Dynamic Output Feedback Guaranteed-Cost Synchronization for Multiagent Networks with Given Cost Budgets

# Dynamic Output Feedback Guaranteed-Cost Synchronization for Multiagent Networks with Given Cost Budgets

## Abstract

The current paper addresses the distributed guaranteed-cost synchronization problems for general high-order linear multiagent networks. Existing works on the guaranteed-cost synchronization usually require all state information of neighboring agents and cannot give the cost budget previously. For both leaderless and leader-following interaction topologies, the current paper firstly proposes a dynamic output feedback synchronization protocol with guaranteed-cost constraints, which can realize the tradeoff design between the energy consumption and the synchronization regulation performance with the given cost budget. Then, according to different structure features of interaction topologies, leaderless and leader-following guaranteed-cost synchronization analysis and design criteria are presented, respectively, and an algorithm is proposed to deal with the impacts of nonlinear terms by using both synchronization analysis and design criteria. Especially, an explicit expression of the synchronization function is shown for leaderless cases, which is independent of protocol states and the given cost budget. Finally, numerical examples are presented to demonstrate theoretical results.

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Multiagent network, guaranteed-cost synchronization, dynamic output feedback, cost budget.

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## 1 Introduction

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In recent years, synchronization of multiagent networks with distributed control protocols has obtained great attention by researchers from different fields, formation and containment control, sensor networks, multiple agent supporting systems, distributed computation, multiple robot systems and network congestion alleviation, et al. [1]-[15]. According to different structures, multiagent networks are usually categorized into two types: leader-following ones and leaderless ones, which are associated with leader-following synchronization and leaderless synchronization, respectively. Moreover, the motions of multiagent networks contain two parts: the whole motion and the relative motions among agents. For leader-following multiagent networks, the whole motion is the motion of the leader. However, for leaderless multiagent networks, the whole motion is associated with the interaction topology and initial states of all agents and is often described by the synchronization function. In [16], some novel conclusions for robust synchronization were given. Sakthivel et al. [17] proposed an inspirational method to deal with stochastic faulty actuator-based reliable synchronization problems. The literatures [18]-[22] also proposed some new results on synchronization. It should be pointed out that the performance optimization was not considered in [16]-[22].

However, in practical multiagent networks, the control energy is usually limited, so it is required to simultaneously consider the following two factors: the synchronization regulation performance and the energy consumption, which can be modeled as certain optimal or suboptimal problems with different cost functions to realize the tradeoff design between them. By optimizing the cost function of each agent, some synchronization control strategies were shown to achieve global goals in [23] and [24]. By constructing the global performance index based on the linear quadratic cost function, Cao and Ren [25] presented an optimal synchronization criteria for first-order linear multiagent networks under the condition that the interaction topology is a complete graph. For first-order nonlinear multiagent networks, optimal synchronization criteria were proposed by convex and coercive properties of the cost function in [26] and [27]. For second-order linear multiagent networks, synchronization regulation performance problems were discussed by hybrid impulsive control approaches in [28] and [29], where the energy consumption was not considered. Cheng et al. [30] dealt with leader-following guaranteed-cost synchronization of second-order multiagent networks, which can realize the suboptimal synchronization tracking, and investigated the applications of theoretical results to interconnected pendulums. In [23]-[30], the dynamics of each agent has a specific structure, which can simplify the synchronization analysis and design problems.

Due to the complex structure of general high-order multiagent networks, optimal synchronization is usually difficult to be achieved and guaranteed-cost synchronization is more challenging than first-order and second-order multiagent networks. Zhao et al. [31] discussed guaranteed-cost synchronization for general high-order linear multiagent networks with the linear quadratic cost function based on state errors among neighboring agents and control inputs of all agents. Zhou et al. [32] proposed an event-triggered guaranteed-cost control method to decrease the energy consumption. In [33], sampled-data information was used to design guaranteed-cost synchronization prototols and an input delay approch was applied to give guaranteed-cost synchronization criteria. In [31]-[33], the linear matrix inequality (LMI) synchronization design criteria contain the Laplacian matrix and the dimensions of variables are associated with the number of agents, which cannot ensure the scalability of multiagent networks since the computational complexity greatly increases as the number of agents increases. To overcome this flaw, the state decomposition approach was shown to deal with guaranteed-cost synchronization in [34]-[36], where LMI synchronization design criteria are only dependent on the nonzero eigenvalues of the Laplacian matrix and the dimensions of all the variables are identical with the one of each agent. Moreover, Xie and Yang [37] proposed sufficient conditions for guaranteed-cost fault-tolarant synchronization by introducing a coupling weight larger than the reciprocal of the minimum nonzero eigenvalue of the Laplacian matrix, where the dimension of the variable of the algebraic Riccati equality is independent of the number of agents.

Although some significant research results on guaranteed-cost synchronization were presented, there still exist many very challenging and open problems. The current paper mainly focuses on the following two aspects: (i) The cost budget is given previously. For practical multiagent networks, each agent usually has the limited energy, so the cost budget cannot be infinite and should be a finite value given previously. In [31]-[37], different upper bounds of the guaranteed cost were determined, but they cannot be given previously; (ii) The outputs instead of the states of neighboring agents are used to construct the synchronization protocol. In practical applications, each agent often can only observe its neighbors and obtain output information which may be partial states or linear combinations of states. It is well-known that output feedback synchronization control is more complex and challengeable than state feedback synchronization control. In [31]-[37], all state information of neighboring agents is required to realize the guaranteed-cost synchronization control.

For leaderless and leader-following general high-order linear multiagent networks with the given cost budgets, the current paper proposes a dynamic output feedback synchronization protocol with a specific structure to deal with guaranteed-cost synchronization analysis and design problems. For leaderless cases, the relationship between the given cost budget and the LMI variable is constructed by initial states of all agents and the Laplacian matrix of a complete graph, guaranteed-cost synchronization analysis and design criteria are proposed, respectively, and the synchronization function is determined. For leader-following cases, the relationship between the given cost budget and the LMI variable is determined via initial states of all agents and the Laplacian matrix of a star graph, and sufficient conditions for guaranteed-cost synchronization criteria are presented by LMI tools. Moreover, based on the cone complementarity approach, an algorithm is proposed to check guaranteed-cost synchronization design criteria which contain nonlinear matrix inequality constraints.

Compared with closely related works on guaranteed-cost synchronization, the current paper has two critical innovations. The first one is that the cost budget is given previously in the current paper. The literatures [31]-[37] only determined different upper bounds of the guaranteed cost, but cannot previously give the cost budget. The second one is that the current paper proposes dynamic output feedback synchronization protocols with the linear quadratic optimization index. The literatures [31]-[37] required all state information of neighboring agents to construct guaranteed-cost synchronization protocols.

The remainder of the current paper is organized as follows. In Section 2, some preliminaries on graph theory and the problem description are presented, respectively. Section 3 gives guaranteed-cost synchronization criteria for leaderless multiagent networks with dynamic output feedback synchronization protocols and the given cost budget, and determines an explicit expression of the synchronization function. Section 4 presents leader-following guaranteed-cost synchronization criteria. Section 5 shows numerical examples to illustrate theoretical results. Some concluding remarks are given in Section 6.

Notations: is the -dimensional real column vector space and is the set of dimensional real matrices. represents the -dimensional identity matrix. denotes a column vector with all components 1. 0 and stand for the zero number and the zero column vector with a compatible dimension, respectively. The notation in a symmetric matrix denotes the symmetric term. The symbol represents the Kronecker product. and mean that the symmetric matrix is negative definite and positive definite, respectively. The notation represents a diagonal matrix with the diagonal elements . The notation denotes the trace of the matrix .

## 2 Preliminaries and problem description

### 2.1 Preliminaries on graph theory

The current paper models the interaction topology of a multiagent network with identical agents by a graph , which is composed by a nonempty vertex set and the edge set . The vertex represents agent , the edge denotes the interaction channel from agent to agent , and the edge weight of stands for the interaction strength from agent to agent . The index of the set of all neighbors of vertex is denoted by . A path between vertex and vertex is a sequence of edges . An undirected graph is said to be connected if there at least exists an undirected path between any two vertices. A directed graph has a spanning tree if there exists a root node which has a directed path to any other nodes. Define the Laplacian matrix of the graph as with and . If the undirected graph is connected, then zero is a simple eigenvalue of , and all the other eigenvalues are positive. If the directed graph has a spanning tree, then zero is a simple eigenvalue of , and all the other eigenvalues have positive real parts. More basic concepts and conclusions on graph theory can be found in [38].

### 2.2 Problem description

For multiagent networks consisting of identical high-order linear agents, the dynamics of the th agent is described by

 {˙xj(t)=Axj(t)+Buj(t),yj(t)=Cxj(t), (1)

where , , , and , and are the state, the output and the control input, respectively. For and , a dynamic output feedback synchronization protocol with a linear quadratic optimization index is proposed as follows:

 ⎧⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎩˙ϕj(t)=(A+BKu)ϕj(t)−KϕC∑i∈Njwji(ϕi(t)−ϕj(t))+Kϕ∑i∈Njwji(yi(t)−yj(t)),uj(t)=Kuϕj(t),Js=∫∞0(Ju(t)+Jxϕ(t))dt, (2)

where , with is the protocol state, and are gain matrices with compatible dimensions to be determined, represents the neighbor set of agent and

 Ju(t)=N∑j=1uTj(t)Ruj(t),
 Jxϕ(t)=N∑j=1∑i∈Nj(wji(xi(t)−xj(t)−ϕi(t)+ϕj(t))T
 ×Q(xi(t)−xj(t)−ϕi(t)+ϕj(t))).

Furthermore, and are called the energy consumption term and the synchronization regulation term, respectively, and the tradeoff design between the energy consumption and the synchronization regulation performance can be realized by choosing proper and . It should be pointed out that there also exists the linear quadratic index to realize guaranteed-cost control for isolated systems as shown in [40], but its structure is different with the one in (2). For isolated systems, the linear quadratic index is constructed by state information, which is convergent. For multiagent networks, it is required that state errors among agents are convergent, but states of each agent may be divergent. Hence, the linear quadratic index for multiagent networks should be constructed by state errors as shown in (2), and cannot use state information. Furthermore, guaranteed-cost control can be clarified into two types. The first one is to calculate the upper bound of the linear quadratic index for given gain matrices as shown in [31]-[37]. The second one is to determine gain matrices of synchronization protocols for the given upper bound of the linear quadratic index; that is, the given cost budget. Moreover, it can be shown that , which means that the term directly impacts on the derivative of the protocol state and indirectly impacts on the derivative of the state of each agent. Hence, we choose as the index function of the synchronization regulation performance.

Let be a given cost budget, then the definition of guaranteed-cost synchronization of multiagent networks with the given cost budget is proposed as follows.

Definition 1:  For any given , multiagent network (1) is said to be guaranteed-cost synchronizable by protocol (2) if there exist and such that and for any bounded disagreement initial states , where is said to be the synchronization function.

The main objects of the current paper are to design and such that multiagent network (1) with leaderless and leader-following structures achieves guaranteed-cost synchronization under the condition that the cost budget is given, and to determine the impacts of the state of the synchronization protocol and the given cost budget on the synchronization function for leaderless cases.

Remark 1: Compared with guaranteed-cost synchronization protocols in [31]-[37], protocol (2) has two critical features. The first one is that outputs instead of states of neighboring agents are applied to construct synchronization protocols. For dynamic output feedback synchronization protocols, the key challenge is that the upper bound of the optimization index is difficult to be determined since both the energy consumption term and the synchronization regulation term are dependent on protocol states. The second one is that the cost budget is given previously. In this case, the key challenge is to determine the relationship between the upper bound of the optimization index and the given cost budget and to design gain matrices of synchronization protocols such that the upper bound is less than the given cost budget. Moreover, compared with the traditional dynamic output feedback controller for isolated systems as shown in classic literatures [39] and [40], the key difference is that output errors between one agent and its neighbors are used to construct synchronization protocols for multiagent networks as shown in (2). It should be pointed out that the state of each agent may be not convergent, but it is required that state errors among all agents are convergent under protocol (2). However, it is needed that the states of an isolated system are convergent by designing the dynamic output feedback controller.

## 3 Guaranteed-cost synchronization for leaderless multiagent networks

For high-order linear multiagent networks with leaderless connected topologies, this section gives sufficient conditions for guaranteed-cost synchronization design and analysis with the given cost budget, respectively, where the guaranteed-cost synchronization design criterion contains a nonlinear constraint, so an algorithm is proposed to determine gain matrices on the basis of the cone complementarity approach. Moreover, an explicit expression of the synchronization function is shown, which is independent of the protocol state and the given cost budget.

Let and then the dynamics of multiagent network (1) with protocol (2) can be written as

 ⎧⎪ ⎪⎨⎪ ⎪⎩˙x(t)=(IN⊗A)x(t)+(IN⊗BKu)ϕ(t),˙ϕ(t)=(IN⊗(A+BKu)+(L⊗KϕC))ϕ(t)−(L⊗KϕC)x(t). (3)

Because the interaction topology is undirected, the Laplacian matrix is symmetric and positive semi-definite. Due to , there exists an orthonormal matrix such that , where with . Let

 ^x(t)=(UT⊗In)x(t)=[^xT1(t),^xT2(t),⋯,^xTN(t)]T, (4)
 ^ϕ(t)=(UT⊗In)ϕ(t)=[^ϕT1(t),^ϕT2(t),⋯,^ϕTN(t)]T, (5)

then multiagent network (3) can be transformed into

 {˙^x1(t)=A^x1(t)+BKu^ϕ1(t),˙^ϕ1(t)=(A+BKu)^ϕ1(t), (6)
 ⎧⎪ ⎪ ⎪⎨⎪ ⎪ ⎪⎩˙^xj(t)=A^xj(t)+BKu^ϕj(t),˙^ϕj(t)=(A+BKu+λjKϕC)^ϕj(t)−λjKϕC^xj(t), (7)

where .

The -dimensional column vector with the th element 1 and 0 elsewhere is denoted by . Define

 xe(t)Δ=N∑j=2Uej⊗^xj(t), (8)
 xs(t)Δ=1√N1⊗^x1(t), (9)

then one can show by (8) that

 xe(t)=(U⊗In)[0T,^xT2(t),^xT3(t),⋯,^xTN(t)]T. (10)

By and , it can be derived from (9) that

 xs(t)=(U⊗In)[^xT1(t),0T]T. (11)

Since is nonsingular, and are linearly independent by (10) and (11). From (4), one can obtain that . By the structure of given in (9), multiagent network (3) achieves leaderless synchronization if and only if and is a valid candidate of the synchronization function. Thus, and can be regarded as the error state among agents and the synchronization state of multiagent network (3), which stands for the disagreement part and the agreement part, respectively. Furthermore, one can find by (7) that can guarantee that multiagent network (1) with protocol (2) achieves leaderless synchronization.

Based on the above analysis, the following theorem presents an approach to determine gain matrices and such that multiagent network (1) with protocol (2) achieves leaderless guaranteed-cost synchronization with a given cost budget.

Theorem 1:  For any given , multiagent network (1) is leaderless guaranteed-cost synchronizable by protocol (2) if there exist and such that

 ^Ξ1=xT(0)((IN−N−111T)⊗In)x(0)Px−J∗sIn≤0,
 ^Ξj=⎡⎢ ⎢⎣Ξ11−λj^PxCTC^KTuR∗Ξj220∗0−R⎤⎥ ⎥⎦<0(j=2,N),
 Px^Px=In,

where and In this case, and .

Proof:   First of all, we give sufficient conditions by LMI techniques such that . One can derive that

 [^ϕj(t)^ϕj(t)−^xj(t)]=[In0In−In][^ϕj(t)^xj(t)], (12)

so subsystems (7) can be converted into

 ⎡⎢⎣˙^ϕj(t)˙^ϕj(t)−˙^xj(t)⎤⎥⎦=[A+BKuλjKϕC0A+λjKϕC]
 ×[^ϕj(t)^ϕj(t)−^xj(t)]. (13)

Let and be symmetric and positive definite matrices, then we construct a Lyapunov function candidate as follows

 Vj(t)=Vϕj(t)+Vxj(t), (14)

where and

 Vϕj(t)=^ϕTj(t)Pϕ^ϕj(t),
 Vxj(t)=(^ϕj(t)−^xj(t))TPx(^ϕj(t)−^xj(t)).

From (13) to (14), one can show that

 ˙Vϕj=^ϕTj(t)(Pϕ(A+BKu)+(A+BKu)TPϕ)^ϕj(t)+2λj^ϕTj(t)PϕKϕC(^ϕj(t)−^xj(t)),
 ˙Vxj=(^ϕj(t)−^xj(t))T(Px(A+λjKϕC)+(A+λjKϕC)TPx)(^ϕj(t)−^xj(t)).

Thus, it can be derived that and if

 Θj=[Pϕ(A+BKu)+(A+BKu)TPϕ∗λjPϕKϕCPx(A+λjKϕC)+(A+λjKϕC)TPx]<0, (15)

where , which means that multiagent network (1) with protocol (2) achieves leaderless synchronization due to .

In the following, the guaranteed-cost performance is discussed. Due to , one can show that . By (6), one has . Thus, it can be obtained by (4) and (5) that

 Ju(t)=ϕT(t)(IN⊗KTuRKu)ϕ(t)
 =N∑j=2^ϕTj(t)KTuRKu^ϕj(t), (16)
 Jxϕ(t)=(ϕ(t)−x(t))T(2L⊗Q)(ϕ(t)−x(t))
 =N∑j=22λj(^ϕj(t)−^xj(t))TQ(^ϕj(t)−^xj(t)). (17)

For , we can derive from (15) to (17) that

 JsTΔ=∫T0(Ju(t)+Jxϕ(t))dt=∫T0(Ju(t)+Jxϕ(t))dt+N∑j=2(∫T0˙Vj(t)dt−Vj(T)+Vj(0))=N∑j=2∫T0(^ϕTj(t)Pϕ((A+BKu)P−1ϕ+P−1ϕ(A+BKu)T+P−1ϕKTuRKuP−1ϕ)Pϕ^ϕj(t)+2λj^ϕTj(t)PϕKϕC(^ϕj(t)−^xj(t))+(^ϕj(t)−^xj(t))T×(Px(A+λjKϕC)+(A+λjKϕC)TPx+2λjQ)×(^ϕj(t)−^xj(t)))dt−N∑j=2(Vj(T)−Vj(0)).

Let with and with . By Schur Complement Lemma in [41] , if , then as tends to infinity, one has

 Js≤N∑j=2Vj(0).

Due to  , one has   by (5), which means that and . Thus, one can find that

 Js≤N∑j=2^xTj(0)Px^xj(0)
 =xT(0)(U⊗In)[0TI(N−1)n](IN−1⊗Px)
 ×[0,I(N−1)n](UT⊗In)x(0). (18)

Since , it can be shown that

 ^U^UT=IN−N−111T. (19)

Due to

 [0,I(N−1)n](UT⊗In)=^UT⊗In,

one can derive by (18) and (19) that

 Js≤xT(0)((IN−N−111T)⊗Px)x(0). (20)

Because are disagreement, there exists some . Thus, one can derive that

 xT(0)((IN−N−111T)⊗In)x(0)=N∑j=2^xTj(0)^xj(0)>0.

Hence, one can set that

 γ=J∗sxT(0)((IN−N−111T)⊗In)x(0);

that is,

 J∗s=xT(0)((IN−N−111T)⊗γIn)x(0). (21)

Since has a simple zero eigenvalue and nonzero eigenvalues, can guarantee that by (20) and (21). Based on the above analysis, by the convex property of LMIs, the conclusion of Theorem 1 can be obtained.

Remark 2: The specific structures of coefficient matrices of protocol (2) make subsystems (7) satisfy some separation principle; that is, their dynamics can transformed into the ones in (13). In this case, and can be independently designed such that and are Hurwitz, which can guarantee that multiagent network (1) with protocol (2) but without the optimization index achieves leaderless synchronization. However, when the guaranteed-cost performance is considered, the impacts of the term in (13) cannot be neglected since can directly influence the derivative of via the term . In this case, by left- and right-multiplying with , can be determined but cannot. Here, by introducing a specific structure , the gain matrices and can be determined simultaneously.

Remark 3: In the associated works about guaranteed-cost control, the value of the Lyapunov function candidate at time zero is used to determine the guaranteed cost. Since and in (7) couple with each other, it seems difficult to construct a Lyapunov function candidate such that the expression of the upper bound of does not contain initial states of synchronization protocols. Based on the separation principle, a Lyapunov function candidate is proposed in (14), which makes an upper bound of only dependent on initial states of all agents under the assumption that initial states of protocol (2) are zero. In this case, the relationship between the upper bound of and can be determined by the property of , which actually is the Laplacian matrix of a complete graph with edge weights equal to . It should be pointed out that it will become very difficult to determine the relationship between and if initial states of protocol (2) are nonzero, and the assumption that initial states of protocol (2) are zero is reasonable for practical multiagent networks.

In the proof of Theorem 1, the changing variable method is used to determine gain matrices and , which makes the guaranteed-cost synchronization design criterion contain the nonlinear constraint . However, if and are given previously, then this nonlinear constraint can be eliminated. The following corollary gives a leaderless guaranteed-cost synchronization analysis criterion.

Corollary 1:  For any given , and , multiagent network (1) with protocol (2) achieves leaderless guaranteed-cost synchronization if there exist and such that

 ^Θ1=xT(0)((IN−N−111T)⊗In)x(0)Px−J∗sIn≤0,
 ^Θj=⎡⎢ ⎢⎣Θ11λjPϕKϕCKTuR∗Θj220∗0−R⎤⎥ ⎥⎦<0(j=2,N),

where and .

In Theorem 1, the leaderless guaranteed-cost synchronization criterion contains a nonlinear constraint, which cannot be directly checked by LMI tools. Based on Corollary 1, the cone complementarity approach proposed by Ghaoui et al. in [42] can deal with this nonlinear constraint by minimizing the trace of . The feasibility problem of matrix inequalities in Theorem 1 can be transformed into the following minimization one:

 mintr(Px^Px)sbuject to^Ξ1<0,^Ξj<0(j=2,N),^Ξ3=[PxI∗^Px]≥0.

The following algorithm is presented to solve the above minimization problem.

Algorithm 1:

Step 1:  Set . Check the feasibility of and , and give and .

Step 2:  Minimize the trace of subject to and . Let and .

Step 3:  Let and . If and in Corollary 1 are feasible and for some sufficiently small scalar , then stop and give and .

Step 4:  If is larger than the maximum allowed iteration number, then stop.

Step 5:  Set and go to Step 2.

By the above analysis, is a valid candidate of the synchronization function. Due to , one can obtain that and by (6), which means that protocol states do not influence the synchronization function when initial protocol states are equal to zero. Moreover, it can be shown that