Leader–follower consensus of uncertain variable-order fractional multi-agent systems

The leader–follower consensus of a class of variable-order fractional (VOF) uncertain linear multi-agent systems is studied in this paper. Firstly, a new general lemma is proposed to find Lyapunov candidate functions for the stability of VOF systems. Secondly, according to the proposed lemma and the stability theorem of VOF systems, and by using the singular value decomposition of matrices and some related lemmas, a sufficient condition to obtain leader–follower consensus of uncertain VOF linear systems is given in the form of a linear matrix inequality. The proposed control protocol is dependent on the order of the VOF multi-agent system and is applicable to fixed-order and nonlinear VOF multi-agent systems. Finally, the feasibility and effectiveness of the approach are verified by using some numerical simulations.

Finding suitable tools for modeling real-world phenomena is the foundation of scientific research. In recent decades, fractional order calculus has gradually been recognized as an excellent mathematical tool that can effectively describe complex physical phenomena or structures. In comparison with integer-order calculus, fractional order calculus has been proven more suitable for describing the long memory of systems, while requiring a reduced number of model parameters. As pointed out in many references [4,5,7,14,19,29,34,50,55], fractional-order differential equations are used to accurately describe the dynamics of several systems. Concerning the coordinated dynamics of agents, such as unmanned aerial vehicles flying in snow, rain, or dust storm, and underwater vehicles operating in lentic lakes, their dynamics cannot be interpreted by means of integer-order models, but can be well described using fractional-order models. However, achieving the con-sensus for FO multi-agent systems (FOMAS) is still challenging. Previous work focused mainly on the control of FOMAS. We can mention the consensus for FOMAS with external disturbances and input saturation [6], the bipartite [11,33], the output [13,44], the event-triggered [15,31,54], the adaptive [25], the faulttolerant [12] and the observer-based [8] consensus, just to cite a few. On the other hand, due to the complex dynamics of FOMAS, parameterized and unstructured uncertainties are inevitable. The consensus control of FOMAS with uncertainty, and the design of the corresponding consensus control protocol based on output feedback was addressed in [10], the consensus control of singular FOMAS with uncertainty under the fixed topology was investigated in [28], the event triggering consensus control of linear FOMAS was studied in [37], and the adaptive fuzzy consensus tracking control with event-triggered input for a class of uncertain FOMAS was discussed in [43].
We should note that all references above consider fixed-order systems. In contrast, VOF operators allow the order to be a function of an independent variable, such as time or space, which offer a novel tool to reflect the evolution of the process. The VOF models have shown great potential for applications in many time-varying physical processes. For example, thermo-viscoelastic rheological phenomena, biochemical tumorous bone remodeling, strain softening and strain hardening behaviors of metals, equivalent circuit of lithium-ion batteries, Hopfield-like neural networks, HIV/AIDS spreading, vibration of a membrane in a viscoelastic environment, aggregation of particles in living cells, and others [35,35,40,42]. However, the lack of intuitive physical explanation has slowed the development of VOF calculus over the past few decades. More recently, with the development of research on VOF calculus, some relevant advances emerged. The basic mathematical properties of VOF differential and integral operators were discussed in [32]. In [36], by conducting experimental studies, the physical significance and application of VOF operators in physical processes were presented. In [17], stability and control of VOF systems were addressed. Therefore, the question "how to obtain the consensus of MAS with agents' states described by VOF operators?" arises. Despite there has been some research on consensus of fixed-order FOMAS, most results and control methods cannot be applied to VOFMAS and, to the best of authors' knowledge, few research is available address-ing the leader-follower consensus of uncertain VOF-MAS.
Motivated by the previous discussion, in this paper we present a consensus protocol for uncertain VOF-MAS with a leader. The main research contributions are: (1) the study of leader-follower consensus of uncertain VOFMAS; (2) a new more general lemma is proposed, which allows to find Lyapunov candidate functions for the stability of VOF systems; (3) the establishment of an order-dependent sufficient condition to obtain consensus, formulated in terms of a linear matrix inequality (LMI); the proposal of consensus criteria that can be applied to fixed-order and nonlinear VOF-MAS.
The paper comprises five sections. Section 2 introduces preliminary concepts and lemmas. Section 3 describes the proposed approach. Section 4 presents some numerical examples to illustrate the effectiveness of the novel procedure. Finally, Sect. 5 summarizes the main conclusions.

Preliminary concepts and fundamental tools
In the follow-up the notation ⊗, R n×m and · will be adopted to represent the Kronecker product, the set of real matrices, and the Euclidean norm of a vector (or 2norm of a matrix), respectively. The symbol L 2 [0, ∞) will stand for the set of squared integrable functions on the interval [0, ∞), I N will represent an identity matrix, diag{·} will denote a diagonal matrix, A > 0 (A ≥ 0) will stand for symmetric positive definite (semi-definite) matrices, and A T and A −1 will correspond to the transpose and the inverse of a matrix A, respectively.

Some notions of graph theory
We classify graphs as undirected or directed. Undirected graphs are represented by an ordered array G = (V, E), with V and E denoting the sets of nodes and pairs of edges of the graph, respectively. The degree of a node v i , deg(v i ), corresponds to the number of edges that are associated with v i . Directed graphs are described identically, but now E includes also information about the direction of the edges. An edge starting and ending at nodes v i and v j is represented by (v i , v j ). The out and in degree of a node refer to the number of edges starting and ending at that node, respectively. A directed spanning tree of a digraph G is a subgraph of G, such that the root node can reach every other nodes following directed edges.
Let us denote by L the Laplace matrix of a graph. Moreover, let the matrices A and D represent the adjacency and the degree matrices of the graph. Therefore, we can obtain L = D − A, and we express the elements in L as: Herein, we consider VOFMAS with only one leader, represented by V 0 . The graph containing V 0 is denoted byḠ, and its corresponding matrix is If the ith agent can obtain data directly from the leader, then we set c i = 1, otherwise we set c i = 0.

Lemma 1 [27] For an undirected graph, the matrix H has only non-negative eigenvalues. If the graphḠ is connected, then the matrix H has only positive eigenvalues.
Lemma 2 [51] For a directed graph, the matrix H has only eigenvalues with positive real parts, if and only if, it has a spanning tree with the leader rooted.

Formulation of the problem
A VOFMAS with N followers and one leader can be represented by: with a control protocol: where x i (t) ∈ R n and x 0 (t) ∈ R n stand for the ith agent and leader states, respectively. The variable u i (t) corresponds to the control input (to be defined), and A ∈ R n×n , B ∈ R n×m , C ∈ R p×n and K ∈ R m×n denote constant and known matrices. Moreover, The matrices ΔA, ΔB are uncertain, meaning: where E, F 1 , and F 2 are constant and known real matrices with suitable dimensions, and H (t) is an unknown time-varying matrix obeying the condition (1), then the N followers and one leader model is: The consensus control of (3) was never addressed in the literature, thus being an open problem.
Definition 1 [27] The consensus of MAS with a leader may be achieved if and only if: To simplify the analysis, we define the error e i (t) = x i (t) − x 0 (t), which dynamics is: The demonstration of the consensus of the VOFMAS (1) with the control protocol (2) can be accomplished by proving that the dynamics of the error (4) is asymptotically stable.

Some useful lemmas
Lemma 3 [17] Let x(t) be a continuous and differentiable function. Therefore, at any time instant t ≥ 0, it holds: Lemma 4 When x(t) ∈ R n is a differentiable vectorvalued function, P = P T > 0 and ∀q(t) ∈ (0, 1), one get; Proof From the definition of VOF derivative, it follows that Inequality (5) is equivalent to have: From (6), it yields which is equivalent to Denoting y(s) = x(t) − x(s), one has y (s) = dy(s) ds = −x (s). In this case, (9) can be rewritten as Let P = p i j n×n and note that P is a symmetrical positive matrix. One has That is y T (s)Py (s)ds = d y T (s) P 2 y(s) . Combining (10) with (11) will result in The left-hand side of (12) is equivalent to In view of the L'Hospital rule, one has where lim s→t y(s) = 0 and y T (s) P 2 y(s) ≥ 0, for ∀s ∈ [t 0 , t]. One has and Form (14), (15) and (16), inequality (12) holds, which means that (6) is true. It is obvious that (7) is also true by the same way. The proof is completed.
Lemma 5 [22] For an undirected graph, we have that the Laplacian L satisfies: and L = L T ≥ 0.
Lemma 8 [26] For any matrix Π ∈ R q×n with q < n and full row rank (rank(Π = q)), there exists a singular value decomposition (SVD) of Π : where S ∈ R q×q is diagonal, with non-negative elements in decreasing order, and U ∈ R q×q and V ∈ R n×n are unitary.
Lemma 9 [26] Given a matrix Π ∈ R q×n with rank(Π )=q, q < n, and assuming that X ∈ R n×n is symmetric, then there exists a matrixX ∈ R q×q satisfying Π X =X Π , if and only if, X is given by: where X 11 ∈ R q×q , X 22 ∈ R (n−q)×(n−q) and V ∈ R n×n is unitary from the SVD of Π .

Order-dependent sufficient condition for the leader-follower consensus protocol of uncertain linear VOFMAS
We establish an order-dependent sufficient condition for the leader-follower consensus protocol of uncertain linear VOFMAS in terms of a LMI.  (2), is attained. The control protocol gain K is:

then, the leader-follower consensus of the VOFMAS (1), with the control protocol
The designed control protocol (2) can be rewritten as . (17) According to (17) and (4), one obtains: Expression (18) yields the following form by using the Kronecker product: Using the function: considering the q(t) derivative on V (t), and Lemma 4 results in: For the symmetric real matrices X = X T and Z = Z T , and the matrix W , we obtain: Therefore, it holds: where ) T ] T . Let X = Z = I N ⊗I n and W = 0 N n for simplifying. Using (20) into (19), yields: with We verify that (21) can be represented as: Via Lemma 6, there is a positive constant λ such that where By means of Lemma 7 and (22), we get that expression (22) is equivalent to: where

By Lemma 7, (23) is equivalent to:
where Pre-and post-multiplying expression (24) by the matrix diag{I N ⊗ P −1 , I N ⊗ I, I N ⊗ I, I N ⊗ I, I N ⊗ I } it yields: where Let P −1 =P. It results by Lemma 9 that there exists Then, the inequality (25) can be represented as the inequality in Theorem 1, meaning: From Lemma 2.6 in [17], it follows that (4) is asymptotically stable, and, from Definition 1, the consensus of the original system (1) is obtained. This completes the proof.
If variable-order fractional multi-agent systems degenerates into fixed-order fractional-order ones, the following corollary can be used to solve the control design problem of system (3). Corollary 1 For a given constant L and the SVD of the output matrix C = U S 1 0 V T , if there exist symmetric positive-definite matricesP,P 1 ,P 2 , plus a matrix Y and a constant positive scalar λ satisfying: (2) is obtained. Moreover, the gain K is:

then, the consensus of (3) with control protocol
The proof follows the steps used to prove Theorem 1.

Remark 2
We should note that Theorem 1 and Corollary 1 apply to both undirected and directed graphs satisfying Lemmas 1 or 2.

Remark 3
In recent decades, some consensus results emerged regarding FOMAS with constant order [1,6,10,11,24,25,28,33,37,43,44]. However, VOFMAS receives little attention, and no studies concerning the consensus of VOFMAS have been reported. Besides being valid for VOFMAS, the control algorithm proposed in this paper, is still valid for consensus of constant order FOMAS.

Remark 4
In [41], it was proven that the variable fractional order was as an indicator for off-line and on-line performance evaluation of lithium-ion batteries, and in [20] it was discussed the co-estimation of the state of charge and state of power of lithium-ion batteries based on VOF models. Both references showed that lithiumion batteries can be efficiently characterized using VOF equivalent circuit models, that can be expressed as in equation (1). Therefore, the consensus protocol proposed herein can be adopted to control battery equalization.

Some numerical illustrative examples
Three examples illustrate Theorem 1 and Corollary 1. The generalized Adams-Bashforth-Moulton method [18] is used to obtain the numerical solution of variableorder dynamical models.

Example 1
We consider the undirected graph of Fig. 1. The matrices L and H are: and the parameters of (1) are:  Figure 6 shows the control input, where u(t) = n i u 2 i . We verify that in the first case the consensus does not occur, while in the second case e 1i (t) and e 2i (t) approach 0 very fast, meaning that the consensus of the VOF system (1) is obtained.

Example 2
We consider the directed topology depicted in Fig. 7. The matrices L and H are given by:  while the parameters of the system are: By means of Corollary 1 and solving the LMI, we obtain the matricesP and K , as well the constant λ:  Figure 12 depicts the control input. As in the previous example, we verify that in the first case the consensus does not occur, while in the second case e 1i (t) and e 2i (t) approach 0 very fast, meaning that the consensus of the VOF system (1) is obtained.
Example 3 Finally, we consider the undirected graph depicted in Fig. 13, where the matrices L and H are:  The matrices and the parameters of the system (3) are: Solving the LMI in Corollary 1, we obtain the matrices P and K , plus the constant λ: As in the previous examples, we verify that in the first case the consensus does not occur, while in the second case e 1i (t) and e 2i (t) approach 0 very fast, meaning that the consensus of the VOF system (3) is obtained.

Conclusion
This paper investigated the consensus of uncertain VOFMAS. Based on the stability theory of VOF sys- tems and the proposed new and general lemma to find Lyapunov candidate functions, and by using matrix SVD and LMI techniques, an order-dependent sufficient condition to achieve the control consensus of VOFMAS was obtained. The feasibility and effectiveness of the method were verified by simulations. The proposed control method can be applied not only to the consensus control of FOMAS with fixed-order, but also be extended to the consensus control of VOFMAS with nonlinear terms. It is well known that time-delays often appear in dynamic systems, thus, further efforts should be devoted to designing consensus control methods of VOF delayed MAS.