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

Authors: Shang Shi; Xin Yu; Guohai Liu
Journal: Mathematical Problems in Engineering
 (2015)

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## Abstract

This paper deals with the problem of finite-time consensus of multiple nonholonomic disturbed systems.
To accomplish this problem, the multiple nonholonomic systems are transformed into two multiple subsystems,
and these two multiple 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.


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## Body

## 1. 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., [1–3] 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 [4–7] and the references therein. Since many practical systems can be transformed into the first-order or second-order 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., [10–12]).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., [14–20]). 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., [21–27]). 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 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. Section2 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.


## 2. Problem and Preliminaries

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


### 2.1. Graph Theory

Consider a leaderless multiagent system. Suppose that there aren nodes whose information can be exchanged with the single agent undirected neighbors. Define an undirected graph G={V,E,A}, where V={1,…,n} is the set of nodes, E is the set of edges, and A=[aij]∈Rn×n is the weighted adjacency matrix of G with nonnegative elements. Compared to the directed graph, the edge (i,j) denotes that ith and jth node can obtain information from each other. If there is an edge between the ith agent and the jth agent, then aij=aji>1. We define specially aii=0 for all i=1,…,n. Let D=diag{d1,…,dn} as a diagonal matrix, where di=∑j=1n‍aij=∑j∈Ni‍aij. Then, the Laplacian matrix of digraph G is defined as L=D-A. Similar to the directed graph, the undirected graph G is connected if there is an undirected path between any two agents.


### 2.2. Some Lemmas

Lemma 1 (see [36]).
IfL is the Laplacian matrix of a connected undirected graph G, one has(1)xTLx=12∑i,j=1naijxi-xj2=12∑i=1n∑j∈Niaijxi-xj2,for any x=[x1,…,xn]T. It is easy to conclude that 0 is a simple eigenvalue of L whose associated eigenvector is 1. Assume that λ1=0≤λ2≤⋯≤λn is the eigenvalues of L. Then one has λ2>0. If 1Tx=0, then xTLx≥λ2xTx.The lemma below presents a Lyapunov-like theorem for finite-time stability of nonlinear systems.Lemma 2 (see [14]).
Consider the system(2)x˙=Fx,F0=0,x∈Rn.Suppose there is C1 function V(x) defined on a neighborhood U^⊂Rn of the origin. If the function V(x) satisfies the following conditions where c>0 and 0<α<1 are real numbers,(1)
V(x) is positive definite on U^,(2)
V˙(x)+cVα(x)≤0, ∀x∈U^,then, the origin of system is locally finite-time stable. The settling time, depending on the initial state x(0)=x0, satisfies (3)Txx0≤Vx01-αc1-α.If V(x) is also radially unbounded and U^=Rn, the system is globally finite-time stable.In order to construct the finite-time controller, the following threelemmas about some useful inequalities are needed.Lemma 3 (see [17]).
For anya∈R, b∈R, the following inequalities hold:(4)a+bq≤2q-1aq+bq,a+b1/q≤a1/q+b1/q,when q≥1 is a constant. If q≥1 is odd, then(5)a-bq≤2q-1aq-bq,aq-bq≤qa-baq-1+bq-1.Lemma 4 (see [17]).
n and m are two positive real numbers, and a≥0, b≥0, and π≥0 are continuous functions. Then, for any constant c0>0,(6)anbmπ≤c0an+m+mn+mnc0n+mn/mbn+mπn+m/m.


### 2.3. Problem Formulation

For the leaderless systems of multiple nonholonomic agents, the graphG is connected and the dynamics of the ith agent is presented as(7)x˙i1=ui1+di1,x˙i2=ui1+di1xi3,x˙i3=ui2+di2,where xi1, xi2, and xi3 and ui1, ui2 are the states and control inputs of the ith nonholonomic agent, respectively, and di1(t), di2(t) 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 disturbancesdi1,di2 are bounded with a known positive constant d0.


## 2.1. Graph Theory

Consider a leaderless multiagent system. Suppose that there aren nodes whose information can be exchanged with the single agent undirected neighbors. Define an undirected graph G={V,E,A}, where V={1,…,n} is the set of nodes, E is the set of edges, and A=[aij]∈Rn×n is the weighted adjacency matrix of G with nonnegative elements. Compared to the directed graph, the edge (i,j) denotes that ith and jth node can obtain information from each other. If there is an edge between the ith agent and the jth agent, then aij=aji>1. We define specially aii=0 for all i=1,…,n. Let D=diag{d1,…,dn} as a diagonal matrix, where di=∑j=1n‍aij=∑j∈Ni‍aij. Then, the Laplacian matrix of digraph G is defined as L=D-A. Similar to the directed graph, the undirected graph G is connected if there is an undirected path between any two agents.


## 2.2. Some Lemmas

Lemma 1 (see [36]).
IfL is the Laplacian matrix of a connected undirected graph G, one has(1)xTLx=12∑i,j=1naijxi-xj2=12∑i=1n∑j∈Niaijxi-xj2,for any x=[x1,…,xn]T. It is easy to conclude that 0 is a simple eigenvalue of L whose associated eigenvector is 1. Assume that λ1=0≤λ2≤⋯≤λn is the eigenvalues of L. Then one has λ2>0. If 1Tx=0, then xTLx≥λ2xTx.The lemma below presents a Lyapunov-like theorem for finite-time stability of nonlinear systems.Lemma 2 (see [14]).
Consider the system(2)x˙=Fx,F0=0,x∈Rn.Suppose there is C1 function V(x) defined on a neighborhood U^⊂Rn of the origin. If the function V(x) satisfies the following conditions where c>0 and 0<α<1 are real numbers,(1)
V(x) is positive definite on U^,(2)
V˙(x)+cVα(x)≤0, ∀x∈U^,then, the origin of system is locally finite-time stable. The settling time, depending on the initial state x(0)=x0, satisfies (3)Txx0≤Vx01-αc1-α.If V(x) is also radially unbounded and U^=Rn, the system is globally finite-time stable.In order to construct the finite-time controller, the following threelemmas about some useful inequalities are needed.Lemma 3 (see [17]).
For anya∈R, b∈R, the following inequalities hold:(4)a+bq≤2q-1aq+bq,a+b1/q≤a1/q+b1/q,when q≥1 is a constant. If q≥1 is odd, then(5)a-bq≤2q-1aq-bq,aq-bq≤qa-baq-1+bq-1.Lemma 4 (see [17]).
n and m are two positive real numbers, and a≥0, b≥0, and π≥0 are continuous functions. Then, for any constant c0>0,(6)anbmπ≤c0an+m+mn+mnc0n+mn/mbn+mπn+m/m.


## 2.3. Problem Formulation

For the leaderless systems of multiple nonholonomic agents, the graphG is connected and the dynamics of the ith agent is presented as(7)x˙i1=ui1+di1,x˙i2=ui1+di1xi3,x˙i3=ui2+di2,where xi1, xi2, and xi3 and ui1, ui2 are the states and control inputs of the ith nonholonomic agent, respectively, and di1(t), di2(t) 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 disturbancesdi1,di2 are bounded with a known positive constant d0.


## 3. Main Results

To facilitate our design, the dynamics of theith agent is divided into two cascaded subsystems, which is composed of a first-order subsystem(8)x˙i1=ui1+di1and a second-order subsystem(9)x˙i2=ui1+di1xi3,x˙i3=ui2+di2.We define that(10)x1=x11,…,xn1,ei1=∑j∈Niaijxi1-xj1,x2=x12,…,xn1,ei2=∑j∈Niaijxi2-xj2,x3=x13,…,xn3.


### 3.1. 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 ui1 as(11)ui1=k1,where k1>d0 is a positive constant. In this case, the multiple subsystems (9) reduce to(12)x˙i2=k1+di1xi3,x˙i3=ui2+di2.Remark 6.
In [30], when di1=di2=0, a finite-time consensus algorithm is designed for multiagent systems (12). If the condition di1=0, |di2|<d0 is considered, the algorithm in [30] can only guarantee that the steady-state errors of any two agents will converge to a region Q about d0. In the following theorem, when di1 and di2 are all bounded by a known positive constant d0, 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 Assumption5, for the multiple subsystems (12), the finite-time consensus protocol based on nonsingular TSM control is designed as(13)ui2=-k4sgn⁡Si2-k2xi3p+k3p∑j∈Niaijxi2-xj22-p/pwith(14)Si2=xi3+∫0τk2xi3p+k3p∑j∈Niaijxi2-xj22-p/pdτ,where 1<p=p1/p2<2, p1,p2 are positive odd integers, k4>d0 is a positive constant, and k2, k3 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 manifoldSi2≡0 can be established in a finite-time t1 with protocol (13). Then, we show that, once the ideal sliding mode Si2≡0 is obtained, the state consensus for multiple subsystems (12) is achieved in a finite-time t2.
DefiningU1(t)=1/2∑i=1nSi22, with ui2 and Si2 given in Theorem 7, the derivative of U1(t) is (15)U˙1t=∑i=1nSi2-k4sign⁡Si2+di2≤-k4-d0∑i=1nSi2.By Lemma 3, one gets(16)U11/2t≤12∑i=1nSi2.For ∀α1∈0,(k4-d0)2, one can verify that(17)U˙1+α1U11/2=α12-k4-d0∑i=1nSi2≤0.By Lemma 2, we can conclude that U1 and so the manifold Si2 will reach zero in a finite-time t1 satisfying(18)t1≤2U11/20α1.
Oncet≥t1, we have Si2=0. Then system (12) will behave in an identical fashion; namely,(19)x˙i2=k1+di1xi3,x˙i3=-k2xi3p+k3p∑j∈Niaijxi2-xj22-p/p.Next, we prove that there exist positive constants k2,k3 such that the states of multiple systems (19) can reach a consensus in a finite-time t2. 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 V0 as(20)V0t=12x2TLx2=14∑i=1n∑j∈Niaijxi2-xj22.With Lemma 1, the derivation of V0 along system (19) satisfies(21)V˙0t=x˙2TLx2=k1+di1∑i=1nxi3∑j∈Niaijxi2-xj2.The virtual controller xi3∗=-k3ei21/p renders(22)V˙0≤-k1-d0k3∑i=1nei21+p/p+k1+di1∑i=1nei2xi3-xi3∗,where k3 is a positive constant to be determined later.
Step  2. To continue the proof, we construct another Lyapunov function as(23)Vt=V0t+∑i=1nVitwith(24)Vit=∫xi3∗xi3sp-xi3∗p2-1/pds.Taking the derivative of V(t), we have(25)V˙t=V˙0+∑i=1nV˙it.In the following, the terms in the right hand of (25) are discussed, respectively.
Firstly,∑i=1nV˙i(t) is estimated. Define ξi=xi3p-xi3∗p; the derivative of Vi(t) is(26)V˙it=-2-1pdxi3∗pdt∫xi3∗xi3sp-xi3∗p1-1/pds+ξi2-1/px˙i3.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 constantc1 such that(27)dxi3∗pdt≤k3pc1∑m=1nxm3,∫xi3∗xi3sp-xi3∗p1-1/pds≤xi3-xi3∗ξi1-1/p.Putting (27) into (26), one has(28)V˙it≤-2-1pc1k3p∑m=1nxm3xi3-xi3∗ξi1-1/p+ξi2-1/px˙i3.Let d=1+1/p; to continue the proof, the following proposition whose proof is given in the appendix is presented.Proposition 9.
There exists a positive constantc2, c3 such that(29)xm3xi3-xi3∗ξi1-1/p≤k3+1ξid+c2ξmd+k3c3em2d.Substituting (29) into (28), one has(30)V˙it≤n2-pk3+1k3ppξid+c22-pk3pp∑m=1nξmd+c32-pk3p+1p∑m=1nem2d+ξi2-1/px˙i3.By (19) and the definition of ξi, one has x˙i3=-k2ξi(2-p)/p. Then, with (30), one gets(31)∑i=1nV˙it≤n2-pk3+1k3pp+nc22-pk3pp-k2·∑i=1nξid+nc32-pk3p+1p∑i=1nei2d.
Secondly,V˙0 is estimated. From (22), one obtains(32)V˙0≤-k1-d0k3∑i=1nei2d+k1+d0∑i=1nei2xi3-xi3∗.To estimate the last term of (32), one introduces the following proposition whose proof is given in the appendix.Proposition 10.
There exists a positive constantc4 such that(33)∑i=1nei2xi3-xi3∗≤c4∑i=1nei2d+ξid.Using Proposition 11, one has(34)V˙0t≤k1+d0c4-k1-d0k3∑i=1nei2d+k1+d0c4∑i=1nξid.Putting (31) and (34) into (25) yields(35)V˙t≤nc32-pk3p+1p+k1+d0c4-k1-d0k3∑i=1nei2d+n2-pk3+1k3pp+nc22-pk3pp+k1+d0c4-k2∑i=1nξid.
Now, the parametersk2,k3 are determined in turn. Let η be as any given positive constant; one first selects the parameter k3. Define(36)fk3=nc32-pk3p+1p+k1+d0c4-k1-d0k3≜a1k3p+1-a2k3+a3.It is easy to conclude that, for ∀a1,a2,a3∈(0,∝) and p∈(1,2), one can select k3 large enough to satisfy(37)fk3=nc32-pk3p+1p+k1+d0c4-k1-d0k3>η.Once the parameter k3 is fixed, one can determine the parameter k2 satisfying(38)k2>η+n2-pk3+1k3pp+nc22-pk3pp+k1+d0c4.With k2, k3 satisfying (38) and (37), it is easy to verify that(39)V˙t≤-η∑i=1nei2d-η∑i=1nξid.To estimate V(t), two useful propositions are introduced in the following.Proposition 11.
Withλ2 given in Lemma 1, the following inequality holds:(40)∑i=1nei22≥2λ2V0.The proof of Proposition 11 is straightforward by Lemma 1 and thus is omitted here.Proposition 12.
The following inequality always holds:(41)∑i=1nVit≤12-1/pk31+p∑i=1nξi2.With Propositions 11 and 12, one gets(42)Vt≤c5∑i=1nei22+∑i=1nξi2with(43)c5=max⁡12λ2,12-1/pk31+p.For 0<d/2<1, using Lemma 3, one has(44)Vd/2t≤c5d/2∑i=1nei2d+∑i=1nξid.Putting (39) and (44) together, for ∀α2∈(0,η/c5d/2), one has(45)V˙t+α2Vd/2t≤α2c5d/2-η∑i=1nei2d+ξid≤0.By Lemma 2, one concludes that V(t) will reach zero in a finite-time t2 satisfying(46)t2≤V1-d/2t1α21-d/2.
Hence, under control (13) with parameters k2,k3 presented in (38) and (37), the state consensus for multiple subsystems (12) can be achieved in a finite-time T1≥t1+t2.Remark 13.
It is easy to know thatV(t)=0 means V0(t)=Vi(t)=0. Since the graph G is connected, we obtain that xi2=xj2, xi3=xi3∗, for ∀i,j=1,…,n. It is worth noting that xi3∗=0 once xi2=xj2. Thus, for ∀i,j=1,…,n, the following equations,(47)xi2=xj2=x2∗,xi3=0,ui2=0,will be established before T1, where x2∗ is a constant. Once (47) is established, it is easy to conclude that, for ∀t≥T1, i=1,…,n, if xi3=0 is kept with the control ui2, the state consensus of xi2, xi3 is kept no matter what the control ui1 is taken as.


### 3.2. The Design of Finite-Time Consensus Protocol for Multiple Subsystems (8)

Motivated by Theorem7, for the multiple subsystems (8), the following theorem is obtained as the main result of this subsection.Theorem 14.
Based on Assumption5, for the multiple subsystems (8), the finite-time consensus protocol based on nonsingular TSM control is designed as(48)ui1=-k6∑j∈Niaijxi1-xj11/p~-k7sgn⁡Si1with(49)Si1=xi1+∫0τk6∑j∈Niaijxi1-xj11/p~dτ,where 1<p~=p~1/p~2<2, p~1,p~2 are positive odd integers, and k6>0, k7>d0 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 manifoldSi1 can reach zero in a finite time with protocol (48). In the second step, we show that, once the ideal sliding mode Si1=0 is established, the state consensus for multiagent subsystems (8) is achieved in a finite time.
Step  1. Define a Lyapunov function U2=1/2∑i=1n‍Si12; taking the derivative of U2 yields(50)U˙2≤-k7sgn⁡Si1+di1Si1≤-k7-d0Si1.Following the same line to obtain (17), we have(51)U˙2+α3U21/2≤-k7-d0-α32∑i=1nSi1≤0for ∀α3∈(0,2(k7-d0)). Then, by Lemma 2, U2 and so the manifold Si1 can reach zero in a finite time.
Step  2. Once the ideal sliding mode Si1=0 is established, multiple systems (8) will behave in an identical fashion; namely,(52)x˙i1=-k6∑j∈Niaijxi1-xj11/p~with k6>0. In the following, we will prove that the finite-time consensus for multiple systems (52) can be achieved.
Define a Lyapunov functionU3(t) as(53)U3t=12x1TLx1=14∑i=1n∑j∈Niaijxi1-xj12.With Lemma 1 and the definition of ei1, the derivation of U3 along system (52) satisfies(54)U˙3t=x˙1TLx1=∑i=1nx˙i1∑j∈Niaijxi1-xj1=-k6∑i=1nei1d~,where d~=1+1/p~. From Lemma 3, one has(55)U˙3t≤-k6∑i=1nei12d~/2.Following the same line to obtain Proposition 8, by Lemma 1, we get(56)∑i=1nei12≥2λ2U3t.Putting (55) and (56) together, we obtain that(57)U˙3t≤-k62λ2d~/2U3d~/2t.Using Lemma 2, we conclude that U3 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 Theorem14 is completed with the two steps above.


### 3.3. The Overall Design of Finite-Time Consensus Protocol for Multiple Systems (7)

Based on Theorems7 and 14, we are ready to state our main result of this subsection for multiple systems (7).Theorem 15.
Based on Assumption5, for the leaderless multiagent systems (7), the finite-time consensus protocol based on a switched strategy is designed as(58)ui1=k1,t≤T1,ui1=-k6∑j∈Niaijxi1-xj11/p~-k7sgn⁡Si1,t>T1,(59)ui2=-k4sgn⁡Si2-k2xi3p+k3p∑j∈Niaijxi2-xj22-p/p,t≤T1,ui2=-k5sgn⁡xi3,t>T1,where k1,k4,k5,k7>d0, k6>0, 1<p=p1/p2<2, 1<p~=p~1/p~2<2, p1,p2,p~1,p~2 are positive odd integers, k2,k3 are designed parameters satisfying (38) and (37), T1≥t1+t2 are switched time, and Si1,Si2 are ideal sliding surface selected as (49) and (14). Under this protocol, all the states of multiple systems (7) can reach a consensus in a finite time.Proof.
From Theorems7 and 14, it is easy to conclude that the consensus of xi2, xi3 can be achieved with a finite-time T1, and when t≥T1, the consensus of xi1 can be achieved in finite time. Therefore, here we only need to prove that the consensus of xi2, xi3 achieved within T1 can be kept after t≥T1.
From (58), when t>T1, we have ui2=-k5signxi3. Define a Lyapunov function U4(t)=(1/2)∑i=1n‍xi32; the derivation of U4(t) satisfies(60)U˙4t=∑i=1n-k5xi3+di2xi3≤-k5-d0∑i=1nxi3.From Lemma 3, we get that ∀α4∈(0,2(k5-d0))(61)U˙4t+α4U41/2t≤-k5-d0-α42∑i=1nxi3≤0.By Lemma 2, we conclude that U4 will converge to zero in a finite time t¯≤U41/2(xi3(T1))/(α4/2). From Remark 13, we know that xi3(T1)=0. Therefore, it can be concluded that the convergence time t¯=0; that is, xi3=0 is always kept after t>T1.
Hence, we complete the proof of Theorem15.


## 3.1. 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 ui1 as(11)ui1=k1,where k1>d0 is a positive constant. In this case, the multiple subsystems (9) reduce to(12)x˙i2=k1+di1xi3,x˙i3=ui2+di2.Remark 6.
In [30], when di1=di2=0, a finite-time consensus algorithm is designed for multiagent systems (12). If the condition di1=0, |di2|<d0 is considered, the algorithm in [30] can only guarantee that the steady-state errors of any two agents will converge to a region Q about d0. In the following theorem, when di1 and di2 are all bounded by a known positive constant d0, 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 Assumption5, for the multiple subsystems (12), the finite-time consensus protocol based on nonsingular TSM control is designed as(13)ui2=-k4sgn⁡Si2-k2xi3p+k3p∑j∈Niaijxi2-xj22-p/pwith(14)Si2=xi3+∫0τk2xi3p+k3p∑j∈Niaijxi2-xj22-p/pdτ,where 1<p=p1/p2<2, p1,p2 are positive odd integers, k4>d0 is a positive constant, and k2, k3 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 manifoldSi2≡0 can be established in a finite-time t1 with protocol (13). Then, we show that, once the ideal sliding mode Si2≡0 is obtained, the state consensus for multiple subsystems (12) is achieved in a finite-time t2.
DefiningU1(t)=1/2∑i=1nSi22, with ui2 and Si2 given in Theorem 7, the derivative of U1(t) is (15)U˙1t=∑i=1nSi2-k4sign⁡Si2+di2≤-k4-d0∑i=1nSi2.By Lemma 3, one gets(16)U11/2t≤12∑i=1nSi2.For ∀α1∈0,(k4-d0)2, one can verify that(17)U˙1+α1U11/2=α12-k4-d0∑i=1nSi2≤0.By Lemma 2, we can conclude that U1 and so the manifold Si2 will reach zero in a finite-time t1 satisfying(18)t1≤2U11/20α1.
Oncet≥t1, we have Si2=0. Then system (12) will behave in an identical fashion; namely,(19)x˙i2=k1+di1xi3,x˙i3=-k2xi3p+k3p∑j∈Niaijxi2-xj22-p/p.Next, we prove that there exist positive constants k2,k3 such that the states of multiple systems (19) can reach a consensus in a finite-time t2. 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 V0 as(20)V0t=12x2TLx2=14∑i=1n∑j∈Niaijxi2-xj22.With Lemma 1, the derivation of V0 along system (19) satisfies(21)V˙0t=x˙2TLx2=k1+di1∑i=1nxi3∑j∈Niaijxi2-xj2.The virtual controller xi3∗=-k3ei21/p renders(22)V˙0≤-k1-d0k3∑i=1nei21+p/p+k1+di1∑i=1nei2xi3-xi3∗,where k3 is a positive constant to be determined later.
Step  2. To continue the proof, we construct another Lyapunov function as(23)Vt=V0t+∑i=1nVitwith(24)Vit=∫xi3∗xi3sp-xi3∗p2-1/pds.Taking the derivative of V(t), we have(25)V˙t=V˙0+∑i=1nV˙it.In the following, the terms in the right hand of (25) are discussed, respectively.
Firstly,∑i=1nV˙i(t) is estimated. Define ξi=xi3p-xi3∗p; the derivative of Vi(t) is(26)V˙it=-2-1pdxi3∗pdt∫xi3∗xi3sp-xi3∗p1-1/pds+ξi2-1/px˙i3.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 constantc1 such that(27)dxi3∗pdt≤k3pc1∑m=1nxm3,∫xi3∗xi3sp-xi3∗p1-1/pds≤xi3-xi3∗ξi1-1/p.Putting (27) into (26), one has(28)V˙it≤-2-1pc1k3p∑m=1nxm3xi3-xi3∗ξi1-1/p+ξi2-1/px˙i3.Let d=1+1/p; to continue the proof, the following proposition whose proof is given in the appendix is presented.Proposition 9.
There exists a positive constantc2, c3 such that(29)xm3xi3-xi3∗ξi1-1/p≤k3+1ξid+c2ξmd+k3c3em2d.Substituting (29) into (28), one has(30)V˙it≤n2-pk3+1k3ppξid+c22-pk3pp∑m=1nξmd+c32-pk3p+1p∑m=1nem2d+ξi2-1/px˙i3.By (19) and the definition of ξi, one has x˙i3=-k2ξi(2-p)/p. Then, with (30), one gets(31)∑i=1nV˙it≤n2-pk3+1k3pp+nc22-pk3pp-k2·∑i=1nξid+nc32-pk3p+1p∑i=1nei2d.
Secondly,V˙0 is estimated. From (22), one obtains(32)V˙0≤-k1-d0k3∑i=1nei2d+k1+d0∑i=1nei2xi3-xi3∗.To estimate the last term of (32), one introduces the following proposition whose proof is given in the appendix.Proposition 10.
There exists a positive constantc4 such that(33)∑i=1nei2xi3-xi3∗≤c4∑i=1nei2d+ξid.Using Proposition 11, one has(34)V˙0t≤k1+d0c4-k1-d0k3∑i=1nei2d+k1+d0c4∑i=1nξid.Putting (31) and (34) into (25) yields(35)V˙t≤nc32-pk3p+1p+k1+d0c4-k1-d0k3∑i=1nei2d+n2-pk3+1k3pp+nc22-pk3pp+k1+d0c4-k2∑i=1nξid.
Now, the parametersk2,k3 are determined in turn. Let η be as any given positive constant; one first selects the parameter k3. Define(36)fk3=nc32-pk3p+1p+k1+d0c4-k1-d0k3≜a1k3p+1-a2k3+a3.It is easy to conclude that, for ∀a1,a2,a3∈(0,∝) and p∈(1,2), one can select k3 large enough to satisfy(37)fk3=nc32-pk3p+1p+k1+d0c4-k1-d0k3>η.Once the parameter k3 is fixed, one can determine the parameter k2 satisfying(38)k2>η+n2-pk3+1k3pp+nc22-pk3pp+k1+d0c4.With k2, k3 satisfying (38) and (37), it is easy to verify that(39)V˙t≤-η∑i=1nei2d-η∑i=1nξid.To estimate V(t), two useful propositions are introduced in the following.Proposition 11.
Withλ2 given in Lemma 1, the following inequality holds:(40)∑i=1nei22≥2λ2V0.The proof of Proposition 11 is straightforward by Lemma 1 and thus is omitted here.Proposition 12.
The following inequality always holds:(41)∑i=1nVit≤12-1/pk31+p∑i=1nξi2.With Propositions 11 and 12, one gets(42)Vt≤c5∑i=1nei22+∑i=1nξi2with(43)c5=max⁡12λ2,12-1/pk31+p.For 0<d/2<1, using Lemma 3, one has(44)Vd/2t≤c5d/2∑i=1nei2d+∑i=1nξid.Putting (39) and (44) together, for ∀α2∈(0,η/c5d/2), one has(45)V˙t+α2Vd/2t≤α2c5d/2-η∑i=1nei2d+ξid≤0.By Lemma 2, one concludes that V(t) will reach zero in a finite-time t2 satisfying(46)t2≤V1-d/2t1α21-d/2.
Hence, under control (13) with parameters k2,k3 presented in (38) and (37), the state consensus for multiple subsystems (12) can be achieved in a finite-time T1≥t1+t2.Remark 13.
It is easy to know thatV(t)=0 means V0(t)=Vi(t)=0. Since the graph G is connected, we obtain that xi2=xj2, xi3=xi3∗, for ∀i,j=1,…,n. It is worth noting that xi3∗=0 once xi2=xj2. Thus, for ∀i,j=1,…,n, the following equations,(47)xi2=xj2=x2∗,xi3=0,ui2=0,will be established before T1, where x2∗ is a constant. Once (47) is established, it is easy to conclude that, for ∀t≥T1, i=1,…,n, if xi3=0 is kept with the control ui2, the state consensus of xi2, xi3 is kept no matter what the control ui1 is taken as.


## 3.2. The Design of Finite-Time Consensus Protocol for Multiple Subsystems (8)

Motivated by Theorem7, for the multiple subsystems (8), the following theorem is obtained as the main result of this subsection.Theorem 14.
Based on Assumption5, for the multiple subsystems (8), the finite-time consensus protocol based on nonsingular TSM control is designed as(48)ui1=-k6∑j∈Niaijxi1-xj11/p~-k7sgn⁡Si1with(49)Si1=xi1+∫0τk6∑j∈Niaijxi1-xj11/p~dτ,where 1<p~=p~1/p~2<2, p~1,p~2 are positive odd integers, and k6>0, k7>d0 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 manifoldSi1 can reach zero in a finite time with protocol (48). In the second step, we show that, once the ideal sliding mode Si1=0 is established, the state consensus for multiagent subsystems (8) is achieved in a finite time.
Step  1. Define a Lyapunov function U2=1/2∑i=1n‍Si12; taking the derivative of U2 yields(50)U˙2≤-k7sgn⁡Si1+di1Si1≤-k7-d0Si1.Following the same line to obtain (17), we have(51)U˙2+α3U21/2≤-k7-d0-α32∑i=1nSi1≤0for ∀α3∈(0,2(k7-d0)). Then, by Lemma 2, U2 and so the manifold Si1 can reach zero in a finite time.
Step  2. Once the ideal sliding mode Si1=0 is established, multiple systems (8) will behave in an identical fashion; namely,(52)x˙i1=-k6∑j∈Niaijxi1-xj11/p~with k6>0. In the following, we will prove that the finite-time consensus for multiple systems (52) can be achieved.
Define a Lyapunov functionU3(t) as(53)U3t=12x1TLx1=14∑i=1n∑j∈Niaijxi1-xj12.With Lemma 1 and the definition of ei1, the derivation of U3 along system (52) satisfies(54)U˙3t=x˙1TLx1=∑i=1nx˙i1∑j∈Niaijxi1-xj1=-k6∑i=1nei1d~,where d~=1+1/p~. From Lemma 3, one has(55)U˙3t≤-k6∑i=1nei12d~/2.Following the same line to obtain Proposition 8, by Lemma 1, we get(56)∑i=1nei12≥2λ2U3t.Putting (55) and (56) together, we obtain that(57)U˙3t≤-k62λ2d~/2U3d~/2t.Using Lemma 2, we conclude that U3 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 Theorem14 is completed with the two steps above.


## 3.3. The Overall Design of Finite-Time Consensus Protocol for Multiple Systems (7)

Based on Theorems7 and 14, we are ready to state our main result of this subsection for multiple systems (7).Theorem 15.
Based on Assumption5, for the leaderless multiagent systems (7), the finite-time consensus protocol based on a switched strategy is designed as(58)ui1=k1,t≤T1,ui1=-k6∑j∈Niaijxi1-xj11/p~-k7sgn⁡Si1,t>T1,(59)ui2=-k4sgn⁡Si2-k2xi3p+k3p∑j∈Niaijxi2-xj22-p/p,t≤T1,ui2=-k5sgn⁡xi3,t>T1,where k1,k4,k5,k7>d0, k6>0, 1<p=p1/p2<2, 1<p~=p~1/p~2<2, p1,p2,p~1,p~2 are positive odd integers, k2,k3 are designed parameters satisfying (38) and (37), T1≥t1+t2 are switched time, and Si1,Si2 are ideal sliding surface selected as (49) and (14). Under this protocol, all the states of multiple systems (7) can reach a consensus in a finite time.Proof.
From Theorems7 and 14, it is easy to conclude that the consensus of xi2, xi3 can be achieved with a finite-time T1, and when t≥T1, the consensus of xi1 can be achieved in finite time. Therefore, here we only need to prove that the consensus of xi2, xi3 achieved within T1 can be kept after t≥T1.
From (58), when t>T1, we have ui2=-k5signxi3. Define a Lyapunov function U4(t)=(1/2)∑i=1n‍xi32; the derivation of U4(t) satisfies(60)U˙4t=∑i=1n-k5xi3+di2xi3≤-k5-d0∑i=1nxi3.From Lemma 3, we get that ∀α4∈(0,2(k5-d0))(61)U˙4t+α4U41/2t≤-k5-d0-α42∑i=1nxi3≤0.By Lemma 2, we conclude that U4 will converge to zero in a finite time t¯≤U41/2(xi3(T1))/(α4/2). From Remark 13, we know that xi3(T1)=0. Therefore, it can be concluded that the convergence time t¯=0; that is, xi3=0 is always kept after t>T1.
Hence, we complete the proof of Theorem15.


## 4. Simulation Results

In this section, we consider the leaderless multiagent systems with 5 mobile robots whose schematic is shown in Figure1. The ith robot is illustrated with two driven rear wheels and a front freewheel. The two rear wheels are driven, respectively, with fixed orientation. The generalized coordinates are q=(xi,yi,θi), where xi,yi are the Cartesian coordinates of the middle point of the rear wheel and θi measures the orientation of the car body with respect to x-axis. Under the assumption of no wheel slips, the kinetic model of the ith robot can be derived as(62)x˙i=vi+di2cos⁡θi,y˙i=vi+di2sin⁡θi,θ˙i=wi+di1,where vi, wi are the linear and angular velocities of the ith mobile robot and di1=0, di2=0.9sin(t) are disturbances. With the coordinate transformation(63)xi1=-θi,xi2=-xisin⁡θi+yicos⁡θi,xi3=xicos⁡θi+yisin⁡θi,(64)vi=xi3ui1+ui2=xicos⁡θi+yisin⁡θiui1+ui2,wi=-ui1,system (62) is transformed as(65)x˙i1=ui1-di1,x˙i2=ui1-di1xi3,x˙i3=ui2+di2.Then, the consensus protocols presented in Theorem 15 can be applied.Figure 1
Generalized coordinates of theith car-like robot.The communication among the 5 agents is given in Figure2. The initial states are given as x(0)=(1,1.1,1.2,1.3,0.9), y(0)=(1,1.1,1.2,1.3,0.9), and θ(0)=(5,6,7,4,3). The parameters are selected as(66)k1=1,k2=40,k3=2.7,k4=1,k5=1,k6=2.7,k7=0,η=1,p=97,T1=10s.Figures 3, 4, and 5 show the consensus of the positions and orientations, respectively.Figure 2
The communication among agents.Figure 3
Agents positionsx.Figure 4
Agents positionsy.Figure 5
Agents orientationθ.


## 5. 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.



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*Source: 358161-2015-05-24.xml*