (t )dt +
j=2
N j=2
j=2 N ∞
0
∞
0
2 2 ηλ j k2 + 2λ j γ2 v˜2j (t )dt
2ηλ2j k1 k2 x˜ j (t )v˜ j (t )dt.
(13)
∞ ∞ Let Ju = 0 Ju (t )dt , Jx = 0 Jx (t )dt , V (t ) = Nj=2 V j (t ), j11 = ηλ2j k12 + 2λ j γ1 − k1 k2 λ2j , j12 = ηλ2j k1 k2 and j22 = ηλ2j k22 + 2λ j γ2 − k2 λ j , j = 2, 3, . . . , N . Due to Js = 0
∞
(Ju (t ) + Jx (t ) )dt +
N j=2
0
∞
V˙ j (t )dt − V (t )|t→∞ + V (0)
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=
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+
N 2 2 ηλ j k1 + 2λ j γ1 − k1 k2 λ2j x˜2j (t )dt +
∞ 0
j=2
N j=2
∞ 0
0
∞
2 2 ηλ j k2 + 2λ j γ2 − k2 λ j v˜2j (t )dt
2ηλ2j k1 k2 x˜ j (t )v˜ j (t )dt − V (t )|t→∞ + V (0),
(14)
one can obtain that Js =
N j=2
∞ 0
j11 x˜ j (t ), v˜ j (t ) j12
j12 j22
T x˜ j (t ), v˜ j (t ) dt − V (t )|t→∞ + V (0).
(15)
Set ηλj k12 + 2γ1 − k1 k2 λ j < 0, ηλ j k22 + 2γ2 − k2 < 0 and ηλ j k12 (2γ2 − k2 )+ (2γ1 −k1 k2 λ j ηλ j k22 + 2γ1 − k1 k2 λ j (2γ2 − k2 ) > 0, then sufficient conditions are proposed: ηλ j k12 + 2γ1 − k1 k2 λ j < 0, ηλ j k22 + 2γ2 − k2 < 0, 2ηλ j k12 < −2γ1 + k1 k2 λ j 2 and 2ηλ j k2 < k2 − 2γ2 .Hence, one can design the value ranges of k1√ and k2 : 2 2 k2 /4η − k2 /16η − γ1 /ηλ2 < k1 < k2 /2η + k22 /4η2 − 2γ1 /ηλ2 and max 16ηγ1 /λ2 , 1/4ηλ j − 1/16η2 λ2j − γ2 /ηλ j < k2 < 1/2ηλN + 1/4η2 λ2N − 2γ2 /ηλN . Let y = 1/4ηλ − 1/16η2 λ2 − γ2 /ηλ, where λ > 0. By the derivation of the function, one can obtain the point of extreme value and get the maximum value of the function y. Hence, one gets the maximum value of the function which is 4γ 2 when λ = 1/16ηγ2 . As a result, it can be √ obtained that max 16ηγ1 /λ2 , 4γ2 < k2 < 1/2ηλN + 1/4η2 λ2N − 2γ2 /ηλN . The designed j11 j12 is negative definite and value ranges of k1 and k2 can guarantee that matrix j12 j22 from Eqs. (13) to (15), one can obtain that Js ≤ V (0).
(16)
As a result, it can guarantee that the wireless sensor networks achieve leader-following guaranteed cost synchronization. Based on the above analysis, the conclusion of Theorem 1 can be obtained. Remark 2. In this case, k1 and k2 are designed such that the wireless sensor networks achieve leader-following guaranteed cost synchronization without the cost budget given previously. One can see that the wireless sensor networks can achieve leader-following synchronization when k1 and k2 are designed as positive values. However, when the guaranteed cost synchronization is taken into consideration, it is difficult to design feasible control gains k1 and k2 to achieve synchronization. In this case, control gains k1 and k2 are separated from the quadratic form which contains two correlated variables k1 and k2 . One can see that k2 can be determined independently but k1 is dependent on k2 . Here, based on the inequality of the quadratic form and the function monotony, control gains k1 and k2 can be determined to the ranges of values simultaneously. Control gains k1 and k2 are dependent on the minimum nonzero eigenvalue λ2 and the maximum eigenvalue λN . To decrease the computational complexity, λ2 and λN can be estimated by using the approach in [36]. Moreover, λN ≤ 2d with
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d = max{l j j = i∈N j w ji , j = 2, 3, . . ., N } is estimated based on the Gersgorin disc theorem in [37]. It is not necessary to get all the eigenvalues of the interaction topology. Remark 3. In the associated work about achieving guaranteed cost synchronization, LMI techniques by the feasp solver are used to determine the feasible control gains matrix. In certain situations, a feasp solver may not obtain a feasible control gains matrix for multinode networks to achieve guaranteed cost synchronization. In this case, the value ranges of existence of control gains k1 and k2 are required to be designed, which can guarantee that the wireless sensor networks achieve leader-following guaranteed cost synchronization. One can choose the values of the control gains within the value ranges without the verification of feasibility. In this case, by the inequality of the quadratic form and the function monotony, control gains k1 and k2 can be determined to the value ranges simultaneously. Moreover, when one designs the control gain k2 , a simplified value range of k2 is determined by finding the maximum point based on the function monotony. Therefore, it reduces the range of value range of control gain k2 convergence. Based on the above analysis, Theorem 2 presents an approach to determine the control gains k1 and k2 with the cost budget given previously such that the wireless sensor networks achieve leader-following guaranteed cost synchronization. Set x(0) = (x2 (0), x3 (0), . . . , xN (0))T and v(0) = (v2 (0), v3 (0), . . . , vN (0))T . ff 2 T It can be obtained f l x(0) ≤ λ2N x T (0) x(0), x T (0) L f f + f l ! T that x ! (0) L + v(0) ≤ λN !x (0) v(0)! and x T (0) L f f + f l x(0) ≤ λN x T (0)x(0). Theorem 2 is proposed as follows. Theorem 2. For any given Js∗ > 0, wireless sensor network (1) achieves leader-following guaranteed cost synchronization by protocol (2) if there exist control gains k1 and k2 such that 16ηγ1 1 1 2 γ2 max , 4 γ2 < k 2 < + − , 2 2 λ2 2ηλN ηλN 4 η λN ⎛ k22 k22 k γ1 2 γ1 2 + − < k1 < min ⎝ − , 2 2 16η ηλ2 2η 4η ηλ2 ! ! −k22 λ2N x T (0)x(0) − 2k2 λN !x T (0) v(0)! + 2Js∗ − 2vT (0) v(0) . 2λN x T (0)x(0)
k2 − 4η
Proof. Firstly, by Eqs. (6) and (7), one can see that N 0.5λ2j k22 + λ j k1 x˜2j (t ) + v˜2j (t ) + λ j k2 x˜ j (t )v˜ j (t ) V (t ) = j=2
2 = 0.5k22 x T (t ) L f f + f l x(t ) + k2 x T (t ) L f f + f l v(t ) +k1 x T (t ) L f f + f l x(t ) + vT (t )v(t ).
(17)
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Let t = 0, then 2 V (0 ) = 0.5k22 x T (0) L f f + f l x(0) + k2 x T (0) L f f + f l v(0) +k1 x T (0) L f f + f l x(0) + vT (0)v(0) ! ! ≤ 0.5k 2 λ2 x T (0)x(0) + k2 λN λN !x T (0) v(0)! + k1 λN x T (0)x(0) 2 N
+ vT (0)v(0).
(18)
By Eqs. (16) and! (18), one can see that V (0 ) ≤ 0.5k22 λ2N x T (0)x(0) + ! k2 λN λN !x T (0) v(0)! + k1 λN x T (0)x(0) + vT (0)v(0) ≤ Js∗ guarantee √can that Js ≤ Js∗ . In this case, one can design control gains: max 16ηγ1 /λ2 , 4γ2 ) < k2 < 1/2ηλN + 1/4η2 λ2N − 2γ2 /ηλN and k2 /4η − k22 /16η2 − γ1 /ηλ2 < k1 ! ! < min x(0) − 2k2 λN !x T (0)v(0)! (k2 /2η + k22 /4η2 − 2γ1 /ηλ2 , −k22 λ2N x T (0) + 2Js∗ − 2vT (0)v(0) / 2λN x T (0)x(0) , which can ensure the wireless sensor networks achieve leader-following guaranteed cost synchronization with the given cost budget Js∗ . Hence, the proof is completed. Remark 4. In Theorem 1, the guaranteed cost is determined by the value of the Lyapunov function candidate at time zero. One can see that the expression of the guaranteed cost contains initial states of all the nodes. In Theorem 2, due to the practical energy limitation, one gives the cost budget previously. Hence, it should take the energy limitation requirement into consideration to choose proper values of the control gains k1 and k2 . There are two difficulties in obtaining Theorems 2. The first one is to construct the relationship between the linear quadratic optimization index and the interaction weights matrix. The second one is to construct the relationship between the given cost budget and the control gains k1 and k2 . By an upper ! bound of the ! guaranteed cost, one can see that V (0 ) ≤ 0.5k22 λ2N x T (0)x(0) + k2 λN !x T (0)v(0)! + k1 λN x T (0)x(0) + vT (0) v(0) ≤ Js∗ can guarantee that the guaranteed cost is less than ! the cost bud! get given previously. Hence, one adds 0.5k22 λ2N x T (0)x(0) + k2 λN !x T (0)v(0)! + k1 λN x T (0)x(0) + vT (0) v(0) ≤ Js∗ as a condition for selecting value ranges of control gains k1 and k2 for leader-following guaranteed cost synchronization with the cost budget given previously. One can see that the added condition is only dependent on the maximum eigenvalue λN without other eigenvalues. Hence, control gains k1 and k2 are still only determined by λ2 and λN without further computational complexity which can also ensure the scalability of the wireless sensor networks. Moreover, Js∗ should be a proper value to be given according to the practical situations so that the control gains k1 and k2 both exist. For example, the cost budget is too small so that the wireless sensor networks cannot work to achieve the synchronization. 4. Numerical simulation In following section, a simulation example is provided to illustrate the effectiveness of the theoretical results obtained in the previous sections. A 2-dimensional wireless sensor network is considered, which is composed of six nodes labeled from 1 to 6. One sets that node 1 as the leader and the other nodes are followers. The dynamics of each node is described as (Eq. 1). This case considers the fixed interaction
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Fig. 1. Interaction topology G.
Fig. 2. State error trajectories of xi (t ) (i = 2, 3, . . . , 6).
topology, which is given as the interaction topology G in Fig. 1, where the weights of edges are 1. The Laplacian matrix associated with G is ⎡ ⎤ 0 0 0 0 0 0 ⎢−1 2 −1 0 0 0⎥ ⎢ ⎥ ⎢0 −1 2 −1 0 0⎥ ⎢ ⎥. L=⎢ 0 −1 2 −1 0⎥ ⎢0 ⎥ ⎣0 0 0 −1 2 −1⎦ −1 0 0 0 −1 2 Moreover, the given cost budget is set as Js∗ = 600, which is an upper bound of cost function in (Eq. 2) including both the energy consumption and the synchronization regulation
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Fig. 3. State error trajectories of vi (t ) (i = 2, 3, . . . , 6).
Fig. 4. Trajectories of cost function.
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performance. The initial states of all nodes are as follows: x1 (0) = [4.0, 2.1]T , x2 (0) = [5.6, 2.1]T , x3 (0) = [3.1, 1.2]T , x4 (0) = [5.9, 2.7]T , x5 (0) = [4.7, 0.9]T , x6 (0) = [8.0, 1.7]T . In cost function (2), positive scalars η = 0.08, γ1 = 0.3 and γ2 = 0.2 are given previously. According to Theorem 2, one can obtain the value ranges of control gains: 3.49 < k1 < 6.12 and 1.19 < k2 < 2.88, so one can choose k1 = 6 and k2 = 1.2. In Figs. 2 and 3, the state errors trajectories of xi (t ) − x1 (t ) and vi (t ) − v1 (t ) (i = 2, 3, . . . , 6) of the wireless sensor networks are shown. Fig. 4 gives the trajectories of the linear quadratic optimization index. Thus, it can be found that the wireless sensor networks achieve leader-following guaranteed cost synchronization with the cost budget given previously. 5. Conclusions Based on the structure property of the second-order wireless sensor networks, both secondorder leader-following guaranteed cost synchronization without and with the given cost budget design problems for wireless sensor networks were investigated by using state errors information between the leader and follower nodes. The second-order leader-following guaranteed cost synchronization design criteria associated with the minimum nonzero and the maximum eigenvalue were proposed by constructing guaranteed cost synchronization protocols and the relationships between the given cost budget and the control gains, where the synchronization protocol can realize the tradeoff design between the battery power consumption and the synchronization regulation performance. Moreover, the inequality of the quadratic form and the function monotony were used to determine the relationships between the given cost budget and the control gains. Especially, the value ranges of control gains are determined to replace control gain matrices solved by LMIs in synchronization criteria. Furthermore, the future research topic can focus on dealing with the impacts of time-varying delays and directed interaction topologies on guaranteed cost synchronization of wireless sensor networks with the given cost budget. Acknowledgments This work was supported by the National Natural Science Foundation of China under Grants 61867005, 61503012, 61703411, 61503009, 61333011 and 61421063, Innovation Foundation of High-Tech Institute of Xi’an (2015ZZDJJ03) and Youth Foundation of HighTech Institute of Xi’an (2016QNJJ004), also supported by Innovation Zone Project under Grants 17-163-11-ZT-004-017-01. References [1] Y.Y. Hu, Z.W. Jin, S. Qi, C.Y. Sun, Estimation fusion for networked systems with multiple asynchronous sensors and stochastic packet dropouts, J. Frankl. Inst. 354 (2017) 145–159. [2] J.Y. Wang, J.W. Feng, C. Xu, Y. Zhao, Synchronization of linear dynamical networks under stochastic impulsive coupling protocols, J. Frankl. Inst. 354 (12) (2017) 4882–4895. [3] J.P. He, H. Li, J.M. Chen, P. Cheng, Study of synchronization-based time synchronization in wireless sensor networks, ISA Trans. 53 (2) (2014) 347–357.
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Journal of the Franklin Institute 356 (2019) 4061–4075 www.elsevier.com/locate/jfranklin
Guaranteed cost synchronization for second-order wireless sensor networks with given cost budgets Xinli Yin a,b, Bailong Yang b, Jianxiang Xi b, Xiaojun Yang a,∗ a The
School of Information Engineering, Guangdong University of Technology, Guangzhou 510006, PR China b Rocket Force University of Engineering, Xi’an 710025, PR China Received 16 June 2018; received in revised form 6 December 2018; accepted 6 January 2019 Available online 19 March 2019
Abstract The current paper addresses leader-following guaranteed cost synchronization with the cost budget given previously for the second-order wireless sensor networks. The published researches on guaranteed cost synchronization design criteria usually are based on the linear matrix inequality (LMI) techniques and cannot take the cost budget given previously into consideration. Firstly, the current paper proposes a guaranteed cost synchronization protocol, which can realize the tradeoff design between the battery power consumption and the synchronization regulation performance. Secondly, for the case without the given cost budget, sufficient conditions for leader-following guaranteed cost synchronization are presented and an upper bound of the cost function is shown. Thirdly, for the case that the cost budget is given previously, the criterion for leader-following guaranteed cost synchronization is proposed. Especially, the value ranges of control gains in these criteria are determined, which means that the existence of control gains in synchronization criteria can be guaranteed, but the LMI techniques can only determine the gain matrix and cannot give the value ranges of control gains. Moreover, these criteria are only associated with the minimum nonzero eigenvalue and the maximum eigenvalue, which can ensure the scalability of the wireless sensor networks. Finally, numerical simulations are given to illustrate theoretical results. © 2019 Published by Elsevier Ltd on behalf of The Franklin Institute.
∗ Corresponding author at: The School of Information Engineering, Guangdong University of Technology, Guangzhou 510006, PR China. E-mail address: [email protected] (X. Yang).
https://doi.org/10.1016/j.jfranklin.2019.01.048 0016-0032/© 2019 Published by Elsevier Ltd on behalf of The Franklin Institute.
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1. Introduction In the past decade, the synchronization of the wireless sensor networks has obtained considerable attention from researchers due to performance degradation caused by network congestion and increasingly higher requirement for the transmission speed and transmission performance [1–7]. As the information communication medium, the wireless sensor networks generally contain one base station (sink node) and many source nodes which can collect, process and transfer the information to the destination. In the process of the information transmission, source nodes send the collected information to the sink node directly or by other source nodes indirectly, which can be modeled as the type of leader-following structure. In this case, the one-many communication mode of the wireless sensor networks easily leads to the network congestion due to the various data transmission rates of bandwidth limited channels and the different queue length of limited buffer space of sensors in [8–12]. Hence, in many practical applications of wireless sensor networks, certain specific performance indexes of groups of nodes need to be synchronizable to suppress network congestion. Furthermore, the state of the leader in the wireless sensor networks is acted as the synchronization state. When the wireless sensor networks achieve synchronization, each sensor on certain specific performance indexes is identical. The resources of sensor nodes in the practical multi-node networks are usually limited such as battery power, memory, and process capabilities, so it is necessary and challenging to achieve the tradeoff design between the system energy consumption and the synchronization regulation performance. In certain published researches, different cost functions of multiple nodes were modeled as the optimization or suboptimization problem. In [13–18], the optimal estimated strategies of synchronization states for multi-node networks were proposed without taking the control energy consumption into consideration. Cao and Ren [18] studied the optimal synchronization problem based on the linear quadratic regulator, where the analysis approach was only applied to the special first-order multi-node networks. In [19,20], the designed suboptimal controllers can guarantee to provide a certain level of performance for the multi-node networks, where only the synchronization regulation performance was discussed. Guo et al. [21] investigated a minimum energy cooperation control problem for the multi-node networks to achieve synchronization, where the mobility energy cost and communication energy cost were considered. In [22], event-triggering containment control strategies for the multi-node networks were discussed, where the communication resources were mitigated. In [23], the synchronization problem with sampled data and packet losses for a class of multi-node systems was studied and some important and interesting results were proposed. In [13–24], these methods cannot achieve guaranteed cost synchronization with the given cost budget; that is, they cannot achieve the tradeoff design between the system energy consumption and the synchronization regulation performance with the cost budget given previously for the multi-node networks. When to explore the conditions of the guaranteed cost synchronization, the linear matrix inequality (LMI) technique is a powerful tool to solve the synchronization criteria for multinode networks. In [25], the minimization of the cost function subject to the synchronization constraint was solved through a set of LMIs, where the guaranteed cost synchronization was not taken into consideration. In [26], Sun proposed several sufficient conditions for average synchronization of the multi-node networks by using the LMI method, which cannot ensure the scalability of multi-node networks since the computational complexity greatly increases when the number of nodes increases. Wang et al. [28] discussed guaranteed cost synchronization for
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general second-order multi-node networks and proposed the synchronization criteria, where the computational complexity was decreased since those criteria are dependent on the minimum nonzero eigenvalue and the maximum eigenvalue of the Laplacian matrix. In [25–29], the synchronization design criteria for the multi-node networks are based on the LMI techniques, which cannot ensure the existence of the gain matrix and cannot determine the value ranges of control gains in synchronization criteria. In this case, based on the structure property of the second-order multi-node networks, it is meaningful to determine the value ranges of control gains in the synchronization design criteria, which can ensure the networks achieve leaderfollowing guaranteed cost synchronization with the cost budget given previously. Moreover, to the best of our knowledge, as the special type of multi-node networks, there are still many open problems to be further investigated for the wireless sensor networks. For the leader-following second-order wireless sensor networks, the current paper proposes a synchronization protocol to address guaranteed cost synchronization design problems, which can realize the tradeoff design between the battery power consumption and the synchronization regulation performance. For the case without giving the cost budget previously, the value ranges of control gains in the guaranteed cost synchronization criterion are determined by the inequality of the quadratic form and the function monotony. An upper bound of the sum of the system energy consumption term and the synchronization regulation term is obtained. For the case with the given cost budget, the relationship between the given cost budget and control gains is determined, which depends on the initial states, the minimum nonzero eigenvalue and the maximum eigenvalue and can ensure the scalability of the wireless sensor networks. The synchronization design criterion is proposed for the wireless sensor networks to achieve leader-following guaranteed cost synchronization. Moreover, the relationship between the given cost budget and control gains can ensure that the cost consumption of the wireless sensor networks is limited. Compared with the published results [28–30] about guaranteed cost synchronization for the multi-node networks, the current paper has the following contributions. Firstly, the value ranges of control gains are determined to achieve guaranteed cost synchronization, which can ensure the existence of control gains in the synchronization design criteria. The synchronization criteria in [28,29] are obtained by the gain matrix using LMI tools. In certain situations, the synchronization criteria may not obtain the feasible gain matrix. Secondly, the cost budget can be given previously, which can limit the energy consumption of the wireless sensor networks to extend the battery lifetime of sensors. For practical wireless sensor networks, each sensor usually has the limited energy, so the cost budget should be a finite value given previously. The research results in [28–30] determined different upper bounds of the guaranteed cost without considering the limited resources of the wireless sensor networks. The current paper takes the limited resources into account and gives the cost budget previously. The remainder of the current paper is organized as follows. In Section 2, some preliminaries and the problem description are presented, respectively. Section 3 gives a sufficient condition of second-order guaranteed cost synchronization for the leader-following wireless sensor networks without the given cost budget, obtains an upper bound of the guaranteed cost, and determines second-order guaranteed cost synchronization design criteria for the leaderfollowing wireless sensor networks with the given cost budget. Section 4 shows the numerical example to demonstrate theoretical results. Some concluding remarks are given in Section 5. Notations: Rn is the n-dimensional real column vector space and Rn×n is the set of n × n dimensional real matrices. In represents the n-dimensional identity matrix. PT = P < 0 and PT = P > 0 mean that the symmetric matrix P is negative definite and positive definite,
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respectively. The notation diag{d1 , d2 , . . . , dN } represents a diagonal matrix with diagonal elements d1 , d2 , . . . , dN . 2. Preliminaries and problem description In following section, the concepts and modeling interaction topology on graph theory are given firstly. Then, the problem description is presented. 2.1. Modeling interaction topology on graph theory The current paper models the interaction topology of the wireless sensor networks with N identical nodes by a graph G = (V (G ), E (G )), which is composed of a nonempty vertex set V (G ) = {v1 , v2 , . . . , vN } and the edge set E = {ei j = {(vi , v j )} ∈ V × V . The vertex vi represents sensor node i, the edge eij denotes the interaction channel from sensor node i to sensor node j, and the edge weight w ji of eij stands for the interaction strength from sensor node i to sensor node j. The index of the set of all neighbors of vertex v j is denoted by N between vertex vi1 and vertex vil is a sequence of edges j = i : (vi , v j ) ∈ E (G ) . A path v i 1 , v i 2 , v i 2 , v i 3 , . . . , vil−1 , vil . Define the Laplacian matrix of the graph G as L = l ji ∈ RN×N with l j j = i∈N j w ji and l ji = −w ji ( j = i ). A directed graph has a spanning tree if there exists a root node which has a directed path to any other nodes. It is assumed that the whole interaction topology of the wireless sensor networks has a spanning tree. The root node denotes the leader, where the leader does not obtain any state information from followers, only some follower nodes can obtain the state information from the leader node, and the local interaction topology among followers is undirected. More basic concepts and conclusions on graph theory can be found in [36]. 2.2. Problem description For the wireless sensor networks consisting of N identical second-order linear sensor nodes, one sets that sink node 1 is the leader and the other N − 1 nodes are followers. The dynamics of the jth sensor node is described by x˙ j (t ) = v j (t ), (1) v˙ j (t ) = u j (t ), where j = 1, 2, . . . , N, xj (t) ∈ R, v j (t ) ∈ R and uj (t) ∈ R are the queue length, the transmission rate and the control input, respectively. Since the leader does not receive any state information from other nodes, one can set that u1 (t) ≡ 0. For the given γ 1 , γ 2 ∈ R > 0 and η ∈ R > 0, a guaranteed cost synchronization protocol with a cost function is proposed as follows:
u j (t ) = w ji k1 xi (t ) − x j (t ) + k2 vi (t ) − v j (t ) , (2) ∞i∈N j Js = 0 (Ju (t ) + Jx (t ) )dt, where j = 2, 3, . . . , N, Nj denotes the neighbor set of node j , and Ju (t ) =
N j=1
ηu2j (t ),
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Jx (t ) =
N
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2 2 w ji γ1 xi (t ) − x j (t ) + γ2 vi (t ) − v j (t ) .
j=1 i∈N j
Ju (t) is constructed by the control inputs while Jx (t) is constructed by state errors. Furthermore, from the perspective of engineering, Ju (t) and Jx (t) represent the battery power consumption of wireless sensor nodes and the system synchronization performance on the queue length and the transmission rate, respectively. Positive parameters γ 1 , γ 2 and η can be chosen freely based on the demand for synchronization problems and all of them are independent of each other. Choosing different γ 1 , γ 2 and η mean different tradeoff design between Ju (t) and Jx (t). In this paper, our object is to determine control gains of synchronization protocols for the given upper bound of the linear quadratic index which means the given cost budget. One can set Js∗ > 0 as a given cost budget, then the definition of leader-following guaranteed cost synchronization with the given cost budget for the wireless sensor networks is proposed as follows. Definition 1. For any given Js∗ > 0, wireless sensor network (1) is said to be leaderfollowing by protocol (2) if there exist k1 and k2 such that guaranteed cost synchronizable limt→∞ x j (t ) − x1 (t ) = 0, limt→∞ v j (t ) − v1 (t ) = 0 ( j = 2, 3, . . . , N ) and Js ≤ Js∗ for any bounded disagreement initial states xj (0) and v j (0) ( j = 2, 3, . . . , N ). The main aim of this paper is to design the feasible value ranges of control gains k1 > 0 and k2 > 0 such that wireless sensor network (1) with the leader-following structure achieves guaranteed cost synchronization under the condition that the cost budget is previously given. Remark 1. Compared with protocols in [31–35] about the synchronization for networks, there are two critical features of protocol (2). The first one is that the relevant research results achieve guaranteed cost synchronization based on the LMI techniques, which need a feasp solver to obtain the feasible control gain matrix. In this case, the control gain matrix may not exist as shown in [28]. It should be pointed out that the feasp solver is a numerical algorithm, which cannot give the analytic solutions. This paper presents an approach to give the value ranges of k1 > 0 and k2 > 0 which can ensure the wireless sensor networks achieve leader-following guaranteed cost synchronization. The second one is that protocol (2) gives the cost budget previously. The main challenge is to determine the relationship between the cost budget given previously and an upper bound of the sum of battery power consumption and the synchronization regulation performance. It requires to design the specific k1 > 0 and k2 > 0 such that the existence of control gains can be ensured and the sum of battery power consumption and the synchronization regulation performance are less than the cost budget given previously. Moreover, our methods cannot deal with high-order multi-node networks. 3. Guaranteed cost synchronization for wireless sensor networks This section mainly proposes a sufficient condition for guaranteed cost synchronization of the second-order wireless sensor networks modeled as the leader-following structure without giving the cost budget previously, obtains an upper bound of the sum of the energy consumption term and the synchronization regulation term, and gives a sufficient condition to achieve the leader-following guaranteed cost synchronization with the cost budget given previously.
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Let Lff be the Laplacian matrix of the interaction topology among follower nodes, l f l = [w21 , w31 , . . . , wN1 ]T and fl represent the interaction weightsfrom the leader to followers. 0 0 . One can The whole Laplacian matrix L of the interaction topology is −l f l L f f + f l obtain that L f f + f l is symmetric and positive. Since the network interaction topology has a spanning tree and the local interaction topology among follower nodes is undirected, there exists an orthonormal matrix U˜ such that T ff fl ˜ ˜ U L + U = diag{λ2 , λ3 , . . . , λN } with 0 < λ2 ≤ λ3 ≤ . . . ≤ λN being eigenvalues of L f f + f l . Let x j (t ) = x j (t ) − x1 (t ), v j (t ) = v j (t ) − v1 (t )( j = 2, 3, . . . , N ), X (t ) = [x2 (t ), v2 (t ), x3 (t ), v3 (t ), . . . , xN (t ), vN (t )]T , and
T U˜ T [x2 (t ), x3 (t ), . . . , xN (t )]T = x˜2 (t ), x˜3 (t ), . . . , x˜N (t ) , T (3) U˜ T [v2 (t ), v3 (t ), · · · , vN (t )]T = v˜2 (t ), v˜3 (t ), · · · , v˜N (t ) . In this case, one can obtain by Eqs. (1)–(3) that ⎧ ⎨x˙ j (t ) = x˙ j (t ) − x˙1 (t ) = v j (t ) − v1 (t ) = v j (t ), w ji k1 xi (t ) − x j (t ) + k2 vi (t ) − v j (t ) , ⎩v˙ j (t ) = v˙ j (t ) − v˙1 (t ) =
(4)
i∈N j
⎧ T T ⎪ ⎨U˜ T [x˙2 (t ), x˙3 (t ), . . . , x˙N (t )] = v˜2 (t ), v˜3 (t ), . . . , v˜N (t ) , U˜ T [v˙2 (t ), v˙3 (t ), . . . , v˙N (t )]T = −k1 λ2 x˜2 (t ) − k2 λ2 v˜2 (t ), −k1 λ3 x˜3 (t ) − k2 λ3 v˜3 (t ), T ⎪ ⎩ . . . , −k1 λN x˜N (t ) − k2 λN v˜N (t ) . (5) Hence, one can obtain by Eqs. (3)–(5) that x˜˙ j (t ) = v˜ j (t ), v˙˜ j (t ) = −k1 λ j x˜ j (t ) − k2 λ j v˜ j (t ),
(6)
where j = 2, 3, . . . , N . Due to U˜ is an orthonormal matrix, one can obtain that x j (t ) = 0 and v j (t ) = 0 when x˜ j (t ) = 0 and v˜ j (t ) = 0 ( j = 2, 3, . . . , N ), which mean that the wireless sensor networks achieve synchronization. Theorem 1. Wireless sensor network (1) is leader-following guaranteed cost synchronizable by protocol (2) if there exist k1 and k2 such that k22 k22 k2 k2 γ1 2 γ1 − + − < k < − , 1 2 2 4η 16η ηλ2 2η 4η ηλ2 16ηγ1 1 1 2 γ2 max , 4 γ2 < k 2 < + − . 2 2 λ2 2ηλN ηλN 4 η λN Proof. Firstly, we give sufficient conditions such that limt→∞ x˜ j (t ) = 0 and limt→∞ v˜ j (t ) = 0 ( j = 2, 3, . . . , N ), which can obtain the leader-following guaranteed cost synchronization. One can adopt a Lyapunov function candidate as follows: V j (t ) = 0.5λ2j k22 + λ j k1 x˜2j (t ) + v˜2j (t ) + λ j k2 x˜ j (t )v˜ j (t ), (7)
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where j = 2, 3, . . . , N. Since k1 > 0, k2 > 0 and λj > 0 ( j = 2, 3, . . . , N ), one has 0.5λ2j k22 + 0.5λ2j k22 + λ j k1 0.5λ j k2 > 0. Hence, one can get that Vj (t) ≥ 0. Moreλ j k1 > 0 and det 0.5λ j k2 1 over, the time derivative of Vj (t) with Eq. (6) is V˙ j (t ) = −k1 k2 λ2j x˜2j (t ) − k2 λ j v˜2j (t ).
(8)
Due to k1 k2 λ2j > 0 and k2 λj > 0, one can get that V˙ j (t ) ≤ 0 and V˙ j (t ) = 0 if and only if x˜ j (t ) = 0 and v˜ j (t ) = 0. Hence, the wireless sensor networks can achieve synchronization. In the following, an upper bound of the guaranteed cost is determined. It can be obtained by Eq. (2) that 2 ∞ ff k1 k2 k1 T fl 2 X (t )dt , Ju = η X (t ) L + (9) k1 k2 k22 0 Jx = 2
∞
X T (t )
0
ff γ L + fl 1 0
0 γ2
X (t )dt .
(10)
Then 2 ff k1 fl 2 X (t ) L + k1 k2
k1 k2 k22
T
X T (t )
ff γ L + fl 1 0
0 γ2
N 2 X (t ) = λ2j k1 x˜ j (t ) + k2 v˜ j (t ) ,
(11)
j=2
N X (t ) = λ j γ1 x˜2j (t ) + γ2 v˜2j (t ) .
(12)
j=2
Hence, it can be obtained that ∞ Js = (Ju (t ) + Jx (t ) )dt 0 ⎞ ⎛ ∞ N N 2 ⎝η = λ2j k1 x˜ j (t ) + k2 v˜ j (t ) + 2 λ j γ1 x˜2j (t ) + γ2 v˜2j (t ) ⎠dt 0
=
j=2
N ∞ j=2
+
0
2 2 ηλ j k1 + 2λ j γ1 x˜2j (t )dt +
j=2
N j=2
j=2 N ∞
0
∞
0
2 2 ηλ j k2 + 2λ j γ2 v˜2j (t )dt
2ηλ2j k1 k2 x˜ j (t )v˜ j (t )dt.
(13)
∞ ∞ Let Ju = 0 Ju (t )dt , Jx = 0 Jx (t )dt , V (t ) = Nj=2 V j (t ), j11 = ηλ2j k12 + 2λ j γ1 − k1 k2 λ2j , j12 = ηλ2j k1 k2 and j22 = ηλ2j k22 + 2λ j γ2 − k2 λ j , j = 2, 3, . . . , N . Due to Js = 0
∞
(Ju (t ) + Jx (t ) )dt +
N j=2
0
∞
V˙ j (t )dt − V (t )|t→∞ + V (0)
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=
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+
N 2 2 ηλ j k1 + 2λ j γ1 − k1 k2 λ2j x˜2j (t )dt +
∞ 0
j=2
N j=2
∞ 0
0
∞
2 2 ηλ j k2 + 2λ j γ2 − k2 λ j v˜2j (t )dt
2ηλ2j k1 k2 x˜ j (t )v˜ j (t )dt − V (t )|t→∞ + V (0),
(14)
one can obtain that Js =
N j=2
∞ 0
j11 x˜ j (t ), v˜ j (t ) j12
j12 j22
T x˜ j (t ), v˜ j (t ) dt − V (t )|t→∞ + V (0).
(15)
Set ηλj k12 + 2γ1 − k1 k2 λ j < 0, ηλ j k22 + 2γ2 − k2 < 0 and ηλ j k12 (2γ2 − k2 )+ (2γ1 −k1 k2 λ j ηλ j k22 + 2γ1 − k1 k2 λ j (2γ2 − k2 ) > 0, then sufficient conditions are proposed: ηλ j k12 + 2γ1 − k1 k2 λ j < 0, ηλ j k22 + 2γ2 − k2 < 0, 2ηλ j k12 < −2γ1 + k1 k2 λ j 2 and 2ηλ j k2 < k2 − 2γ2 .Hence, one can design the value ranges of k1√ and k2 : 2 2 k2 /4η − k2 /16η − γ1 /ηλ2 < k1 < k2 /2η + k22 /4η2 − 2γ1 /ηλ2 and max 16ηγ1 /λ2 , 1/4ηλ j − 1/16η2 λ2j − γ2 /ηλ j < k2 < 1/2ηλN + 1/4η2 λ2N − 2γ2 /ηλN . Let y = 1/4ηλ − 1/16η2 λ2 − γ2 /ηλ, where λ > 0. By the derivation of the function, one can obtain the point of extreme value and get the maximum value of the function y. Hence, one gets the maximum value of the function which is 4γ 2 when λ = 1/16ηγ2 . As a result, it can be √ obtained that max 16ηγ1 /λ2 , 4γ2 < k2 < 1/2ηλN + 1/4η2 λ2N − 2γ2 /ηλN . The designed j11 j12 is negative definite and value ranges of k1 and k2 can guarantee that matrix j12 j22 from Eqs. (13) to (15), one can obtain that Js ≤ V (0).
(16)
As a result, it can guarantee that the wireless sensor networks achieve leader-following guaranteed cost synchronization. Based on the above analysis, the conclusion of Theorem 1 can be obtained. Remark 2. In this case, k1 and k2 are designed such that the wireless sensor networks achieve leader-following guaranteed cost synchronization without the cost budget given previously. One can see that the wireless sensor networks can achieve leader-following synchronization when k1 and k2 are designed as positive values. However, when the guaranteed cost synchronization is taken into consideration, it is difficult to design feasible control gains k1 and k2 to achieve synchronization. In this case, control gains k1 and k2 are separated from the quadratic form which contains two correlated variables k1 and k2 . One can see that k2 can be determined independently but k1 is dependent on k2 . Here, based on the inequality of the quadratic form and the function monotony, control gains k1 and k2 can be determined to the ranges of values simultaneously. Control gains k1 and k2 are dependent on the minimum nonzero eigenvalue λ2 and the maximum eigenvalue λN . To decrease the computational complexity, λ2 and λN can be estimated by using the approach in [36]. Moreover, λN ≤ 2d with
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d = max{l j j = i∈N j w ji , j = 2, 3, . . ., N } is estimated based on the Gersgorin disc theorem in [37]. It is not necessary to get all the eigenvalues of the interaction topology. Remark 3. In the associated work about achieving guaranteed cost synchronization, LMI techniques by the feasp solver are used to determine the feasible control gains matrix. In certain situations, a feasp solver may not obtain a feasible control gains matrix for multinode networks to achieve guaranteed cost synchronization. In this case, the value ranges of existence of control gains k1 and k2 are required to be designed, which can guarantee that the wireless sensor networks achieve leader-following guaranteed cost synchronization. One can choose the values of the control gains within the value ranges without the verification of feasibility. In this case, by the inequality of the quadratic form and the function monotony, control gains k1 and k2 can be determined to the value ranges simultaneously. Moreover, when one designs the control gain k2 , a simplified value range of k2 is determined by finding the maximum point based on the function monotony. Therefore, it reduces the range of value range of control gain k2 convergence. Based on the above analysis, Theorem 2 presents an approach to determine the control gains k1 and k2 with the cost budget given previously such that the wireless sensor networks achieve leader-following guaranteed cost synchronization. Set x(0) = (x2 (0), x3 (0), . . . , xN (0))T and v(0) = (v2 (0), v3 (0), . . . , vN (0))T . ff 2 T It can be obtained f l x(0) ≤ λ2N x T (0) x(0), x T (0) L f f + f l ! T that x ! (0) L + v(0) ≤ λN !x (0) v(0)! and x T (0) L f f + f l x(0) ≤ λN x T (0)x(0). Theorem 2 is proposed as follows. Theorem 2. For any given Js∗ > 0, wireless sensor network (1) achieves leader-following guaranteed cost synchronization by protocol (2) if there exist control gains k1 and k2 such that 16ηγ1 1 1 2 γ2 max , 4 γ2 < k 2 < + − , 2 2 λ2 2ηλN ηλN 4 η λN ⎛ k22 k22 k γ1 2 γ1 2 + − < k1 < min ⎝ − , 2 2 16η ηλ2 2η 4η ηλ2 ! ! −k22 λ2N x T (0)x(0) − 2k2 λN !x T (0) v(0)! + 2Js∗ − 2vT (0) v(0) . 2λN x T (0)x(0)
k2 − 4η
Proof. Firstly, by Eqs. (6) and (7), one can see that N 0.5λ2j k22 + λ j k1 x˜2j (t ) + v˜2j (t ) + λ j k2 x˜ j (t )v˜ j (t ) V (t ) = j=2
2 = 0.5k22 x T (t ) L f f + f l x(t ) + k2 x T (t ) L f f + f l v(t ) +k1 x T (t ) L f f + f l x(t ) + vT (t )v(t ).
(17)
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Let t = 0, then 2 V (0 ) = 0.5k22 x T (0) L f f + f l x(0) + k2 x T (0) L f f + f l v(0) +k1 x T (0) L f f + f l x(0) + vT (0)v(0) ! ! ≤ 0.5k 2 λ2 x T (0)x(0) + k2 λN λN !x T (0) v(0)! + k1 λN x T (0)x(0) 2 N
+ vT (0)v(0).
(18)
By Eqs. (16) and! (18), one can see that V (0 ) ≤ 0.5k22 λ2N x T (0)x(0) + ! k2 λN λN !x T (0) v(0)! + k1 λN x T (0)x(0) + vT (0)v(0) ≤ Js∗ guarantee √can that Js ≤ Js∗ . In this case, one can design control gains: max 16ηγ1 /λ2 , 4γ2 ) < k2 < 1/2ηλN + 1/4η2 λ2N − 2γ2 /ηλN and k2 /4η − k22 /16η2 − γ1 /ηλ2 < k1 ! ! < min x(0) − 2k2 λN !x T (0)v(0)! (k2 /2η + k22 /4η2 − 2γ1 /ηλ2 , −k22 λ2N x T (0) + 2Js∗ − 2vT (0)v(0) / 2λN x T (0)x(0) , which can ensure the wireless sensor networks achieve leader-following guaranteed cost synchronization with the given cost budget Js∗ . Hence, the proof is completed. Remark 4. In Theorem 1, the guaranteed cost is determined by the value of the Lyapunov function candidate at time zero. One can see that the expression of the guaranteed cost contains initial states of all the nodes. In Theorem 2, due to the practical energy limitation, one gives the cost budget previously. Hence, it should take the energy limitation requirement into consideration to choose proper values of the control gains k1 and k2 . There are two difficulties in obtaining Theorems 2. The first one is to construct the relationship between the linear quadratic optimization index and the interaction weights matrix. The second one is to construct the relationship between the given cost budget and the control gains k1 and k2 . By an upper ! bound of the ! guaranteed cost, one can see that V (0 ) ≤ 0.5k22 λ2N x T (0)x(0) + k2 λN !x T (0)v(0)! + k1 λN x T (0)x(0) + vT (0) v(0) ≤ Js∗ can guarantee that the guaranteed cost is less than ! the cost bud! get given previously. Hence, one adds 0.5k22 λ2N x T (0)x(0) + k2 λN !x T (0)v(0)! + k1 λN x T (0)x(0) + vT (0) v(0) ≤ Js∗ as a condition for selecting value ranges of control gains k1 and k2 for leader-following guaranteed cost synchronization with the cost budget given previously. One can see that the added condition is only dependent on the maximum eigenvalue λN without other eigenvalues. Hence, control gains k1 and k2 are still only determined by λ2 and λN without further computational complexity which can also ensure the scalability of the wireless sensor networks. Moreover, Js∗ should be a proper value to be given according to the practical situations so that the control gains k1 and k2 both exist. For example, the cost budget is too small so that the wireless sensor networks cannot work to achieve the synchronization. 4. Numerical simulation In following section, a simulation example is provided to illustrate the effectiveness of the theoretical results obtained in the previous sections. A 2-dimensional wireless sensor network is considered, which is composed of six nodes labeled from 1 to 6. One sets that node 1 as the leader and the other nodes are followers. The dynamics of each node is described as (Eq. 1). This case considers the fixed interaction
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Fig. 1. Interaction topology G.
Fig. 2. State error trajectories of xi (t ) (i = 2, 3, . . . , 6).
topology, which is given as the interaction topology G in Fig. 1, where the weights of edges are 1. The Laplacian matrix associated with G is ⎡ ⎤ 0 0 0 0 0 0 ⎢−1 2 −1 0 0 0⎥ ⎢ ⎥ ⎢0 −1 2 −1 0 0⎥ ⎢ ⎥. L=⎢ 0 −1 2 −1 0⎥ ⎢0 ⎥ ⎣0 0 0 −1 2 −1⎦ −1 0 0 0 −1 2 Moreover, the given cost budget is set as Js∗ = 600, which is an upper bound of cost function in (Eq. 2) including both the energy consumption and the synchronization regulation
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Fig. 3. State error trajectories of vi (t ) (i = 2, 3, . . . , 6).
Fig. 4. Trajectories of cost function.
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performance. The initial states of all nodes are as follows: x1 (0) = [4.0, 2.1]T , x2 (0) = [5.6, 2.1]T , x3 (0) = [3.1, 1.2]T , x4 (0) = [5.9, 2.7]T , x5 (0) = [4.7, 0.9]T , x6 (0) = [8.0, 1.7]T . In cost function (2), positive scalars η = 0.08, γ1 = 0.3 and γ2 = 0.2 are given previously. According to Theorem 2, one can obtain the value ranges of control gains: 3.49 < k1 < 6.12 and 1.19 < k2 < 2.88, so one can choose k1 = 6 and k2 = 1.2. In Figs. 2 and 3, the state errors trajectories of xi (t ) − x1 (t ) and vi (t ) − v1 (t ) (i = 2, 3, . . . , 6) of the wireless sensor networks are shown. Fig. 4 gives the trajectories of the linear quadratic optimization index. Thus, it can be found that the wireless sensor networks achieve leader-following guaranteed cost synchronization with the cost budget given previously. 5. Conclusions Based on the structure property of the second-order wireless sensor networks, both secondorder leader-following guaranteed cost synchronization without and with the given cost budget design problems for wireless sensor networks were investigated by using state errors information between the leader and follower nodes. The second-order leader-following guaranteed cost synchronization design criteria associated with the minimum nonzero and the maximum eigenvalue were proposed by constructing guaranteed cost synchronization protocols and the relationships between the given cost budget and the control gains, where the synchronization protocol can realize the tradeoff design between the battery power consumption and the synchronization regulation performance. Moreover, the inequality of the quadratic form and the function monotony were used to determine the relationships between the given cost budget and the control gains. Especially, the value ranges of control gains are determined to replace control gain matrices solved by LMIs in synchronization criteria. Furthermore, the future research topic can focus on dealing with the impacts of time-varying delays and directed interaction topologies on guaranteed cost synchronization of wireless sensor networks with the given cost budget. Acknowledgments This work was supported by the National Natural Science Foundation of China under Grants 61867005, 61503012, 61703411, 61503009, 61333011 and 61421063, Innovation Foundation of High-Tech Institute of Xi’an (2015ZZDJJ03) and Youth Foundation of HighTech Institute of Xi’an (2016QNJJ004), also supported by Innovation Zone Project under Grants 17-163-11-ZT-004-017-01. References [1] Y.Y. Hu, Z.W. Jin, S. Qi, C.Y. Sun, Estimation fusion for networked systems with multiple asynchronous sensors and stochastic packet dropouts, J. Frankl. Inst. 354 (2017) 145–159. [2] J.Y. Wang, J.W. Feng, C. Xu, Y. Zhao, Synchronization of linear dynamical networks under stochastic impulsive coupling protocols, J. Frankl. Inst. 354 (12) (2017) 4882–4895. [3] J.P. He, H. Li, J.M. Chen, P. Cheng, Study of synchronization-based time synchronization in wireless sensor networks, ISA Trans. 53 (2) (2014) 347–357.
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