Distributed tracking control problem of Lipschitz multi-agent systems with external disturbances and input delay

ABSTRACT This paper addresses the distributed tracking control problem for Lipschitz nonlinear multiagent systems (LNMASs) in the presence of external disturbances and input delay. For this end, a distributed tracking control algorithm is proposed for LNMASs that guarantee each agent can estimate and track a nonlinear target. The suggested algorithm is developed based on a future state predictor and a finite time estimator to cope with the tracking control problem for LNMASs with input delay and external disturbances. Finally, the efficiency of the proposed algorithms is illustrated by simulation results.


Introduction
In the past decade, distributed tracking control (DTC) of a group of mobile agents has attracted researchers' attention due to its wide applications in many fields including formation control, leader-follower problem, target tracking, flocking, intelligent transportation and so on (Fang, Wei, Chen, & Xin, 2017;Hong, Yu, Yu, Wen, & Alsaedi, 2017;Miao, Wang, & Fierro, 2017;Miranbeigi, Moshiri, & Rahimi Kian, 2016;Tian, Zuo, & Wang, 2017;Wang, Gao, Alsaadi, & Hayat, 2014;Yu, Yan, & Li, 2017;Zhao & Jia, 2016). DTC of multiagent systems includes two aspects of control and estimation. The control aspect plans to design a distributed controller for each agent to track the states of the target. The estimation aspect aims to propose a distributed estimator for each agent to estimate the states of the target.
The DTC problem can be investigated for multiagent systems with linear and nonlinear dynamics. Most of the existing algorithms on DTC focus on multiagent systems with linear dynamics (such as Olfati-Saber & Jalalkamali, 2012;Su, Chen, Chen, & Wang, 2016;Su, Li, & Chen, 2017), while practical multiagent systems and targets usually have nonlinear dynamics. Therefore, it is important to consider the DTC algorithms of nonlinear multiagent systems. In Yu, Dong, et al. (2017), a distributed control law has been proposed for second order nonlinear multiagent systems to cope with DTC problem. Li, Dong, and Nguang (2017) has studied the DTC problem for third order nonlinear multiagent systems and presented CONTACT Behrouz Safarinejadian safarinejad@sutech.ac.ir some algorithms for every agent to track a target with third order nonlinear dynamics. In Li (2015), some distributed controllers have been designed that ensure multiagent systems with high order dynamics can track a target under a directed graph. In Zhao, Guan, Li, Zhang, and Chen (2017), two control laws have been presented for multiagent systems with nonholonomic dynamics to guarantee collision avoidance among agents and tracking a target with unknown dynamics under fixed and switching topologies.
In the recent years, study of LNMASs has received great attention due to the fact that this kind of multiagent systems includes many practical multiagent systems. Li, Liu, Fu, and Xie (2012) presented a distributed control law for LNMAS to ensure each follower (agent) can track a leader (target) with Lipschitz nonlinear dynamics. The suggested controller has been developed for LNMAS with switching topologies (Wen, Duan, Chen, & Yu, 2014). In Wen, Yu, Xia, Yu, and Hu (2017), a DTC algorithm has been introduced for LNMAS to estimate and track the leader's states with Lipschitz nonlinearities. It is worth noting that the DTC algorithms in Wen et al. (2017), Wen et al. (2014), andLi et al. (2012) have been designed based on the assumption that the states of the targets are available or at least one of the agents is connected to the target. This assumption does not hold in the DTC problem and the states of the target should be estimated by a distributed estimator. Therefore, the algorithms of Wen et al. (2017), Wen et al. (2014), Li et al. (2012) cannot be used for the DTC of LNMAS. It should be noted that there are some distributed estimators which can estimate the states of the targets with nonlinear dynamics Jenabzadeh, Safarinejadian, & Mohammadnia, 2017;Li, Shen, Wang, & Alsaadi, 2017;Li, Shen, Wang, & Alsaadi, 2018;Shen, Wang, & Liu, 2011;Wei, Liu, Wang, & Wang, 2016). In Shen et al. (2011), Li, Shen, et al. (2017, , , the authors have proposed some estimators for the targets whose nonlinear part satisfy a special constraint and cannot be applied to the target with Lipschitz nonlinearities. Furthermore, Wei et al. (2016) has suggested a distributed filter based on an event-triggered mechanism to estimate the states of a class of time varying discrete-time nonlinear systems with Lipschitz-like condition. Moreover, the distributed state estimators of Li et al. (2018) like the presented filter in Wei et al. (2016) have been presented to estimate the states of a class of discrete systems and cannot be used for the targets with continuous-time dynamics. Based on these explanations, it is necessary to introduce a DTC algorithm for LNMASs in which the states are not available to any of the agents explicitly.
In reality, some inevitable destructive factors affect on system performance and may lead to instability. For instance, external disturbances are the source of poor performance and instability of multiagent systems. Moreover, time delay exists in practical multiagent systems and degrades the system performance. One type of time delay in multiagent systems is the input delay which is caused by processing and connecting time for the packets arriving at each agent. Based on these explanations, it is important to design some algorithms to decrease the effect of external disturbances and input delay on the performance of multiagent systems. To the best of the authors' knowledge, the DTC problem in LNMASs subject to external disturbances and input delay in which the target's states are not available is still open in the literature. This motivates us for this study.
In this paper, the DTC problem of LNMASs is solved in the presence of external disturbances and input delay. For this goal, a DTC algorithm is firstly established and analyzed for LNMASs in the absence of external disturbances and input delay. Then, the proposed algorithm is extended for LNMASs with external disturbances and input delay. The main difficulties in the design of the distributed estimator and controller in this paper are as follows: (1) The agents should track a target whose states are not available. It means that the distributed controller requires the estimated states of the target. Consequently, stability analysis of estimation errors and tracking errors should be carried out simultaneously. Since the target as well as the multiagent system have nonlinear dynamics, stability analysis is much more difficult than that of existing work in which multiagent systems have linear dynamics.
(2) Due to input delay, the proposed controllers cannot track the target. It means that the controllers must be modified such that the states of agents with input delay converge to the states of the target. Since there is no controller for the DTC of LNMASs in the literature, a novel distributed controller is required for LNMASs with input delay. This controller is designed based on a future state predictor. The proposed predictor is one of the first predictors which can estimate the future states of the multiagent systems with nonlinear dynamics. (3) In this paper, the deign of distributed estimator and controller are also carried out in the presence of external disturbances. Most of the existing works have solved the DTC problem in the presence of external disturbances for multiagent system with linear dynamics. Therefore, solving the DTC problem in the presence of external disturbances for LNMASs can be one of the first attempts for DTC of multiagent systems with nonlinear dynamics. The main contributions of this paper are as follows: • Compared with the distributed control laws of Yu, Dong, et al. (2017) (2018) can estimate the states of some limited targets with some special constraints or with discrete dynamics. • A DTC algorithm is suggested for multiagent systems with Lipschitz nonlinear dynamics that can be used for many practical multiagent systems, whereas the results of Olfati-Saber and Jalalkamali (2012), Su et al. (2016) and Su et al. (2017) are limited to linear multiagent systems that cannot be applied to nonlinear multiagent systems. • A DTC algorithm is obtained for LNMASs so that they can work appropriately in the presence of external disturbance and input delay.
The rest of this paper is organized as follows: the problem definition is presented in Section 2. Some DTC algorithms are designed for LNMASs in the absence and presence of input delay and external disturbances in Section 3. The simulation results are given in Section 4. Finally, Section 5 concludes the paper.
Notation: I υ represents an υ × υ identity matrix. R υ denotes the υ dimensional Euclidean space. R υ×p represents the set of all υ × p real matrices. ||.|| denotes the Euclidean vector norm. λ 2 (L) indicates the second smallest eigenvalue of the matrix L. The matrix inequality υ > 0 expresses that υ is a symmetric positive definite matrix.
Graph Theory. To model a multiagent system, an F, j = i} expresses the neighbours set of the agent i. If agents iand jare connected, the agent iis the neighbour of agent jand q ij = q ji > 0. Moreover, if there exists at least one path between every two arbitrary agents, then G is called a connected graph.

Problem definition
Consider a LNMAS with the ith agent dynamics given as: where r i (t) ∈ R m and u i (t) ∈ R S are state and input of the ith agent, respectively. A ∈ R m×m and B ∈ R m×S are constant matrices with (A, B) being controllable. h is a given . . ,D im are positive constants. The target dynamics is described by: where s(t) ∈ R m is the state of the target. The ith agent has the following sensing model: where w and v i are Gaussian white noises. We assume that there are positive constants ζ 1 and ζ 2 such that for q, p ∈ R m : The main goals of this paper are twofold: • To obtain a DTC algorithm such that the LNMAS (1) with external disturbances (h = 0) can estimate and track the states of the target (2). • To design a DTC algorithm such that the LNMAS (1) with input delay (D i (t) = 0) can estimate and track the states of the target (2).
Before designing a DTC algorithm for the above two cases, it will be derived for external disturbances and input delay free case (D i (t) = 0, h = 0). In this case, a distributed estimator and a control law are needed for every agent to design a DTC algorithm for LNMAS (1). The following distributed estimator (DE) is proposed for the LNMAS (1) to estimate the states of the target (2): whereŝ i (t) ∈ R m is the estimation of s(t), X i ∈ R m is the gain matrix, i > 0 ∈ R m×m is the estimator matrix which is used to derive the sufficient conditions for the existence of the DE (5), and γ i > 0 is the consensus gain which is used to adjust the consensus time of the states of DE (5) when the estimator stability holds. It is worth noting that the DE (5) includes three terms. The term Aŝ i (t) + ψ(ŝ i (t)) is used to reconstruct the dynamics of the target (2). Furthermore, the term (y i (t) − h i (ŝ i (t))) is the part of the measurement that contains new information about the target's states and is used to update the state estimate. Since the estimator (5) should be a distributed estimator which uses the information of the estimators of the neighbouring agents to estimate the states of the target (2), the term  2017) is extended for targets with Lipschitz condition that can be used for many practical targets.
To track the states of the target (2), a distributed controller (DC) is suggested for LNMAS (1) as follows:

Main results
In this section, the stability of the proposed DTC algorithm including DE (5) and DC (6) is analyzed. Then, the proposed algorithm is developed for the DTC problem in the presence of input delay and external disturbances.
where ρ i is a positive scalar which satisfies ρ i ≤ 2γ i λ 2 (L).
Then, one gets (without noise): A Lyapunov function is selected as follows: One obtains the derivative of V as follows: Utilizing inequalities and conditions (3) and (4), one can obtain Substituting (12) in (9) and usingŝ Applying (10) and (11) Using graph theory and defining By choosing ρ i so that ρ i ≤ 2γ i λ 2 (L) and utilizing Assumption 1, (7) and (8), one can obtain One can conclude from (13) and (14) thatV < 0. It implies that ε i and σ i converge to zero asymptotically for i = 1, . . . , N and LNMAS (1) with (5) and (6) can asymptotically estimate and track the target (2) in the absence of external disturbances and input delay.

Distributed tracking control problem in the presence of input delay
In the presence of input delay, DC (6) is transformed into LNMAS (1) with DC (15) cannot track the target (2) because of the delay h. To solve this problem, the following DC is proposed: where (r j (t),r i (t),s i (t)) are estimations of (r j (t + h), r i (t + h),ŝ i (t + h)). When (r j (t),r i (t),s i (t)) becomes equal to (r j (t + h), r i (t + h),ŝ i (t + h)), (16) with delay h has the same performance of DC (6) and DTC is achieved in the presence of input delay. DC (15) includes the states of the agents and the target. Therefore, two predictors are needed to predict the future states of the agents and the target. Since agents and target have the same dynamics, a predictor is designed to predict future states of the agents and the target. The local future states predictor (FSP) is described by:

Theorem 3.2: Consider FSP (17). The state ofr i (t) asymptotically converges to r i (t + h) if the following condition holds:
The time derivative of V 1 (e i (t)) is as follows: Using (3) and (4), we have Putting (20) into (19), one can obtaiṅ where (18) and (21) thatV 1 (e i (t)) < 0. Thus, one can conclude that e i (t) converges to zero andr i (t) converges to r i (t + h) asymptotically.
where i = i X i and H i = i φ i . The '*' symbol represents the symmetric entry of the matrix. K io is obtained by solving LMI (22). The LMIs (22-24) are solved to compute the parameters K i , i , X i and φ i .

Remark 3.2:
The existence of controller (6) depends on the feasibility of LMI (22). Using Finsler's Lemma (Iwasaki and Skelton (1994)), it can be concluded that there is a K i > 0 such that LMI (22) holds if and only if there is a K such that (A + BK)K i +K i (A + BK) T + I m + ζ 2 1K 2 i < 0 whereK i = K i /ζ 2 1 , which is dual of the observer design problem for the Lipschitz systems in Rajamani and Cho (1998) and Rajamani (1998). It implies that LMI (22) is feasible and also there exists controller (6) if and only if (A, B) is controllable. Similarly, it can be proved that LMI (23) is feasible when λ 2 (L) is positive, i.e. the graph related to the multiagent systems (1) is connected. Furthermore, LMI (24) is feasible if the matrix A is Herwitz according to the Boyd et al. (1994). Based on these explanations, if the mentioned conditions hold, then the LMIs (22), (23) and (24) are feasible and the parameters of DE (5) and DC (6) can be computed. Therefore, the multiagent systems (1) can estimate and track the target (2) using the DE (5) and DC (6) with the computed parameters.

Distributed tracking control problem in the presence of external disturbance
It is evident that LNMAS (1) with DC (6) do not track the target (2) in the presence of external disturbance D i (t).
To solve this problem, the following finite time estimator (FTE) is used to estimate D i (t) for each agent in a finite time (Shtessel, Shkolnikov, & Levant, 2007):  r i (t)), . . . ,g m (r i (t))) are elements of the vector Ar i (t) + ψ(r i (t)).
Under Assumption 1, the agents can estimate and track the states of the target (2) asymptotically in the presence of external disturbance D i (t).
Proof: We should prove that DC (26) in the presence of D i (t) has the same performance of DC (6). To this end, assume that the estimation errors of FTE (25) are as follows: Then, one obtainṡ Shtessel et al. (2007) proved that there exists a finite time T 1 such that the states of the system (27) converge to zero. Thus, for t ≥ T 1 , one hasD i (t) ≡ D i (t). It implies that the DC (26) has the same performance as DC (6) in the presence of external disturbance D i (t). Therefore, one can conclude that LNMAS (1) with DC (26) and DE (5) in the presence of D i (t) can estimate and track the states of the target (2) asymptotically.

Remark 3.3:
In this paper, the tracking control problem is solved for LNMASs with input delay or external disturbances. The proposed DTC algorithms cannot be used when there exist input delay and external disturbances simultaneously in the dynamics of LNMASs. The reason for this is the structure of DTC algorithms in both cases. Actually, in the presence of external disturbances, controller (26) is proposed based on the finite time estimator (25) (Shtessel et al. (2007)) which can estimate external disturbances of the systems in the absence of input delay. This means that the controller (26) cannot be used in the presence of input delay. On the other hand, in the presence of input delay, the control law (16) is designed based on the FSP (17). In this case, the existence of external disturbances make us not to be able to prove asymptotically converges of the FSP (17). Therefore, the condition (18) and the parameters of FSP (17) cannot be obtained. This implies that the controller (16) cannot be applied to a LNMAS with external disturbances.
At first, using the above parameters, LNMAS (28) with DC (6) and DE (5) are simulated in the absence of external disturbance and input delay (D i (t) = 0, h = 0). Figures  2 and 3 show tracking and estimation errors of LNMAS (28). These results verify the desired performance of DC (6) and DE (5) where the agents of LNMAS (28) can estimate and track the states of the target (29) in the absence of external disturbance and input delay. In the following, to validate DC (16) and DE (5), LNMAS (28) with DC (16) and DE (5)

Conclusions
In this paper, a distributed tracking control problem has been investigated for LNMASs with external disturbances and input delay. Some novel DTC algorithms have been suggested and proved that every agent can estimate and track a Lipschitz target. The simulations have shown the effectiveness of the presented algorithms in distributed tracking control of a LNMAS.

Disclosure statement
No potential conflict of interest was reported by the authors.