Leader-following consensus for networks with single- and double-integrator dynamics

Nonlinear Analysis: Hybrid Systems 31 (2019) 302–316

Contents lists available at ScienceDirect

Nonlinear Analysis: Hybrid Systems journal homepage: www.elsevier.com/locate/nahs

Leader-following consensus for networks with single- and double-integrator dynamics ∗

Ewa Girejko , Agnieszka B. Malinowska Bialystok University of Technology,15-351 Białystok, Poland

article

info

Article history: Available online 3 November 2018 MSC: 34N05 26E70 Keywords: Consensus problem Time scales Multiagent system

a b s t r a c t In the paper, consensus protocols for networks of dynamic agents with single- and doubleintegrator dynamics are proposed. We provide necessary and sufficient conditions for leader-following consensus problem of the presented models. In order to examine the studied cases in a general way the time scale calculus is employed to get the results. The method used in the proofs is based on spectral characterization of stability for time invariant linear systems on time scales. All the results are enriched in illustrative examples and numerical simulations. © 2018 Elsevier Ltd. All rights reserved.

1. Introduction Consensus/synchronization problems for networks of dynamic multiagent systems have attracted many researchers from diverse fields of science, such as biology, physics, sociology [1–4], and engineering disciplines where examples include: formation control, flocking, networked multi-motor system, sensor networks, fractional or time delay systems [5–13]. The main idea of a consensus algorithm is to drive, using certain controls, a team of agents to reach an agreement on a specific issue. In a cooperative multiagent system (without a leader) each agent adjusts its state, based on the information received from other agents, in such a way that all agents eventually reach an agreement. A leader-following consensus means that there exists a special agent, called the leader, whose motion is independent on all other agents. Both leaderless and leaderfollowing consensus control protocols for continuous-time or discrete-time multiagent systems have been widely studied in the literature. However, in the case of many applications there appear hybrid systems (with nonuniform time domain) or systems with nonuniform sampling. Thus, it is relevant to establish a theory that covers also these types of systems. A powerful tool that can be used for this purpose is the theory of time scales [14]. As a time scale is an arbitrary nonempty closed subset of real numbers, problems with continuous, discrete or nonuniform time domain can be simultaneously studied by employing the calculus on time scales. Over the last decade an increasing interest in consensus algorithms for multiagent systems on time scales has been observed [15–21]. These papers generally refer to consensus in systems with single-integrator dynamics, and the analysis of consensus algorithms is conducted by using graph theory and Lyapunov functions. In our work, we study a leader-following consensus problem in two types of multiagent systems, with single- and double-integrator dynamics. Observer-type consensus protocols based on agents’ information exchange are proposed. We introduce a positive scalar parameter indicating the coupling strength between neighboring agents. We specify the range of this parameter, which ensures the consensus in the system. The idea of introducing a leader to the system has roots in realworld phenomena such as the relations between a sheepdog and sheep [22], and the influence of mass media on opinions of the society members. In addition, controlling the system through the leader can be justified in practice, e.g., crowd evacuation ∗ Corresponding author. E-mail addresses: [email protected] (E. Girejko), [email protected] (A.B. Malinowska). https://doi.org/10.1016/j.nahs.2018.10.007 1751-570X/© 2018 Elsevier Ltd. All rights reserved.

E. Girejko, A.B. Malinowska / Nonlinear Analysis: Hybrid Systems 31 (2019) 302–316

303

in case of panic situation [23], designing the reference trajectories for the master robot to guide slave robots [5] or in the seller–buyers networks [24]. The virtual leader considered in this paper can be treated as the reference state of a target to follow (see for example [25,26]). The current work covers two cases: constant and time-varying reference states. The paper is organized as follows. In the second section preliminary definitions and facts on time scale calculus, in particular results on a spectral characterization of stability for time invariant linear systems on time scales, are gathered. Section 3 concerns consensus problem in systems with single-integrator dynamics. Necessary and sufficient conditions for the leader-following consensus of the group of agents are obtained. Moreover, we applied the consensus based control strategy to the tracking problem in order to achieve a desired formation shape. For all the results illustrative examples and numerical simulations are provided. Further, in Section 4, we examine the consensus problem for systems with doubleintegrator dynamics. Also in this case, necessary and sufficient conditions under which leader-following consensus is achieved by the system are given, and a formation tracking problem is analyzed. Theoretical results are illustrated by examples. The paper ends with conclusions. 2. Preliminaries Since information exchange among agents in multiagent system can be naturally modeled by a graph, firstly we recall some simple notions from graph theory. Let G = (V , E) be a weighted communication graph of n agents, with the set of nodes V = {v1 , v2 , . . . , vn } and the set of edges E ⊆ V × V . Each edge denoted by (vi , vj ) means that agent i obtains information from agent j. Accordingly, there are two types of matrices to represent the communication graph G: one is the adjacency matrix n×n A = [aij ] ∈ with aij > 0 if (vi , vj ) ∈ E and aij = 0 otherwise. The other is the Laplacian matrix L = [lij ] ∈ Rn×n ∑R with lii = a i̸ =j ij and lij = −aij , i ̸ = j. It means that L has at least one zero eigenvalue with a corresponding eigenvector (1, . . . , 1)T . If for a certain graph we have that (υj , υi ) ∈ E for every (υi , υj ) ∈ E, the graph is called undirected. If a graph is not undirected it is referred to as a directed. Clearly, for an undirected graph A and L are symmetric. An undirected graph is connected if there exists a path between any two distinct nodes. Next we review those facts from the calculus on time scales that will be needed in the sequel. For a more in-depth study we refer the reader to books [27,28]. A time scale, denoted by T, is an arbitrary nonempty closed subset of real numbers R. We define the forward jump operator σ : T → T, and the graininess function µ : T → [0, ∞) by

σ (t) = inf {s ∈ T : s > t } and µ(t) = σ (t) − t . One can also define the backward jump operator ρ : T → T and the backward graininess function ν : T → [0, +∞) by

ρ (t) = sup{s ∈ T : s < t } and ν (t) = t − ρ (t). A time scale T, which is unbounded above, is called homogeneous if the graininess is constant, otherwise T is said to be nonhomogeneous. The following classification of points is used within the theory: a point t ∈ T is called right-dense, rightscattered, left-dense and left-scattered if σ (t) = t, σ (t) > t, ρ (t) = t and ρ (t) < t, respectively. In the continuous-time case, when T = R, we have that for all t ∈ R: σ (t) = ρ (t) = t and µ(t) = ν (t) = 0. In the discrete-time case, for each t ∈ T = hZ, h > 0: σ (t) = t + h, ρ (t) = t − h, µ(t) = ν (t) = h, while for T = {qn : n ∈ Z}∪{0} with q > 1, we have σ (t) = qt, ρ (t) = qt and µ(t) = (q − 1)t. Definition 1. A function f : T → R is called rd-continuous provided it is continuous at right-dense points and its left-sided limits exist (finite) at left-dense points. Let Tκ := T \ {sup T} , if sup T is finite and left-scattered; otherwise Tκ := T. Definition 2. Let f : T → R and t ∈ Tκ . The delta derivative of f at t is the real number f ∆ (t) with the property that for any ε there is a neighborhood U of t such that

|(f (σ (t)) − f (s)) − f ∆ (t)(σ (t) − s)| ≤ ε|σ (t) − s| for all s ∈ U. We say that f is delta differentiable on T provided f ∆ (t) exists for all t ∈ Tκ . Remark 3. If σ (t) = t, then f ∆ (t) = lims→t ∆

If σ (t) > t, then f (t) =

f (σ (t))−f (t) , µ(t)

f (t)−f (s) , t −s

provided this limit exists.

provided f is continuous at t.

Our main results are based on a spectral characterization of stability for time invariant linear systems on time scales. For the convenience of the reader we recall some facts from this theory (see [29]), thus making our exposition self-contained. Let us consider dynamic initial value problem on time scale T: x∆ = Ax, x(t0 ) = x0 ,

(1) κ

κ

where A ∈ R . By ΦA : {(t , t0 ) ∈ T × T : t ≥ t0 } → R ϕ (t , t0 , x0 ) = ΦA (t , t0 )x0 solves initial value problem (1). n×n

n× n

we denote transition matrix corresponding to (1), that is,

304

E. Girejko, A.B. Malinowska / Nonlinear Analysis: Hybrid Systems 31 (2019) 302–316

Definition 4. Let T be a time scale, which is unbounded above. We call the linear system x∆ = Ax:

• exponentially stable if there exists a constant α > 0 such that for every t0 ∈ T there exists K = K (t0 ) ≥ 1 with ∥ΦA (t , t0 )∥ ≤ Ke−α(t −t0 ) for t ≥ t0 , • uniformly exponentially stable if K can be chosen independently of t0 in the definition of exponential stability. Definition 5. [29] Given a time scale T unbounded above, define for arbitrary t0 ∈ T SC (T) := {λ ∈ C : lim sup T →∞

1 T − t0

∫

T

lim

t0 s↘µ(t)

log |1 + sλ| s

∆t < 0}

and SR (T) := {λ ∈ R : ∀T ∈ T, ∃t ∈ T with t > T such that 1 + µ(t)λ = 0} ,

where the integral given above is the time scale integral (see, e.g., [27]). Then the region of exponential stability for the time scale T is defined by S (T) = SC (T) ∪ SR (T) .

In general, the set SC (T) is hard to calculate because of the limit superior involved in the definition. Thus, we present some examples to make this notion more intuitive. Example 6 ([29]). If T = R, then SR (T) = ∅ and SC (T) = {λ ∈ C : Re(λ) < 0}. If T = hZ, then SR (T) = {− 1h } and S (T) = {λ ∈ C : |λ + 1h | < 1h }. If the time scale is the scale of harmonic numbers T = {tn := Σkn=1 1k }, n ∈ N, which is unbounded 1 above, then the graininess function is µ(tn ) = n+ and we obtain SR (T) = ∅, S (T) = SC (T) = {λ ∈ C : Re(λ) < 0}. 1 Definition 7. A complex number λ is regressive if 1 + µ(t)λ ̸ = 0 for all t ∈ T, uniformly regressive if there exists a γ > 0 such that

γ −1 ≤ |1 + µ(t)λ|

(2)

for all t ∈ T. A square matrix is (uniformly) regressive if all of its eigenvalues are (uniformly) regressive. For matrix A ∈ Rn×n let us define the set spec(A) := {λ ∈ C : λ is an eigenvalue of matrix A}. The following theorems characterize exponential stability of linear time invariant systems. Theorem 8. [29] Let T be a time scale that is unbounded above and let A ∈ Rn×n be regressive. Then the following hold: 1. If system (1) is exponentially stable, then spec(A) ⊂ SC (T). 2. If matrix A is uniformly regressive and spec(A) ⊂ SC (T), then system (1) is exponentially stable. Theorem 9. [29] Let T be a time scale that is unbounded above. Let A ∈ Rn×n and consider linear system (1). Then the following hold: 1. If system (1) is exponentially stable, then spec(A) ⊂ S (T). 2. If spec(A) ⊂ S (T), the time scale T has bounded graininess and for all defective λ ∈ spec(A) the scalar equation x∆ (t) = λx(t) is uniformly exponentially stable, then system (1) is exponentially stable. 3. If matrix A is diagonalizable, then system (1) is exponentially stable if and only if spec(A) ⊂ S (T). As we mentioned before, in general, the set S (T) is complicated to calculate. Therefore, in some of our main results we will rather use the notion of Hilger circle defined below. Definition 10. For each t ∈ T, the set

⏐ ⏐ } ⏐ 1 ⏐ ⏐ ⏐< 1 H(t) := z ∈ C : ⏐z + µ(t) ⏐ µ(t) {

is called the Hilger circle at t. For T with the bounded graininess the smallest Hilger circle, denoted Hmin , is the Hilger circle associated with µ(t) = − µmax := supt ∈T {µ(t)}. When µ(t) ⏐= 0 we ⏐ define H0 := C = S (R), the open left-half complex plane, while when µ(t) = h > 0, then Hh := {z ∈ C : ⏐z + 1h ⏐ < 1h } = S (hZ). Not all Hilger circles H are subsets of the region of exponential stability S (T), but Hmin ⊂ S (T).

E. Girejko, A.B. Malinowska / Nonlinear Analysis: Hybrid Systems 31 (2019) 302–316

305

3. Consensus in multiagent systems with single-integrator dynamics Consider the multiagent system consisting of n agents from the set N = {1, . . . , n}. The single-integrator dynamics of each agent is given by x∆ i (t) = ui (t) , i ∈ N ,

(3)

where xi : T → R is the state function and ui : T → R is the control input function for the ith agent. The virtual leader for multiagent system (3) is an isolated agent described by x∆ r (t) = f (t) ,

(4)

where f : T → R is rd-continuous function. Observe that when f (t) ≡ 0, then xr (t) ≡ constant (constant reference state). In the opposite case we get time-varying reference state. For the simplicity of presentation we assume that all agents are in the one-dimensional space. By introducing the Kronecker product all results can be rewritten for the m-dimensional (m > 1) case. Definition 11. Multiagent system (3)–(4) is said to achieve exponentially leader-following consensus if there exists a constant p > 0 such that for every t0 ∈ T there exists K = K (to ) ≥ 1 that a solution to the system satisfies |xi (t) − xr (t)| < Ke−p(t −t0 ) for t ≥ t0 , all i ∈ N and any initial conditions x(t0 ) = (x1 (t0 ), . . . , xn (t0 ))T ∈ Rn , xr (t0 ) ∈ R. Remark 12. Let us note that Definition 11 is based on the exponential stability. One could propose other definitions using notions of uniform exponential or asymptotic stability. In the classical cases, continuous or discrete ones, all those concepts of stability for the autonomous systems coincide. This fails to be true on inhomogeneous time scales (for a fuller treatment of the problem we refer the reader to [29]). Taking into account the interaction between agents we propose the consensus control law as follows:

⎤ ⎡ n ∑ ( ) ui (t) = f (t) − β ⎣ aij xi (t) − xj (t) + bi (xi (t) − xr (t))⎦ ,

(5)

j=1

where aij (i, j = 1, 2, . . . , n) is the (i, j)th entry of the adjacency matrix A, bi = 1 if the virtual leader is available to agent i and bi = 0 otherwise, β > 0 is the coupling strength. Let L be the Laplacian matrix of A and B := diag{b1 , . . . , bn } ∈ Rn×n be a diagonal matrix with nonzero trace. Set M := L + B and Mβ := β M. Theorem 13. Let spec(−Mβ ) ⊂ S (T) and one of the following conditions holds: 1. matrix −Mβ is uniformly regressive; 2. the time scale T has bounded graininess and for all defective λ ∈ spec(−Mβ ) scalar equation x∆ (t) = λx(t) is uniformly exponentially stable. Then control law (5) solves the consensus problem for multiagent system (3)–(4). Proof. Let us observe that using control law (5) system (3) can be written as follows: x∆ (t) = f (t)1 − Mβ x(t) + β Bxr (t)1 ,

(6)

where x(t) = (x1 (t), . . . , xn (t)) and 1 = (1, 1, . . . , 1) . Let x˜ i (t) := xi (t) − xr (t). Then one gets T

T

x˜ ∆ (t) =x∆ (t) − x∆ r (t)1

=f (t)1 − Mβ x(t) + β Bxr (t)1 − x∆ r (t)1 =f (t)1 − Mβ (x(t) − xr (t)1) − Mβ xr (t)1

(7)

+ β Bxr (t)1 − x∆ r (t)1 =f (t)1 − Mβ x˜ (t) − β Lxr (t)1 − x∆ r (t)1 . where x˜ (t) := (x˜ 1 (t), . . . , x˜ n (t))T . Since Lxr (t)1 = 0 and x∆ r (t) = f (t) , we obtain the following system x˜ ∆ (t) = − Mβ x˜ (t) .

(8)

By assumption spec(−Mβ ) ⊂ S (T). Hence, by Theorem 8, case 1. and Theorem 8, case 2., we get that system (8) is exponentially stable, that is there exists a constant p > 0 such that for every t0 ∈ T there exists K = K (to ) ≥ 1 that solution to (8) satisfies |˜xi (t)| < Ke−p(t −t0 ) for i ∈ N and any initial conditions. This is the desired conclusion. □ In view of the above proof, by Theorem 9, a necessary condition for a leader-following consensus is as follows.

306

E. Girejko, A.B. Malinowska / Nonlinear Analysis: Hybrid Systems 31 (2019) 302–316

Fig. 1. Solutions to multiagent systems in Example 15 with initial condition x(0) = (1, 1/2, 2).

Theorem 14. If control law (5) solves the consensus problem for multiagent system (3)–(4), then spec(−Mβ ) ⊂ S (T). Example 15. Let us exploit the time scale proposed by authors in [29]:

T := {0, 1, 4, 5, 8, 11, 12, . . . , sk , sk + 3, . . . , sk + 3k, sk+1 , . . .}, where s0 := 0, sk+1 := sk + 3k + 1, k ∈ N. t Then we consider multiagent system (3)–(4) with the leader dynamics x∆ r (t) = sin( 6 ), β = 1 and

( L=

1 0 0

−1 1 0

0 −1 0

1

0

0

B=⎝ 0

1

0 ⎠,

0

0

2

⎛

) ,

⎜

⎞ ⎟

on T1 = 12 T. Observe that {−2} = spec(−Mβ ) ⊂ S (T) but conditions 1 and 2 of Theorem 13 fail to be true. However, we get a leader-following consensus in finite time. Moreover, it is worth pointing out that, in dependence on the initial time the behavior of solutions to the system varies (see Fig. 1). Namely, for t0 = 35 the trajectories explode for a certain time and 2 only after we get a leader-following consensus, what is not the case for t0 = 0. This is because the system is exponentially stable but is not uniformly exponentially stable. Recall that Hmin ⊂ S (T). As a consequence of Theorem 13 we get the next result. Corollary 16. Assume that T has bounded graininess and all the eigenvalues λi , i ∈ N, of matrix M = L + B, are real and positive. ( ) Set λmax = maxi {λi }. If β ≤

1

µmax λmax

1−

1

γ

for some γ > 1, then control law (5) solves the consensus problem for multiagent

system (3)–(4). Proof. Indeed, if µmax = 0 (what means that T = R), then spec(−Mβ ) ⊂ S(R) and −Mβ is uniformly regressive. For µmax ̸ = 0, by assumptions, we get 1 − βλi | < µmax , 1 2. |1 − µ(t)βλi | ≥ γ .

1. | µ 1

max

The first condition is equivalent to spec(−Mβ ) ⊂ Hmin and the second one means that the eigenvalues of matrix −Mβ are uniformly regressive. □ The following example illustrates Theorem 13 with the use of Corollary 16.

E. Girejko, A.B. Malinowska / Nonlinear Analysis: Hybrid Systems 31 (2019) 302–316

307

Fig. 2. Solutions to multiagent system given in Example 17 with initial condition x(0) = (1, 1/2, 2, 3, 3/2, 1/3).

Example 17. Let us consider a multiagent system with matrices:

⎛ ⎜ ⎜ ⎜ L=⎜ ⎜ ⎝

2 −2 0 0 0 0

−2 3 0 −1 0 0

0 0 1 /2 0 −1/2 0

0 −1 0 1 0 0

0 0 −1/2 0 7/2 −3

0 0 0 0 −3 3

⎞ ⎟ ⎟ ⎟ ⎟, ⎟ ⎠

B = diag {1, 1, 1, 0, 0, 0} ,

∑k

i−1 on nonhomogeneous time scale T = {tk = i=1 i : k ∈ N}, and the virtual leader with a constant state xr (tk ) = 5 for all k ∈ N. What follows, f (tk ) = 0 for all k ∈ N, and µmax = limk→∞ µ(tk ) = 1. In Fig. 2(a) a consensus is reached with β = 14 and in Fig. 2(b) with β = 31 is not.

Next we consider the case when the interaction topology in the multiagent system is modeled by undirected and connected graph. If T = R, then it is known that control law (5) solves the consensus problem for multiagent system (3)–(4) (see, e.g., [30,31]). For a time scale with 0 < µmax < ∞ we have the following result. Theorem 18. Let A = AT be irreducible with entries aij = 1 or aij = 0. If β < problem for multiagent system (3)–(4).

2

µmax (n+1)

, then control law (5) solves the consensus

Proof. Since A is symmetric with entries {0, 1}, it implies that Laplacian L is also symmetric and the eigenvalues of L fulfill the inequalities 0 ≤ λ1 ≤ λ2 ≤ · · · ≤ λn ≤ n. Putting γi as an eigenvalue of matrix L + B we get, by [32], the following inequalities: 0 ≥ −βγi ≥ −β (n + 1). By assumption, A is irreducible, thus βγi > 0 (see [30,31]). On the other hand, since β < µ 2(n+1) and γi ≤ n + 1, we get 0 < βγi < µ 2 . It means that spec(−Mβ ) ⊂ Hmin . Finally, by symmetry of matrix A max

max

and by Theorem 9 we get the thesis. □ In order to illustrate theoretical results presented in Theorem 18 we apply the proposed control law (5) to the problem of the buyer–seller network (see [24] and the references therein). In this context agents are buyers and a leader is a seller. Example 19. Let us consider a multiagent system consisting of six buyers and a seller. The buyer–seller interaction topology is represented by the following matrices:

⎛ ⎜ ⎜ ⎜ L=⎜ ⎜ ⎝

1 −1 0 0 0 0

−1 2 0 −1 0 0

0 0 1 0 −1 0

0 −1 0 2 0 −1

0 0 −1 0 2 −1

0 0 0 −1 −1 2

⎞ ⎟ ⎟ ⎟ ⎟, ⎟ ⎠

B = diag {1, 0, 0, 0, 0, 0} .

308

E. Girejko, A.B. Malinowska / Nonlinear Analysis: Hybrid Systems 31 (2019) 302–316

Fig. 3. Solutions to multiagent system given in Example 19 with initial conditions x(0) = (1, 0, 1, 1, 0, 0).

The reference state is xr (t) = 0 if the high product price is offered, and xr (t) = 1 otherwise. As the initial state of the ith buyer we can choose 1 or 0. If xi (t0 ) = 0, then buyer i is not active (he does not want to buy a product). If xi (t0 ) = 1, then buyer i is active (he wants to buy a product). The leader-following (seller-following) consensus means the following (see [24]): ‘‘When the seller offers a low product price xr (t) = 1, the purchase demand of buyers is increased. The nonactive buyers xi (t) = 0 change their status to be the active buyers xi (t) = 1, thus, all buyers become active, and then, the seller can increase the production to get higher profit. On the other hand, when the high product price xr (t) = 0, is offered, the active buyers xi (t) = 1 deny to buy the product then their status is changed to be non-active, then the seller can decrease the investment to avoid the loss’’. In other words, the seller can control the purchase behaviors of buyers through the product price. In the paper [24], the authors have studied a discrete buyer–seller network and have concluded that the seller-following consensus depends on the agents’ interaction topology and the change period of product price (if bi = 0 for some i). Discrete buyer–seller networks follow a specific time scale, namely T = Z. However, in the real-life situations, it is not always the case that the time scale, on which interactions between agents take place, matches the commonly known discrete-time scale (T = Z). Thus it is important to study dynamics of buyer–seller networks on general time scales. As we mentioned before Theorem 18, in the case T = R the leader-following consensus is reached in the considered buyer–seller network. However, if we change the time scale it is not necessarily the truth. Let us consider T = {tk =

∑k

i−1 i

: k ∈ N} and the constant reference state of the seller (xr (t) = 1 for all k ∈ N). In Fig. 3, we see that a consensus is reached with β = 71 and with β = 23 is not. This evidently shows that in order to control the global purchase behavior i=1

of the buyers’ network it is necessary to take into account the time scale on which interactions between agents take place. Then, we can specify the coupling strength between agents (parameter β ) and ensure a seller-following consensus in the considered buyer–seller network. Remark 20. Let us notice that we can choose matrix B in such a way that we put bi = 1 for those i for which Jordan block of matrix L is made of zeros. Then, spec(M) ⊂ R+ × R, as by Geršgorin’s Theorem it is known that all eigenvalues of L have their real part larger or equal to 0 (see, for example, [33]). 3.1. Formation tracking problem Having established conditions ensuring a leader following consensus in multiagent system (3)–(4), we now apply the consensus based control strategy to the tracking problem with a desired formation shape. For this problem the following control law is proposed:

⎡ ui (t) = f (t) − β ⎣

n ∑

⎤ aij xi (t) − xj (t) + δj − δi + bi (xi (t) − xr (t) − δi )⎦ ,

(

)

j=1

for i ∈ N, where δi denotes a desired state deviation between the agent i and the leader.

(9)

E. Girejko, A.B. Malinowska / Nonlinear Analysis: Hybrid Systems 31 (2019) 302–316

309

Definition 21. The formation tracking of multiagent system (3)–(4) is achieved by the control law (9) if lim (xi (t) − xr (t)) = δi , i ∈ N ,

(10)

t →∞

for any initial conditions x(t0 ) = (x1 (t0 ), . . . , xn (t0 ))T ∈ Rn , xr (t0 ) ∈ R. Theorem 22. Let spec(−Mβ ) ⊂ S (T) and let one of the following conditions hold: 1. matrix −Mβ is uniformly regressive; 2. the time scale T has bounded graininess and for all defective λ ∈ spec(−Mβ ) scalar equation x∆ (t) = λx(t) is uniformly exponentially stable. Then control law (9) solves formation tracking problem for multiagent system (3)–(4). Proof. First let us observe that using control law (9), system (3) can be written as x¯ ∆ (t) = −Mβ x¯ (t) , where x¯ (t) = [¯x1 (t), . . . , x¯ n (t)]T , x¯ i (t) = xi (t) − δi − xr (t) for i ∈ N. The same reasoning as in the proof of Theorem 13 implies that |¯xi (t)| = |xi (t) − δi − xr (t)| ≤ Ke−α (t −t0 ) . Now taking a limit as t → ∞ we get limt →∞ (xi (t) − xr (t)) = δi for i ∈ N, what finishes the proof. □ In order to verify effectiveness of the conditions for the formation tracking problem simulations in two-dimensional space are presented in the next example. Example 23. Let √ us consider control law √ (9) for six agents with the Laplacian √ matrix and matrix B√the same as in Example 17. Set δ1 = [−1/2, 3/2]T , δ2 = [1/2, 3/2]T , δ3 = [1, 0]T , δ4 = [1/2, − 3/2]T , δ5 = [−1/2, − 3/2]T , δ6 = [−1, 0]T , and the leader state [2, 1]T . It implies that the desired formation geometry is a regular hexagon. We analyze the system on ∑k xir−= 1 T = {tk = i=1 i : k ∈ N} and with β = 1/7. Fig. 4 shows that six agents follow the constant leader state and achieve the desired formation geometry. 4. Consensus in multiagent systems with double-integrator dynamics In this section we study the leader-following consensus in multiagent system with double-integrator dynamics. Namely, the dynamics of each agent is given by x∆ i (t) = vi (t) , i ∈ N ,

(11)

vi∆ (t) = ui (t) , i ∈ N ,

where xi : T → R is the state function and vi : T → R is the consensus parameter of the ith agent, ui : T → R is the control input function. The virtual leader for multiagent system (11) is an agent described by x∆ r (t) = vr (t) ,

(12)

vr∆ (t) = f (t) , where xr and vr are the state and the consensus parameter of the leader, respectively. Function f is rd-continuous.

Definition 24. The multiagent system (11)–(12) is said to achieve exponentially leader-following consensus if there exist constants p, q > 0 such that for every t0 ∈ T there exist K1 = K1 (to ) ≥ 1 and K2 = K2 (to ) ≥ 1 that a solution to the system satisfies

|xi (t) − xr (t)| < K1 e−p(t −t0 ) ,

t ≥ t0

(13)

|vi (t) − vr (t)| < K2 e−q(t −t0 ) ,

t ≥ t0 ,

(14)

and

for all i ∈ N and any initial conditions x(t0 ) = (x1 (t0 ), . . . , xn (t0 ))T ∈ Rn , v (t0 ) = (v1 (t0 ), . . . , vn (t0 ))T ∈ Rn , xr (t0 ) ∈ R, vr (t0 ) ∈ R. The following control input is considered to implement a leader-following consensus to multiagent system (11)–(12): ui (t) = f (t) −

n ∑

aij (xi (t) − xj (t)) + β (vi (t) − vj (t))

[

j=1

− bi [(xi (t) − xr (t)) + β (vi (t) − vr (t))] ,

] (15)

310

E. Girejko, A.B. Malinowska / Nonlinear Analysis: Hybrid Systems 31 (2019) 302–316

Fig. 4. Verification of the effectiveness of the extended control law for six agents that follow the constant leader state and lead to the desired formation geometry.

where aij is the (i, j)th entry of the adjacency matrix A ∈ Rn×n associated with graph G that models interaction topology among agents, bi = 1 if the virtual leader state is available to agent i and bi = 0 otherwise, β > 0 is the coupling strength. Similarly as for the single-integrator dynamics let B = diag{b1 , b2 , . . . , bn }, M = L + B, where L is the Laplacian matrix of A. Then using (15), system (11) can be rewritten as follows x∆ (t) =v (t)

v ∆ (t) = − M(x(t) + βv (t)) + B(xr (t) + βvr (t))1 + f (t)1 ,

(16)

E. Girejko, A.B. Malinowska / Nonlinear Analysis: Hybrid Systems 31 (2019) 302–316

311

where x(t) = (x1 (t), . . . , xn (t))T , v (t) = (v1 (t), . . . , vn (t))T and 1 = (1, 1, . . . , 1)T ∈ Rn . Next we define

( M :=

0n×n M

−In×n βM

)

∈ R2n×2n ,

where In×n is the n × n identity matrix, 0n×n denotes the n × n zero matrix. Putting y(t) := (x(t), v (t))T , we can rewrite (16) as y∆ (t) = −My(t) +

(

)

0n×1

B(xr (t) + βvr (t))1

( +

)

0n×1 f (t)1

.

(17)

Theorem 25. Let spec(−M) ⊂ S(T) and one of the following conditions is fulfilled 1. matrix −M is uniformly regressive; 2. the time scale T has bounded graininess and for all defective λ ∈ spec(−M) scalar equation x∆ (t) = λx(t) is uniformly exponentially stable. Then control law (15) solves the consensus problem for multiagents system (11)–(12). Proof. Let us introduce the following changes of coordinates: x˜ i := xi − xr , v˜ i = vi − vr . Putting y˜ (t) := (x˜ (t), v˜ (t))T = (x(t) − xr (t)1, v (t) − vr (t)1)T ∈ R2n in system (11), using (12) and the facts that Lvr (t)1 = 0, Lxr (t)1 = 0 and vr∆ (t) = f (t) , one gets: y˜ ∆ (t) =

(

x∆ (t) − x∆ r (t) · 1 v ∆ (t) − vr∆ (t) · 1

)

(

)

x(t) v (t)

=−M

( +

)

0n×1

B(xr (t) + βvr (t))1

(

)

( +

)

0n×1 f (t)1

( −

f (t)1

)(

(

) vr (t)1

)

xr (t)1 x(t) − xr (t)1 0n×n −In×n =−M − L + B β (L + B) vr (t)1 v (t) − vr (t)1 ( ) ( ) ( ) 0n×1 vr (t)1 x˜ (t) + − = −M = −My˜ (t) . Bxr (t)1 + β Bvr (t)1 01 v˜ (t) Therefore in the coordinates y˜ we get the following linear system y˜ ∆ (t) = −My˜ (t) .

(18)

The rest of the proof follows the same pattern as the proof of Theorem 13. □ According to the proof of Theorem 25, we have that the fact that control law (15) solves the consensus problem for double-integrator system (11) with the time-varying dynamics of the virtual leader given by (12) is equivalent to exponential stability of system (18). Therefore, applying Theorem 9 we get the next result. Theorem 26. If control law (15) solves the consensus problem for multiagent system (11)–(12), then spec(−M) ⊂ S(T). Based on the range of parameter β , we can provide two sufficient conditions for achieving a leader-following consensus. Theorem 27. Let λi ∈ R+ , i ∈ N, be the eigenvalues of matrix M and for γ ≥

√

√

λmin , λmin −2µmax λmax

[ ( )) √ λmin − 2µmax λmax > 0. If β ∈ √λ2 , µmax1λmax 1 − γ1 min

then control law (15) solves the consensus problem for double-integrator system (11) with the virtual

leader given by (12).

( Proof. Let us first recall that if M := k ≤ n, j = 1, 2}, then

si1 =

λi β +

√

λ2i β 2 − 4λi 2

0 M

) −I , spec(M) = {λi , i = 1, . . . , k, k ≤ n} and spec(M) = {sij , i = 1, . . . , k, βM

and si2 =

(for details see [13]). Assumption β ≥

√2 λmin

λi β −

√

λ2i β 2 − 4λi

(19)

2

implies that sij ∈ R for i = 1, . . . , k, k ≤ n, j = 1, 2. We analyze two cases. First

suppose that µmax = 0. Then matrix −M is uniformly regressive and Hmin is the open left-half complex plane. Since λi ∈ R+ for i ∈ N, we have −sij < 0. Therefore, problem. Now, √ suppose that µmax ̸ = 0. Then, ( ) control law (15) solves( the consensus ) by assumption β <

1

µmax λmax

1−

1

γ

, it follows 2βλi <

1

µmax

1−

1

γ

for all i ∈ N. As λi β +

λ2i β 2 − 4λi < 2λi β we get:

312

E. Girejko, A.B. Malinowska / Nonlinear Analysis: Hybrid Systems 31 (2019) 302–316

Fig. 5. Solutions to multiagent system given in Example 28 with initial conditions x(0) = (5, 1, 3, 6, 8, 2), v (0) = (1, 1, 1, 1, 1, 1) and β = 3, 5.

1. si1 = 2. si1 =

√ λi β+ λ2i β 2 −4λi 2

√ λi β+ λ2i β 2 −4λi 2

1

(

2

.

<

µ(t)

<

µmax

1−

1

)

γ

∀t ∈T , µ(t) ̸= 0,

Therefore, the first condition above implies |1 − si1 µ(t)| > γ1 for all i ∈ N, while the second gives | µ 1 − si1 | < max reasoning holds true also for si2 . Hence, by Theorem 25 we get the thesis. □

1

µmax

. This

Example 28. Let consider a system with six agents and the consensus parameter of virtual leader vr = 3, on time scale ∑us k T = {tk = 0, 1 · i=1 |sin i| : k ∈ N}. For matrices

⎛ ⎜ ⎜ ⎜ L=⎜ ⎜ ⎝

0 −1 0 0 0 0

0 2 0 0 −1 0

0 0 1 0 0 0

0 0 0 1 0 0

0 −1 0 −1 1 −1

0 0 −1 0 0 1

⎞ ⎟ ⎟ ⎟ ⎟, ⎟ ⎠

B = diag {1, 0, 0, 0, 0, 0},

one can calculate that λmax ≈ 2, 62, λmin ≈ 0, 38. Clearly, µ(tk ) ≤ 0, 1 and since vr∆ (tk ) = 0, f (tk ) = 0 for all k ∈ N. 1. Fig. 5 illustrates the case when a leader following consensus is achieved with β = 3, 5. 2. Fig. 6 illustrates the case when a consensus is not achieved with β = 14. Now we consider the case when all eigenvalues of matrix −M are complex. Theorem 29. Let λi ∈ R+ be the eigenvalues of matrix M = L + B and µmax < β < √λ2 . Then control law (15) solves the max consensus problem for double-integrator system (11) with the virtual leader given by (12). Proof. By assumption β <

si1 =

√2 λmax

√ λi β − j 4λi − λ2i β 2 2

we can write the eigenvalues of matrix M in the following way:

and si2 =

√ λi β + j 4λi − λ2i β 2 2

,

(20)

E. Girejko, A.B. Malinowska / Nonlinear Analysis: Hybrid Systems 31 (2019) 302–316

313

Fig. 6. Solutions to multiagent system given in Example 28 with initial conditions x(0) = (5, 1, 3, 6, 8, 2), v (0) = (1, 1, 1, 1, 1, 1) and β = 14.

with 4λi − λ2i β 2 ̸ = 0 (see the proof of Theorem 27). Therefore, there exists a constant 1 ≥ c > 0 such that

( 1−

βλi µ(t)

)2

2

√

⎛ + ⎝µ(t)

4λi − λ2i β 2 2

⎞2 ⎠ ≥c,

for all t with µ(t) ̸ = 0. What follows, there exists γ > 0 such that

√ ⏐ ⏐ ⏐ ⏐ 4λi − λ2i β 2 ⏐ ⏐ βλ µ (t) i ⏐≥ 1 , ⏐1 − + jµ(t) ⏐ γ ⏐ 2 2 ⏐ ⏐ for γ1 = c. It implies uniform regressivity of −si1 , −si2 . Since µmax < β and λi > 0 for all i ∈ N, we get λi (µ(t) − β ) < 0. Going further,

) ( λ2 β 2 λ2 β 2 1 1 − i + < −βλi + µ(t) λi + i 4 4 µ(t) µ(t) ⏐ ⏐ ⏐ 1 ⏐ 1 ⏐ ⏐ ⏐ µ(t) − sij ⏐ < µ(t) . Thus spec(−M) ⊂ Hmin and the proof is finished. □ Example 30. Let us consider a system with three agents and the consensus parameter of virtual leader vr = 3, on a hybrid ⋃∞ time scale T = k=0 [2k, 2k + 1]. For matrices

( L=

1 0 0

−1 1 0

0 −1 0

) ,

B = diag {1, 1, 2},

one can calculate that λmax √ = 2. Due to Theorem 29, in order to get a leader-following consensus the range of the coupling strength β should be (1, 2). Clearly, µmax = 1 and since vr∆ (t) = 0, f (t) = 0 for all t ∈ T. 1. Fig. 7 illustrates the case when a leader following consensus is achieved with β = 1, 2. 2. Fig. 8 illustrates the case when a consensus is not achieved with β = 12 .

314

E. Girejko, A.B. Malinowska / Nonlinear Analysis: Hybrid Systems 31 (2019) 302–316

Fig. 7. Solutions to multiagent system given in Example 30 with initial conditions x(0) = (5, 1, 3), xr (0) = 3, v (0) = (1, 1, 1), vr (0) = 3 and β = 1, 2.

4.1. Formation tracking problem Finally, we give conditions ensuring that the formation tracking is achieved by double-integrator multiagent system (11)–(12). The control law for this problem is proposed as follows:

ui (t) =f (t) −

n ∑

aij (xi (t) − xj (kt) + δj − δi ) + β (vi (t) − vj (t) + ρj − ρi )

[

]

j=1

(21)

− bi [(xi (t) − xr (t) − δi ) + β (vi (t) − vr (t) − ρi )] , for i ∈ N, where δi , ρi denote desired state and consensus parameter deviations between the agent i and the leader. Definition 31. The formation tracking of multiagent system (11)–(12) is achieved by the control law (21) if lim (xi (t) − xr (t)) = δi

t →∞

lim (vi (t) − vr (t)) = ρi ,

(22)

t →∞

for any initial conditions x(t0 ) = (x1 (t0 ), . . . , xn (t0 ))T ∈ Rn , v (t0 ) = (v1 (t0 ), . . . , vn (t0 ))T ∈ Rn , xr (t0 ) ∈ R, vr (t0 ) ∈ R . Theorem 32. Let spec(−M) ⊂ S(T) and one of the following conditions is fulfilled 1. matrix −M is uniformly regressive; 2. the time scale T has bounded graininess and for all defective λ ∈ spec(−M) scalar equation x∆ (t) = λx(t) is uniformly exponentially stable. Then control law (21) solves the formation tracking problem for multiagents system (11)–(12). Proof. The proof follows from proofs of Theorems 22 and 25. □ 5. Conclusions In this paper, we have studied a leader-following consensus problem for systems with single- and double-integrator dynamics on an arbitrary time scale. A time scale is a model of time. Besides commonly known cases of continuous-time (T = R) and discrete-time (T = hZ), there are many useful and interesting time scales, like nonhomogeneous or hybrid. Necessary and sufficient conditions were proved for both considered cases of systems. Moreover, we analyzed the formation

E. Girejko, A.B. Malinowska / Nonlinear Analysis: Hybrid Systems 31 (2019) 302–316

Fig. 8. Solutions to multiagent system given in Example 28 with initial conditions x(0) = (5, 1, 3), xr (0) = 3, v (0) = (1, 1, 1), vr (0) = 3 and β =

315

1 . 2

tracking problem. The method used in the analysis was based on stability characterization for time invariant linear systems on time scales. If the time scale is nonhomogeneous, the stability properties of such systems differ from stability properties of systems on homogeneous time scales, and the graininess function has an essential importance (see, e.g., Example 15). Moreover, the study of multiagent systems on homogeneous time scales (like T = R or T = hZ) is incomplete. Namely, in the real-life situations, it is not always the case that the time scale on which interactions between agents take place, matches the commonly known continuous- or discrete-time scale (see, e.g., Example 19). Those facts show that it is important to study a leader-following consensus problem for multiagent systems on arbitrary time scales. We have shown that in order to ensure a leader-following consensus it is necessary to know the time scale on which interactions between agents take place. Then, it is possible to specify the coupling strength between agents (parameter β ), and ensure a leader-following consensus. Examples and numerical simulations made the consideration more intelligible and clear. Acknowledgments The work was supported by Polish founds of the National Science Center, Poland, granted on the basis of decision DEC2014/15/B/ST7/05270. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14]

I. Couzin, J. Krause, N. Franks, S. Levin, Effective leadership and decision-making in animal groups on the move, Nature 433 (2005) 513–516. N.E. Friedkin, E.C. Johnsen, Social influence networks and opinion change, Adv. Group Process. 16 (1999) 1–29. K. Lehrer, C. Wagner, Rational Consensus in Science and Society, D. Reidel Publishing Company, Dordrecht, Holland, 1981. T. Vicsek, A. Czirók, E. Ben-Jacob, O. Shochet, Novel type of phase transition in a system of self-driven particles, Phys. Rev. Lett. 75 (1995) 1226–1229. L. Cheng, Y. Wang, W. Ren, Z.-G. Hou, M. Tan, Containment control of multiagent systems with dynamic leaders based on a PI n-type approach, IEEE Trans. Cybern. 46 (2016) 3004–3017. L. Cheng-Lin, L. Fei, Consensus Problem of Delayed Linear Multi-agent Systems: Analysis and Design, SpringerBriefs in Electrical and Computer Engineering, Singapore, 2017. J. Desai, J. Otrowski, V. Kumar, Modeling and control of formations of nonholonomic robots, IEEE Trans. Robot. Autom. 17 (6) (2001) 905–908. R. Dong, Z. Geng, Consensus based formation control laws for systems on lie groups, Systems Control Lett. 62 (2013) 104–111. S. Kar, M.F. Moura Jose, Distributed concensus algorithms in sensor networks: quantized data and random link failures, IEEE Trans. Signal Process. 58 (3) (2010) 1383–1400. S. Masroor, C. Peng, Z.A. Ali, Event triggered multi-agent consensus of DC motors to regulate speed by LQR scheme, Math. Comput. Appl. 22 (1) (2017) 1–14. D. Mozyrska, M. Wyrwas, Fractional discrete-time of Hegselmann–Krause’s type consensus model with numerical simulations, Neurocomputing 216 (C) (2016) 381–392. R. Olfati-Saber, J.A. Fax, R.M. Murray, Consensus and cooperation in networked multi-agent systems, Proc. IEEE 95 (1) (2007) 215–233. G. Ren, Y. Yu, S. Zhang, Leader-following consensus of fractional nonlinear multiagent systems, Math. Probl. Eng. 2015, pp. 8, Art. no. 919757. S. Hilger, Ein Maßkettenkalkül mit Anwendung auf Zentrumsmanningfaltigkeiten (Ph.D. thesis), Universität Würzburg, 1988.

316

E. Girejko, A.B. Malinowska / Nonlinear Analysis: Hybrid Systems 31 (2019) 302–316

[15] G. Eisenbarth, J.M. Davis, I. Gravagne, Stability of simultaneously triangularizable switched systems on hybrid domains, Electron. J. Differ. Equ. 2014 (63) (2014) 1–19. [16] E. Girejko, L. Machado, A.B. Malinowska, N. Martins, Krause’s model of opinion dynamics on isolated time scales, Math. Methods Appl. Sci. 39 (18) (2016) 5302–5314. [17] E. Girejko, L. Machado, A.B. Malinowska, N. Martins, On consensus in the Cucker–Smale type model on isolated time scales, Discrete Contin. Dyn. Syst. Ser. S 11 (1) (2018) 77–89, http://dx.doi.org/10.3934/dedss.2018005. [18] J. Shen, J. Cao, Consensus of multi-agent systems on time scales, IMA J. Math. Control Inform. 29 (2012) 507–517. [19] F.Z. Taousser, M. Defoort, M. Djemai, Stability analysis of a class of switched linear systems on non-uniform time domain, Syst. Control Lett. 74 (2014) 24–31. [20] F.Z. Taousser, M. Defoort, M. Djemai, Stability analysis of a class of uncertain switched systems on time scale using Lyapunov functions, Nonlinear Anal. Hybrid Syst. 16 (2015) 13–23. [21] Q. Xiao, Z. Huang, Consensus of multi-agent systems with distributed control on time scales, Appl. Math. Comput. 277 (2016) 54–71. [22] R. Vaughan, N. Sumpter, J. Henderson, A. Frost, S. Cameron, Experiments in automatic flock control, Robot. Auton. Syst. 30 (2000) 109–117. [23] Y. Yang, D.V. Dimarogonas, X. Hu, Optimal leader-follower control for crowd evacuation, in: Proc. 52nd IEEE Conf. Decision Control, 2013, pp. 2769– 2774. [24] S. Sodsee, M. Komkhao, Z. Li, W.A. Halang, P. Meesad, Leader-following Discrete–time Consensus Protocol in a Buyer-seller Network, 2010, https://www.semanticscholar.org/paper/Leader-following-Discrete-time-Consensus-Protocol-a-Sodsee-Komkhao/68e5d176ba3ef0dd8464df c08d813098fb056423. [25] J. Bai, G. Wen, A. Rahmani, X. Chu, Y. Yu, Consensus with a reference state for fractional-order multi-agent systems, Int. J. Syst. Sci. 47 (1) (2016) 222–234. [26] H. Yang, H. Zou, H. Liu, F. Liu, M. Zhao, F. Han, Consensus tracking of multiagent systems with time-varying reference state and exogenous disturbances, Math. Probl. Eng. 2014 (2014) 213862, http://dx.doi.org/10.1155/2014/213862, 7 pages. [27] M. Bohner, A. Peterson, Dynamic Equations on Time Scales, Birkhäuser Boston, Boston, MA, 2001. [28] M. Bohner, A. Peterson, Advances in Dynamic Equations on Time Scales, Birkhäuser Boston, Boston, MA, 2003. [29] C. Pötzsche, S. Sigmund, F. Wirth, A spectral characterization of exponential stability for linear time–invariant systems on time scales, Discrete Contin. Dyn. Syst. 9 (2003) 98–103. [30] I.M. Chakravarti, On a characterization of irreducibility of a non–negative matrix, Linear Algebra Appl. 10 (1975) 103–109. [31] L. Chen, Y. Chai, R. Wu, J. Sun, T. Ma, Cluster synchronization in fractional–order complex dynamical networks, Phys. Lett. A 376 (2012) 2381–2388. [32] J.H. Wilkinson, The Algebraic Eigenvalue Problem, Oxford University Press, London, 1965. [33] R.S. Varga, Geršgorin and His Circles, Springer-Verlag, Berlin, Berlin, 2004.