Distributed Consensus Design for a Class of Uncertain Linear Multiagent Systems under Unbalanced Randomly Switching Directed Topologies

This paper proposes a novel approach to design fully distributed consensus controllers for heterogeneous linearMultiagent Systems subjected to randomly switching directed topologies andmodel uncertainties.The appealing features of this approach are as follows. First, it uses the mildest assumption for the randomly switching topologies that the union of switched graphs has a spanning tree. Second, the consensus is achieved under a class of state multiplicative uncertainties. Moreover, the proposed consensus controllers are low-rank and have nonconservative coupling strengths. Finally, a numerical example is presented to illustrate the effectiveness of the proposed theoretical approach.


Introduction
Multiagent systems (MASs) have been widely studying from both theoretical viewpoints and practical applications, e.g., power networks, transportation systems, and autonomous vehicle groups, where a set of dynamical systems collaborate with their few neighbors to achieve their own targets as well as the whole system's objective.One of the most important and extensively investigated area in MASs is the consensus problem where agents' states or outputs reach a nonzero agreement [1][2][3][4][5].
Most of the studies so far on consensus of linear MASs with generic models of agents require solving Linear Matrix Inequality (LMI) problems and have a lower bound on the coupling strength depending on the eigenvalue with smallest nonzero real part of the graph Laplacian matrix, e.g., [6][7][8].Therefore, the associated consensus algorithms are not fully distributed, and other algorithms to approximate eigenvalues of Laplacian matrix or to adjust the coupling strength in an adaptive way are needed.This unexpectedly increases the complexity and the computational efforts of consensus algorithms and possibly slows down the convergence to consensus.Another direction is to treat the agents from the passivity framework, e.g., [9], but additional conditions on the system matrices are needed, which limits the class of agents.Other works may utilize higher derivatives of agents' states, e.g., [10], which are not possible in many practical situations.
The studies for consensus of MASs under switching topologies often employ strong assumptions on each switched graph, such as strongly connectedness (e.g., [7]), or balance (e.g., [8,11]), or both (e.g., [2,9]).Those assumptions for deterministically switching graphs can be relaxed to be weaker, e.g., the switched undirected graphs are jointly connected [1]; the union of switched directed graphs remains strongly connected after sometime [12] or has a spanning tree frequently enough [10,13].On the other hand, the relaxed assumptions for randomly switching graphs still require the balance of the union of switched graphs, e.g., [14].In fact, consensus can be achieved under randomly switching topologies without that assumption.Thus, further investigations are emergent to overcome this limitation.
Linear Quadratic Regulator (LQR), a classical control design method, has been proven to be an effective approach for developing consensus algorithms [15][16][17][18][19][20][21], which usually results in solving structured Riccati equations.As previously shown, e.g., in [19,21] and references therein, low-rank consensus design can reduce the computational cost and possibly increases the consensus speed.Furthermore, by employing the idea of selective pole shift to LQR, we have recently proposed a fully distributed approach in [21] to achieve not only a low-rank consensus controller but also a nonconservative consensus coupling strength for general linear MASs.Unfortunately, the communication graph in [21] is limited to be undirected and fixed.These results were then extended in [14] to both fixed and randomly switching directed graphs and also to handle state matrix uncertainty.However, union of switching graphs in [14] was assumed to be balanced and a single zero eigenvalue of system matrix of agents' dynamics was also assumed.This research will relax those assumptions to be less restrictive.
Most importantly, the current study proposes a fully distributed consensus design for heterogeneous linear MASs with multiplicative state uncertainty and randomly switching topologies where none of them is balanced or has a spanning tree and their union is unbalanced.This problem has not been addressed in the literature, to the best of our knowledge, and hence clearly constitutes the main contribution of the current research.Another contribution is that the proposed consensus controller has low rank and nonconservative coupling strength.
The following notations and symbols will be used in the paper.The notations R, C, C 0 − , and C − stand for the sets of real, complex, complex with nonpositive real part, and complex with negative real part numbers.Moreover, 1  and 0  denote the  × 1 vector with all elements equal to 1 and 0, respectively, and   denotes the  ×  identity matrix.Next, Re() denotes the real part of a complex number  and ⊗ stands for the Kronecker product.On the other hand, () and  min () denote the eigenvalue set and the eigenvalue with smallest, nonzero real part of , respectively.Lastly, ≻ and ⪰ denote the positive definiteness and positive semidefiniteness of a matrix.A directed path connecting vertices  and  in G is a set of consecutive edges starting from  and stopping at .Then G is said to have a spanning tree if there exists a node from which there are directed paths to every other node.

Problem Description
Lemma 1 (see [13]).The Laplacian matrix L always has a zero eigenvalue with associated eigenvector 1  , and all nonzero eigenvalues of L have positive real parts.Furthermore, L has only one zero eigenvalue if and only if G has a spanning tree.
The communication among agents considered in this research is randomly switched, which can be described by a continuous-time Markov process defined as follows.A continuous-time Markov process {  :  ≥ 0}, which takes values on the set N = {1, 2, . . ., }, is defined on the probability space (Ω, F, P) where Ω is the space of elementary events, F is the underlying -field on Ω, and P is a probability measure on F. Let the infinitesimal generator be defined as follows [8,22]: where (⋅) stands for the Little-o notation and   represents the transition rate from state  to state  with   ≥ 0 as  ̸ =  and It can be easily seen that the row sums of Q are zero.
Next, denote G() the randomly switching graph which switches among the elements of a finite set S G ≜ {G 1 , . . ., G ℓ }, with a switching signal () ∈ {1, . . ., ℓ}, and L() the Laplacian matrix associated with G().Also, the time index  is used for all parameters related to the timevarying graph G() such as   () and N  ().The transition rate matrix Q = [  ] ∈ R ℓ×ℓ .For  = 1, . . ., ℓ, let L  be the Laplacian matrix corresponding to the graph G  .Denote  = [ 1 , . . .,  ℓ ]  the stationary distribution of this Markov process which is assumed to be ergodic.Thus,  is unique,   > 0 ∀ = 1, . . ., ℓ, and each state of the Markov chain can be reached from any other state.Furthermore, we can also assume that the Markov process starts from  [8].As a result, the distribution of () is equal to  for all  ≥ 0. Let  * = min =1,...,ℓ   .Finally, let G ∪ ≜ ⋃ =1,...,ℓ G  denote the union of all possible switching topologies and L ∪ denote its Laplacian matrix.

Multiagent System Dynamics. Consider a class of MASs composing of 𝑁 linear agents subjected to multiplicative parametric uncertainties,
where   ∈ R  ,   ∈ R  are vectors of states and control inputs of agent ;  ∈ R × ,  ∈ R × are nominal state matrices; Δ  ∈ R × represent the state uncertainties.Note that due to the uncertainties Δ  agents' dynamics is heterogeneous.
In literature, a widely used consensus protocol for the linear MASs in (2) without uncertainties (see, e.g., [2-4, 20, 21]) is where  ∈ R × is a consensus controller gain matrix and  > 0 is called the coupling strength.In this research, we also employ (3) for the uncertain MAS ( 2) and aim at finding  and  such that the consensus of agents is achieved, which is defined below.
Next, the following assumptions are employed, where assumptions (A2) and (A4) are for a finite and nonzero consensus state and (A2) is inspired by [23].

Remark 3. Assumption (A3
) is the mildest one for consensus with randomly switching directed graphs, which significantly and obviously advances the existing results in the literature that require stronger assumptions such as the balance of every switched graph G  ∀ = 1, . . ., ℓ (see, e.g., [6,8]), or the balance of G ∪ [14,24].On the other hand, assumption (A4) is made to be consistent with assumption (A2) and the latter consensus design and proof.This is an assumption to simplify the consensus design, and if it is not satisfied then another consensus design must be sought.Further, a bound of the uncertainty Δ  is usually known; hence tools from robust control theory can be utilized to express the assumption (A4) in the form of an LMI.

Consensus Design for Unbalanced Randomly Switching Graphs and Model Uncertainty
3.1.Leaderless Consensus Design.Let  ∈ R ×  ⪰ 0 be the unique solution of the following Riccati equation: where  ∈ R × ,  ≻ 0 and  ∈ R × , and  ⪰ 0 are design parameters and ( 1/2 , ) is detectable.Let V ∈ R × be the matrix whose rows are left eigenvectors associated with the zero eigenvalue of .Due to assumption (A2), rank(V) = .Consequently, it can be shown that  must have the form  = V  PV, where P ∈ R × , P ⪰ 0, with  selected to be  = V  QV, Q ∈ R × , Q ⪰ 0. This can be proved using the similar method in [21], so we skip here for brevity.Then the main result of this research is presented in the following theorem.

Under assumptions (A1)-(A4), the uncertain MAS (2) reaches a mean-square consensus by the distributed rank-𝛼 controller,
for any  > 0,  = 1, . . ., , () (  () −   ()) .( 6) Furthermore, the consensus speed is lower bounded by Proof.The basic idea here is similar to that in [8], but significant difficulties need to be resolved upon system uncertainties and unbalanced switched graphs.Moreover, the consensus state is also different from that in [8] due to the unbalanced union graph.
Let  ∈ R  be such that   L ∪ = 0 and 1 where 1 {•} is the Dirac measure.Obviously, () = ∑ ℓ =1   ().Then using Lemma 4.2 in [22], we have Mathematical Problems in Engineering Therefore, Accordingly, because   ≥  * ∀ = 1, . . ., ℓ, ∑ ℓ =1 L  = L ∪ and ML ∪ = L ∪ .In fact, L ∪ +L  ∪ can be regarded as a Laplacian matrix of a connected undirected graph, so there exists an orthogonal matrix  ∈ R × such that L ∪ + L  ∪ = Λ  , where Λ = diag{  } =1,..., and   are eigenvalues of L ∪ + L  ∪ , and This finally leads to Thus, () converges to a stationary point at which V() = 0 due to Lasalle's invariance principle.It can be easily seen that V() = 0 if and only if () = 0. Hence, the meansquare consensus is achieved with the speed lower bounded by  *  min (L ∪ + L  ∪ ), i.e., (7).Theorem 4 provides a powerful consensus design for (i) the most general class of randomly switching graphs where none of them or their union is balanced; (ii) a class of heterogeneous uncertain linear MASs with state multiplicative uncertainties.These distinguish the current research from the existing literature, and hence, clearly shows its contribution.
Remark 5.The result of Theorem 4 also covers the scenario of consensus for homogeneous linear MASs under switching directed topologies where none of switched topologies nor their union is balanced and the circumstance of heterogeneous linear MASs with multiplicative uncertainties with fixed unbalanced directed topology.

Leader-Follower Consensus Design.
In the leaderless consensus control, all agents converge to an unknown state due to uncertainties.This motivates the control of uncertain agents (2) to track their known nominal dynamics, The following assumption is utilized: (A5): At least one follower is connected to the leader in at least one switched graph.
Then the tracking consensus design is as follows.
Thus, according to Lasalle's invariance principle, () converges to a stationary point at which V() = 0 which occurs if and only if () = 0. Therefore, the mean-square consensus is achieved and the speed is lower bounded by  *  min ( L∪ + L ∪ ), i.e., (18).

Illustrative Example
To illustrate the proposed consensus designs, let us consider an MAS composing of 6 linear agents (2) with nominal state matrices satisfying assumptions (A1) and (A2), First, the leaderless low-rank consensus design is demonstrated where the MAS topology randomly switches between two directed graphs G 1 and G 2 shown in Figure 1.
Here, neither G 1 , G 2 , nor G ∪ is balanced and G ∪ has a spanning tree; i.e., assumption (A3) is satisfied.The random process for switched graphs is described by a continuous-time Markov chain with generator matrix Q = [ −1 1 1 −1 ] and the invariant distribution  = [1/2, 1/2].Model uncertainties of agents in (2) are randomly generated where Δ  are distinct diagonal random matrices such that assumption (A4) is satisfied.The design parameters are chosen as Q = 1,  = 0.5.Then the simulation results are displayed in Figure 2 with two different sets of uncertainties.Obviously, agents reach consensus but the responses of agents and the consensus state are affected by uncertainties.Next, the proposed leader-follower low-rank consensus design is illustrated where the leader called agent 0 is connected to one follower in only one switched graph as seen in Figure 3.
Subsequently, the uncertainties are also randomly generated but design parameters are the same as above.The simulation result is then shown in Figure 4. Clearly, the consensus tracking is achieved even though the communications among them are randomly switched; none of switched graphs is connected or balanced and the union graph is unbalanced, and the leader is connected to only one follower in only one switched graph.

Conclusions
A novel approach has been proposed in this paper to design fully distributed consensus controllers for heterogeneous linear MASs with state multiplicative uncertainties and randomly switching graphs where none of them or their union is balanced.In addition, the proposed consensus controllers are low rank and have nonconservative coupling strengths.Numerical simulations were carried out to illustrate the effectiveness of the theoretical results.
The future studies should investigate more complex scenarios with more general classes of uncertainties, and when time delays, input, or state constraints exist.
agents, where V and E represent the set of vertices and edges of G, respectively.There is an edge   ∈ E if agent  receives information from agent .The neighboring set of a vertex  is denoted by N  ≜ { :   ∈ E}.Moreover, let   be elements of the adjacency matrix A of G; i.e.,   > 0 if   ∈ E and   = 0 if   ∉ E.