Finite-Time Consensus Algorithm for Multiple Nonholonomic Disturbed Systems with Its Application

This paper deals with the problem of finite-time consensus of multiple nonholonomic disturbed systems. To accomplish this problem, themultiple nonholonomic systems are transformed into twomultiple subsystems, and these twomultiple subsystems are studied, respectively. For these two multiple subsystems, the terminal sliding mode (TSM) algorithms are designed, respectively, which achieve the finite-time reaching of sliding surface. Next, a switching control strategy is proposed to guarantee the finite-time consensus of all the states for multiple nonholonomic systems with disturbances. Finally, we demonstrate the effectiveness of the proposed consensus algorithms with application to multiple nonholonomic mobile robots.


Introduction
Over the last ten years, there has been increasing interest in the control problem of multiagent systems.Driven by its application to various scientific communities, such as the control of formations, optimization, and task assignment, many results have been given in recent years (see, e.g., [1][2][3] and the references therein).
As one of the most fundamental and important issues, consensus problem means that all the states of agents are required to agree with a common desired value.Since the exchange of information can only occur between the agent and its neighbours, the state consensus control of multiagent systems becomes difficult and challenging.We can classify the multiagent systems into leaderless and leader-follower systems.Many consensus protocols have been proposed for the leaderless and leader-follower multiagent systems in [4][5][6][7] and the references therein.Since many practical systems can be transformed into the first-order or secondorder dynamics, it is significant and necessary to study the consensus algorithms for the first-order and second-order multiagent systems.The first-order multiagent systems with fixed or switching interconnection topology have been studied deeply in [8,9] and the references therein.However, for the second-order multiagent systems, the problem becomes complicated and many existing consensus algorithms of the first-order multiagent systems can not be extended to it.In the last few years, lots of consensus algorithms have been proposed to solve the consensus problem of second-order multiagent systems (e.g., [10][11][12]).
Most existing consensus algorithms mentioned above for multiagent systems are usually asymptotic consensus algorithms.Since the finite-time stability appeared in [13], finite-time control has received compelling attention for its faster convergence and better robustness (e.g., [14][15][16][17][18][19][20]).To further investigate the control problems of multiagent systems with the property of finite-time convergence, some results have been given for the first-order and second-order multiagent systems (e.g., [21][22][23][24][25][26][27]).The work in [28] proposed a finite-time consensus via binary control protocols for the first-order systems which only requires sign information of the relative state measurements.In [29], the terminal sliding mode technique was used to design finite-time consensus algorithms for second-order systems.The work in [30] proposed a finite-time consensus protocol for leaderless and leader-follower second-order multiagent systems.
Most of the existing consensus algorithms were designed only for the simple single or double integrators.Since 2 Mathematical Problems in Engineering many existing nonholonomic systems with nonholonomic constraints can not be transformed as simple single or double integrators, such as mobile robots (see, e.g., [31]), these existing finite-time consensus algorithms are generally not applicable.There have been several results of finite-time consensus algorithms for the multiple nonholonomic systems in recent years.The paper [32] discusses the cooperative control of multiple nonholonomic agents under different communication scenarios.In [33,34], finite-time consensus protocols are presented for nonholonomic mobile agents with and without communication delay, respectively.In paper [35], the finite-time leader-following consensus problem for multiple nonholonomic mobile agents is discussed.
It is worth noting that these papers do not consider the problem of finite-time consensus of multiple nonholonomic uncertain systems.In this paper, the main contribution is that the finite-time consensus problem for a class of multiple nonholonomic uncertain systems is solved.To address this problem, we first proposed a finite-time consensus protocol for the second-order integral systems with uncertainties, which can be seen as a more general case of that in [30].In addition, a finite-time consensus protocol is obtained for the first-order multiple integral systems in the presence of disturbances.Then, a switched strategy is designed to guarantee that the state consensus for nonholonomic multiagent systems with disturbances can be achieved in finite time.
The rest of this paper is arranged as follows.Section 2 presents the control objective and some preliminary results.Section 3 gives the finite-time consensus protocols design for multiple nonholonomic systems with disturbances, which are the main results of the paper.Simulation results of application to multiple nonholonomic mobile robots are given in Section 4 and finally conclusion is drawn in Section 5.

Problem and Preliminaries
In this section, a basic concept about graph theory, some useful lemmas, and control objective are introduced.

Graph Theory.
Consider a leaderless multiagent system.Suppose that there are  nodes whose information can be exchanged with the single agent undirected neighbors.Define an undirected graph  = {, , }, where  = {1, . . ., } is the set of nodes,  is the set of edges, and  = [  ] ∈  × is the weighted adjacency matrix of  with nonnegative elements.Compared to the directed graph, the edge (, ) denotes that th and th node can obtain information from each other.If there is an edge between the th agent and the th agent, then   =   > 1.We define specially   = 0 for all  = 1, . . ., .Let  = diag{ 1 , . . .,   } as a diagonal matrix, where   = ∑  =1   = ∑ ∈    .Then, the Laplacian matrix of digraph  is defined as  =  − .Similar to the directed graph, the undirected graph  is connected if there is an undirected path between any two agents.

Some Lemmas
Lemma 1 (see [36]).If  is the Laplacian matrix of a connected undirected graph , one has for any  = [ 1 , . . .,   ]  .It is easy to conclude that 0 is a simple eigenvalue of  whose associated eigenvector is 1.Assume that The lemma below presents a Lyapunov-like theorem for finite-time stability of nonlinear systems.
Lemma 2 (see [14]).Consider the system ẋ =  () , Suppose there is  1 function () defined on a neighborhood Û ⊂   of the origin.If the function () satisfies the following conditions where  > 0 and 0 <  < 1 are real numbers, then, the origin of system is locally finite-time stable.The settling time, depending on the initial state (0) =  0 , satisfies If () is also radially unbounded and Û =   , the system is globally finite-time stable.
In order to construct the finite-time controller, the following three lemmas about some useful inequalities are needed.Lemma 3 (see [17]).For any  ∈ ,  ∈ , the following inequalities hold: Lemma 4 (see [17]). and  are two positive real numbers, and  ≥ 0,  ≥ 0, and  ≥ 0 are continuous functions.Then, for any constant  0 > 0,

Problem Formulation.
For the leaderless systems of multiple nonholonomic agents, the graph  is connected and the dynamics of the th agent is presented as where  1 ,  2 , and  3 and  1 ,  2 are the states and control inputs of the th nonholonomic agent, respectively, and  1 (),  2 () are disturbances.
In this paper, finite-time consensus protocols for leaderless multiple nonholonomic agents with disturbances are designed to achieve the state consensus in finite time.To achieve this objective, the following assumption is given first.
Assumption 5.The disturbances  1 ,  2 are bounded with a known positive constant  0 .

Main Results
To facilitate our design, the dynamics of the th agent is divided into two cascaded subsystems, which is composed of a first-order subsystem and a second-order subsystem We define that

The Design of Finite-Time Consensus Protocol for Multiple
Subsystems (9).To design the finite-time consensus algorithm for the second-order multiple subsystems (9), we first choose the control  1 as where  1 >  0 is a positive constant.In this case, the multiple subsystems (9) reduce to Remark 6.In [30], when  1 =  2 = 0, a finite-time consensus algorithm is designed for multiagent systems (12).
If the condition  1 = 0, | 2 | <  0 is considered, the algorithm in [30] can only guarantee that the steady-state errors of any two agents will converge to a region  about  0 .
In the following theorem, when  1 and  2 are all bounded by a known positive constant  0 , we extend the result in [30] to achieve the finite-time consensus for multiple subsystems (12).Now, we are ready to state our main result in this subsection.

Theorem 7. Based on Assumption 5, for the multiple subsystems (12), the finite-time consensus protocol based on nonsingular TSM control is designed as
(2−)/ (13) with where 1 <  =  1 / 2 < 2,  1 ,  2 are positive odd integers,  4 >  0 is a positive constant, and  2 ,  3 are two designed positive constants to be determined later.Under the finite-time consensus protocol (13), the states of all subsystems (12) can reach a consensus in a finite time.
Proof.Firstly, we prove that the manifold  2 ≡ 0 can be established in a finite-time  1 with protocol (13).Then, we show that, once the ideal sliding mode  2 ≡ 0 is obtained, the state consensus for multiple subsystems (12) is achieved in a finite-time By Lemma 3, one gets For ∀ 1 ∈ (0, ( 4 −  0 ) √ 2), one can verify that By Lemma 2, we can conclude that  1 and so the manifold  2 will reach zero in a finite-time  1 satisfying Once  ≥  1 , we have  2 = 0. Then system (12) will behave in an identical fashion; namely, Next, we prove that there exist positive constants  2 ,  3 such that the states of multiple systems (19) can reach a consensus in a finite-time  2 .The stability analysis of system ( 19) is based on that of [30], but the perturbations are not considered there.The proof is divided into two steps.
Step 1. Define a Lyapunov function  0 as With Lemma 1, the derivation of  0 along system (19) satisfies The virtual controller where  3 is a positive constant to be determined later.
Step 2. To continue the proof, we construct another Lyapunov function as with Taking the derivative of (), we have In the following, the terms in the right hand of (25) are discussed, respectively.Firstly, To estimate the terms in the right hand of ( 26), we introduce the following proposition, whose proof is given in the appendix.
Proposition 8.There exists a positive constant  1 such that Putting ( 27) into (26), one has Let  = 1 + 1/; to continue the proof, the following proposition whose proof is given in the appendix is presented.
Proposition 9.There exists a positive constant  2 ,  3 such that Substituting ( 29) into (28), one has By (19) and the definition of   , one has ẋ 3 = − 2 (2−)/  .Then, with (30), one gets Secondly, V 0 is estimated.From (22), one obtains To estimate the last term of (32), one introduces the following proposition whose proof is given in the appendix.
Using Proposition 11, one has Putting ( 31) and ( 34) into (25) yields Now, the parameters  2 ,  3 are determined in turn.Let  be as any given positive constant; one first selects the parameter  3 .Define It is easy to conclude that, for ∀ 1 ,  2 ,  3 ∈ (0, ∝) and  ∈ (1, 2), one can select  3 large enough to satisfy (37) Once the parameter  3 is fixed, one can determine the parameter  2 satisfying With  2 ,  3 satisfying ( 38) and ( 37), it is easy to verify that To estimate (), two useful propositions are introduced in the following.
Proposition 11.With  2 given in Lemma 1, the following inequality holds: The proof of Proposition 11 is straightforward by Lemma 1 and thus is omitted here.

The Design of Finite-Time Consensus Protocol for Multiple
Subsystems (8).Motivated by Theorem 7, for the multiple subsystems (8), the following theorem is obtained as the main result of this subsection.

Theorem 14. Based on Assumption 5, for the multiple subsystems (8), the finite-time consensus protocol based on nonsingular TSM control is designed as
with where 1 < p = p1 / p2 < 2, p1 , p2 are positive odd integers, and  6 > 0,  7 >  0 are positive constants.Under the protocol, all the states of multiple systems (8) can reach a consensus in a finite time.
Proof.The proof is divided into two steps.At the first step, we prove that the manifold  1 can reach zero in a finite time with protocol (48).In the second step, we show that, once the ideal sliding mode  1 = 0 is established, the state consensus for multiagent subsystems ( 8) is achieved in a finite time.
Step 2. Once the ideal sliding mode  1 = 0 is established, multiple systems (8) will behave in an identical fashion; namely, with  6 > 0. In the following, we will prove that the finitetime consensus for multiple systems (52) can be achieved.Define a Lyapunov function  3 () as With Lemma 1 and the definition of  1 , the derivation of  3 along system (52) satisfies where d = 1 + 1/ p. From Lemma 3, one has Following the same line to obtain Proposition 8, by Lemma 1, we get Putting ( 55) and (56) together, we obtain that Using Lemma 2, we conclude that  3 converges to zero with a finite time; that is, the states of multiple systems (52) can reach a consensus in a finite time.
Hence, the proof of Theorem 14 is completed with the two steps above.

The Overall Design of Finite-Time Consensus Protocol for
Multiple Systems (7).Based on Theorems 7 and 14, we are ready to state our main result of this subsection for multiple systems (7).
Theorem 15.Based on Assumption 5, for the leaderless multiagent systems (7), the finite-time consensus protocol based on a switched strategy is designed as where  1 ,   49) and (14).Under this protocol, all the states of multiple systems (7) can reach a consensus in a finite time.
Hence, we complete the proof of Theorem 15.

Simulation Results
In this section, we consider the leaderless multiagent systems with 5 mobile robots whose schematic is shown in Figure 1.
The th robot is illustrated with two driven rear wheels and a Y X 0 (x i , y i )  i front freewheel.The two rear wheels are driven, respectively, with fixed orientation.The generalized coordinates are  = (  ,   ,   ), where   ,   are the Cartesian coordinates of the middle point of the rear wheel and   measures the orientation of the car body with respect to -axis.Under the assumption of no wheel slips, the kinetic model of the th robot can be derived as where V  ,   are the linear and angular velocities of the th mobile robot and  1 = 0,  2 = 0.9sin() are disturbances.With the coordinate transformation system (62) is transformed as Then, the consensus protocols presented in Theorem 15 can be applied.The communication among the 5 agents is given in Figure 2. The initial states are given as (0) = (1, 1.1, 1.2, 1.3, 0.9), (0) = (1, 1.1, 1.2, 1.3, 0.9), and (0) = (5,6,7,4,3).The parameters are selected as

Conclusion
This paper studies the problem of finite-time consensus problem of multiple nonholonomic systems in the presence of disturbances.At first, the triple chained form agent system is divided into two agent subsystems and these two agent subsystems are discussed, respectively.Then, two integral terminal sliding mode controllers are designed for the two subsystems.Finally, we demonstrate the effectiveness of the proposed consensus algorithms with application to multiple nonholonomic mobile robots.(A.5)

Figure 1 :
Figure 1: Generalized coordinates of the th car-like robot.