Robust Time-Varying Output Formation Control for Swarm Systems with Nonlinear Uncertainties

Time-varying output formation control problems for high-order time-invariant swarm systems are studied with nonlinear uncertainties and directed network topology in this paper. A robust controller which consists of a nominal controller and a robust compensator is applied to achieve formation control. #e nominal controller based on the output feedback is designed to achieve desired time-varying formation properties for the nominal system. And the robust compensator based on the robust signal compensator technology is constructed to restrain nonlinear uncertainties. #e time-varying formation problem is transformed into the stability problem. And the formation errors can be arbitrarily small with expected convergence rate. Numerical examples are provided to illustrate the effectiveness of the proposed strategy.


Introduction
e past decades have witnessed an increasing attention in distributed cooperative control of large-scale swarm systems due to their broad applications in various fields, such as physics, biology, mobile robots, sensor networks, and unmanned aerial vehicles [1][2][3][4][5]. Compared with the individual agent, swarm systems have great benefits in high efficiency, low cost, robust, and easy maintenance. e research involves a variety of branches, including consensus, flocking, synchronization, formation, and containment [6][7][8][9][10][11]. As one of the critical problems in distributed cooperative control systems, the formation problem is to find control laws that drive states or outputs of all agents to reach a predefined configuration. Many approaches have been proposed to achieve formation control [12], to name a few, leader-follower [13], virtual structure [14], and behaviorbased approaches [15].
Consensus is one of the fundamental problems of swarm systems [16,17]. Inspired by the development of consensus control theory, more and more researchers are interested in realizing the predefined formation via consensus approach without a central controller [18]. A consensus protocol was applied to achieve distributed formation control by Ren [16]. Olfati-Saber, Fax, and Murray [19] proposed a consensusbased theoretical framework for swarm systems with fixed or dynamic network topology. Xie and Wang [20] presented sufficient conditions for second-order swarm systems to realize time-invariant formation via local neighboring information interaction. Lafferriere et al. [21] studied the timeinvariant formation stability problem for integrator-type high-order multiagent systems. e formation mentioned above in [19][20][21] is time-invariant. In practical applications, the formation may change with the environment and tasks. Dong et al. [22] presented necessary and sufficient conditions for linear swarm systems with time delay to achieve time-varying formation. Xiao et al. [23] proposed a finite-time formation control framework which can realize a variety of complex formations and greatly reduce the data exchange. Generally speaking, only the output information of each agent can be obtained rather than state information. As a result, it is significant to study the formation control problem only using output information. Fax and Murry [23] studied output formation stability problems of high-order linear swarm systems. In [24], a dynamic output approach was applied to solve a fully distributed time-varying formationtracking problem for linear swarm systems. Zuo et al. [25] studied the adaptive output formation-tracking problem of linear heterogeneous swarm systems whose followers only received relative output information from their neighbors, and leaders' dynamics were only known to their neighboring followers. Dong et al. [26] proposed a consensus-based approach for swarm systems with directed interaction topologies to achieve time-varying output formations.
All these results listed above were for linear swarm systems. Practically, there are many cases where each agent involves uncertainties such as unknown time-varying parameters, unknown functions, and bounded external disturbances. Considering uncertain leader dynamics and uncertain local dynamics, Peng et al. [27] presented a neural network-based adaptive formation control approach for swarm systems. Li et al. [28] provided sufficient and necessary conditions to achieve the leader-following formation for second-order swarm systems with time-varying delay and nonlinear dynamics. Liu et al. [29] proposed a iterative learning-based method to deal with the formation control problem of swarm systems with unknown dynamics. Lu [30] proposed a robust controller consisting of a nominal controller and a robust compensator which is linear, time-invariant, and easy to implement. Because of great advantages, robust control can well deal with nonsmooth and discontinuous uncertainties compared to other approaches.
Compared with the approaches on formation control mentioned above, the main contributions of this paper are as follows. First, the dynamics of each agent is high-order with nonlinear uncertainties, and an output feedback control approach is proposed to deal with the time-varying formation problem. Second, the output formation error can be as small as desired with arbitrarily specified convergence rate under the proposed controller. Meanwhile, the proposed controller can be easily applied to practical situations. e rest of the paper are arranged as follows. In Section 2, some basic concepts on graph theory and the problem description are introduced, respectively. Meanwhile, some assumptions, definitions, and useful lemmas are presented. In Section 3, sufficient and necessary conditions for swarm systems to achieve time-varying formation are given, and the robust compensator is introduced to suppress the nonlinear part. In Section 4, simulation examples are shown to verify the analytical results. Finally, Section 5 concludes the whole work.
Notation 1. Let 0 N and 1 denote the matrix with all elements being 0 and column vectors of ones with dimension N, respectively. Use the superscripts T and H to represent the transpose and Hermitian adjoint of a matrix, respectively. ‖ · ‖ and ⊗ denote the Euclidean norm and Kronecker product, respectively. L and L − 1 represent Laplace transform and inverse transformation, respectively. And * represents the convolution operator.

Basic Concept on Graph eory. A directed graph of order N can be denoted by
node v i is the parent node, node v j is the child node, and v i is a neighbour of v j . e set of neighbours of node v i is denoted by If at least one node has a directed path to all the other nodes, the graph G is said to have a spanning tree. More details on graph theory can be found in [31].

Problem Description.
A swarm system consists of N agents with the directed graph G which is used to describe the interaction topology. e dynamics of each agent and y i (t) ∈ R q denote the control input and output of agent i with u(t) � [u T 1 (t), u T 2 (t), . . . , u T n (t)] T and y(t) � [y T 1 (t), y T 2 (t), . . . , y T n (t) T ]. Δ i is the external disturbance, and ω i (Δ i , x i ) is the nonlinear uncertainty. Simply, the nonlinear uncertainty Assumption 1. e matrix B 1 is of full-column rank, i.e., rank(B 1 ) � m. e matrix C is of full-row rank, i.e., rank(C) � q. Moreover, the output dimension q and input dimension m satisfy q ≥ m.
Definition 2. (see [26]). If the swarm system can achieve time-varying output formation h(t) under control input Definition 3. (see [26]). Swarm system (1) is said to achieve output consensus if there exists a function c(t) ∈ R q satisfying where c(t) is called an output consensus function.
Remark 2. From Definitions 1 and 3, one sees that the output formation reference function is equal to the output consensus function if h(t) ≡ 0, and output formation problems are converted into state formation problems if C � I.
Lemma 1 (see [32]). L ∈ R N×N is the Laplacian matrix of a directed graph G; then, (i) L has at least one zero eigenvalue, and 1 is the associated eigenvector, that is, L1 � 0 N (ii) If G has a spanning tree, then 0 is a simple eigenvalue of L, and all the other N − 1 eigenvalues have positive real parts Consider the following system: Lemma 2 (see [33]). If (A 22 , A 12 ) is completely observable, then system (2) is asymptotically stable with respect to y(t) if and only if A is Hurwitz.

Main Results
In this section, a robust control approach is introduced to deal with time-varying output formation problems with nonlinear uncertainties in directed networks. e robust controller consists of a nominal controller and a robust compensator. Section 3.1 designs the robust compensator, and Section 3.2 is to design the nominal controller and provide time-varying output formation analysis.

Robust Compensator Design.
e robust controller is constructed with two parts: where u nom i (t) stands for the nominal controller for the nominal model to obtain desired time-varying formation and u rob i (t) stands for the robust compensator to restrain the influence of nonlinear uncertainty ω i (t).
Based on Laplace transformation, the following frequency domain equation can be found from (1): where To restrain uncertainties, a robust compensator is designed as Substituting (5) and (7) into (6), we can get which means robust compensator (7) can restrain the nonlinear uncertainties perfectly. As a matter of fact that nonlinear uncertainty ω i (s) cannot be measured directly, substituting ω i (s) into (7), one can further express the robust compensator by To avoid the high-order derivative terms of the output, a robust filter is considered which can be described as where the robust filter is .F(s) � (f/(s + f)) d . f is a positive constant and d is a positive integer greater than or equal to the relative degree of the elements in N − 1 0 (s)D 0 (s). Substituting (5) into (10), the robust compensator us, one can obtain that

Nominal Controller Design and Feasibility
Analysis. e nominal system without uncertainties is shown as Complexity 3 To achieve time-varying output formation, the nominal controller can be designed as Under protocol (14), swarm system (13) can be written as follows: According to Assumption 1, one can find Using nonsingular transformation I N ⊗ T, swarm system (15) can be rewritten as According to Definition 1, time-varying output formation means swarm system (18) achieves output consensus. Let λ i (i � 1, 2, . . . , N) be the eigenvalues of the Laplacian matrix L and λ 1 � 0 with the associated eigenvector is an output consensus subspace, and the subspace C(U) spanned by p (q+1) , p (q+2) , . . . , p qN is a complement output consensus subspace. Because p j (j � 1, 2, . . . , qN) are linearly independent, Lemma 3 can be obtained as follows.
J can be denoted as J � diag 0, J by Lemma 1, and J consists of Jordan blocks corresponding to , and ζ(t) � (U ⊗ I)y n (t); then, swarm system (18) can be converted into

achieves timevarying output formation h(t) if and only if
Lemma 4 shows that the two subsystems with states ξ(t) and ζ(t) determine the output consensus and complement output consensus parts of system (18), respectively. From Lemma 2, it is easy to find that only observable components of (A 22 , A 12 ) affect the subsystem with state ζ(t). Let T be a nonsingular matrix, and the observability decomposition of (A 22 , A 12 ) is given as follows: where en, system (20) can be transformed into For B 11 and B 121 , there exist nonsingular matrices T � T 11 T 12 T 21 T 22 and Theorem 1. Suppose that the system described by (1), satisfying Assumptions 1-3. If the controller given by (5), (12), and (14) is applied, for any given positive constant ϵ and any initial conditions, one can find T ≥ t 0 that ‖e(t)‖ 2 ≤ ϵ, t ≥ T if and only if the following conditions hold simultaneously: (ii) e following N − 1 matrices are Hurwitz: where i � 2, 3, . . . , N.
Proof. From Corollary 1 in [26], one can see that (28) and (29) are sufficient and necessity conditions for nominal system (13) with nominal controller (14) to achieve timevarying output formation h(t). en, u nom (t) � [u nom 1 (t) T , u nom 2 (t) T , . . . , u nom n (t) T ] T . And u nom (t) can be described as From (1), (5), and (10), one has From (15), (24), (26), and (31), one can obtain the error system given by Combining (26), (27), and (32), one can get that Moreover, from (5), (10), (31), and Assumption 3, one gets Combining (34) and Assumption 3, one can obtain that where ζ x and ζ u are positive constants. From Assumption 2, (24), and (35), positive constants ζ and μ can be found satisfying where η � max(η 1 , η 2 ). From (26), (27), and (35), positive constants ζ q and μ q can be found such that where If f is large enough and satisfies ζ q ‖q‖ η− 1 ∞ ≤ (f/2), from (37), one can get Choose the Lyapunov function candidate where P is a symmetric positive definite constant matrix and satisfies en, the derivative of V is as follows: where Noticing that ‖q(t)‖ 2 ≤ � n √ ‖q(t)‖ ∞ ≤ � n √ ‖q‖ ∞ , ∀q(t) ∈ R p , from (39) and (42), one can obtain (43) If f is large enough to satisfy then one can obtain From (45), one can get that if V(z(0)) ≤ ((λ max (P))/f), then V(z(t)) ≤ ((λ max (P))/f); hence, ‖z(t)‖ 2 ≤ ������������������ � (λ max (P))/(fλ min (P)). Else, if V(z(0)) > (λ max (P)/f), then V(z(t)) ≤ V(z(0)) so that ‖z(t)‖ 2 ≤ ����������������� � (V(z(0)))/(λ min (P)). at means z(t) is bounded. 6 Complexity Because (39) and z(t) are bounded, it follows that if f is sufficiently large, then where λ is a positive constant. As a result, x e (t) is bounded and converges for any initial conditions. And for any given positive constant ϵ, where T is a positive constant depending on initial conditions. erefore, the time-varying output formation is achieved. □ Remark 3. From equation (28), we can see that the condition is very conservative and restrictive. Only a few formations h(t) can satisfy the condition. e contribution of this paper is to find sufficient and necessary conditions to achieve time-varying formation for general high-order linear systems. Especially, when the formation h(t) can be described as sin function, cos function, or exponential function, it is very likely to satisfy condition (28). Moreover, the trial-and-error method can be used to find K 1 . C and T can be adjusted to find proper K 1 . (11) and (14), one can see that only gain matrices K 1 , K 2 , and K 3 and positive constants f and d are required. K 1 and K 2 can be found based on the pole place method. According to He and Wang [34], iterative linear matrix inequality algorithm can be applied to find K 3 . d is the relative degree of the elements in N − 1 0 (s)D 0 (s). e value of f can be chosen by using the trial-and-error method. In general, the proposed controller can be easily applied to practical situations.

Numerical Simulations
In this section, a numerical simulation is presented to illustrate the effectiveness of the proposed control method. e designing process of nominal controller (12) is presented for swarm system (13) to achieve time-varying output formation.
Step 1: solving feasible condition (28) for K 1 Step 2: choosing K 2 to assign the eigenvalues of A + BK 1 C + BK 2 C at desired locations in the complex plane Step 3: designing K 3 to make condition (29) satisfied en, Γ i (i � 2, 3, . . . , N) in condition (ii) can be rewritten as 2, 3, . . . , N). (49) From (49), one can see that if and only if K 3 can stabilize subsystems (A 0 , λ i B 0 , C 0 ) through the static output feedback (SOF), then Γ i are Hurwitz. To find the gain matrix K 3 , an improved iterative linear matrix inequality (ILMI) algorithm [34] is employed which can solve the SOF problem without introducing any additional variables. And the dimensions of the LMIs need not be increased.
Time-varying output formation control for a fifth-order system with eight agents is considered. e directed interaction topology is shown in Figure 1, and its adjacency matrix is assumed to be 0-1 matrix. e dynamics of each agent described by (1) are Obviously, (A, B) is stabilizable, and (A 22 , A 12 ) is not observable. e nonsingular matrix T is selected as It can be obtained that According to the designing process of protocol (5), choose K 1 to satisfy condition (i) in eorem 1. e initial state is 8 Complexity e eight agents need to achieve predefined time-varying output formations which are defined as h i (t) � [10sin (t + ((i− 1)π/4))10cos(t+((i− 1)π/4))− 10sin(t+((i− 1) π/4))] T (i � 1,2,..., 8).
e nonlinear uncertainty is denoted as 8)which is related to external disturbances and agents' states. e robust compensator parameters are f � 60 and d � 2.
Considering nonlinear uncertainty, Figures 2 and 3 show the output formation snapshot of eight agents with the        Complexity nominal controller and robust controller, respectively. e outputs of agents are denoted by the plus, triangle, diamond, square, asterisk, x-mark, circle, and point, and the output formation reference function is denoted by the pentagram. From the comparison of two figures, one can see that the robust controller has better control effect under nonlinear uncertainties. Figure 3 achieves the specified formation in ten seconds while Figure 2 does not. Figures 4 and 5 show the output formation error of eight agents with the nominal controller and robust controller, respectively. From Figure 4, we can see that output errors converge. Because of the existence of nonlinear uncertainty ω i (t), output errors can only converge to the small neighborhood of zero. Compared with Figure 4, output errors in Figure 5 can also converge to the small neighborhood of zero. However, the convergence speed is faster, and the convergence domain is smaller in Figure 5. From the comparison of two figures, one can see that the robust controller can restrain the nonlinear uncertainties well. Meanwhile, the formation errors can be made as small as desired with arbitrarily specified convergence rate. Figure 6 shows the input signal of eight agents including ‖u norm i (t)‖ 2 and ‖u rob i (t)‖ 2 . From Figure 6, we can see that two controllers tend to be stable. e robust compensator outputs change over time and are similar to nonlinear uncertainty ω i (t) which also verifies the effectiveness of the proposed controller. Meanwhile, input constrain may be encountered considering actuator saturation in practical situations. As a result, it is meaningful to study the influence of input constrain on multiagent systems in future work.

Conclusion
Time-varying output formation control problems for multiagent systems with directed interaction topologies and nonlinear uncertainties are studied. e nonlinear uncertainties are related to external disturbances, parameter uncertainties, nonlinearities, and couplings. By extending the dimensions of the observation matrix, the formation control problem is transformed into a consensus problem and then into a stability problem of the subsystem. A novel robust controller is proposed which includes a nominal controller and a robust compensator. e nominal controller based on relative outputs of neighbour agents is designed to achieve expected performance for the nominal system. And the robust compensator is to restrain nonlinear uncertainties. Based on the proposed controller, the output formation error can be as small as desired. Future work will focus on swarm systems with switching topologies, input constrain, and time-delay group formation control problems.
Some related works can be found in [35]. Compared with previous works, different problems are addressed using the similar method. First, time-varying formation is studied in this paper. However, it is time-invariant formation that was studied. Second, most of the formulas are different. And more assumptions and remarks are added to compare with other works. ird, the proof of eorem 1 is different. e process of proof in this paper is more rigorous and complete.
Data Availability e data used in this paper have been given in the simulation section.

Conflicts of Interest
e authors declare that they have no conflicts of interest. 12 Complexity