Distributed Coordination for a Class of High-Order Multiagent Systems Subject to Actuator Saturations by Iterative Learning Control

1is paper investigates a distributed coordination control for a class of high-order uncertain multiagent systems. Under the framework of iterative learning control, a novel fully distributed learning protocol is devised for the coordination problem of MASs including time-varying parameter uncertainties as well as actuator saturations. Meanwhile, the learning updating laws of various parameters are proposed. Utilizing Lyapunov theory and combining with Graph theory, the proposed algorithm canmake each follower track a leader completely over a limited time interval even though each follower is without knowing the dynamics of the leader. Moreover, the extension to formation control is made. 1e validity and feasibility of the algorithm are verified conclusively by two examples.


Introduction
Coordination of multiagent systems (MASs) draws a lot of attention because of its wide applications in UAV, biological systems, sensor networks, and so forth. An important problem in coordination is to develop the distributed protocols, which specifies information exchange among agents, such that the group as a whole can agree on a common quantity. is kind of problem is called consensus [1][2][3][4][5]. For this consensus realization, a leader-follower consensus [6,7] has been extensively studied by many scholars. Another problem of coordination is formation control; the fault-tolerant leader-following formation control and cluster formation control were discussed in [8,9], respectively. However, in the real world, the dynamics of systems are usually uncertain. erefore, the consensus or/ and formation control for uncertain MASs become a hot topic in the field of control.
It is noteworthy that the abovementioned studies did not consider actuator saturations in the MASs. Actually, due to the limited actuator capability, actuator saturations may exist in most physical systems and cause the system instability. Consequently, it is useful to take into account actuator saturations in system analysis. Recently, a large number of work about MASs with actuator saturations emerge [25][26][27][28][29][30][31][32]. References [25][26][27][28][29] studied consensus algorithms under one-dimensional system framework when time goes to infinity and [30][31][32] tackled the consensus problem by iterative learning control (ILC). In [30], authors addressed the neural network consensus for the second-order MASs subject to saturation input; authors in [31] proposed the fully distributed learning scheme and made use of the properties of saturation function to solve actuator saturations; the leaderfollower consensus for the first-order MASs with input saturation was researched in [32]. ereinto, the dynamic of the leader was known to each follower in [31,32].
Based on the aforementioned observations, we have successfully combined the fully distributed algorithm applying ILC over the finite time interval to address the coordination for a class of high-order MASs.
e major contributions are highlighted as follows: (i) Differing from the adaptive consensus of MASs with actuator saturations, the learning consensus researched in this paper can make each follower track the leader absolutely over, where the parameter uncertainties are time-varying (ii) In contrast with the adaptive ILC consensus algorithm available, the algorithm in the paper is more complicated as well as each follower is without knowing the dynamic of the leader (iii) e consensus problem is extended to the formation control for a class of high-order MASs e rest is arranged as follows. Some useful preliminary results and problem formulation are presented in Section 2. Section 3 is the protocols design and consensus analysis. e extension to formation control is given in Section 4. Two examples for illustration are taken in Sections 5 and conclusion is drawn in Sections 6, respectively.

Problem Formulation
Here, the MASs with N followers and one leader are considered under a repetitive environment. e models are where x k i (t) ∈ R n is the state vector of the i th follower and u k i (t) ∈ R n is the input vector of the i th follower; θ i (t) is an unknown continuous time-varying parameter, which shows the uncertainty in system models for each follower agent; ξ i (x k i ) ∈ R n is a known smooth nonlinear vector valued function; A ∈ R n×n ; sat(u k ij , u j * ) is the saturation function [31], j � 1, 2, . . . , n, and sat(u k i , u * ) � [sat(u k i1 , u * 1 ), sat (u k i2 , u * 2 ), . . . , sat(u k in , u * n )] T is a saturation vector function; x 0 (t) ∈ R n is the state vector of the leader, and f(x 0 (t), t) � [f 01 (x 0 (t), t), f 02 (x 0 (t), t), . . . , f 0n (x 0 (t), t)] T ∈ R n is an unknown but bounded vector valued nonlinear function. Assumption 1. It is assumed that |f 0j (x 0 (t))|⩽η j with η j being an unknown positive constant. Denote η � [η 1 , η 2 , . . . , η n ] T . Remark 1. From the above, it is known that each follower is without knowing the dynamic of the leader.
For the i th follower, the consensus error is In this paper, we aim to find suitable protocols u k i , i � 1, 2, . . . , N, 0 ≤ t ≤ T and the updating laws of parametric uncertainties so that all the followers can uniformly track the leader over [0, T] as k approaches to infinity, that is to say, Assumption 2. e state vector of each follower and the leader satisfy x k i (0) � x k− 1 i (t) and x 0 (0) � x 0 (t).
To design the distributed protocols, the distribute error is e compact forms are where e k � [(e k 1 ) T , (e k 2 ) T , . . . , (e k N ) T ] T ∈ R Nn and δ k � [(δ k 1 ) T , (δ k 2 ) T , . . . , (δ k N ) T ] T ∈ R Nn ; H � L + B is a symmetric positive definite matrix, and the communication topology of the paper is the same as that in [31]. where Remark 4. It is obvious that η k i (t) is nonnegative, which is guaranteed from the updating law (12). e i th error dynamic becomes where θ where Proof. e proof falls into three parts. At the k th iteration, a Lyapunov candidate is established as where From the error dynamic (14), we have where Simultaneously, 4 Complexity In addition, Substituting (18)-(23) into (17) yields Due to (8) and Property 1 in [20], erefore, On account of the positiveness of H, C, and P, set δ k � (F T ⊗ I n )δ k and F is an orthogonal matrix: where λ i (H) > 0 shows the eigenvalue of H and c min � min 1≤i≤N c i . For the sufficient large constant c min > 0, the inequality PA + A T P − 2(c min λ min (H)λ min (P) − λ max 2 (H) λ max 2 (P)) ≤ − σI with σ > 0 always holds. When t � T, it follows that which results in

Complexity
In the second place, let us prove boundednesses of signals involved. On the basis of the definition of V k results, at is, It can be obtained from (29) and (31) that As lim k⟶∞ 2NTε k+1 l�2 Δ l ⩽4NTεa [20], 2NTε k+1 l�2 Δ l is uniformly bounded, ∀k ∈ Z + . Denote 2NTε k+1 l�2 Δ l ⩽S with S > 0. Hence, If the finiteness of V 0 (T) is attained, the uniform boundedness of V k (t) is followed. And then, we will show the finiteness of V 0 . It obtained that and where us, Since θ i (t) is continuous, the boundedness of it obtained [0, T]. en, where V 0 (t) is finite; it is followed that V 0 (T) is bounded. erefore, the uniformly boundedness of V k (t) is obtained over [0, T], ∀k ∈ Z + . Furthermore, it is inspired by the definition of V k ; it obtained the uniform boundednesses of δ k (t), μ k i (t), c k i (t), and η k i (t). e updating law (11) indicates that θ k i (t) is bounded. From (7), we can calculate the uniform boundedness of u k i (t). So, the boundednesses of signals involved are gained.
At last, we prove the property of learning consensus. As we know, From (28), it can yield Since V k (T) is positive, V 0 (T) is bounded, and the series 2NTε k+1 l�2 Δ l is convergent, and the series k l�1  e condition PA + A T P − 2(c min λ min (H) λ min (P) − λ max 2 (H)λ max 2 (P))I ≤ − σI with σ > 0 is only for the analysis purpose; as a matter of fact, it is not utilized in the design of protocols. Accordingly, the distributed learning control protocols are fully distributed and the consensus for the MASs is solved faultlessly even if each follower is without knowing the dynamic of the leader. (1) is concerned here. If the followers and leader form a formation at a certain distance over [0, T], we can say that the formation control is achieved.

Formation Control of the MASs e formation control of the MASs
Let us define where Δ i is the expected formation vector for the i th follower relative to the leader. e formation error is And, δ k il (t) is the same as δ k il (t) defined in (2), l � 2, 3, . . . , n.
Like that, the problem of formation can be reformulated as the consensus problem, i.e., lim k⟶∞ ‖δ k i ‖ � 0. Simultaneously, the neighborhood formation errors are (44) N and l � 2, 3, . . . , n; for the leader,

Theorem 2. For the MASs with graph G, under assumptions 1 and 3, N followers represented by (1) under the protocols (7)-(8) with learning-based updating laws (9)-(12) with the local neighborhood formation errors (44) can make the followers form the desired formation in the iteration domain on
t ∈ [0, T]. e variables involved are bounded.

Simulation
In this part, two examples are provided to validate the validity and practicability for the fully distributed learning protocol of this paper. As mentioned in [33], the LC 8 Complexity oscillator system can be considered as a MASs. erefore, here, let us consider the MASs consisting of six followers as well as one leader. Figure 1 shows the communication graph. en, we have    x i2 x 11 x 21 x 31 x 41 x 51 x 61 x 01 x 12 x 22 x 32 x 42 x 52 x 62 x 02    Complexity After 30 cycles, Figures 2-4 show the simulation results. Even if the dynamic of the leader is unknown to each follower and there exit actuator saturations in the dynamic of system, it can be seen from Figures 2 and 3 that six followers can perfectly track the leader, and in Figure 4, it is evident the signals involved are bounded. e results fit into eorem 1.  Figure 6 shows that the errors converge to zero over [0, 2π], and Figure 7 illustrates that the signals involved in the closed-loop system are bounded. e results obtained align with eorem 2.
Example 2. In the LC oscillator system [33] with six follower oscillators and one leader oscillator, each LC oscillator is governed by x 11 x 21 x 31 x 41 x 51 x 61 x 01 x 12 x 22 x 32 x 42 x 52 x 62 x 02 where L, C, c i (t), and v i (t) denote the inductance, capacitor, current, and voltage; i � 1, 2, . . . , 6. Under the repetitive environment, we study the fully distributed adaptive ILC for the LC oscillator system (47); hence, the input is applied to each oscillator (47) and affected by actuator saturations. Furthermore, assume that that the disturbances are time-varying linearly parameterized uncertainties θ i (t)ξ i (x k i ). Under these circumstances, system (47) is rewritten as ,     x i2 x 11 x 21 x 31 x 41 x 51 x 61 x 01 x 12 x 22 x 32 x 42 x 52 x 62 x 02 Figures 8-10 demonstrate the results of consensus control for 60 iterations, respectively. We can see that six follower oscillators can perfectly track the leader oscillator from Figures 8 and 9, even if each follower oscillator is without knowing the dynamic of the leader oscillator, and the signals involved are bounded. e results fit into eorem 1.

Conclusions
We have solved the fully distributed learning coordination problem of a class of high-order nonlinear MASs by adaptive ILC in this study. With the help of algebraic graph theory, Barlat-like lemma, and Lyapunov theory, the perfect consensus tracking as well as the formation control problem has been resolved over [0, T]. At last, two examples testify the effectiveness and efficiency of the algorithm devised in the paper.

Data Availability
No data were used to support this study.

Disclosure
is article has been submitted as a pre-print.

Conflicts of Interest
e authors declare that they have no conflicts of interest.