2.1 Graph Theory and Communication Topology

The exchange of information among the agents of a system is commonly described by a graph 𝒢 = (ϑ,Φ, 𝒜), where ϑ is a set of nodes (agents), Φ ∈ ϑ x ϑ is a set of edges that connect a node to another (self edges are not allowed), and 𝒜= [α _i,j ] ∈ ℝ ^NxN is its adjacency matrix containing positive weights describing the relationships among nodes. The Laplacian 𝓛 will be defined as 𝓛= diag {Σj=1N α_{1j ...}Σj=1N α_Nj}𝒜. An edge (Vi, 𝒱j) ∈ Φ means that node V _j can get information from node _Vi. If an edge (Vi, Vj) is contained in Φ, this implies that the term α _i,j of the adjacency matrix is different from zero and vice versa, nevertheless, it does not imply that the edge (Vi, Vj) is also contained in Φ. The set of neighbors of node i will be denoted by Θ _i = {Vj: (Vj Vi,) ∈ Φ , j = 1,..., N} and by ρ_i = | Θ_i | its cardinality.

Let 𝒢 be a connected graph in which no cycles exist; then it is called a tree. A graph 𝒢 is said to have a spanning tree if every one of its nodes and a subset of its edges form a tree, which means that at least one of the nodes has a communication path to every other node.

In this work, given that switching communication topologies are considered, a switching graph becomes necessary. A switching graph will be defined as 𝒢ᵨ_t = (ϑ,Φᵨ_t, 𝒜ᵨ_t ), where ϱ_t : [t ₀ , ∞) → {1,...,γ } is the switching signal that determines the communication topology at time t. It will be considered that the set of nodes ϑ is constant and that the condition (Vj Vi,) ∈ Φᵨ_t ⇒ (Vj Vi,) ∉ Φᵨ_t is met. The set of neighbors of node i at time t, Θ _iᵨt , will be defined analogously as Θ _iᵨt = {Vj: (Vj Vi,) ∈ Φᵨ_t , j = l,...,N} and ρ _iᵨt =|Θ _iᵨt | its cardinality.

2.2 Switched Linear Systems

A Switched Linear System (SLS) V = 〈𝓕, σ_t〉 is a hybrid dynamical system where 𝓕 ={Σ ₁, Σ ₂, ..., Σ _k } is a collection of linear systems of the form

where σ _t : [t₀ ∞) →{1,..., k} is the switching signal that determines the evolving linear dynamics, x(t) ∈ ℝ ⁿ, u(t) ∈ ℝ ^m, y(t) ℝ _^q, are the state, input and output variables, respectively, and Α _σt , B _σt and C, σt ∈ {1,...,k} are constant matrices of appropriate dimensions.

Remark 1 In systems as (1), it is often considered that y(t) = C _σt x(t), but for the objective of this work we restrict ourselves to consider systems of the form (1) with C _σt = C for all σ_t, ∈ {1 ... k}.

2.2.1 Lyapunov Stability

The switched linear system (1) under a state feedback control

is uniformly asymptotically stable if there exist a positive constant g and a class K𝓛 function ƒ such that for all switching signals σ _t the solutions of (1) with |x(0)| ≤ g satisfy

If the inequality (3) is valid for all switching signals and all initial conditions, the system (1) under a state feedback control (2) is globally uniformly asymptotically stable (GUAS).

A positive definite continuously differentiate function V : ℝ ⁿ → ℝis a common Lyapunov function for the family of systems 𝓕 if there exists a positive definite continuous function Q : ℝ ⁿ → ℝ such that

Then, the following result on stability can be stated.

Lemma 1 (Theorem 2.1 ^{( 8 ₎₎} If all systems in the family 𝓕 with control (2) share a radially unbounded common Lyapunov function V(x(t)), then the switched system (1) is globally uniformly asymptotically stable (GUAS).

This result assures the stability of the SLS system under any switching sequence. It is important to mention that if a common Lyapunov function for the SLS is not found, it is not enough proof to assert that a SLS is unstable.

The following result from ¹³ will be used afterwards.

Lemma 2 If there exist a symmetric positive definite matrix Ρ ∈ ℝ ^{n x n} and matrices Z_σt ∈ ℝ ^{m x n} , σt = {1, ··· ,k}, such that

then the switched control law (2) with

K_σt = Z _{σt ^P-1,}

assures the closed-loop stability of the switched system (1).

In this way, it is possible to state the problem of finding a common Lyapunov function for a SLS system as a Linear Matrix Inequality (LMI) problem to compute a switched state feedback.

2.2.2 Output Regulation

Let the system

where w(t) ∈ℝ^p and S ∈ℝ _{^{p x p,}} be an exosystem from which a reference y_r(t) = Rw(t) will be defined for an SLS system and consider the input to the SLS as

Then, from ¹⁰, the problem of Output Regulation via Full Information for SLS (ORFI) consists in, given Α_σι, B_σι, C, R, S and having full access to x(t), and ω(T), finding a feedback control law of the form (6) such that

Consider the following hypotheses.

Hypothesis 1 (H1) System (5) is antistable, i.e. all the eigenvalues of S have nonnegative real part.

Hence, it is guaranteed that the reference will never tend to zero and will be continuously changing.

Hypothesis 2 (H2) The pair (R, S) is detectable.

Hypothesis 3 (H3)

Hypothesis 4 (H4) It will be considered that the evolving dynamics of system (1) and its switching instants are always known.

Lemma 3 ^{₍₍ 10 )}, Theorem 1) Assume (H1) and (H3). Then the ORFI of switched system (1) is solvable only if ∀h, σt = 1, 2, · ,k, the following condition holds:

where s is the index of the pair(C, A _σt - B_σt K _σt ).

A more restrictive condition is given if (A_σt -B_σtK_σt, C) is observable, as in the following result.

Lemma 4 ^{₍₍ 10 _),} Theorem 2) Assume (H1), (H3), and that (A_σt - B_σtK_σt, C), σt = 1,2,···,k is observable. Then the ORFI of (1) is solvable if and only if Π_h = Π _σt, for h, σ_t = 1,2, · ,k.

From Lemma 3, it is clear that under Hypotheses (H1) and (H3) a sufficient condition for the problem of ORFI to be solvable is the following: if the subset of observable states of every subsystem ∑_h ∈ 𝓕 is the same, the elements of Π _h corresponding to observable states have to be equal to those of Π_σt for h, σt = 1,2, ··,k.

A class of SLS systems that satisfies the previous condition is the class of SLS systems of the form

for σt = 1, 2, ··· ,k, where x₁(t) ∈ ℝ ^{η -q}, x₂(t) ∈ ℝ^q, x₃(t) ∈ ℝ^{n- η}, u(t) ∈ ℝ^m, q ≤ m, B _{1, σt} is a full row rank matrix. The submatrices A₁ and A₂ represent the non-switching part of the system.

Additionally, let

the subsystem

must be controllable.

The values of Πi corresponding to the observable part of the linear subsystems are determined by their non-switching part, therefore they are equal for every linear subsystem. The subsystem ẋ₃(t) = A_{7, σt} x₃ (t) + B₂, _σt u(t) represents the unobservable part of the system.

The previous systems (7) are introduced to be used in the simulation example. It is worth noting that the results presented are not restricted to such systems.

3 Problem Statement

Consider a MAS consisting of Ν agents with different SLS dynamics of class (7) described by

where, _Xi(t) ∈ ℝ ⁿⁱ is the state, u_i(t) ∈ ℝ ^mi the control, and y_i(t) ∈ ℝ^q the output of the i-th agent, and a virtual reference agent whose dynamics is represented by the exosystem (5) with output

where R ∈ ℝ^{q x p}

Remark 2 It is considered that rank {C_i} = q, ∀i.

Let 𝒢ᵨ _t be a switching graph representing communication among the SLS agents of system (9) together with the exosystem (5) where ᵨ _t: [t₀, ∞) → {1,..., υ} is the switching signal that determines the evolving communication topology, then the following hypothesis is assumed.

Hypothesis 5 (H5) The graph 𝒢ᵨ_t does not contain loops, (v_i, v_j) ∈ Φᵨ _t ⇒ (v_j,v_i) ∉ Φᵨ_t, and incorporates a spanning tree with root on, the reference agent for a given ᵨt.

For a MAS, the consensus is achieved by the states of the agent if all of them converge to the same value. In this work only the output is considered in the consensus, i.e., it is required that limt→∞ |yi - yj| = 0 for all i, j = 0,1,...,N. Additionally, the output regulation deals with the problem of a system following an exosystem where lim t→∞ |Cx(t) - Rw(t) | = 0. Now, we define the switching i - th agent output consensus and output regulation error as

Note that the weight αi,j,ᵨt, is zero if agent i does not receive information from agent j , similarly, the weight αi,0,ᵨt applies to the difference between the i-th agent output and the virtual agent (reference) output (10). Remember that these weights depend on ᵨt (the evolving communication topology).

Thus, the problem of distributed output regulation and output consensus of SLS multi-agent systems under switching interaction topologies consists in obtaining a distributed control law such that every agent output consensus and tracking error ζiᵨt (t) tends to zero asymptotically under any communication switching sequence of ᵨt.

This problem is addressed and solved in the next section.

4 Distributed Output Regulation and Output Consensus Control

For ease of notation and without loss of generality, consider the controllable part (8) of Ν SLS agents in the form (7) as in (9).

Given that only a subset of the SLS agents will have access to reference (10) and none of them to the exosystem state (5), the following observers are proposed:

where k_i is such that the matrix (S - _KiR) is Hurwitz. Now, set the distributed output regulation control law as

for σ _it ∈ {1, · · · h _i}. Note that control (13) is of form (6) and depends on the exosystem observer state (t).

Define the i-th observer error as

hence, its dynamics will be described by

Consider that Hypothesis (H3) is met, then the i-th agent regulation error can be defined as

and its dynamics is given by

From (14) and (16), we have that

In order to save space, let us work with the second term in (15) substituting _Vj as in (18):

Hence,

and

Using (21) we can restate the i-th observer and regulation error dynamics (15), (17) as

Define the i-th agent observer-regulation error as

and SLS multi-agent observer-regulation error as

Consider that Hypothesis (H5) is met, then given any communication topology 𝒢_γ0, there exists an order for the systems (O_γ0), such that the Laplacian 𝓛_γ0 is a lower triangular matrix. Thus, reordering the agents according to the order O_γ0, the SLS multi-agent observer-regulation error dynamics will have a block lower triangular form

where the matrix Ā_γ0 can be described as follows: where

with indexes according to the order O_γ0 and switched matrix blocks

on its diagonal, where ξ _ᵨt (t) = T _ᵨt ξ(t) for ᵨt = γ0 and T_ᵨt ∈ ℝ^{Σ ni x Σ ni} · is a transformation matrix, which changes the order of the agents.

Hence, it is clear that the SLS multi-agent observer-regulation error will have switched linear dynamics

where Ã_ᵨt ∈ ℝ^{Σ ni x Σ _ni.}

Now the main result of this work can be stated.

Theorem 1 Let Ν SLS agents be of form (7), with a switching communication topology such that Hypothesis (H5) is met. Consider also, that Hypotheses (H1)-(H4) are met, then the distributed output regulation and output consensus of the SLS multi-agent system can be solved by a control law of form (13).

Proof: Consider an SLS multi-agent system with agents of the form (7), an exosystem (5) with output (10), and control law (13).

Then, under Hypothesis (H5) and a given order O_γ0, the corresponding SLS multi-agent observer-regulation error will have form (26). Given Hypothesis (HZ), the switched block matrices Ω _i,σit are such that there exist positive definite matrices P _i and Q _i,σit , for which the equations

are met.

A common Lyapunov function for system (26) is selected as

where

onsider V₁(t), whose derivative

is clearly negative, independent from the value of ξ₁(t). Thus, the observer-regulation error ξ ₁(t) is stable. For its V₂ (t)derivative is

hence, given the stability of ξ₁(t) the terms with β_2,1 will become zero and the derivative of V₂(t) will be negative, therefore, the observer-regulation error ξ₂(t) is stable. The same reasoning can be applied to the rest of the V_i (t)'s:

In this way, we conclude that the derivative of the common Lyapunov function (31) is negative, and as a result, the observer-regulation error (26) is stable.

Given that the diagonal blocks are always the same for a given agent, the previous procedure can be carried out for any order O_ᵨt, then function (31) is a common Lyapunov function for the observer-regulation error system (29), which is thus ultimately bounded.

Finally, given the stability of the observer-regulation error (29), the following holds in steady state

thus, the i-th output consensus and tracking error ζ _iᵨt (t) = 0 for i = 1,2,··· ,N.