# Sign-Consensus of Linear Multiagent Systems under a State Observer Protocol

Authors: Wei-Lu Diao; Cui-Qin Ma
Journal: Complexity
 (2019)

---

## Abstract

The sign-consensus problem for linear time-invariant systems under signed digraph is considered. The information of the agents’ states is reconstructed, and then, a state observer-type sign-consensus protocol is proposed, whose performance is analyzed using matrix analysis and ordinary differential equation theory. Sufficient conditions for ensuring sign-consensus are given. It is proven that if the adjacency matrix of the signed digraph has strong Perron–Frobenius property or is eventually positive, sign-consensus can be achieved under the proposed protocol. In particular, conventional consensus is a special case of sign-consensus under mild conditions.


---

## Body

## 1. Introduction

Consensus, as the key to coordination of multiagent systems (MASs), has been investigated extensively in recent years [1–9]. Most existing studies on consensus of MASs usually assume cooperative interactions among the agents, while in many cases, the agents can not only cooperate but also compete with each other, resulting in the coexistence of cooperation and competition in MASs, e.g., the two-party political system and the business alliance of competitors. To describe this scenario, the concept of bipartite consensus is proposed in [10], which means that all agents converge to the same value as required by conventional consensus but with different signs in bipartite consensus. Altafini [10] proves that, under the assumption of signed digraph being strongly connected, single-integrator MASs reach bipartite consensus if and only if the signed digraph is structurally balanced. Following this line, the strongly connected requirement on signed digraph in [10] is relaxed in [11], bipartite consensus under switched signed digraph is investigated in [12], and bipartite consensus for general linear MASs has also been studied in [13]. In [14–18], measurement noise is further considered. As pointed out in [15], the signed digraph being structurally balanced is one of the necessary and sufficient communication conditions to guarantee bipartite consensus regardless of measurement noise. This means that structural balance plays a crucial role in bipartite consensus. Structural balance is fragile and can be easily broken by changing the sign of some edge weight or by adding or deleting an edge in a signed digraph, meaning that structural unbalance can be more often seen in practice.In fact, studies on structurally unbalanced signed digraph have already been reported in [14, 15, 19–24], etc. In [14, 15], agents converge to zero in mean square under arbitrary initial conditions. In [19], the agents’ states are proved to lie in between the polarized values, which are called the interval bipartite consensus. Another topic closely related to structural unbalance is the unanimity of opinion [21]; that is, all agents achieve an agreement or disagreement on a certain subject but with different extents. This concept is further extended in [23] to sign-consensus, meaning that agents reach values with the same sign but different modulus. The concept of sign-consensus is rooted in reality. For example, in a social network, people rarely have the same comment on a topic, but may have the same tendency. In the pioneering work [23], it is shown that, for the linear time-invariant (LTI) MASs, if the graph adjacency matrix is eventually positive, sign-consensus is achieved under a state feedback sign-consensus protocol and a fully distributed sign-consensus protocol, respectively. The fixed signed digraph in [23] is extended in [24] to being switched over time. Sign-consensus protocols are proposed for three types of time-varying graphs, respectively. It is shown that sign-consensus can be reached if the graph adjacency matrix is frequently eventually positive. We notice that both works [23, 24] propose state feedback-type sign-consensus protocols. However, in reality, due to constraints on measurement, it is usually hard to directly measure agents’ states, but only the state estimates are available. Hence, it would be more convenient to synthesize consensus protocols based on the agents’ state estimates.With these observations, in this work, we investigate sign-consensus for LTI MASs under signed digraphs. The agents’ states are reconstructed, and state observer-type protocols based on them are given. By using tools from matrix analysis and ordinary differential equation theory, the closed-loop system is analyzed. It is shown that if the graph adjacency matrix has strong Perron–Frobenius property or is eventually positive, sign-consensus for LTI MASs can be achieved. Our main contributions are as follows. First, the agent dynamics are extended to be LTI system, not limited to integrators [21, 22]. Second, state observer-type consensus protocols based on state estimates between neighboring agents are proposed, which are more practical than most existing protocols based on the agents’ states [23, 24].


### 1.1. Organization

The state observer-type consensus protocol is proposed in Section2. The main results with respect to sign-consensus are given in Section 3. A simulation example is given to verify the proposed theoretical results in Section 4. The paper is concluded in Section 5.


### 1.2. Notations and Preliminaries

For a matrix or vectorA, if all its elements are positive, A is said to be positive, denoted as A>0. For A∈ℝn×n, λiAi=1,…,n is the eigenvalue of A, ReλiA is the real part of λiA, and ρA is the spectral radius, which is the smallest positive real number satisfying ρA≥λiA,∀i=1,…,n. A∈ℝn×n has the strong Perron–Frobenius property if ρA is a simple positive eigenvalue of A, and its corresponding right eigenvector vr>0. A∈ℝn×n is eventually positive if ∃l0∈ℕ satisfying Al>0, ∀l≥l0. For a given vector v=v1,…,vnT∈ℝn, signv=signv1,…,signvnT, where sign is the sign function. 1n=1,…,1T∈ℝn, ⊗ is the Kronecker product.G
=
V
,
ɛ
,
ℱ is a signed digraph with V=1,…,N, ɛ⊂V×V, and the adjacency matrix ℱ=fij∈ℝN×N. Ni=j,i∈ℰ. Generally, ℱ=fij∈ℝN×N is defined by fii=0, fij≠0⟺j,i∈ε; otherwise, fij=0. Denote Dℱ=ρℱIN−ℱ with ρℱ being the spectral radius of ℱ.Lemma 1 (see [23]).
AssumeG=V,ɛ,ℱ is a signed digraph. If ℱ is eventually positive, then G is structurally unbalanced.Lemma 2 (see [25]).
ForA∈ℝn×n, A has the strong Perron–Frobenius property ⟺ A is eventually positive.


## 1.1. Organization

The state observer-type consensus protocol is proposed in Section2. The main results with respect to sign-consensus are given in Section 3. A simulation example is given to verify the proposed theoretical results in Section 4. The paper is concluded in Section 5.


## 1.2. Notations and Preliminaries

For a matrix or vectorA, if all its elements are positive, A is said to be positive, denoted as A>0. For A∈ℝn×n, λiAi=1,…,n is the eigenvalue of A, ReλiA is the real part of λiA, and ρA is the spectral radius, which is the smallest positive real number satisfying ρA≥λiA,∀i=1,…,n. A∈ℝn×n has the strong Perron–Frobenius property if ρA is a simple positive eigenvalue of A, and its corresponding right eigenvector vr>0. A∈ℝn×n is eventually positive if ∃l0∈ℕ satisfying Al>0, ∀l≥l0. For a given vector v=v1,…,vnT∈ℝn, signv=signv1,…,signvnT, where sign is the sign function. 1n=1,…,1T∈ℝn, ⊗ is the Kronecker product.G
=
V
,
ɛ
,
ℱ is a signed digraph with V=1,…,N, ɛ⊂V×V, and the adjacency matrix ℱ=fij∈ℝN×N. Ni=j,i∈ℰ. Generally, ℱ=fij∈ℝN×N is defined by fii=0, fij≠0⟺j,i∈ε; otherwise, fij=0. Denote Dℱ=ρℱIN−ℱ with ρℱ being the spectral radius of ℱ.Lemma 1 (see [23]).
AssumeG=V,ɛ,ℱ is a signed digraph. If ℱ is eventually positive, then G is structurally unbalanced.Lemma 2 (see [25]).
ForA∈ℝn×n, A has the strong Perron–Frobenius property ⟺ A is eventually positive.


## 2. Problem Formulation

Consider an MAS withN agents with each agent being described by(1)x˙i=Axi+Bui,yi=Cxi,∀i=1,…,N,where xi∈ℝn,ui∈ℝm,andyi∈ℝq are the state, input, and output, respectively. It is assumed that A,B,C is controllable and observable.The communication topology among agents is represented by a signed digraphG=V,ɛ,ℱ, where V=1,…,N and ℱ=fijN×N. This paper is to synthesize a state observer-type control for each agent such that agents in (1) reach sign-consensus. Firstly, we give the definition of sign-consensus for the system in (1).Definition 1 (see [24]).
The system in (1) is said to achieve sign-consensus if there exists a consensus protocol ui,i=1,…,N such that for any given initial states xi0i=1,…,N,(2)limt⟶∞signxit−signxjt=0,i,j∈1,…,N.
From Definition 1, given any initial value, the states of agents in (1) converge to values with the same sign but different modulus.
Letx¯i and y¯i be the estimates of agent i’s state and output, respectively. Denote y^i=yi−y¯i=Cxi−Cx¯i as agent i’s output estimate error. In practice, it is usually hard to measure the agent’s state. Hence, it would be more convenient to design consensus protocols based on the estimates of the agents’ states. The useful information for agent i is synthesized by other agents’ state estimates as(3)δi¯=−σix¯i+∑j∈Nifijx¯j,i=1,…,N,where σi>0 denotes the degradation rate of the system. A state observer-type consensus protocol is given as follows:(4)ui=aHδi¯=aH−σix¯i+∑j∈Nifijx¯j,x¯˙i=Ax¯i+Bui−aGy^i,i=1,…,N,where(i)
a∈ℝ satisfies a≥1/2mini∈ΔReλiDℱ with Δ=iReλiDℱ>0,i=1,…,N;(ii)
H is designed as H=R−1BTP, where P is the positive definite solution of the algebraic Riccati equation(5)ATP+PA+Q−PBR−1BTP=0,whereQandR are positive definite matrices;(iii)
SinceA,C is observable, G can be chosen such that A+aGC is Hurwitz.
Next, we will demonstrate that the system in (1) achieves sign-consensus under the protocol in (4).


## 3. Main Results

Theorem 1.
Suppose thatℱ has the strong Perron–Frobenius property. Then, the system in (1) can reach sign-consensus under the protocol in (4) with σi=ρℱ∀i=1,…,N. Moreover, the state estimate error will converge to 0, i.e., limt⟶∞xi−x¯i=0 and ∀i=1,…,N.Proof.
DefineX=x1T,…,xNTT and X¯=x¯1T,…,x¯NTT. Applying the protocol in (4) to the system in (1), we obtain(6)X˙=IN⊗AX−aDℱ⊗BHX¯.By direct calculation, we obtain(7)X¯˙=IN⊗A−aDℱ⊗BHX¯−aIN⊗GCX^,where X^=X−X¯. Clearly,(8)X^˙=IN⊗A+aGCX^.
SinceG is chosen such that A+aGC is Hurwitz, IN⊗A+aGC is Hurwitz. Immediately, one has(9)limt⟶∞X^=0,i.e.,limt⟶∞xi−x¯i=0,i=1,…,N.
By assumption,ℱ has the strong Perron–Frobenius property. This implies that ρℱ is a simple positive eigenvalue of ℱ, and its corresponding right eigenvector vr>0. Noting that Dℱ=ρℱIN−ℱ. Hence, λiDℱ=ρℱ−λiℱ, i=1,…,N. Clearly, 0 is a simple eigenvalue of Dℱ with corresponding right eigenvector vr. For simplicity, we assume λ1Dℱ=0. Then, ReλiDℱ>0, i=2,…,N. There must exist a nonsingular matrix S=vrS2∈ℝN×N with S2∈ℝN×N−1 such that(10)S−1DℱS=J=000JN−1,where JN−1∈ℝN−1×N−1 is a Jordan block with λiDℱi=2,…,N on its diagonal. Let Wt=S−1⊗InX≜W1Tt,…,WNTtT=W1TtΩTtT and W^t=S−1⊗InX^≜W^1TtΩ^TtT. Then, by (6)–(8), we have(11)W˙t=IN⊗A−aJ⊗BHWt+aJ⊗BHW^t,or equivalently,(12)W˙1t=AW1t,Ω˙t=IN−1⊗A−aJN−1⊗BHΩt+aJN−1⊗BHΩ^t.
Obviously,IN−1⊗A−aJN−1⊗BH is an upper triangle block matrix with diagonal blocks A−aλiDℱBH, i=2,…,N. Notice that a≥1/2mini∈ΔReλiDℱ and H=R−1BTP, where P is positive definite solution of (5). Then, by adopting the similar arguments as in [26], we obtain that A−aλiDℱBHi=2,…,N is Hurwitz. By (9), we know that limt⟶∞W^t=limt⟶∞S−1⊗InX^=0, and hence, limt⟶∞Ω^t=0. According to (12) and the ordinary differential equation theory, we obtain(13)limt⟶∞W1t=limt⟶∞eAtW10,limt⟶∞Ωt=0.
SinceX=S⊗InWt=vr⊗InW1t+S2⊗InΩt,(14)limt⟶∞Xt=vr⊗Inlimt⟶∞eAtW10,or equivalently,(15)limt⟶∞xit=vrilimt⟶∞eAtW10,where vri is the ith component of vr. Note that vri>0,∀i=1,…,N. Thus, by Definition 1, the system in (1) can achieve sign-consensus under the protocol in (4).Remark 1.
In Theorem 1, sign-consensus is achieved based on a state observer-type protocol. This is different from protocols in [23, 24] where state feedback protocols are proposed. Communication topology G=V,ɛ,ℱ is directed, and it is assumed that ℱ has the strong Perron–Frobenius property. Such an assumption together with Lemma 1 implies that G is structurally unbalanced.Theorem 2.
For the system in (1), suppose that ℱ is eventually positive. Then, the system in (1) can achieve sign-consensus under the protocol in (4) with σi=ρℱ∀i=1,…,N.Proof.
By assumption,ℱ is eventually positive. This together with Lemma 2 gives that ℱ has the strong Perron–Frobenius property. Applying the same procedure as in Theorem 1, one sees that the system in (1) can achieve sign-consensus under the protocol in (4).
From (14), we know that vr is vital to the collective behavior of MASs. In particular, if vr=1n, then conventional consensus is achieved.Corollary 1.
For the system in (1), suppose that ℱ has the strong Perron–Frobenius property. Then, the system in (1) can achieve conventional consensus under the protocol in (4) with vr=1n and σi=ρℱ∀i=1,…,N.Remark 2.
In Corollary 1, if the requirement onℱ is changed to being eventually positive, Corollary 1 still holds.


## 4. Simulation

Example 1.
Consider the system in (1) with 6 agents, where(16)A=1100,B=01,C=2101.
Obviously,A,B,C is controllable and observable. Assume that interactions among the 6 agents are represented by G=V,ɛ,ℱ, where V=1,…,6 and ℱ=fij∈ℝ6×6 with f12=f25=f35=2, f13=f36=−1, f14=1, f24=f42=5, f26=f51=f63=3, and f61=4. From Figure 1, G is structurally unbalanced. By direct calculation, one has ℱl>0,∀l≥7. This implies that ℱ is eventually positive. Therefore, by Lemma 2, one sees that ℱ has strong Perron–Frobenius property, and hence, communication conditions in Theorem 1 and Theorem 2 are satisfied.
Note thatλ1Dℱ=0, λ2Dℱ=10.4347, λ3Dℱ=6.7544−1.4418jj2=−1, λ4Dℱ=6.7544+1.4418j, λ5Dℱ=5.5158−2.1144j, and λ6Dℱ=5.5158+2.1144j. Therefore, a≥1/2min2≤i≤6ReλiDℱ=0.0906, and hence, we assume a=1. Let G=−100−1. By calculation, one obtains that A+aGC is Hurwitz. Choose Q=CCT and R=1. Then, one can get P=12.34855.44955.44953.4495 which is the positive definite solution of (5), and H=R−1BTP=5.44953.4495∈ℝ1×2. Applying the protocol in (4) to the system in (1), we obtain that sign-consensus is achieved. The simulation results are given in Figures 2 and 3, where xi1 and xi2 are the first and second components of agent i, respectively. Moreover, from Figures 4 and 5, we see that state estimate error converges to zero. Thus, the simulation example verifies the validity of Theorem 1 and Theorem 2.Figure 1
Communication topology among 6 agents.Figure 2
State trajectories ofxi1 for 6 agents.Figure 3
State trajectories ofxi2 for 6 agents.Figure 4
State estimate error trajectories ofxi1.Figure 5
State estimate error trajectories ofxi2.


## 5. Conclusions

The sign-consensus of MASs is investigated where each agent is described by an LTI system. A state observer-type protocol is designed, which is more practical than the usual state feedback protocols. It is shown that if the adjacency matrix has strong Perron–Frobenius property or is eventually positive, then sign-consensus can be achieved based on the proposed protocols.



---

*Source: 3010465-2019-11-03.xml*