Distributed tracking control of multiple nonholonomic mobile agents with input delay

Abstract

This paper considers the distributed control problem of nonholonomic mobile agents with input delay to track a target. In contrast to the existing works in which some controllers have been designed for ideal conditions, the destructive factor of input delay is considered in the dynamics of the agents in this work. At first, a distributed controller is suggested for each agent to track the target in the absence of input delay. In this case, each nonholonomic mobile agent is divided into two subsystems and two terminal sliding mode controllers are designed for the two subsystems. Based on the proposed controllers, a switching control strategy is obtained to guarantee the finite time tracking control of nonholonomic mobile agents. Then, this controller is extended based on a future state estimator for tracking control in the presence of input delay. The stability analysis of distributed controllers and estimator is also provided. Simulation results illustrate the effectiveness of the suggested algorithms.

Keywords

Distributed control multi-agent systems nonholonomic systems target tracking delay

Introduction

In the recent decade, the tracking control problem of multi-agent systems (MASs) has attracted researchers’ interest because of its successful applications in different areas including formation control, consensus control, flocking, cooperative control and so on (Chen et al., 2016; Franchi et al., 2016; Hausman et al., 2015; Jiang et al., 2015; Li, 2015; Wang et al., 2014; Xia et al., 2014). Tracking control of MASs includes two aspects of estimation and control. The estimation aspect plans to design a distributed estimator for each agent to estimate the states of the target. The control aspect aims to design distributed controller for every agent to track the estimated states of the target. In both aspects, distributed algorithms should be designed so that the states of every agent can agree on a common value with their neighbors.

The tracking control problem of MASs can be considered in two categories of linear and nonlinear dynamics. Most of the existing works on tracking control focus on MASs with linear dynamics, while practical MASs usually have nonlinear dynamics with nonholonomic constraints. Thus, it is important to study tracking control of nonholonomic agents. In Yu and Liu (2016), the formation control problem of nonholonomic MASs with directed communication graph has been solved. The suggested controller in Yu and Liu (2016) ensures that all agents move along a common circle. Dong and Djapic (2016) proposed distributed state feedback control laws for nonholonomic MASs such that states of each agent converge to the states of a target with an exponential rate. This paper has also extended the results for MASs with time-varying communication graph. Cao et al. (2017) has introduced some distributed controllers for multiple nonholonomic mobile robots such that the mobile robots can move along a desired trajectory with some desired formation. In Zhao et al. (2017), two controllers have been presented so that nonholonomic MAS can track a target with unknown dynamics in a flocking manner for both fixed and switching topologies. The proposed controllers guaranteed collision avoidance among agents. Yang et al. (2017) has solved two problems of tracking control and rendezvous for multiple nonholonomic agents in the presence of unknown disturbances. In this work, a disturbance estimator is used in each agent to degrade the effect of unknown disturbances. In Liu et al. (2017), a sampled-data control law has been introduced for agents with nonholonomic dynamics to track a mobile target. This paper designed a dwell time for the feasibility of processing and information sensing to prevent chattering caused by abrupt changes of the neighbor relations.

In the recent decade, study of the chained-form of nonholonomic systems has absorbed great attentions because any kinematic model of nonholonomic systems is included in this form. In fact, Murray and Sastry (1993) has showed that nonholonomic systems can be transformed into the chained-form by input and state transformations. For instance, unicycles and cars with trailers that have three states and two inputs can be converted into the first order chained-form nonholonomic system. There exist some papers that have studied the tracking control problem of chained-form nonholonomic systems (Cao et al., 2015; Du et al., 2015; Mobayen, 2015). Mobayen (2015) has presented a recursive terminal sliding mode strategy to solve finite time tracking control problem of chained-form nonholonomic systems. In Du et al. (2015), a finite time observer-based controller has been proposed for high order nonholonomic MASs to ensure finite time convergence of the agents to a target with nonholonomic dynamics. Cao et al. (2015) has suggested some controllers for chained-form nonholonomic MASs with directed communication graph so that agents can track a target with chained-form nonholonomic dynamics. It should be noted that Mobayen (2015) has introduced a tracking control algorithm for a single nonholonomic system that cannot be used for chained-form nonholonomic MASs. Moreover, Du et al. (2015) and Cao et al. (2015) have assumed that at least one of the agents is connected to the targets. This assumption does not hold in the tracking control problem and hence the proposed algorithms in Du et al. (2015) and Cao et al. (2015) cannot be applied for tracking control of chained-form nonholonomic MASs.

In reality, delay is inevitable in practical MASs and degrades performance of the system and may lead to instability. There exist two types of delays in MASs: communication delay and input delay. Communication delay is related to communication among agents while input delay is related to processing and connecting time for the packets arriving at each agent. In Shang et al. (2016), the stabilization problem has been solved for chained-form nonholonomic systems with input delay. In this work, a controller has been suggested based on input-state-scaling technique and the backstepping approach to ensure asymptotically stability of the closed-loop system. The suggested control law in Shang et al. (2016) is not a distributed controller and cannot be used for tracking control of chained-form nonholonomic MASs. It is worth noting that although the presented algorithm in Dong and Djapic (2016) is a distributed control law, it has been suggested for dynamic form of nonholonomic MASs with input delay and cannot be utilized for the chained form of nonholonomic MASs with input delay. Based on these explanations, it is necessary to propose some algorithms to solve the tracking control problem of chained-form nonholonomic MASs in the presence of input delay. To the best of the authors’ knowledge, this problem has not been considered in the literature. This motivates us for this study.

In this paper, a finite time control law is firstly obtained for nonholonomic MASs in which the states of each agent track the states of the target in the absence of input delay by using the results of Shi et al. (2015). Actually, Shi et al. (2015) proposed some controllers for a nonholonomic MAS to ensure that the agents can agree on a common value with their neighbors. Moreover, the agents do not track any target in Shi et al. (2015). In this paper, we consider a target whose states must be tracked by the states of each agent. Existence of states and input of the target and tracking errors in distributed controllers in this paper results in a much more difficult stability analysis than that of Shi et al. (2015). Furthermore, input delay causes the existing controllers fail to track the target. It means that the existing controllers should be modified such that the states of each agent with input delay converge to the states of the target in a finite time. Since there is not any finite time controller for the tracking control of chained-form nonholonomic MASs in the literature and the finite time convergence is considered in this paper, a novel finite time distributed controller is needed for chained-form nonholonomic MASs with input delay. This controller is designed based on the terminal sliding mode method, a switching control strategy and a future state estimator (FSE). The dynamics of agents and also the target is given by a third order nonholonomic system in the chained form.

Compared with the previous works, the contributions of this paper are at least fourfold. First, in contrast to the works of Yu and Liu (2016), Dong and Djapic (2016), Cao et al. (2015), Zhao et al. (2017), Yang et al. (2017) and Liu et al. (2017), which suggested some control laws for the dynamic form of nonholonomic MASs and cannot be used for chained form of nonholonomic MASs, this paper will propose some controllers for chained-form nonholonomic MASs that can be applied to any nonholonomic MASs with three states and two inputs. Second, it is assumed in this paper that the target is not connected with any agent in the design of the distributed control law for tracking control, whereas the previous results in Du et al. (2015) and Cao et al. (2015) have assumed that at least one of the agents is connected to the target. Third, in contrast to the works of Mobayen (2015) and Shang et al. (2016), in which the proposed controllers are not distributed and cannot be utilized for MASs, in this paper some distributed controllers are introduced to solve the tracking control of chained form nonholonomic MASs. Fourth, in Yu and Liu (2016), Cao et al. (2015), Zhao et al. (2017), Yang et al. (2017) and Liu et al. (2017), Mobayen (2015), Du et al. (2015) and Cao et al. (2015), the control laws have been designed for nonholonomic systems in the ideal conditions, whereas the suggested controllers in this paper are obtained for nonholonomic MASs with input delay that mean these algorithms can be used in practical environments and work appropriately in the presence of the destructive factor of input delay.

The rest of this paper is organized as follows: Section 2 provides the preliminaries and problem definition. Sections 3 and 4 introduce distributed controllers for nonholonomic MASs with and without input delay. Section 5 studies the effectiveness of the suggested controllers and estimator with simulations. Finally, the conclusions are drawn in Section 6.

Notation: $R$ presents the set of real numbers. $R^{n}$ denotes the $n$ dimensional Euclidean space. $R^{n \times m}$ denotes the set of all $n \times m$ real matrices. $I_{M}$ expresses an $M \times M$ identity matrix. $λ_{min} (.)$ denotes the smallest eigenvalue.

Preliminaries and problem definition

Graph Theory: To model MASs, we use an undirected graph $G = (ω, μ, A)$ , where $ω = {1, 2, \dots, N}$ , $μ \in ω \times ω = {(i, j) : i, j \in ω}$ and $A = [a_{ij}] \in R^{N \times N}$ denote set of agents, set of links among agents and the adjacent matrix, respectively. The Laplacian matrix of the graph $G$ is defined as $L = R - A$ , in which $R$ is a diagonal matrix with $R_{ii} = \sum_{j \in N_{i}} a_{ij}$ . $N_{i} = {j \in ω : (i, j) \in μ, j \neq i}$ denotes the set of neighbors of the agent i. If nodes $i$ and $j$ are connected, then the node $i$ is the neighbor of node $j$ and $a_{ij} = a_{ji} > 0$ . Moreover, if there is at least one path between every two arbitrary agents, $G$ is called a connected graph.

Assumption 1: There exist positive constants $u_{1 d}^{min}$ , $u_{1 d}^{max}$ such that $0 < u_{1 d}^{min} \leq u_{1 d} \leq u_{1 d}^{max}$ .

Assumption 2: The graph G related to the nonholonomic mobile agents is connected and undirected.

Lemma 1 (Hong et al., 2006): If Assumption 2 holds, then the matrix $L + I$ is positive definite where $I$ is an identity matrix with proper dimensions.

Lemma 2 (Olfati-Saber and Murray, 2004): If L is the Laplacian matrix of a connected undirected graph G, we have

x^{T} Lx = \frac{1}{2} \sum_{i, j = 1}^{N} a_{ij} {(x_{i} - x_{j})}^{2} = \frac{1}{2} \sum_{i = 1}^{N} \sum_{j \in N_{i}} a_{ij} {(x_{i} - x_{j})}^{2}

for any $x = {[x_{1}, \dots, x_{N}]}^{T}$ . 0 is a simple eigenvalue of L and 1 is the associated eigenvector, where 1= ${[1, \dots, 1]}^{T}$ with appropriate dimensions.

Lemma 3 (Bhat and Bernstein, 2000): Consider the following nonlinear system

\overset{\cdot}{x} = F (x), F (0) = 0, x \in R^{n} .

Suppose that there exists a positive definite continuous function $V (x)$ such that $\overset{\cdot}{V} (x) + c V^{α} (x) \leq 0$ , where $c > 0$ and $0 < α < 1$ , then $V (x)$ approaches 0 in a finite time. Furthermore, the finite convergence time $T$ satisfies $T_{x} (x_{0}) \leq \frac{V {(x_{0})}^{1 - α}}{c (1 - α)}$ .

Lemma 4 (Huang et al., 2005): For any $a \in R$ and $s \in R$ , the following inequalities hold

\begin{matrix} {| a + s |}^{z} \leq 2^{z - 1} | a^{z} + s^{z} | \\ (| a | + | s |)^{1 / z} \leq {| a |}^{1 / z} + {| s |}^{1 / z} \end{matrix}

when $z \geq 1$ is a constant. If $z \geq 1$ is odd, then

\begin{matrix} {| a - s |}^{z} \leq 2^{z - 1} | a^{z} - s^{z} | \\ ({| a |}^{z} - {| s |}^{z}) \leq z | a - s | | a^{z - 1} + s^{z - 1} | . \end{matrix}

Problem definition

Consider N nonholonomic mobile agents with the following dynamic equations

{\begin{matrix} {\overset{\cdot}{x}}_{1 i} (t) = u_{1 i} (t - t_{D 1}) \\ {\overset{\cdot}{x}}_{2 i} (t) = u_{2 i} (t - t_{D 2}) \\ {\overset{\cdot}{x}}_{3 i} (t) = u_{1 i} (t - t_{D 1}) x_{2 i} (t) \end{matrix}, i = 1, \dots, N

(1)

where $(x_{1 i}, x_{2 i}, x_{3 i})$ are states of the ith agent and $(u_{1 i}, u_{2 i})$ are control inputs of agent i. $t_{D 1}$ and $t_{D 2}$ are positive constants. A nonholonomic mobile target is given as

{\overset{\cdot}{x}}_{d} (t) = (\begin{matrix} u_{1 d} (t) \\ u_{2 d} (t) \\ x_{2 d} (t) u_{1 d} (t) \end{matrix}), x_{d} (t) \in R^{3}

(2)

where $x_{d} (t)$ are states and $(u_{1 d}, u_{2 d})$ are control inputs of the target. The main problem is to design a distributed control algorithm such that the states of agents with dynamics (1) track the states of the target with dynamics (2) in a finite time. Before the design of distributed control algorithm in the presence of input delay, we will derive this algorithm in the absence of input delay.

The distributed tracking control in the absence of input delay

To design the distributed controller for the ith agent, the tracking error between the states and inputs of the agent i and the target are defined as ${\bar{x}}_{1 i} = x_{1 i} - x_{1 d}$ , ${\bar{x}}_{2 i} = x_{2 i} - x_{2 d}$ , ${\bar{x}}_{3 i} = x_{3 i} - x_{3 d}$ , ${\bar{u}}_{1 i} = u_{1 i} - u_{1 d}$ and ${\bar{u}}_{2 i} = u_{2 i} - u_{2 d}$ . The error dynamics equation is given as

{\begin{matrix} {\overset{\cdot}{\bar{x}}}_{1 i} = {\bar{u}}_{1 i} \\ {\overset{\cdot}{\bar{x}}}_{2 i} = {\bar{u}}_{2 i} \\ {\overset{\cdot}{\bar{x}}}_{3 i} = x_{2 i} u_{1 i} - x_{2 d} u_{1 d} . \end{matrix}

(3)

To solve the finite-time tracking problem, control laws ${\bar{u}}_{1 i}$ and ${\bar{u}}_{2 i}$ are designed such that ${\bar{x}}_{1 i} = {\bar{x}}_{2 i} = {\bar{x}}_{3 i} \to 0$ in a finite time. At first, the system (3) is divided into two subsystems, a first order subsystem as

{\overset{\cdot}{\bar{x}}}_{1 i} = {\bar{u}}_{1 i}

(4)

and a second order subsystem

{\begin{matrix} {\overset{\cdot}{\bar{x}}}_{2 i} = {\bar{u}}_{2 i} \\ {\overset{\cdot}{\bar{x}}}_{3 i} = x_{2 i} u_{1 i} - x_{2 d} u_{1 d} . \end{matrix}

(5)

In the following, controllers ${\bar{u}}_{1 i}$ and ${\bar{u}}_{2 i}$ are provided.

Theorem 1: Consider the system (3). Under Assumptions 1–2, the following finite time control law is proposed

\begin{matrix} {\bar{u}}_{1 i} = 0, t \leq T^{*}, \\ {\bar{u}}_{1 i} = - ρ_{4} [\sum_{j \in N_{i}} a_{ij} ({\bar{x}}_{1 i} - {\bar{x}}_{1 j}) + ({\bar{x}}_{1 i})]^{\frac{1}{\tilde{p}}}, t > T^{*}, \end{matrix}

(6)

{\bar{u}}_{2 i} = - ρ_{3} sgn S_{i 1} - ρ_{1} [({\bar{x}}_{2 i})^{p} + ρ_{2}^{p} (\sum_{j \in N_{i}} a_{ij} ({\bar{x}}_{3 i} - {\bar{x}}_{3 j}) + ({\bar{x}}_{3 i}))]^{\frac{2}{p} - 1}

(7)

with

S_{i 1} = {\bar{x}}_{2 i} + \int_{0}^{t} (ρ_{1} {[{({\bar{x}}_{2 i})}^{p} + ρ_{2}^{p} (\sum_{j \in N_{i}} a_{ij} ({\bar{x}}_{3 i} - {\bar{x}}_{3 j}) + ({\bar{x}}_{3 i}))]}^{\frac{2}{p} - 1}) d τ

where $1 < p = p_{1} / p_{1} p_{2} p_{2} < 2$ , $1 < \tilde{p} = {\tilde{p}}_{1} / {\tilde{p}}_{1} {\tilde{p}}_{2} {\tilde{p}}_{2} < 2$ , $p_{1}$ , $p_{2}$ , ${\tilde{p}}_{1}$ , ${\tilde{p}}_{2}$ are positive odd integers, $ρ_{1}, ρ_{2}, ρ_{3}, ρ_{4}$ are positive constants, $T^{*} \geq (t_{1} + t_{2})$ is switched time, $S_{i 1}$ is the ideal sliding surface, and $sgn (\cdot)$ is the signum function. Under the control laws (6) and (7), the states of the system (3) can converge to zero in a finite time.

Proof: For $t \leq T^{*}$ , since ${\bar{u}}_{1 i} = 0$ , the subsystem (5) can be rewritten as

{\begin{matrix} {\overset{\cdot}{\bar{x}}}_{2 i} = {\bar{u}}_{2 i} \\ {\overset{\cdot}{\bar{x}}}_{3 i} = {\bar{x}}_{2 i} u_{1 d} . \end{matrix}

(8)

The states of the system (8) can reach zero in a finite time by applying controller (7). To prove this fact, the two following proof steps must be considered:

There is a finite time $t_{1}$ such that $S_{i 1} \equiv 0$ for any $t > t_{1}$ .

When $S_{i 1} \equiv 0$ is obtained, the states of subsystem (8) reach zero in finite time $t_{2}$ .

Step 1: Define the Lyapunov function $U_{1} (t) = \frac{1}{2} \sum_{i = 1}^{N} S_{i 1}^{2}$ . Taking the derivative of $U_{1} (t)$ yields

{\overset{\cdot}{U}}_{1} (t) = \sum_{i = 1}^{N} S_{i 1} (- ρ_{3} sgn S_{i 1}) \leq - ρ_{3} \sum_{i = 1}^{N} | S_{i 1} | .

(9)

By Lemma 4, we have

{\overset{\cdot}{U}}_{1} (t) + α_{1} U_{1}^{1 / 2} = (\frac{α_{1}}{\sqrt{2}} - ρ_{3}) \sum_{i = 1}^{N} | S_{i 1} | \leq 0

(10)

where $α_{1} \in (0, \sqrt{2} ρ_{3})$ . By using Lemma 3, one can obtain from (10) that $S_{i 1} \equiv 0$ in a finite time $t_{1}$ satisfying $t_{1} \leq \frac{2 U_{1}^{1 / 2} (0)}{α_{1}}$ .

Step 2: For $t > t_{1}$ , we have $S_{i 1} \equiv 0$ and the system (8) is converted to

{\begin{matrix} {\overset{\cdot}{\bar{x}}}_{2 i} = - ρ_{1} [({\bar{x}}_{2 i})^{p} + ρ_{2}^{p} (\sum_{j \in N_{i}} a_{ij} ({\bar{x}}_{3 i} - {\bar{x}}_{3 j}) + ({\bar{x}}_{3 i}))]^{\frac{2}{p} - 1} \\ {\overset{\cdot}{\bar{x}}}_{3 i} = {\bar{x}}_{2 i} u_{1 d} . \end{matrix}

(11)

We should prove that the states of the system (11) can reach zero in a finite time $t_{2}$ . Choose the following Lyapunov function

V (t) = V_{0} (t) + \sum_{i = 1}^{N} V_{i} (t)

(12)

where $V_{0} (t) = \sum_{i = 1}^{N} \frac{1}{2} {\bar{x}}_{3 i}^{T} (L + I) {\bar{x}}_{3 i}$ and

V_{i} (t) = \int_{{\bar{x}}_{2 i}^{*}}^{{\bar{x}}_{2 i}} {(s^{p} - {\bar{x}}_{2 i}^{*}^{p})}^{2 - 1 / p} ds .

(13)

Taking derivation of $V (t)$ , one gets

\overset{\cdot}{V} (t) = {\overset{\cdot}{V}}_{0} (t) + \sum_{i = 1}^{N} {\overset{\cdot}{V}}_{i} (t) .

(14)

By using Lemma 2, one can obtain

{\overset{\cdot}{V}}_{0} (t) = \sum_{i = 1}^{N} {\overset{\cdot}{\bar{x}}}_{3 i}^{T} (L + I) {\bar{x}}_{3 i} = \sum_{i = 1}^{N} u_{1 d} {\bar{x}}_{2 i} (\sum_{j \in N_{i}} a_{ij} ({\bar{x}}_{3 i} - {\bar{x}}_{3 j}) + {\bar{x}}_{3 i}) .

(15)

Defining $e_{3 i} = \sum_{j \in N_{i}} a_{ij} ({\bar{x}}_{3 i} - {\bar{x}}_{3 j}) + {\bar{x}}_{3 i}$ and taking ${\bar{x}}_{2 i}^{*} = - ρ_{2} e_{3 i}^{1 / p}$ as a virtual control law, it follows that

{\overset{\cdot}{V}}_{0} (t) \leq - ρ_{2} \sum_{i = 1}^{N} u_{1 d} e_{3 i}^{(1 + p) / p} + \sum_{i = 1}^{N} u_{1 d} e_{3 i} ({\bar{x}}_{2 i} - {\bar{x}}_{2 i}^{*}) .

(16)

To continue proof, we must obtain the $\sum_{i = 1}^{N} {\overset{\cdot}{V}}_{i} (t)$ . Defining $ξ_{i} = {\bar{x}}_{2 i}^{p} - {\bar{x}}_{2 i}^{* p}$ , one gets

{\overset{\cdot}{V}}_{i} (t) = - (2 - \frac{1}{p}) \frac{d {\bar{x}}_{2 i}^{*}^{p}}{dt} \int_{{\bar{x}}_{2 i}^{*}}^{{\bar{x}}_{2 i}} {(s^{p} - {\bar{x}}_{2 i}^{*}^{p})}^{1 - 1 / p} ds + ξ_{i}^{2 - 1 / p} {\overset{\cdot}{\bar{x}}}_{2 i} .

(17)

Note that

\begin{matrix} \frac{d {\bar{x}}_{2 i}^{*}^{p}}{dt} = - ρ_{2}^{p} {\overset{\cdot}{e}}_{3 i} = \\ - ρ_{2}^{p} [(\sum_{j \in N_{i}} a_{ij} ({\bar{x}}_{2 i} u_{1 d} - {\bar{x}}_{2 j} u_{1 d}) + {\bar{x}}_{2 i} u_{1 d})] \\ \leq ρ_{2}^{p} ((\sum_{j \in N_{i}} a_{ij} + 1) | {\bar{x}}_{2 i} | | u_{1 d} | + \sum_{j \in N_{i}} a_{ij} | {\bar{x}}_{2 j} | | u_{1 d} |) \\ \leq ρ_{2}^{p} (β_{1} | {\bar{x}}_{2 i} | | u_{1 d} | + β_{2} \sum_{m = 1}^{N} | {\bar{x}}_{2 m} | | u_{1 d} |) \leq ρ_{2}^{p} c_{1} \sum_{m = 1}^{N} | {\bar{x}}_{2 m} | | u_{1 d} | . \end{matrix}

(18)

where $β_{1} = max_{\forall i} {\sum_{j \in N_{i}} a_{ij} + 1}$ , $β_{2} = max_{\forall i} {\sum_{j \in N_{i}} a_{ij}}$ and $c_{1}$ is a positive constant. It can be also verified from Shi et al. (2015) that

\begin{matrix} | {\int_{{\bar{x}}_{2 i}^{*}}^{{\bar{x}}_{2 i}} (s^{p} - {\bar{x}}_{2 i}^{*}^{p})}^{1 - 1 / p} ds | & \leq | {\bar{x}}_{2 i} - {\bar{x}}_{2 i}^{*} | {| {\bar{x}}_{2 i}^{p} - {\bar{x}}_{2 i}^{*}^{p} |}^{1 - 1 / p} \\ \leq | {\bar{x}}_{2 i} - {\bar{x}}_{2 i}^{*} | {| ξ_{i} |}^{1 - 1 / p} . \end{matrix}

(19)

Using (17)–(19) yields

{\overset{\cdot}{V}}_{i} (t) \leq (2 - \frac{1}{p}) (ρ_{2}^{p} c_{1} \sum_{m = 1}^{N} | u_{1 d} | | {\bar{x}}_{2 m} |) | {\bar{x}}_{2 i} - {\bar{x}}_{2 i}^{*} | {| ξ_{i} |}^{1 - 1 / p} + ξ_{i}^{2 - 1 / p} {\overset{\cdot}{\bar{x}}}_{2 i} .

(20)

By using Shi et al. (2015), one can obtain

| {\bar{x}}_{2 m} | | {\bar{x}}_{2 i} - {\bar{x}}_{2 i}^{*} | {| ξ_{i} |}^{1 - 1 / p} \leq (ρ_{2} + 1) {| ξ_{i} |}^{d} + c_{2} {| ξ_{m} |}^{d} + ρ_{2} c_{3} {| e_{3 m} |}^{d}

(21)

where $c_{2}$ and $c_{3}$ are positive constants and $d = 1 + 1 / p$ . Substituting (21) into (20), it follows that

\begin{matrix} {\overset{\cdot}{V}}_{i} (t) \leq \frac{c_{1} (2 p - 1) (ρ_{2} + 1) ρ_{2}^{p}}{p} {| ξ_{i} |}^{d} \\ \sum_{m = 1}^{N} | u_{1 d} | + \frac{c_{1} c_{2} (2 p - 1) ρ_{2}^{p}}{p} \sum_{m = 1}^{N} {| ξ_{m} |}^{d} | u_{1 d} | \\ + \frac{c_{1} c_{3} (2 p - 1) ρ_{2}^{p + 1}}{p} \sum_{m = 1}^{N} {| e_{3 m} |}^{d} | u_{1 d} | + ξ_{i}^{2 - 1 / p} {\overset{\cdot}{\bar{x}}}_{2 i} . \end{matrix}

(22)

By the definition of ξi and (21), we have ${\overset{\cdot}{\bar{x}}}_{2 i} = - ρ_{1} ξ_{i}^{(2 - p) / p}$ and it is concluded that

\begin{matrix} \sum_{i = 1}^{N} {\overset{\cdot}{V}}_{i} (t) \leq (\frac{c_{1} | u_{1 d} | N (2 p - 1) (ρ_{2} + 1) ρ_{2}^{p}}{p} \\ + \frac{| u_{1 d} | c_{1} c_{2} (2 p - 1) ρ_{2}^{p}}{p} - ρ_{1}) \sum_{i = 1}^{N} {| ξ_{i} |}^{d} \\ + \frac{| u_{1 d} | c_{1} c_{3} (2 p - 1) ρ_{2}^{p + 1}}{p} \sum_{i = 1}^{N} {| e_{3 i} |}^{d} . \end{matrix}

(23)

In the following, from (16), one gets

{\overset{\cdot}{V}}_{0} (t) \leq - ρ_{2} \sum_{i = 1}^{N} u_{1 d} e_{3 i}^{d} + | \sum_{i = 1}^{N} u_{1 d} e_{3 i} ({\bar{x}}_{2 i} - {\bar{x}}_{2 i}^{*}) | .

(24)

By Lemma 4, we have

\begin{matrix} | \sum_{i = 1}^{N} u_{1 d} e_{3 i} ({\bar{x}}_{2 i} - {\bar{x}}_{2 i}^{*}) | \leq \sum_{i = 1}^{N} | u_{1 d} | | e_{3 i} | 2^{1 - \frac{1}{p}} {| {\bar{x}}_{2 i}^{p} - {\bar{x}}_{2 i}^{*}^{p} |}^{\frac{1}{p}} \\ = 2^{1 - \frac{1}{p}} \sum_{i = 1}^{N} | u_{1 d} | | e_{3 i} | {| ξ_{i} |}^{\frac{1}{p}} \\ \leq \sum_{i = 1}^{N} | u_{1 d} | (β_{3} {| e_{3 i} |}^{d} + β_{4} {| ξ_{i} |}^{d}) \\ \leq c_{4} \sum_{i = 1}^{N} | u_{1 d} | ({| e_{3 i} |}^{d} + {| ξ_{i} |}^{d}) \end{matrix}

(25)

where $β_{3}$ , $β_{4}$ and $c_{4}$ are positive constants. Putting (25) into (24), it follows that

{\overset{\cdot}{V}}_{0} (t) \leq - ρ_{2} \sum_{i = 1}^{N} u_{1 d} e_{3 i}^{d} + c_{4} | u_{1 d} | \sum_{i = 1}^{N} {| e_{3 i} |}^{d} + c_{4} | u_{1 d} | \sum_{i = 1}^{N} {| ξ_{i} |}^{d} .

(26)

Substituting (26) and (23) into (14) leads to

\begin{matrix} \overset{\cdot}{V} (t) \leq (\frac{| u_{1 d} | c_{1} c_{3} (2 p - 1) ρ_{2}^{p + 1}}{p} + c_{4} | u_{1 d} |) \sum_{i = 1}^{N} e_{3 i}^{d} - ρ_{2} \sum_{i = 1}^{N} u_{1 d} e_{3 i}^{d} \\ + (\frac{c_{1} | u_{1 d} | N (2 p - 1) (ρ_{2} + 1) ρ_{2}^{p}}{p} + \frac{| u_{1 d} | c_{1} c_{2} (2 p - 1) ρ_{2}^{p}}{p} \\ + c_{4} | u_{1 d} | - ρ_{1}) \sum_{i = 1}^{N} ξ_{i}^{d} . \end{matrix}

(27)

From (27) and using Assumption 1, one gets

\begin{matrix} \overset{\cdot}{V} (t) \leq - (ρ_{2} u_{1 d}^{\min} - ((u_{1 d}^{\max}) (\frac{c_{1} c_{3} (2 p - 1) ρ_{2}^{p + 1}}{p} + c_{4})))) \sum_{i = 1}^{N} e_{3 i}^{d} \\ - (ρ_{1} - ((u_{1 d}^{\max}) (\frac{c_{1} N (2 p - 1) (ρ_{2} + 1) ρ_{2}^{p}}{p} + \frac{c_{1} c_{2} (2 p - 1) ρ_{2}^{p}}{p} + c_{4}))) \sum_{i = 1}^{N} ξ_{i}^{d} . \end{matrix}

(28)

Parameters $ρ_{1}$ and $ρ_{2}$ must be chosen such that

\overset{\cdot}{V} (t) \leq - η \sum_{i = 1}^{N} e_{3 i}^{d} - η \sum_{i = 1}^{N} ξ_{i}^{d}

(29)

for positive constant $η$ . Using Assumption 2, Lemmas 1, 4 and Shi et al. (2015), it follows that

\sum_{i = 1}^{N} e_{3 i}^{2} \geq 2 λ_{min} (L + I) V_{0}

(30)

and

\begin{matrix} \sum_{i = 1}^{N} V_{i} (t) \leq \frac{1}{(2 - 1 / p) 2^{1 - 1 / p} ρ_{2}^{p + 1}} \sum_{i = 1}^{N} | {\bar{x}}_{2 i} - {\bar{x}}_{2 i}^{*} | {| ξ_{i} |}^{2 - 1 / p} \\ \leq \frac{1}{(2 - 1 / p) ρ_{2}^{p + 1}} \sum_{i = 1}^{N} {| ξ_{i} |}^{1 / p} {| ξ_{i} |}^{2 - 1 / p} \\ \leq \frac{1}{(2 - 1 / p) ρ_{2}^{p + 1}} \sum_{i = 1}^{N} {| ξ_{i} |}^{2} . \end{matrix}

(31)

Putting (30) and (31) into (29), we have

V (t) \leq c_{5} (\sum_{i = 1}^{N} e_{3 i}^{2} + \sum_{i = 1}^{N} ξ_{i}^{2})

(32)

where $c_{5} = max {\frac{1}{2 λ_{min} (L + I)}, \frac{1}{(2 - 1 / p) ρ_{2}^{p + 1}}}$ . By using Lemma 4 and for $0 < d / 2 < 1$ , one can obtain

V^{d / 2} (t) \leq c_{5}^{d / 2} (\sum_{i = 1}^{N} e_{3 i}^{d} + \sum_{i = 1}^{N} ξ_{i}^{d}) .

(33)

By using (33), (29) and for $α_{2} \in (0, η / c_{5}^{d / 2})$ , one gets

\overset{\cdot}{V} (t) + α_{2} V^{d / 2} (t) \leq (α_{2} c_{5}^{d / 2} - η) (\sum_{i = 1}^{N} e_{3 i}^{d} + \sum_{i = 1}^{N} ξ_{i}^{d}) \leq 0 .

(34)

By Lemma 3, it is concluded that $V (t)$ converges to zero in the following finite time

t_{2} \leq \frac{V^{1 - d / 2} (t_{1})}{α_{2} (1 - d / 2)} .

(35)

Thus, by using (10) and (34), we have $V (t) = 0$ for $T^{*} \geq t_{1} + t_{2}$ . It means that $V_{0} (t) = \sum_{i = 1}^{N} V_{i} (t) = 0$ and we have

{\bar{x}}_{2 i} = {\bar{x}}_{3 i} = 0, i = 1, \dots, N

(36)

for $T^{*} \geq (t_{1} + t_{2})$ . Therefore, the states of the system (8) under finite-time controller (7) reach zero in a finite time.

When $t \geq T^{*}$ , $x_{2 i}$ and $x_{3 i}$ have converged to $x_{2 d}$ and $x_{3 d}$ . Therefore, the subsystem (4) should be only considered. By applying ${\bar{u}}_{1 i}$ in Theorem 1 into subsystem (4), we have

{\overset{\cdot}{\bar{x}}}_{1 i} = - ρ_{4} [\sum_{j \in N_{i}} a_{ij} ({\bar{x}}_{1 i} - {\bar{x}}_{1 j}) + ({\bar{x}}_{1 i})]^{\frac{1}{\tilde{p}}} .

(37)

Choose the Lyapunov function $U_{2} (t) = \sum_{i = 1}^{N} \frac{1}{2} {\bar{x}}_{1 i}^{T} (L + I) {\bar{x}}_{1 i} .$ Taking the derivative of $U_{2} (t)$ along system (37) yields

{\overset{\cdot}{U}}_{2} (t) = \sum_{i = 1}^{N} {\overset{\cdot}{\bar{x}}}_{1 i}^{T} (L + I) {\bar{x}}_{1 i} = - \sum_{i = 1}^{N} {\overset{\cdot}{\bar{x}}}_{1 i} (\sum_{j \in N_{i}} a_{ij} ({\bar{x}}_{1 i} - {\bar{x}}_{1 j}) + ({\bar{x}}_{1 i})) = - ρ_{4} \sum_{i = 1}^{N} e_{i 1}^{\tilde{d}}

where $e_{i 1} = \sum_{j \in N_{i}} a_{ij} ({\bar{x}}_{1 i} - {\bar{x}}_{1 j}) + ({\bar{x}}_{1 i})$ and $\tilde{d} = 1 + 1 / \tilde{p}$ . By using Lemmas 1 and 4, one gets

{\overset{\cdot}{U}}_{2} (t) \leq - ρ_{4} {(\sum_{i = 1}^{N} e_{i 1}^{2})}^{\tilde{d} / 2}

(38)

and

\sum_{i = 1}^{N} e_{i 1}^{2} \geq 2 λ_{min} (L + I) U_{3} (t) .

(39)

From (38) and (39), it is concluded that

{\overset{\cdot}{U}}_{2} (t) + ρ_{4} (2 λ_{min} (L + I))^{\tilde{d} / 2} U_{2}^{\tilde{d} / 2} (t) \leq 0 .

(40)

By Lemma 3 and (40), one can conclude that $U_{2} (t)$ converges to zero within a finite time

t_{3} = \frac{U_{2}^{\tilde{d} / 2} (0)}{ρ_{4} {(2 λ_{min} (L + I))}^{\tilde{d} / 2} (1 - \tilde{d} / 2)} .

When $U_{2} (t) = 0$ , we have

{\bar{x}}_{1 i} = 0, i = 1, \dots, N .

(41)

Thus, it follows from (36) and (41) that ${\bar{x}}_{1 i} = {\bar{x}}_{2 i} = {\bar{x}}_{3 i} = 0$ after the time $T = T^{*} + t_{3}$ . This completes the proof.

Remark 1: Theorem 1 proved that the states of tracking error system (3) using controllers ${\bar{u}}_{1 i}$ and ${\bar{u}}_{2 i}$ can reach zero in a finite time. It implies that the states of MAS (1) with the following controllers can converge to the states of the target (2) in a finite time.

\begin{matrix} u_{1 i} = u_{1 d}, t \leq T^{*}, \\ u_{1 i} = - ρ_{4} [\sum_{j \in N_{i}} a_{ij} ({\bar{x}}_{1 i} - {\bar{x}}_{1 j}) + ({\bar{x}}_{1 i})]^{\frac{1}{\tilde{p}}} + u_{1 d}, t > T^{*}, \end{matrix}

(42)

u_{2 i} = - ρ_{3} sgn S_{i 1} - ρ_{1} [({\bar{x}}_{2 i})^{p} + ρ_{2}^{p} (\sum_{j \in N_{i}} a_{ij} ({\bar{x}}_{3 i} - {\bar{x}}_{3 j}) + ({\bar{x}}_{3 i}))]^{\frac{2}{p} - 1} + u_{2 d}

(43)

The distributed tracking control in the presence of input delay

In the presence of input delay, proposed control laws (42) and (43) for each agent are converted to

\begin{matrix} u_{1 i} (t - t_{D 1}) = u_{1 d} (t - t_{D 1}) t \leq T^{*}, \\ u_{1 i} (t - t_{D 1}) = - ρ_{4} [\sum_{j \in N_{i}} a_{ij} ({\bar{x}}_{1 i} (t - t_{D 1}) - {\bar{x}}_{1 j} (t - t_{D 1})) + ({\bar{x}}_{1 i} (t - t_{D 1}))]^{\frac{1}{\tilde{p}}} + u_{1 d} (t - t_{D 1}), t > T^{*}, \end{matrix}

(44)

\begin{matrix} u_{2 i} (t - t_{D 2}) = - ρ_{3} sgn S_{i 2} - ρ_{1} [({\bar{x}}_{2 i} (t - t_{D 2}))^{p} \\ + ρ_{2}^{p} (\sum_{j \in N_{i}} a_{ij} ({\bar{x}}_{3 i} (t - t_{D 2}) - {\bar{x}}_{3 j} (t - t_{D 2})) \\ + ({\bar{x}}_{3 i} (t - t_{D 2})))]^{\frac{2}{p} - 1} + u_{2 d} (t - t_{D 2}) \end{matrix}

(45)

where

S_{i 2} = {\bar{x}}_{2 i} (t - t_{D 2}) + \int_{0}^{t} (ρ_{1} {[{({\bar{x}}_{2 i} (t - t_{D 2}))}^{p} + ρ_{2}^{p} (\sum_{j \in N_{i}} a_{ij} ({\bar{x}}_{3 i} (t - t_{D 2}) - {\bar{x}}_{3 j} (t - t_{D 2})) + ({\bar{x}}_{3 i} (t - t_{D 2})))]}^{\frac{2}{p} - 1}) d τ .

Controller (44) and (45) fail to track the target in the presence of constant delay $t_{D 1}$ and $t_{D 2}$ . To solve this problem, modified controllers are proposed as

\begin{matrix} u_{1 i} (t) = {\tilde{u}}_{1 d} (t), t \leq T^{*}, \\ u_{1 i} (t) = - ρ_{4} [\sum_{j \in N_{i}} a_{ij} ({\tilde{\bar{x}}}_{1 i} (t) - {\tilde{\bar{x}}}_{1 j} (t)) + ({\tilde{\bar{x}}}_{1 i} (t))]^{\frac{1}{\tilde{p}}} + {\tilde{u}}_{1 d} (t), t > T^{*}, \end{matrix}

(46)

u_{2 i} (t) = - ρ_{3} sgn S_{i 3} - ρ_{1} [({\tilde{\bar{x}}}_{2 i} (t))^{p} + ρ_{2}^{p} (\sum_{j \in N_{i}} a_{ij} ({\tilde{\bar{x}}}_{3 i} (t) - {\tilde{\bar{x}}}_{3 j} (t)) + ({\tilde{\bar{x}}}_{3 i} (t)))]^{\frac{2}{p} - 1} + {\tilde{u}}_{2 d} (t)

(47)

with

S_{i 3} = {\tilde{\bar{x}}}_{2 i} (t) + \int_{0}^{t} (ρ_{1} {[{({\tilde{\bar{x}}}_{2 i} (t))}^{p} + ρ_{2}^{p} (\sum_{j \in N_{i}} a_{ij} ({\tilde{\bar{x}}}_{3 i} (t) - {\tilde{\bar{x}}}_{3 j} (t)) + ({\tilde{\bar{x}}}_{3 i} (t)))]}^{\frac{2}{p} - 1}) d τ

where $({\tilde{\bar{x}}}_{1 i} (t), {\tilde{\bar{x}}}_{2 i} (t), {\tilde{\bar{x}}}_{3 i} (t))$ and $({\tilde{u}}_{1 d} (t), {\tilde{u}}_{2 d} (t))$ are estimates of $({\bar{x}}_{1 i} (t + t_{D 1}), {\bar{x}}_{2 i} (t + t_{D 2}), {\bar{x}}_{3 i} (t + t_{D 2}))$ and $(u_{1 d} (t + t_{D 1}), u_{2 d} (t + t_{D 2}))$ . When $({\tilde{\bar{x}}}_{1 i} (t), {\tilde{\bar{x}}}_{2 i} (t), {\tilde{\bar{x}}}_{3 i} (t))$ and $({\tilde{u}}_{1 d} (t), {\tilde{u}}_{2 d} (t))$ get equal to $({\bar{x}}_{1 i} (t + t_{D 1}), {\bar{x}}_{2 i} (t + t_{D 2}), {\bar{x}}_{3 i} (t + t_{D 2}))$ and $(u_{1 d} (t + t_{D 1}), u_{2 d} (t + t_{D 2}))$ , (46) and (47) in the presence of delay have the same performance of (42) and (43) in the absence of delay and the tracking control is achieved. Since

\begin{matrix} {\bar{x}}_{1 i} (t + t_{D 1}) = x_{1 i} (t + t_{D 1}) - x_{1 d} (t + t_{D 1}) \\ {\bar{x}}_{2 i} (t + t_{D 2}) = x_{2 i} (t + t_{D 2}) - x_{2 d} (t + t_{D 2}) \\ {\bar{x}}_{3 i} (t + t_{D 2}) = x_{3 i} (t + t_{D 2}) - x_{3 d} (t + t_{D 2}) \end{matrix}

then, an estimator should be designed to estimate future states of the agents and the target. It should be noted that we obtain $({\tilde{u}}_{1 d} (t), {\tilde{u}}_{2 d} (t))$ from estimation of future states of the target and do not need to design an estimator for estimation of $({\tilde{u}}_{1 d} (t), {\tilde{u}}_{2 d} (t))$ . The proposed estimator has the following dynamics

{\begin{matrix} {\overset{\cdot}{\tilde{x}}}_{1 i} (t) = u_{1 i} (t) + γ_{1} sgn (x_{1 i} (t) - {\tilde{x}}_{1 i} (t - t_{D 1})) \\ {\overset{\cdot}{\tilde{x}}}_{2 i} (t) = u_{2 i} (t) + γ_{2} sgn (x_{2 i} (t) - {\tilde{x}}_{2 i} (t - t_{D 2})) \\ {\overset{\cdot}{\tilde{x}}}_{3 i} (t) = u_{1 i} (t) {\tilde{x}}_{2 i} (t) + γ_{3} sgn (x_{3 i} (t) - {\tilde{x}}_{3 i} (t - t_{D 2})) \end{matrix}, i = 1, \dots, N

(48)

where $γ_{1}$ , $γ_{2}$ and $γ_{3}$ are positive constants. To analyses the stability of the FSE (48), we assume FSE (48) is used to predict the future states of the agents. It can be shown that $({\tilde{x}}_{1 i} (t), {\tilde{x}}_{2 i} (t), {\tilde{x}}_{3 i} (t))$ converge to $(x_{1 i} (t + t_{D 1}), x_{2 i} (t + t_{D 2}), x_{3 i} (t + t_{D 2})))$ with a hyper-exponential rate. For this purpose, we define three error functions as

\begin{matrix} e_{1 i} (t) = x_{1 i} (t + t_{D 1}) - {\tilde{x}}_{1 i} (t) \\ e_{2 i} (t) = x_{2 i} (t + t_{D 2}) - {\tilde{x}}_{2 i} (t) \\ e_{3 i} (t) = x_{3 i} (t + t_{D 2}) - {\tilde{x}}_{3 i} (t) . \end{matrix}

Then, we have

{\begin{matrix} {\overset{\cdot}{e}}_{1 i} (t) = - γ_{1} sgn (e_{1 i} (t - t_{D 1})) \\ {\overset{\cdot}{e}}_{2 i} (t) = - γ_{2} sgn (e_{2 i} (t - t_{D 2})) \\ {\overset{\cdot}{e}}_{3 i} (t) = u_{1 i} (t) e_{2 i} (t) - γ_{3} sgn (e_{3 i} (t - t_{D 2})) \end{matrix}

(49)

At first, we consider the stability of the following system

{\overset{\cdot}{e}}_{1 i} (t) = - γ_{1} sgn (e_{1 i} (t - t_{D 1}))

Efimov et al. (2016) showed $e_{1 i} (t)$ converges to zero with a hyper-exponential rate. Therefore, it can be concluded that ${\tilde{x}}_{1 i} (t) = x_{1 i} (t + t_{D 1})$ . Since the dynamics of $e_{2 i} (t)$ is the same as $e_{1 i} (t)$ , then we have ${\tilde{x}}_{2 i} (t) = x_{2 i} (t + t_{D 2})$ . After ${\tilde{x}}_{2 i} (t) = x_{2 i} (t + t_{D 2})$ , we replace $e_{2 i} (t) = 0$ in the dynamics of $e_{3 i} (t)$ and it can be obtained from (49) that

{\overset{\cdot}{e}}_{3 i} (t) = - γ_{3} sgn (e_{3 i} (t - t_{D 2}))

(50)

By using the result of Efimov et al. (2016), it is proved that $e_{3 i} (t)$ converge to zero with a hyper-exponential rate and then we can conclude $({\tilde{x}}_{1 i} (t), {\tilde{x}}_{2 i} (t), {\tilde{x}}_{3 i} (t))$ converge to $(x_{1 i} (t + t_{D 1}), x_{2 i} (t + t_{D 2}), x_{3 i} (t + t_{D 2})))$ . It should be noted that we choose $γ_{2}$ bigger than $γ_{3}$ because $e_{2 i} (t)$ converges to zero with a faster rate than $e_{3 i} (t)$ to conclude (50) and to complete the stability analysis. Furthermore, FSE (48) can be used to predict the future estimated states of the target because the dynamics of agents and the target are similar.

Remark 2: Here, the effect of input delay and number of agents on the proposed controllers (46) and (47) is investigated as follows:

Based on the above analysis, it is proved that FSE (48) can estimate future states of MAS (1) and target (2) with a hyper-exponential rate. When the states of the estimator (48) get equal to future states of MAS (1) and target (2), (46) and (47) with input delay have the same performance of (42) and (43) in the absence of delay. So, it is concluded that the effect of input delay on control laws (46) and (47) is related to the performance of FSE (48) and its convergence rate. Efimov et al. (2016) proved that increasing $γ_{1} t_{D 1}$ , $γ_{2} t_{D 2}$ and $γ_{3} t_{D 2}$ results in increasing the convergence rate of FSE (48). Therefore, since coefficients $γ_{1}$ , $γ_{2}$ and $γ_{3}$ are constants, bigger delays (bigger $t_{D 1}$ and $t_{D 2}$ ) cause the tracking control to be achieved in a faster rate.

The effect of number of agents (N) on the controllers (46) and (47) can be considered in two aspects. First, it is clear from the controllers that N has not any effect on obtaining the parameters of controllers (46) and (47). Second, since the technique of the consensus on state is used in the design of the proposed controllers, the graph structure is effective on convergence rate or consensus time of the tracking results (Olfati-saber et al., 2004; Ogiwara et al., 2017). In other words, the more the algebraic connectivity (the second smallest eigenvalue of the Laplacian matrix) of the MAS’s graph, the less is the consensus time of the tracking of the agents. According to Ogiwara et al. (2017), the number of agents has an indirect effect on algebraic connectivity. It implies that increasing the number of agents does not lead to increase of algebraic connectivity. Actually, in addition to the number of agents, the set of links among agents as another factor of graph structure also has an effect on the algebraic connectivity and should be considered. Therefore, the number of agents affects indirectly on the performance of controllers (46) and (47) and it cannot be expressed this effect is positive or negative (other factors such as the set of links among agents should be investigated).

Remark 3: We assume that the states of the target (2) are available to design distributed controllers (42) and (43) while they are not available in practical environments. Moreover, there exists noise in dynamics of target in practical MASs. Therefore, it is necessary to use an estimator for each agent to estimate states of target in the presence of noise. Target (2) with noise has the following dynamics

{\overset{\cdot}{x}}_{d} (t) = (\begin{matrix} u_{1 d} (t) \\ u_{2 d} (t) \\ x_{2 d} (t) u_{1 d} (t) \end{matrix}) + B_{d} w_{d} (t), x_{d} (t), B_{d} \in R^{3}

(51)

The sensing model of the ith agent is described by

y_{i} (t) = h_{i} (x_{d} (t)) + D_{i}^{d} v_{i}^{d} (t), i = 1, \dots, N

(52)

where $w_{d} (t)$ and $v_{i}^{d} (t)$ are Gaussian white noises and $h_{i} (.)$ is a nonlinear function. To estimate the states of the target (51), for every agent, the presented filter in Jenabzadeh et al. (2017) is used with dynamics

{\overset{\cdot}{\hat{x}}}_{i} (t) = (\begin{matrix} u_{1 d} (t) \\ u_{2 d} (t) \\ {\hat{x}}_{2 d}^{i} (t) u_{1 d} (t) \end{matrix}) + φ_{i} (y_{i} (t) - h_{i} ({\hat{x}}_{i} (t))) + T_{i} χ_{i}^{- 1} \sum_{j \in N_{i}} a_{ij} [{\hat{x}}_{j} (t) - {\hat{x}}_{i} (t)], i = 1, \dots, N

(53)

where ${\hat{x}}_{i} (t) \in R^{3}$ is the estimation of $x_{d} (t)$ . $φ_{i} \in R^{3 \times 1}$ , $T_{i} > 0$ and $χ_{i} \in R^{3 \times 3}$ are the estimator gain matrix, the consensus gain and the estimator matrix of agent i, respectively. By using filter (53), we can estimate the states of the target (51) and replace them in controllers (42) and (43). It should be mentioned that to estimate the future states of the target in the presence of delay, the form of FSE (48) is changed as follows

\begin{array}{l} {\overset{\cdot}{\tilde{x}}}_{i t} (t) = (\begin{array}{l} u_{1 d} + γ_{1} sgn ({\hat{x}}_{1 i} (t) - {\tilde{x}}_{1 i t} (t - t_{D 1})) \\ u_{2 d} + γ_{2} sgn ({\hat{x}}_{2 i} (t) - {\tilde{x}}_{2 i t} (t - t_{D 2})) \\ {\tilde{x}}_{2 i t} (t) u_{1 d} + γ_{3} sgn ({\hat{x}}_{3 i} (t) - {\tilde{x}}_{3 i t} (t - t_{D 2})) \end{array}) \\ + φ_{i} (y_{i} (t) - h_{i} ({\hat{x}}_{i} (t))) + T_{i} χ_{i}^{- 1} \sum_{j \in N_{i}} a_{i j} [{\hat{x}}_{j} (t) - {\hat{x}}_{i} (t)], i = 1, …, N \end{array}

(54)

In this case, the state of ${\tilde{x}}_{it} (t)$ converges to the future state of ${\hat{x}}_{i} (t)$ with a hyper-exponential rate. By defining the error functions as

\begin{matrix} e_{1 it} (t) = {\hat{x}}_{1 i} (t + t_{D 1}) - {\tilde{x}}_{1 it} (t) \\ e_{2 it} (t) = {\hat{x}}_{2 i} (t + t_{D 2}) - {\tilde{x}}_{2 it} (t) \\ e_{3 it} (t) = {\hat{x}}_{3 i} (t + t_{D 2}) - {\tilde{x}}_{3 it} (t) \end{matrix}

the error dynamics are the same as system (49). Thus, by using stability analysis of system (49), it can be easily shown that error functions $e_{1 it} (t)$ , $e_{2 it} (t)$ and $e_{3 it} (t)$ converge to zero.

Simulation results

In this section, the performance of the proposed finite time controllers (46) and (47) and FSE (48) is investigated in target tracking of MASs with four nonholonomic mobile agents. The communication among the four agents with $a_{ij} = 0.5$ is shown in Figure 1. The dynamic equations of each agent are described by

\begin{matrix} {\overset{\cdot}{x}}_{i} = v_{i} \cos θ_{i} \\ {\overset{\cdot}{y}}_{i} = v_{i} \sin θ_{i} \\ {\overset{\cdot}{θ}}_{i} = ω_{i} \end{matrix}

(55)

where $x_{i}$ and $y_{i}$ are the Cartesian coordinates of the center of the rear wheel and $θ_{i}$ measures the heading angle from the x-axis. $v_{i}$ and $ω_{i}$ are the linear and angular velocities. With the state and control transformations

\begin{matrix} x_{1 i} = θ_{i} \\ x_{2 i} = x_{i} \cos θ_{i} + y_{i} \sin θ_{i} \\ x_{3 i} = x_{i} \sin θ_{i} - y_{i} \cos θ_{i} \\ v_{i} = x_{3 i} u_{1 i} + u_{2 i} \\ ω_{i} = u_{1 i}, \end{matrix}

and in the presence of input delay, system (55) is transformed as

{\begin{matrix} {\overset{\cdot}{x}}_{1 i} (t) = u_{1 i} (t - t_{D 1}) \\ {\overset{\cdot}{x}}_{2 i} (t) = u_{2 i} (t - t_{D 2}) \\ {\overset{\cdot}{x}}_{3 i} (t) = u_{1 i} (t - t_{D 1}) x_{2 i} (t) . \end{matrix}, i = 1, \dots, 4

(56)

The purpose is to track the states of the target (51) with $u_{1 d} = 1$ and $u_{2 d} = - x_{2 d} - x_{3 d}$ . The sensing model of the ith agent is described by

y_{i} (t) = x_{1 d} (t) + x_{2 d} (t) + x_{3 d} (t) + 0.1 v_{i}^{d} (t),

(57)

where $w_{d} (t)$ and $v_{i}^{d} (t)$ are independent white noises with variances 0.1 and 0.01. The parameters considered for finite time controllers, FSE and filter (53) are

\begin{matrix} χ_{i} = (\begin{matrix} 0.7546 & - 0.2224 & - 0.0407 \\ - 0.2224 & 0.7779 & 0.1534 \\ - 0.0407 & 0.1534 & 0.9467 \end{matrix}), φ_{i} = (\begin{matrix} 0.6039 \\ 0.0162 \\ 0.3352 \end{matrix}), \\ B_{d} = (\begin{matrix} 0.01 \\ 0.1 \\ 0.1 \end{matrix}) \\ T_{i} = 0.5, ρ_{1} = 40, ρ_{2} = 2.7, ρ_{3} = 3, ρ_{4} = 40, \\ ρ_{5} = 1, p = \tilde{p} = 9 / 7, T^{*} = 10, \\ t_{D 1} = 0.5, t_{D 2} = 0.2, γ_{1} = 1.5, γ_{2} = 6, γ_{3} = 5.5 . \end{matrix}

(58)

Figure 1.

MAS topology with $N = 4$ agents.

Simulations are carried out based on two simultaneous steps. At the first step, the future states of MAS (56) and estimated states of the target (51) are predicted by FSEs (48) and (55). Figure 2 illustrates the estimation errors of the FSE (48) of agent 1 in which $e_{1} = x_{11} (t + t_{D 1}) - {\tilde{x}}_{11} (t)$ , $e_{2} = x_{21} (t + t_{D 2}) - {\tilde{x}}_{21} (t)$ and $e_{3} = x_{31} (t + t_{D 2}) - {\tilde{x}}_{31} (t)$ . It follows from Figure 2 that FSE (48) estimates the future states of MAS (56) in almost 4 seconds. At the second step, the states of FSE (48) is used in controllers (46) and (47). According to the explanations in previous sections, the controllers (46) and (47) are designed based on a switching strategy. It means that the controller (47) makes the states of $x_{2 i} (t)$ and $x_{3 i} (t)$ converge to estimates of $x_{2 d} (t)$ and $x_{3 d} (t)$ in a finite time. Then, the controller (46) makes the state of $x_{1 i} (t)$ reach to estimate of $x_{1 d} (t)$ in a finite time. Figure 3 demonstrates the tracking errors of four agents. This figure shows that $x_{2 i} (t)$ and $x_{3 i} (t)$ converge to estimates of $x_{2 d} (t)$ and $x_{3 d} (t)$ in almost 6 seconds. Moreover, $x_{1 i} (t)$ converge to estimate of $x_{1 d} (t)$ in 12 seconds. Therefore, the states of all agents converge to the estimated states of the target (51) in the presence of input delay in a finite time. It is worth noting that the convergence times of tracking cannot be decreased because these times depend on the estimation time of the target’s states. Actually, to track the target, the states of the target (51) should be firstly estimated by estimator (53). Since the parameters of the estimator (53) are obtained by solving some given linear matrix inequalities, so they are given and unchangeable. Thus, the estimation time of target’s states is constant and the convergence times of tracking cannot be reduced.

Figure 2.

FSE estimation errors of agent 1 for $t_{D 1} = 0.5$ and $t_{D 2} = 0.2$ .

Figure 3.

Tracking errors (TEs) of agents for $t_{D 1} = 0.5$ and $t_{D 2} = 0.2$ .

For more investigation, simulations are performed for bigger input delays $t_{D 1} = 1$ and $t_{D 2} = 0.8$ . In this case, simulation parameters are the same as (58) except for $T^{*} = 13, γ_{1} = 0.8, γ_{2} = 1$ and $γ_{3} = 0.8$ . Tracking errors of agents and FSE estimation errors of agent 1 are illustrated in Figures 4 and 5. These figures show the convergence time of agents and estimation time of FSE are more than the previous simulation. This verifies Remark 2 in which it was stated that smaller coefficients $γ_{1} t_{D 1}$ , $γ_{2} t_{D 2}$ and $γ_{3} t_{D 2}$ cause increase of convergence time.

Figure 4.

Tracking errors (TEs) of agents for $t_{D 1} = 1$ and $t_{D 2} = 0.8$ .

Figure 5.

FSE estimation errors of agent 1 for $t_{D 1} = 1$ and $t_{D 2} = 0.8$ .

Conclusions and future work

In this paper, two controllers have been suggested for nonholonomic MASs with and without input delay to track the states of a target with nonholonomic dynamics. The proposed control laws have been obtained based on a switched strategy and an FSE is used to predict the future states of agents. The stability analyses of the proposed controllers and estimator have been also provided. The simulations showed the suitable performance of the suggested algorithms in tracking control of nonholonomic MASs. Future directions of this paper might involve: (1) To design and analyse a distributed tracking control algorithm for discrete-time nonholonomic MASs with time delay using the results of Kokil (2017) and Kokil (2014). (2) To develop the suggested controllers for nonholonomic MASs with directed graphs and jointly connected topologies. (3) Since this paper uses the idea of finite time to design the distributed controllers, where the finite time is dependent on the initial values, the controller design can be improved by considering the fast terminal sliding modes, given in Tie and Cai (2010) that can provide a better convergence time independent of the initial values. (4) To extend the proposed control laws for nonholonomic MASs with external disturbances using the results of Shi et al. (2015).

Footnotes

Acknowledgements

We wish to thank Dr Bahman Gharesifard from Queen’s University, Canada for his valuable comments that helped us to improve this paper.

Declaration of Conflicting Interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

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