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Discrete-time sliding mode protocols for leader-following consensus of discrete multi-agent system with switching graph topology
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Discrete-time sliding mode protocols for leader-following consensus of discrete multi-agent system with switching graph topology Keyurkumar Patel, Axaykumar Mehta∗ Department of Electrical Engineering, Institute of Infrastructure Technology Research and Management, Ahmedabad, Gujarat, 380026 India
a r t i c l e
i n f o
Article history: Received 1 February 2019 Revised 1 May 2019 Accepted 27 June 2019 Available online xxx Recommended by Dr X. Chen Keywords: Discrete- time sliding mode control (DSMC) Discrete multi-agent system (DMAS) 2-DOF helicopter system
a b s t r a c t In this paper, discrete-time sliding mode protocols for consensus of a leader following discrete multiagent system having switching topologies are proposed. The discrete multi-agent system is described with a fixed, undirected switching graph topology as a global system having one leader and other as follower agents. Sliding surface for global consensus of agents for switching graph topology is defined and discrete-time sliding mode protocols using enhanced Gao’s and Power rate reaching laws are derived. Under the influence of switching topology, the graph topology switches in a different no. of step intervals and ensure that consensus protocols asymptotically synchronize follower agents with leader agent. The condition for global stability in both cases is also derived using the Lyapunov function. Further, the algorithms are implemented on the multi-agent system comprise of multiple 2-DOF (Degree of Freedom) helicopter systems where the pitch angle and its velocity & yaw angle and its velocity are synchronized with the leader. It is inferred from simulation results that the consensus using Power rate reaching law give better performance compared with consensus using Gao’s reaching law. Finally, the robustness property of both the algorithms is checked by applying the matched disturbances in follower agents. © 2019 European Control Association. Published by Elsevier Ltd. All rights reserved.
1. Introduction Over the past few decades, cooperative control of multiple systems well known as Multi-Agent System (MAS) has drawn more attention among academician and researchers due to its wide range of applications in various domains such as cooperative control of multiple vehicles, smart grids, surveillance and reconnaissance, Autonomous under water vehicle (AUV) etc... [2,8,13,17,26,30,32,35,37]. The main idea in cooperative control of MAS is to design distributed controllers on each agent by getting local neighbouring information for reaching a certain defined global goal. In other words, under the influence of cooperation, the agents in MAS only share information with their neighbours locally and try to reach an agreement to a certain level of degree. To achieve this task, different cooperative control algorithm such as, consensus, formation, flocking, rendezvous, etc... are proposed in the literature [4,5,24,35]. Among all co-operative control algorithms, the consensus algorithm has been studied widely with different types of control techniques in recent years. The consensus of MAS can be mainly divided into two classes: First one is leaderless consensus algorithm in which individual follower agent share
∗
Corresponding author. E-mail address: [email protected] (A. Mehta).
its information with neighbouring agent [3,19]. The second one is the leader-following consensus in which the states of all the follower agents track the leader states [9,31]. However, the consensus problem is more interesting and challenging with communication topology using graph theory. The communication topology described by a digraph can be static or time-varying. The most common time-varying graph is called a switching graph. Switching topologies consideration in actual application due to unreliable transmission or limited communication and sensing range. Many researchers and academician have conducted research on consensus problems for both first and second-order multi-agent system with switching topologies. For first-order multi-agent system, Guo et al. [7] proposed the leader-following consensus problem of multi-agent systems with switching topologies and time delays in which the leader is static and the controlling effect of each follower depends on its own state. For second-order multi-agent system, Fang et al. [6] proposed about second order lag consensus means the state of follower agent lag behind the leader state algorithms for leader-following multi-agent system with communication constraint and switching topologies. The proposed consensus protocol ensure to tackle the communication delay problem using switching topologies. Wang et al. [27] proposed a consensus disturbance rejection for linear MAS with directed switching topologies. Wang et al. [28] discussed Leader-follower fault-tolerant consensus for linear MAS with switching directed network.
https://doi.org/10.1016/j.ejcon.2019.06.011 0947-3580/© 2019 European Control Association. Published by Elsevier Ltd. All rights reserved.
Please cite this article as: K. Patel and A. Mehta, Discrete-time sliding mode protocols for leader-following consensus of discrete multiagent system with switching graph topology, European Journal of Control, https://doi.org/10.1016/j.ejcon.2019.06.011
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Nomenclature FA − 1 FA − 2 FA − 3 FA − 4 FA − 5 S U
Follower agent-1 Follower agent-2 Follower agent-3 Follower agent-4 Follower agent-5 Sliding surface Control effort (u)
The paper is organized as follows: The brief description of graph theory and DMAS along with problem statement are presented in section-2. The main contribution that is the design of DSMC global consensus protocol using Gao’s reaching law and Power rate reaching laws and global stability analysis are presented in the Sections 3 and 4, respectively. Section 5 presents system description, and modeling of 2-DOF helicopter system used as DMAS configuration and simulation results are discussed in Section 6 followed by conclusion in Section 7. 2. Preliminaries of graph theory and Discrete-time leader-following MAS
Researchers and academician around the globe have proposed different consensus control protocol for MAS. Wen et.al. discussed the H∞ consensus problem for higher order MAS with switching topologies, in which they proposed a distributed protocol and employed to achieve state consensus in the presence of disturbance with switching directed communication topology [29]. Song et al. developed adaptive leader-following state consensus of MAS with switching topologies, the main goal is to design consensus protocol in the presence of system parameter uncertainties and introduced a new uniform nominal controller structure to achieve the consensus [22]. Zhang et al. proposed a leader-following state consensus protocol for the homogeneous system over time-varying feedback graphs using a state-feedback controller [33]. Jiang et.al [11] proposed a robust consensus protocol using integral sliding mode control for multi-agent system for fixed and switching topologies and derived sufficient condition for the existence of sliding surface for global stability. However, due to more reliable communication and saving the bandwidth of the channel, many researchers and scientists have opted consensus in the discrete domain. The consensus problem for discrete-time multi-agent system was mainly studied in [23,36] using switching topologies network. Mahmoud et al. introduced a leader-following discrete consensus protocol of MAS with fixed and switching topologies, in this paper MAS system subjected to external disturbance is considered and dynamic output feedback based consensus controller was designed [14]. Shao et al. designed a leader-follower consensus with switching topologies inspired by pigeon hierarchies. Further, they discussed and investigated the influence of convergence rate of follower agent with self-loop and without self-loop with switching topologies and found that follower without self-loop has a faster convergence rate. They also derived a novel sufficient condition for consensus with some followers having no self loops [21]. Lee et al. [12] proposed a novel krasovskii-LaSalle theorem for exponential consensus of discrete-time MAS. This theory is applied to leaderless and leader-follower network. Shao et al. [20] discussed asynchronous bipartite consensus for discrete-time second-order multiagent system using switching topologies. However, all the consensus protocol defined in the discretetime domain was not robust and achieve the consensus asymptotically. This lead us to design a consensus protocol using DSMC for the leader following consensus with undirected switching topologies which converge all the information state of the DMAS in a fixed time steps. The main contributions of this paper are:
2.1. Graph Theory With switching topologies, the set G = {G1 , G2 , . . .} denotes the collection of all possible communication topologies on V. In this paper, it is assumed that σ (.) is a piecewise constant switching signal in the sense that there exists a subsequence {pi } of p called switching instants, such that σ (p)is a constant for pi ≤ p < pi+1 for any pi ≥ 0. Considering piecewise constant switching σ (.) and T be the sampling period, let us define a dynamic graph Gσ [ p] = (V, Eσ [ p], Aσ [ p] ) ∈ G which stand for the interaction topology of information exchange betweenN followers at time pT, where V = 1, 2, . . . , N and Eσ [p]⊆V × V. The weighted adjacency matrix of graph Gσ [p] is denoted by Aσ [ p] = ai j [ p] ∈ RN×N with nonnegative entries, where aij [p] > 0 if and only if (j, i) ∈ Eσ [p];otherwise, ai j [ p] = 0. Let Lσ [ p] = [li j ( p)] ∈ RN×N , i, j = 0, 1 . . . , N be the Laplacian matrix of the graph Gσ [p]. let us Define the degree matrix as D = diag{di } ∈ RN×N with d = j∈N ai j and the Laplacian mai trix as Lσ [ p] = D − Aσ [ p]. A graph is said to be directed graph (i, j) ∈ Eσ [p] if and only if the agent i can obtain information from the agent j. However in the undirected graph (i, j) ∈ Eσ [p] if and only if (j, i) ∈ E[p], but it is not necessary in the directed graph. it is assumed that there is no repeated edges and no self loops, i.e., aii = 0, ∀i ∈ N , where N = {1, 2, . . . , N}. Define the leader adjacency matrix associated with graph G[p] as a diagonal matrix B[ p] = diag{a10 [ p], a20 [ p], . . . , aN0 [ p]}, where the weight ai0 [p] > 0 if there is a edge from 0 to i agent with leader at time pT and ai0 [ p] = 0 otherwise. Lemma 1 [16]. Zero is a simple eigenvalue of L if and only if graph G has a rooted spanning tree. In order to achieve the leader-following consensus problem, we also take another digraph G, which consist of graph G vertex and edges of leader denoted as 0 from some vertices to vertex 0. we can say that vertex 0 is globally reachable in G if vertex 0 is reachable from any vertex in G. For a given positive integer m, the union of digraph G1 = {V, E1 [ p], A1 [ p]}, . . . , Gm = {V, Em [ p], Am [ p]} m m m is denoted by Gi [ p] = {V, Ei [ p], Ai [ p]}. Same overall neti=1
L
m
=
Gi [ p]
i=1
• Proposed a sliding surface for global consensus of DMAS using undirected switching graph topology. • Design of a global DSMC consensus protocols using enhanced Gao’s and Power rate reaching laws for the global consensus of DMAS. • Condition for global stability of DMAS using Lyapunov function. • Validation and comparison of both the protocols for consensus of 2-DOF helicopter systems.
i=1
i=1
work graph Laplacian matrix is defined as m
L(Gi [ p] )
(1)
i=1
and overall network graph Pinning gain matrix is written as
B
m i=1
Gi [ p]
=
m
B (Gi [ p] )
(2)
i=1
Lemma 2 [18]. If the digraph G has a directed spanning tree, then the matrix (L + B ) is invertible.
Please cite this article as: K. Patel and A. Mehta, Discrete-time sliding mode protocols for leader-following consensus of discrete multiagent system with switching graph topology, European Journal of Control, https://doi.org/10.1016/j.ejcon.2019.06.011
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2.2. Leader following DMAS
3. DSMC for the consensus of DMAS using Gao’s reaching law
Consider the following identical linear DMAS:
xi (k + 1 ) = Axi (k ) + B(ui (k ) + Di (k )) Rn×n
∀i ∈ N,
(3)
Rn×m
where i = 1, . . . , N, A ∈ and B ∈ are the system matrix and input matrix of ith system respectively. State vector xi (k) ∈ Rn and the input vector ui (k) ∈ Rm , Di ∈ Rm is matched disturbance acting on ith system. Assumption 1. The matrix pair(A, B) for ith system in (1) is controllable. We may represent the global DMAS from (1) as
X (k + 1 ) = (IN A )X (k ) + (IN B )(u(k ) + D (k )),
(4)
where state vector X (k ) = [x1 (k ), x2 (k ) . . . , xN (k )]T ∈ RnN and input vector ui (k ) = [u1 (k ), u2 (k ) . . . , uN (k )]T ∈ RmN , disturbance vector Di (k ) = [D1 (k ), D2 (k ), D3 (k ), . . . DN (k )]T ∈ RmN , denotes the Kronecker product. Rewriting (4) in more generic form as
X (k + 1 ) = A˜ X (k ) + B˜(u(k ) + D (k )),
(5)
where, A˜ = (IN A ), B˜ = (IN B ). Let us define the autonomous leader dynamics as
x0 (k + 1 ) = Ax0 (k ). Where x0 (k) ∈
Rn
(6)
is the state vector of the leader.
lim Xi (k ) − x0 (k ) = 0.
k→k∗
Theorem 1. For the given DMAS (5), the synchronization with leader (6) is achieved in fixed time steps using a DMAS consensus protocol given by
u(k ) = −(csT γ B˜ )−1 [csT γ A˜ X (k ) + csT γ (−1N Ax0 (k )) −(1 − qT )s˜(k ) + T sign(s˜(k )) − D (k )].
(14)
Proof. Let us define the sliding surface using (8) for an individual ith agent as
s˜i (k ) = csTi εi (k ),
(15)
where csi is the sliding gain to be obtained using the poleplacement approach [15]. In the leader-follower network if ith agent is connected to the leader then the sliding surface can be written as
s˜(k ) = csT ξ (k ).
(16)
s˜(k ) = csT γ (X (k ) − 1N x0 (k )).
(17)
Further,
(7)
Problem statement: To design a robust DSMC Consensus protocol for global DMAS such that all the follower agents (3) synchronize with the leader (6) using the neighbourhood agent information using switching graph topology. It is assumed that the leader node can be observed from a small subset of nodes in graph G. If ith agent is connected to the leader then particular this edge is said to exist with weighting gain ai0 > 0. The agent with ai0 > 0 is referred as pinning node. Let us define the error function for the leader-follower network
εi ( k ) =
In this section, fixed time steps DMAS consensus protocol for the global DMAS is designed using Gao’s reaching law of DSMC in the form of Theorem 1. And the condition for global stability of the DMAS with the proposed DSMC law is also derived
Using (11), we may write a sliding surface for the global system (5) as
Definition 1. The global system defined in (4) of DMAS is said to achieve consensus from any initial condition in fixed time step k∗ ∈ [0, ∞) such that
3
s˜(k + 1 ) = csT γ (X (k + 1 ) − 1N x0 (k + 1 )).
(18)
Substituting (5) and (6) into (18), we may get
s˜(k + 1 ) = csT γ (A˜ X (k ) + B˜(u(k ) + D (k )) − 1N Ax0 (k )).
(19)
Now motivated from the reaching law proposed in [1], let us define the consensus reaching law for ith agent as
s˜i (k + 1 ) = (1 − qi T )s˜i (k ) − i T sign(s˜i (k )),
(20)
where = [1 , 2 , 3 , . . .N ] ∈ > 0, q = [q1 , q2 , q3 , . . . , qN ] ∈ RnN > 0, 0 < (1 − qT ) < 1. Then global consensus reaching law for global DMAS can be defined as RnN
s˜(k + 1 ) = (1 − qT )s˜(k ) − T sign(s˜(k )).
(21)
Comparing (19) and (21), we may write
a i j ( k )[ x i ( k ) − x j ( k )] + a i 0 ( k )[ x i ( k ) − x 0 ( k )] .
(8)
Using (8) and according to Lemmas 1 and 2 the consensus error in global form can be rewritten as
ξ (k ) = ((IN + DGσ [ p] + BGσ [ p] )−1 )(LGσ [ p] + BGσ [ p] )) In )x˜,
(9)
where x˜ = X (k ) − 1N x0 (k ) ∈ RnN . The eigenvalues of the weighted matrix = ((IN + DGσ [ p] + BGσ [ p] )−1 )(LGσ [ p] + BGσ [ p] )) obtained using Gersgorin circle criteria [25], which is inside the unit circle. We can express (9) as
ξ (k ) = ( In )x˜.
csT γ (A˜ X (k ) + B˜(u(k ) + D (k )) − 1N Ax0 (k )) = (1 − qT )s˜(k ) − T sign(s˜(k )).
j∈N
(10)
consider ( In ) = γ and substituting x˜ we may get
ξ (k ) = γ (X (k ) − 1N x0 (k )),
(11)
ξ (k + 1 ) = γ (X (k + 1 ) − 1N x0 (k + 1 )),
(12)
ξ (k + 1 ) = γ (A˜ X (k ) + B˜(u(k ) + D (k )) − 1N Ax0 (k )).
(13)
(22)
Further (22) can be expressed in terms of control protocol as
u(k ) = −(csT γ B˜ )−1 [csT γ A˜ X (k ) + csT γ (−1N Ax0 (k )) −(1 − qT )s˜(k ) + T sign(s˜(k )) − D (k )].
(23)
As per protocol (23), the consensus error trajectory between leader and ith follower agents reach to the surface in a zig-zag step within specified band called Quasi-Sliding Mode Band(QSMB). So the width of consensus error QSMB for ith agent (i ) is defined using (20)
i =
i T . 2 − qi T
(24)
Stability Analysis: The global stability of leader following consensus using distributed control protocol (23) is guaranteed if the close loop error dynamics in (11) for global DMAS in (5) drives towards the sliding surface (16) and maintain on it for any gain q, > 0, 0 < 1 − qT < 1, and 1 − qT < , provided the following conditions hold true:
Please cite this article as: K. Patel and A. Mehta, Discrete-time sliding mode protocols for leader-following consensus of discrete multiagent system with switching graph topology, European Journal of Control, https://doi.org/10.1016/j.ejcon.2019.06.011
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(i)
−qT s˜(k ) < T sign(s˜(k )), s˜(k + 1 ) < s˜(k )
(28)
The main lacking point of consensus protocol defined in (23) is chattering and continuous consensus control effort is required to enforce the consensus error trajectory to sliding surface during reaching and sliding phase which is access more power and due to chattering serious problem may occur during implementation. To overcome these drawbacks we design the consensus protocol motivated by Power rate reaching law [34] in which the consensus effort is more when the consensus error trajectory is away from the surface and reduce the effort when consensus error trajectory nearer to the surface. Also, this consensus protocol reduce the control effort for the global DMAS defined in (5) under the influence of matched uncertainty.
(29)
4. DSMC for the consensus of DMAS using Power rate reaching law
(25)
(ii)
(2 − qT )s˜(k ) > T sign(s˜(k )), s˜(k + 1 ) > s˜(k )
(26)
Proof. Consider the sliding surface for global consensus given in (17)
s˜(k ) = csT (( In )(X (k ) − 1N x0 (k ))).
(27)
Let us select the Lyapunov function as
Vs (k ) = s˜2 (k ). Writing the forward derivative function of the (28) as
Vs (k ) = v˜ (k + 1 ) − v˜ (k ). Using (28), we may write
Vs (k ) = s˜2 (k + 1 ) − s˜2 (k ),
(30)
Vs (k ) = (s˜(k + 1 ) − s˜(k ))(s˜(k + 1 ) + s˜(k )).
(31)
For stability, it is required to ensure Vs (k ) < 0. For proving the same let us consider two cases: Case 1. s˜(k + 1 ) − s˜(k ) < 0 Now using s˜(k + 1 ) defined in (18), we get
s˜(k + 1 ) − s˜(k ) = [csT γ (X (k + 1 ) − 1N x0 (k + 1 ))] − s˜(k ).
(32)
s˜(k + 1 ) − s˜(k ) = csT γ [(A˜ X (k ) + B˜(u(k ) + D (k )) (33)
Now substituting the consensus protocol (23) into (33), we get
s˜(k + 1 ) − s˜(k ) = [(1 − qT )s˜(k ) − T sign(s˜(k ))] − s˜(k ) < 0, (34) and
s˜(k + 1 ) − s˜(k ) = −qT s˜(k ) − T sign(s˜(k )) < 0.
(35)
(36)
From (36), we tune the value of q and such that s˜(k + 1 ) − s˜(k ) < 0. Case 2. s˜(k + 1 ) + s˜(k ) > 0 Now again considering s˜(k + 1 ) in (18), we get
s˜(k + 1 ) + s˜(k ) = [csT γ (X (k + 1 ) − 1N x0 (k + 1 ))] + s˜(k ).
(37)
Substituting X (k + 1 ) and x(k + 1 ) defined in (5) and (6), we may get
s˜(k + 1 ) + s˜(k ) = csT γ [(A˜ X (k ) + B˜(u(k ) + D (k )) −1N Ax0 (k ))] + s˜(k ).
(38)
Now substituting the consensus protocol (23) in (38), we get
(42)
Proof. Let us define sliding surface of ith agent as
s˜i (k ) = csTi εi (k ),
(43)
where csi is the sliding gain to be obtain using pole-placement approach [15]. In the leader-follower network for global system (5) sliding surface can be written as
s˜(k ) = csT ξ (k ).
(44)
Using (11), we may write
s˜(k ) = csT γ (X (k ) − 1N x0 (k )).
(45)
Further,
(46)
Substituting (5) and (6) into (46), we may get
s˜(k + 1 ) = csT γ (A˜ X (k ) + B˜(u(k ) + D (k )) − 1N Ax0 (k )).
(47)
Now motivated from the reaching law proposed in [34], let us define the consensus reaching law for ith agent as
s˜i (k + 1 ) = (1 − qi T )s˜i (k ) − i T
| s˜i (k ) |η sign(s˜i (k )),
(48)
where = [1 , 2 , 3 , . . . N ] ∈ > 0, q = [q1 , q2 , q3 , . . . , qN ] ∈ RnN > 0, 0 < (1 − qT ) < 1. Then global consensus reaching law for global DMAS can be defined as RnN
s˜(k + 1 ) = (1 − qT )s˜(k ) − T
| s˜(k ) |η sign(s˜(k )).
(49)
csT γ (A˜ X (k ) + B˜(u(k ) + D (k )) − 1N Ax0 (k )) = (1 − qT )s˜(k ) − T
and
(40)
Further, (40) can be written as
(2 − qT )s˜(k ) > T sign(s˜(k )).
| s˜(k ) |η sign(s˜(k ))] − D (k ).
Comparing (47) and (49), we may get
s˜(k + 1 ) + s˜(k ) = [(1 − qT )s˜(k ) − T sign(s˜(k ))] + s˜(k ) > 0, (39) s˜(k + 1 ) + s˜(k ) = [(2 − qT )s˜(k ) − T sign(s˜(k ))] > 0.
u(k ) = −(csT γ B˜ )−1 [csT γ A˜ X (k ) + csT γ (−1N Ax0 (k )) − (1 − qT )s˜(k )
s˜(k + 1 ) = csT γ (X (k + 1 ) − 1N x0 (k + 1 )).
Further, (35) can be written as
−qT s˜(k ) < T sign(s˜(k )).
Theorem 2. For the given DMAS (5), the synchronization with a leader (6) is achieved using a DMAS consensus protocol given by
+ T
Substituting X (k + 1 ) and x(k + 1 ) defined in (5) and (6) respectively, we get
−1N Ax0 (k ))] − s˜(k ).
In this section, fixed time steps DMAS consensus protocol for the global DMAS is designed using modified Power rate reaching law of DSMC in the form of Theorem 2. The condition for global stability of the DMAS with the proposed DSMC law is also derived.
| s˜(k ) |η sign(s˜(k )).
(50)
Further, (50) can be expressed in terms of consensus protocol as
u(k ) = −(csT γ B˜ )−1 [csT γ A˜ X (k ) + csT γ (−1N Ax0 (k )) − (1 −qT )s˜(k ) (41)
From (41) select the parameter value of q and such that s˜(k + 1 ) + s˜(k ) > 0. From the above two cases, it is informed that by choosing appropriate values of tuning parameters q and , the stability of global DMAS is guaranteed.
+ T
| s˜(k ) |η sign(s˜(k ))] − D (k ).
(51)
The consensus of leader and follower is obtained using DMAS consensus protocol based on Power rate reaching law.sliding mode
Please cite this article as: K. Patel and A. Mehta, Discrete-time sliding mode protocols for leader-following consensus of discrete multiagent system with switching graph topology, European Journal of Control, https://doi.org/10.1016/j.ejcon.2019.06.011
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5
Fig. 1. 2-DOF helicopter system. Courtesy : Quanser Inc. Fig. 2. Free body diagram of 2-DOF helicopter system.
band in which system remains steadily is defined for ith agent from (48) as
i =
T 1−1ηi i
(52)
1 − qi T
Proper selection of η reduce the chattering effect and enhance the robustness. Stability Analysis: The global stability of leader following consensus using consensus protocol (51) is guaranteed if the close loop error dynamics in (11) for global DMAS in (5) drives towards the sliding surface (16) and maintain on it for any gain q, > 0, 0 < η < 1, 0 < 1 − qT < 1, and 1 − qT < , provided the following conditions hold true:
−s˜(k ) < 0,
(62)
and
s˜(k + 1 ) − s˜(k ) = −qT s˜(k ) − T
| s˜(k ) |η sign(s˜(k )) < 0.
Further, (63) can be written as
−qT s˜(k ) < T
| s˜(k ) |η sign(s˜(k )).
(64)
From (64), choose the proper value of q and such that s˜(k + 1 ) − s˜(k ) < 0 Case 2:. s˜(k + 1 ) + s˜(k ) > 0 Let us consider s˜(k + 1 ) define in (18) as
s˜(k + 1 ) + s˜(k ) = [csT γ (X (k + 1 ) − 1N x0 (k + 1 ))] + s˜(k ).
(i)
−qT s˜(k ) < T
|
s˜(k ) |η sign(s˜(k )),
s˜(k + 1 ) < s˜(k )
(53)
(63)
(65)
Using X (k + 1 ) and x(k + 1 ) defined in (5) and (6)
s˜(k + 1 ) + s˜(k ) = csT γ [(A˜ X (k ) + B˜(u(k ) + D (k )) − 1N Ax0 (k ))]
(ii)
(2 − qT )s˜(k ) > T | s˜(k ) |η sign(s˜(k )),
Proof. Consider sliding surface given in the (17)
s˜(k ) =
csT
+s˜(k ).
s˜(k + 1 ) > s˜(k ) (54)
(( In )(X (k ) − 1N x0 (k ))).
(55)
Vs (k ) = s˜2 (k ).
Now, substituting consensus protocol define in (51) into (66) we may get
s˜(k + 1 ) + s˜(k ) = [(1 −qT )s˜(k ) − T
Let us select the Lyapunov function as
(66)
| s˜(k ) |η sign(s˜(k ))] + s˜(k ) > 0, (67)
(56)
Writing the forward derivative function and substituting for (56)
Vs (k ) = v˜ (k + 1 ) − v˜ (k ),
(57)
and
s˜(k + 1 ) + s˜(k ) = [(2 − qT )s˜(k ) − T
| s˜(k ) |η sign(s˜(k ))] > 0. (68)
Vs (k ) = s˜ (k + 1 ) − s˜ (k ),
(58)
Vs (k ) = (s˜(k + 1 ) − s˜(k ))(s˜(k + 1 ) + s˜(k )).
(59)
2
2
For stability, it is required to ensure Vs (k ) < 0. For proving the same let us consider two cases: Case 1:. s˜(k + 1 ) − s˜(k ) < 0 Now substituting s˜(k + 1 ) using (18), we get
s˜(k + 1 ) − s˜(k ) = [csT γ (X (k + 1 ) − 1N x0 (k + 1 ))] − s˜(k ).
(60)
considering X (k + 1 ) and x(k + 1 ) using (5) and (6)
s˜(k + 1 ) − s˜(k ) = csT γ [(A˜ X (k ) + B˜(u(k ) + D (k )) −1N Ax0 (k ))] − s˜(k )
(61)
Now substituting consensus protocol define in (51) into (61) we may get
s˜(k + 1 ) − s˜(k ) = [(1 − qT )s˜(k ) − T
| s˜(k ) |η sign(s˜(k ))]
Further (68) can be written as
(2 − qT )s˜(k ) > T | s˜(k ) |η sign(s˜(k )).
(69)
From (69), select the parameter value of q and such that s(k + 1 ) + s ( k ) > 0. From the above two cases, we may conclude that by choosing appropriate values of tuning parameters q and the stability of global DMAS is guaranteed. 5. 2 DOF helicopter systems In order to validate the efficacy of the proposed consensus protocols in Sections 3 and 4, let us consider six number of 2-DOF helicopter systems as agents in a leader-following network shown in (Fig. 3). Out of which one helicopter act as a leader and other five act as follower agent. The 2-DOF helicopter is an important model from the control system engineering point of view due to its wide non-linear
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nose. The center of mass acts at l distance from the pitch axis along the helicopter body length. Where l is the distance between the center of mass and the intersection of the pitch and yaw axes. The notations used are given as under: θ pitch angle, deg ψ yaw angle, deg K pp thrust force constant of yaw motor/propeller, N.m/V Kyy thrust torque constant of yaw axis from yaw motor/propeller, N.m/V K py thrust torque constant acting on pitch axis from yaw motor/propeller, N.m/V Kyp thrust torque constant acting on yaw axis from pitch motor/propeller, N.m/V Bp viscous damping about pitch axis, N/V By viscous damping about yaw axis, N/V mh total moving mass of the helicopter (body, two propeller assemblies, etc.), kg l Center of mass length along helicopter body from pitch axis, meter(m) Jeq , p rotal moment of inertia about pitch axis, kg.m2 Jeq , y total moment of inertia about yaw axis, kg.m2 Vp , Vy respective voltages applied to the front and rear motors g gravitational constant, 9.8 m/s2 The 2-DOF helicopter modeling conventions are: Fig. 3. Switching graph topology.
characteristics, highly cross coupling effects and instability in open loop. The 2-DOF helicopter model (fixed base) with two propellers driven by DC motors is shown in Fig. 1 The elevation of the nose over the pitch axis is controlled by the front propeller and the rotational motion around the yaw axis is controlled by the back propeller. The voltages across the pitch and yaw motors are ± 24V and ± 15V, respectively [10]. The 2-DOF helicopter model dynamics are shown in Fig. 2 The thrust forces Fp and Fy are applied across the pitch and yaw axis, respectively. The torques act at a distance rp and ry from the respective axis. The gravitational force Fg pulls down the helicopter
• The helicopter is horizontal when the pitch angle equals θ = 0. • The pitch angle increases positively, θ˙ > 0, when the nose is moved upwards and the body rotates in the counter-clockwise (CCW) direction. • The yaw angle increases positively, ψ˙ > 0 when the body rotates in the clockwise (CW) direction. • Pitch increases, θ > 0, when the pitch thrust force is positive Fp > 0. • Yaw increases, ψ > 0, when the yaw thrust force is positive, Fy > 0. Consider the dynamics of 2-DOF helicopter acting as ith agent is written as:
Fig. 4. Pitch position and velocity consensus.
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7
Fig. 5. Yaw position and velocity consensus.
Fig. 6. Control effort (u) using Gao’s reaching law.
For the pitch axis angle θ , the dynamic equation is written as
1 θ¨ = [{K ppVp + K pyVy } − {B p θ˙ + mh glcos(θ ) Jeq,p + mh l 2 +mh l 2 sin(θ )cos(θ )ψ˙ 2 }].
(70)
Consider Jt p = Jeq,p + mh l 2 , we may write (70) as
θ¨ =
(72)
Consider Jty = Jeq,y + mh l 2 cos2 (θ ), we can obtain from the above equation
ψ¨ =
1 [{K ppVp + K pyVy } − {B p θ˙ + mh glcos(θ ) Jt p +mh l 2 sin(θ )cos(θ )ψ˙ 2 }].
+2mh l 2 sin(θ )cos(θ )ψ˙ ψ˙ 2 }].
1 [{KypVp + KyyVy } − {By ψ˙ + 2mh l 2 sin(θ )cos(θ )ψ˙ ψ˙ 2 }]. Jty (73)
(71)
Similarly, for the yaw axis it is defined as
The state space model for ith agent system is defined as
1 ψ¨ = [{KypVp + KyyVy } − {By ψ˙ Jeq,y + mh l 2 cos2 (θ )
xi (k + 1 ) = Axi (k ) + B(ui (k ) + Di (k ))
∀i ∈ N.
(74)
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Fig. 7. Control effort (u) using Power rate reaching law.
Fig. 8. Sliding surface of individual agent using Gao’s reaching law.
⎡
0 ⎢0
0 ⎢ 0
⎢ kpp ⎢J ⎣ Tp ky p JTy
1 0
A = ⎢0
0
−J
0
0
0
⎢ ⎣
Where,
⎡
0 0
⎤
Bp Tp
⎤
0 1 ⎥
⎥ ⎦
0 ⎥ B
−J y
Di ∈ Rm are state vector, input vector and matched disturbance respectively acting on ith follower agent. ∈ Rn×n
and
B=
Ty
0 0 ⎥ k py ⎥ ⎥∈ Rn×m are the system matrix input matrix reJTp kyy JTy
⎦
spectively. xi (k ) = [θi (k )ψi (k )θ˙ i (k )ψ˙ i (k )] ∈ Rn , ui (k) ∈ Rm and
6. Simulation and result discussion In this section, Pitch position(θ ) and its velocity (θ˙ ), Yaw position(ψ ) and its velocity (ψ˙ ) states of 2 DOF helicopter system are to be considered as a single agent parameters for consensus. As discussed, it is assumed here that the leader node has same dynamics as the follower node. The simulation is carried out using Matlab R15. The following notations are used for simulation results.
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9
Fig. 9. Sliding surface of individual agent using power rate reaching law.
⎡0
Discretizing the continuous-time system at T = 0.03 sampling period, we obtain the discrete-time system model as
∀i ∈ N.
xi (k + 1 ) = Axi (k ) + B(ui (k ) + Di (k ))
⎡
(75)
⎤
1 0 0.0262 0 0 0.0285⎥ ⎢0 1 Where, A = ⎣ ⎦, 0 0 0.7571 0 0 0 0 0.9004 ⎡ ⎤ 0.0010 0 ⎢0.0 0 01 0.0 0 03⎥ B=⎣ ⎦ 0.0620 0.0021 0.0069 0.0225 As discussed in Section 3, consider MAS with five follower node and one leader node as undirected switching graph topology. The leader node notation is given as 0 while follower node notation is given as 1,2,3,4,5 respectively. Their communication topology is shown in the Fig. 3. Initial states of each follower agent are generated randomly in the span[−1, 1] while the initial state of the leader node is [0.1, 0.5, 0.7, 0.8]T and Laplacian matrix of individual graph topology Gσ [p] are defined as
⎡0 ⎢−1 L1 = ⎢ 0 ⎣
0 1 0 0 0
0 0
⎡0 L3 =
⎢0 ⎢0 ⎣ 0 0
0 1 0 0 0
0 0 0 0 0
0 0 0 0 0
0 −1 1 0 0
0 0 −1 0 0
⎤
⎡
0 0 0⎥ ⎢0 0⎥, L2 = ⎢0 ⎦ ⎣ 0 0 0 0
⎤
0 0 0 0 0
0 0 1 0 0
0 0 −1 1 0
⎤
0 0⎥ 0⎥ ⎦ −1 0
0 0⎥ 0⎥. ⎦ 0 0
(76)
Further, pinning gains matrix are also defined as
⎡1 B1 =
⎢0 ⎢0 ⎣ 0 0
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
⎤
⎡
0 0 0⎥ ⎢0 0⎥, B2 = ⎢0 ⎦ ⎣ 0 0 0 0
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
⎤
0 0⎥ 0⎥ , ⎦ 0 1
B3 =
⎢0 ⎢0 ⎣ 0 0
0 0 0 0 0
0 0 0 0 0
0 0 0 1 0
⎤
0 0⎥ 0⎥. ⎦ 0 0
(77)
The sliding gain for surface define in (16) is calculated using the pole-placement method which comes out to be
csT
93.64 = 126.18
52.39 482.90
13.17 −1.5626
−0.0043 . 45.98
(78)
In order to check the robustness of the derived consensus protocol for the global system, a slow varying disturbance is applied to each DMAS with a magnitude Di (k ) = 0.002 ∗ cos(0.86k ) for individual graph topology. The gain for each follower agent i , qi are chosen as 2 and 1 respectively in the case of Gao’s reaching law and the gain value of η in the case of Power rate reaching law is taken as 0.6. Proper selection of the value η decrease the sliding mode band in Power rate reaching law. Fig. 4 shows the comparison results of the pitch position and velocity states consensus with leader position and velocity states using Gao’s and Power rate reaching law using switching graph topology for the graph G1 , G2 , G3 . Graph G1 has changed to G2 after 50 no. of steps. Simultaneously graph G2 has changed to G3 and remain stable for 49 no. of steps after that G3 graph topology steady for another 49 no. of steps. In this way, we switch the topology and find that consensus begins after 148 no. of steps. However, in this case, Power rate reaching law gives faster convergence speed compared to Gao’s reaching law. Fig. 5 shows the comparative study of yaw position and velocity states consensus of the follower with leader states using Gao’s and Power rate reaching law and found that all the followers synchronize with leader in fixed steps size using switching graph topology. Figs. 6 and 7 show the consensus effort(u) of the follower agents, which further applied to individual agent into the overall network of DMAS. Figs. 8 and 9 show the sliding surface (consider as consensus error trajectories) of each agent of DMAS. It crosses first time sliding surface band called as QSMB from any initial condition in fixed
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Fig. 10. Pitch and yaw position consensus tracking error.
Fig. 11. Pitch and yaw velocity consensus tracking error.
time steps and steady within this band and value of this band is vary from -0.03 to +0.03 (O(T)) calculated using (21) in the case of Gao’s reaching law and similarly in the case of Power rate reaching law sliding surface remains steadily in O(T2 ) band once it is on the sliding surface. Figs. 10 and 11 express the Pitch and yaw Position and velocity tracking consensus error using two different reaching law and observed that reached within fixed time steps. It may be noted that Power rate reaching law quickly reaches to the zero as compared to Gao’s reaching law.
7. Conclusion In this paper, two discrete-time sliding mode control protocols using Gao’s reaching law and Power rate reaching law are proposed for leader following consensus of discrete multi-agent system configured with a fixed, undirected switching graph topology. Both the consensus protocols synchronize leader with the follower agents under the influence of switching topology at different intervals. The condition for global stability is also derived using the Lyapunov function. Finally, both the consensus protocols
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are validated in simulation for the 2-DOF helicopter system. In future, the DSMC algorithms for consensus of DMAS with bipartite and event-triggered control shall be explored. Supplementary material Supplementary material associated with this article can be found, in the online version, at doi:10.1016/j.ejcon.2019.06.011. References [1] W. Gao, Y. Wang, A. Homaifa, Discrete-time variable structure control systems, IEEE Trans. Ind. Electron. 42 (2) (1995) 117–122, doi:10.1109/41.370376. [2] Q. Cao, Y. Song, J.M. Guerrero, S. Tian, Coordinated control for flywheel energy storage matrix systems for wind farm based on charging/discharging ratio consensus algorithms, IEEE Trans. Smart Grid 7 (3) (2016) 1259–1267, doi:10.1109/TSG.2015.2470543. [3] C. Ding, X. Dong, C. Shi, Y. Chen, Z. Liu, Leaderless output consensus of multiagent systems with distinct relative degrees under switching directed topologies, IET Control Theory Appl. 13 (3) (2019) 313–320, doi:10.1049/iet-cta.2018. 5140. [4] Y. Dong, J. Huang, A leader-following rendezvous problem of double integrator multi-agent systems, Automatica 49 (5) (2013) 1386–1391. https://doi.org/10. 1016/j.automatica.2013.02.024. [5] M.-C. Fan, H.-T. Zhang, M. Wang, Bipartite flocking for multi-agent systems, Commun. Nonlinear Sci. Numer. Simulat. 19 (9) (2014) 3313–3322. https://doi. org/10.1016/j.cnsns.2013.10.009. [6] K. Fang, F. Shu, Y. Deng, Lag consensus of second-order leader-following multiagent systems with communication delays and switching topologies, in: Proceedings of the Chinese Automation Congress (CAC), 2017, pp. 4061–4066, doi:10.1109/CAC.2017.8243491. [7] L. Guo, X. Nian, H. Pan, Leader-following consensus of multi-agent systems with switching topologies and time-delays, J. Control Theory Appl. 11 (2) (2013) 306–310, doi:10.1007/s11768- 013- 1191- 2. [8] J. Han, C. Wang, G. Yi, Cooperative control of uav based on multi-agent system, in: Proceedings of the IEEE 8th Conference on Industrial Electronics and Applications (ICIEA), 2013, pp. 96–101, doi:10.1109/ICIEA.2013.6566347. [9] M. Hu, L. Guo, A. Hu, Y. Yang, Leader-following consensus of linear multi-agent systems with randomly occurring nonlinearities and uncertainties and stochastic disturbances, Neurocomputing 149 (2015) 884–890, doi:10.1016/j.neucom. 2014.07.047. [10] Q. Inc, Quanser 2-DOF helicopter control user manual, Technical Report, 2015. [11] Y. Jiang, J. Liu, S. Wang, Robust integral sliding-mode consensus tracking for multi-agent systems with time-varying delay, Asian J. Control 18 (1) (2014) 224–235, doi:10.1002/asjc.1007. [12] T.-C. Lee, W. Xia, Y. Su, J. Huang, Exponential consensus of discrete-time systems based on a novel Krasovskii-LaSalle theorem under directed switching networks, Automatica 97 (2018) 189–199. https://doi.org/10.1016/j.automatica. 2018.07.022. [13] Y. Li, C. Tang, K. Li, X. He, S. Peeta, Y. Wang, Consensus-based cooperative control for multi-platoon under the connected vehicles environment, IEEE Trans. Intell. Transp. Syst. (2018) 1–10, doi:10.1109/TITS.2018.2865575. [14] M.S. Mahmoud, G.D. Khan, Leader-following discrete consensus control of multi-agent systems with fixed and switching topologies, J. Frankl. Inst. 352 (6) (2015) 2504–2525, doi:10.1016/j.jfranklin.2015.03.026. [15] T. Nguyen, Z. Miao, Y. Pan, N.H. Amini, V. Ugrinovskii, M.R. James, Pole placement approach to coherent passive reservoir engineering for storing quantum information, in: Proceedings of the American Control Conference (ACC), 2017, pp. 234–239, doi:10.23919/ACC.2017.7962959. [16] W. Ni, D. Cheng, Leader-following consensus of multi-agent systems under fixed and switching topologies, Syst. Control Lett. 59 (3–4) (2010) 209–217, doi:10.1016/j.sysconle.2010.01.006. [17] M.M. Rana, L. Li, S.W. Su, W. Xiang, Consensus-based smart grid state estimation algorithm, IEEE Trans. Ind. Inf. 14 (8) (2018) 3368–3375, doi:10.1109/TII. 2017.2782750.
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Contents lists available at ScienceDirect
European Journal of Control journal homepage: www.elsevier.com/locate/ejcon
Discrete-time sliding mode protocols for leader-following consensus of discrete multi-agent system with switching graph topology Keyurkumar Patel, Axaykumar Mehta∗ Department of Electrical Engineering, Institute of Infrastructure Technology Research and Management, Ahmedabad, Gujarat, 380026 India
a r t i c l e
i n f o
Article history: Received 1 February 2019 Revised 1 May 2019 Accepted 27 June 2019 Available online xxx Recommended by Dr X. Chen Keywords: Discrete- time sliding mode control (DSMC) Discrete multi-agent system (DMAS) 2-DOF helicopter system
a b s t r a c t In this paper, discrete-time sliding mode protocols for consensus of a leader following discrete multiagent system having switching topologies are proposed. The discrete multi-agent system is described with a fixed, undirected switching graph topology as a global system having one leader and other as follower agents. Sliding surface for global consensus of agents for switching graph topology is defined and discrete-time sliding mode protocols using enhanced Gao’s and Power rate reaching laws are derived. Under the influence of switching topology, the graph topology switches in a different no. of step intervals and ensure that consensus protocols asymptotically synchronize follower agents with leader agent. The condition for global stability in both cases is also derived using the Lyapunov function. Further, the algorithms are implemented on the multi-agent system comprise of multiple 2-DOF (Degree of Freedom) helicopter systems where the pitch angle and its velocity & yaw angle and its velocity are synchronized with the leader. It is inferred from simulation results that the consensus using Power rate reaching law give better performance compared with consensus using Gao’s reaching law. Finally, the robustness property of both the algorithms is checked by applying the matched disturbances in follower agents. © 2019 European Control Association. Published by Elsevier Ltd. All rights reserved.
1. Introduction Over the past few decades, cooperative control of multiple systems well known as Multi-Agent System (MAS) has drawn more attention among academician and researchers due to its wide range of applications in various domains such as cooperative control of multiple vehicles, smart grids, surveillance and reconnaissance, Autonomous under water vehicle (AUV) etc... [2,8,13,17,26,30,32,35,37]. The main idea in cooperative control of MAS is to design distributed controllers on each agent by getting local neighbouring information for reaching a certain defined global goal. In other words, under the influence of cooperation, the agents in MAS only share information with their neighbours locally and try to reach an agreement to a certain level of degree. To achieve this task, different cooperative control algorithm such as, consensus, formation, flocking, rendezvous, etc... are proposed in the literature [4,5,24,35]. Among all co-operative control algorithms, the consensus algorithm has been studied widely with different types of control techniques in recent years. The consensus of MAS can be mainly divided into two classes: First one is leaderless consensus algorithm in which individual follower agent share
∗
Corresponding author. E-mail address: [email protected] (A. Mehta).
its information with neighbouring agent [3,19]. The second one is the leader-following consensus in which the states of all the follower agents track the leader states [9,31]. However, the consensus problem is more interesting and challenging with communication topology using graph theory. The communication topology described by a digraph can be static or time-varying. The most common time-varying graph is called a switching graph. Switching topologies consideration in actual application due to unreliable transmission or limited communication and sensing range. Many researchers and academician have conducted research on consensus problems for both first and second-order multi-agent system with switching topologies. For first-order multi-agent system, Guo et al. [7] proposed the leader-following consensus problem of multi-agent systems with switching topologies and time delays in which the leader is static and the controlling effect of each follower depends on its own state. For second-order multi-agent system, Fang et al. [6] proposed about second order lag consensus means the state of follower agent lag behind the leader state algorithms for leader-following multi-agent system with communication constraint and switching topologies. The proposed consensus protocol ensure to tackle the communication delay problem using switching topologies. Wang et al. [27] proposed a consensus disturbance rejection for linear MAS with directed switching topologies. Wang et al. [28] discussed Leader-follower fault-tolerant consensus for linear MAS with switching directed network.
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Nomenclature FA − 1 FA − 2 FA − 3 FA − 4 FA − 5 S U
Follower agent-1 Follower agent-2 Follower agent-3 Follower agent-4 Follower agent-5 Sliding surface Control effort (u)
The paper is organized as follows: The brief description of graph theory and DMAS along with problem statement are presented in section-2. The main contribution that is the design of DSMC global consensus protocol using Gao’s reaching law and Power rate reaching laws and global stability analysis are presented in the Sections 3 and 4, respectively. Section 5 presents system description, and modeling of 2-DOF helicopter system used as DMAS configuration and simulation results are discussed in Section 6 followed by conclusion in Section 7. 2. Preliminaries of graph theory and Discrete-time leader-following MAS
Researchers and academician around the globe have proposed different consensus control protocol for MAS. Wen et.al. discussed the H∞ consensus problem for higher order MAS with switching topologies, in which they proposed a distributed protocol and employed to achieve state consensus in the presence of disturbance with switching directed communication topology [29]. Song et al. developed adaptive leader-following state consensus of MAS with switching topologies, the main goal is to design consensus protocol in the presence of system parameter uncertainties and introduced a new uniform nominal controller structure to achieve the consensus [22]. Zhang et al. proposed a leader-following state consensus protocol for the homogeneous system over time-varying feedback graphs using a state-feedback controller [33]. Jiang et.al [11] proposed a robust consensus protocol using integral sliding mode control for multi-agent system for fixed and switching topologies and derived sufficient condition for the existence of sliding surface for global stability. However, due to more reliable communication and saving the bandwidth of the channel, many researchers and scientists have opted consensus in the discrete domain. The consensus problem for discrete-time multi-agent system was mainly studied in [23,36] using switching topologies network. Mahmoud et al. introduced a leader-following discrete consensus protocol of MAS with fixed and switching topologies, in this paper MAS system subjected to external disturbance is considered and dynamic output feedback based consensus controller was designed [14]. Shao et al. designed a leader-follower consensus with switching topologies inspired by pigeon hierarchies. Further, they discussed and investigated the influence of convergence rate of follower agent with self-loop and without self-loop with switching topologies and found that follower without self-loop has a faster convergence rate. They also derived a novel sufficient condition for consensus with some followers having no self loops [21]. Lee et al. [12] proposed a novel krasovskii-LaSalle theorem for exponential consensus of discrete-time MAS. This theory is applied to leaderless and leader-follower network. Shao et al. [20] discussed asynchronous bipartite consensus for discrete-time second-order multiagent system using switching topologies. However, all the consensus protocol defined in the discretetime domain was not robust and achieve the consensus asymptotically. This lead us to design a consensus protocol using DSMC for the leader following consensus with undirected switching topologies which converge all the information state of the DMAS in a fixed time steps. The main contributions of this paper are:
2.1. Graph Theory With switching topologies, the set G = {G1 , G2 , . . .} denotes the collection of all possible communication topologies on V. In this paper, it is assumed that σ (.) is a piecewise constant switching signal in the sense that there exists a subsequence {pi } of p called switching instants, such that σ (p)is a constant for pi ≤ p < pi+1 for any pi ≥ 0. Considering piecewise constant switching σ (.) and T be the sampling period, let us define a dynamic graph Gσ [ p] = (V, Eσ [ p], Aσ [ p] ) ∈ G which stand for the interaction topology of information exchange betweenN followers at time pT, where V = 1, 2, . . . , N and Eσ [p]⊆V × V. The weighted adjacency matrix of graph Gσ [p] is denoted by Aσ [ p] = ai j [ p] ∈ RN×N with nonnegative entries, where aij [p] > 0 if and only if (j, i) ∈ Eσ [p];otherwise, ai j [ p] = 0. Let Lσ [ p] = [li j ( p)] ∈ RN×N , i, j = 0, 1 . . . , N be the Laplacian matrix of the graph Gσ [p]. let us Define the degree matrix as D = diag{di } ∈ RN×N with d = j∈N ai j and the Laplacian mai trix as Lσ [ p] = D − Aσ [ p]. A graph is said to be directed graph (i, j) ∈ Eσ [p] if and only if the agent i can obtain information from the agent j. However in the undirected graph (i, j) ∈ Eσ [p] if and only if (j, i) ∈ E[p], but it is not necessary in the directed graph. it is assumed that there is no repeated edges and no self loops, i.e., aii = 0, ∀i ∈ N , where N = {1, 2, . . . , N}. Define the leader adjacency matrix associated with graph G[p] as a diagonal matrix B[ p] = diag{a10 [ p], a20 [ p], . . . , aN0 [ p]}, where the weight ai0 [p] > 0 if there is a edge from 0 to i agent with leader at time pT and ai0 [ p] = 0 otherwise. Lemma 1 [16]. Zero is a simple eigenvalue of L if and only if graph G has a rooted spanning tree. In order to achieve the leader-following consensus problem, we also take another digraph G, which consist of graph G vertex and edges of leader denoted as 0 from some vertices to vertex 0. we can say that vertex 0 is globally reachable in G if vertex 0 is reachable from any vertex in G. For a given positive integer m, the union of digraph G1 = {V, E1 [ p], A1 [ p]}, . . . , Gm = {V, Em [ p], Am [ p]} m m m is denoted by Gi [ p] = {V, Ei [ p], Ai [ p]}. Same overall neti=1
L
m
=
Gi [ p]
i=1
• Proposed a sliding surface for global consensus of DMAS using undirected switching graph topology. • Design of a global DSMC consensus protocols using enhanced Gao’s and Power rate reaching laws for the global consensus of DMAS. • Condition for global stability of DMAS using Lyapunov function. • Validation and comparison of both the protocols for consensus of 2-DOF helicopter systems.
i=1
i=1
work graph Laplacian matrix is defined as m
L(Gi [ p] )
(1)
i=1
and overall network graph Pinning gain matrix is written as
B
m i=1
Gi [ p]
=
m
B (Gi [ p] )
(2)
i=1
Lemma 2 [18]. If the digraph G has a directed spanning tree, then the matrix (L + B ) is invertible.
Please cite this article as: K. Patel and A. Mehta, Discrete-time sliding mode protocols for leader-following consensus of discrete multiagent system with switching graph topology, European Journal of Control, https://doi.org/10.1016/j.ejcon.2019.06.011
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2.2. Leader following DMAS
3. DSMC for the consensus of DMAS using Gao’s reaching law
Consider the following identical linear DMAS:
xi (k + 1 ) = Axi (k ) + B(ui (k ) + Di (k )) Rn×n
∀i ∈ N,
(3)
Rn×m
where i = 1, . . . , N, A ∈ and B ∈ are the system matrix and input matrix of ith system respectively. State vector xi (k) ∈ Rn and the input vector ui (k) ∈ Rm , Di ∈ Rm is matched disturbance acting on ith system. Assumption 1. The matrix pair(A, B) for ith system in (1) is controllable. We may represent the global DMAS from (1) as
X (k + 1 ) = (IN A )X (k ) + (IN B )(u(k ) + D (k )),
(4)
where state vector X (k ) = [x1 (k ), x2 (k ) . . . , xN (k )]T ∈ RnN and input vector ui (k ) = [u1 (k ), u2 (k ) . . . , uN (k )]T ∈ RmN , disturbance vector Di (k ) = [D1 (k ), D2 (k ), D3 (k ), . . . DN (k )]T ∈ RmN , denotes the Kronecker product. Rewriting (4) in more generic form as
X (k + 1 ) = A˜ X (k ) + B˜(u(k ) + D (k )),
(5)
where, A˜ = (IN A ), B˜ = (IN B ). Let us define the autonomous leader dynamics as
x0 (k + 1 ) = Ax0 (k ). Where x0 (k) ∈
Rn
(6)
is the state vector of the leader.
lim Xi (k ) − x0 (k ) = 0.
k→k∗
Theorem 1. For the given DMAS (5), the synchronization with leader (6) is achieved in fixed time steps using a DMAS consensus protocol given by
u(k ) = −(csT γ B˜ )−1 [csT γ A˜ X (k ) + csT γ (−1N Ax0 (k )) −(1 − qT )s˜(k ) + T sign(s˜(k )) − D (k )].
(14)
Proof. Let us define the sliding surface using (8) for an individual ith agent as
s˜i (k ) = csTi εi (k ),
(15)
where csi is the sliding gain to be obtained using the poleplacement approach [15]. In the leader-follower network if ith agent is connected to the leader then the sliding surface can be written as
s˜(k ) = csT ξ (k ).
(16)
s˜(k ) = csT γ (X (k ) − 1N x0 (k )).
(17)
Further,
(7)
Problem statement: To design a robust DSMC Consensus protocol for global DMAS such that all the follower agents (3) synchronize with the leader (6) using the neighbourhood agent information using switching graph topology. It is assumed that the leader node can be observed from a small subset of nodes in graph G. If ith agent is connected to the leader then particular this edge is said to exist with weighting gain ai0 > 0. The agent with ai0 > 0 is referred as pinning node. Let us define the error function for the leader-follower network
εi ( k ) =
In this section, fixed time steps DMAS consensus protocol for the global DMAS is designed using Gao’s reaching law of DSMC in the form of Theorem 1. And the condition for global stability of the DMAS with the proposed DSMC law is also derived
Using (11), we may write a sliding surface for the global system (5) as
Definition 1. The global system defined in (4) of DMAS is said to achieve consensus from any initial condition in fixed time step k∗ ∈ [0, ∞) such that
3
s˜(k + 1 ) = csT γ (X (k + 1 ) − 1N x0 (k + 1 )).
(18)
Substituting (5) and (6) into (18), we may get
s˜(k + 1 ) = csT γ (A˜ X (k ) + B˜(u(k ) + D (k )) − 1N Ax0 (k )).
(19)
Now motivated from the reaching law proposed in [1], let us define the consensus reaching law for ith agent as
s˜i (k + 1 ) = (1 − qi T )s˜i (k ) − i T sign(s˜i (k )),
(20)
where = [1 , 2 , 3 , . . .N ] ∈ > 0, q = [q1 , q2 , q3 , . . . , qN ] ∈ RnN > 0, 0 < (1 − qT ) < 1. Then global consensus reaching law for global DMAS can be defined as RnN
s˜(k + 1 ) = (1 − qT )s˜(k ) − T sign(s˜(k )).
(21)
Comparing (19) and (21), we may write
a i j ( k )[ x i ( k ) − x j ( k )] + a i 0 ( k )[ x i ( k ) − x 0 ( k )] .
(8)
Using (8) and according to Lemmas 1 and 2 the consensus error in global form can be rewritten as
ξ (k ) = ((IN + DGσ [ p] + BGσ [ p] )−1 )(LGσ [ p] + BGσ [ p] )) In )x˜,
(9)
where x˜ = X (k ) − 1N x0 (k ) ∈ RnN . The eigenvalues of the weighted matrix = ((IN + DGσ [ p] + BGσ [ p] )−1 )(LGσ [ p] + BGσ [ p] )) obtained using Gersgorin circle criteria [25], which is inside the unit circle. We can express (9) as
ξ (k ) = ( In )x˜.
csT γ (A˜ X (k ) + B˜(u(k ) + D (k )) − 1N Ax0 (k )) = (1 − qT )s˜(k ) − T sign(s˜(k )).
j∈N
(10)
consider ( In ) = γ and substituting x˜ we may get
ξ (k ) = γ (X (k ) − 1N x0 (k )),
(11)
ξ (k + 1 ) = γ (X (k + 1 ) − 1N x0 (k + 1 )),
(12)
ξ (k + 1 ) = γ (A˜ X (k ) + B˜(u(k ) + D (k )) − 1N Ax0 (k )).
(13)
(22)
Further (22) can be expressed in terms of control protocol as
u(k ) = −(csT γ B˜ )−1 [csT γ A˜ X (k ) + csT γ (−1N Ax0 (k )) −(1 − qT )s˜(k ) + T sign(s˜(k )) − D (k )].
(23)
As per protocol (23), the consensus error trajectory between leader and ith follower agents reach to the surface in a zig-zag step within specified band called Quasi-Sliding Mode Band(QSMB). So the width of consensus error QSMB for ith agent (i ) is defined using (20)
i =
i T . 2 − qi T
(24)
Stability Analysis: The global stability of leader following consensus using distributed control protocol (23) is guaranteed if the close loop error dynamics in (11) for global DMAS in (5) drives towards the sliding surface (16) and maintain on it for any gain q, > 0, 0 < 1 − qT < 1, and 1 − qT < , provided the following conditions hold true:
Please cite this article as: K. Patel and A. Mehta, Discrete-time sliding mode protocols for leader-following consensus of discrete multiagent system with switching graph topology, European Journal of Control, https://doi.org/10.1016/j.ejcon.2019.06.011
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(i)
−qT s˜(k ) < T sign(s˜(k )), s˜(k + 1 ) < s˜(k )
(28)
The main lacking point of consensus protocol defined in (23) is chattering and continuous consensus control effort is required to enforce the consensus error trajectory to sliding surface during reaching and sliding phase which is access more power and due to chattering serious problem may occur during implementation. To overcome these drawbacks we design the consensus protocol motivated by Power rate reaching law [34] in which the consensus effort is more when the consensus error trajectory is away from the surface and reduce the effort when consensus error trajectory nearer to the surface. Also, this consensus protocol reduce the control effort for the global DMAS defined in (5) under the influence of matched uncertainty.
(29)
4. DSMC for the consensus of DMAS using Power rate reaching law
(25)
(ii)
(2 − qT )s˜(k ) > T sign(s˜(k )), s˜(k + 1 ) > s˜(k )
(26)
Proof. Consider the sliding surface for global consensus given in (17)
s˜(k ) = csT (( In )(X (k ) − 1N x0 (k ))).
(27)
Let us select the Lyapunov function as
Vs (k ) = s˜2 (k ). Writing the forward derivative function of the (28) as
Vs (k ) = v˜ (k + 1 ) − v˜ (k ). Using (28), we may write
Vs (k ) = s˜2 (k + 1 ) − s˜2 (k ),
(30)
Vs (k ) = (s˜(k + 1 ) − s˜(k ))(s˜(k + 1 ) + s˜(k )).
(31)
For stability, it is required to ensure Vs (k ) < 0. For proving the same let us consider two cases: Case 1. s˜(k + 1 ) − s˜(k ) < 0 Now using s˜(k + 1 ) defined in (18), we get
s˜(k + 1 ) − s˜(k ) = [csT γ (X (k + 1 ) − 1N x0 (k + 1 ))] − s˜(k ).
(32)
s˜(k + 1 ) − s˜(k ) = csT γ [(A˜ X (k ) + B˜(u(k ) + D (k )) (33)
Now substituting the consensus protocol (23) into (33), we get
s˜(k + 1 ) − s˜(k ) = [(1 − qT )s˜(k ) − T sign(s˜(k ))] − s˜(k ) < 0, (34) and
s˜(k + 1 ) − s˜(k ) = −qT s˜(k ) − T sign(s˜(k )) < 0.
(35)
(36)
From (36), we tune the value of q and such that s˜(k + 1 ) − s˜(k ) < 0. Case 2. s˜(k + 1 ) + s˜(k ) > 0 Now again considering s˜(k + 1 ) in (18), we get
s˜(k + 1 ) + s˜(k ) = [csT γ (X (k + 1 ) − 1N x0 (k + 1 ))] + s˜(k ).
(37)
Substituting X (k + 1 ) and x(k + 1 ) defined in (5) and (6), we may get
s˜(k + 1 ) + s˜(k ) = csT γ [(A˜ X (k ) + B˜(u(k ) + D (k )) −1N Ax0 (k ))] + s˜(k ).
(38)
Now substituting the consensus protocol (23) in (38), we get
(42)
Proof. Let us define sliding surface of ith agent as
s˜i (k ) = csTi εi (k ),
(43)
where csi is the sliding gain to be obtain using pole-placement approach [15]. In the leader-follower network for global system (5) sliding surface can be written as
s˜(k ) = csT ξ (k ).
(44)
Using (11), we may write
s˜(k ) = csT γ (X (k ) − 1N x0 (k )).
(45)
Further,
(46)
Substituting (5) and (6) into (46), we may get
s˜(k + 1 ) = csT γ (A˜ X (k ) + B˜(u(k ) + D (k )) − 1N Ax0 (k )).
(47)
Now motivated from the reaching law proposed in [34], let us define the consensus reaching law for ith agent as
s˜i (k + 1 ) = (1 − qi T )s˜i (k ) − i T
| s˜i (k ) |η sign(s˜i (k )),
(48)
where = [1 , 2 , 3 , . . . N ] ∈ > 0, q = [q1 , q2 , q3 , . . . , qN ] ∈ RnN > 0, 0 < (1 − qT ) < 1. Then global consensus reaching law for global DMAS can be defined as RnN
s˜(k + 1 ) = (1 − qT )s˜(k ) − T
| s˜(k ) |η sign(s˜(k )).
(49)
csT γ (A˜ X (k ) + B˜(u(k ) + D (k )) − 1N Ax0 (k )) = (1 − qT )s˜(k ) − T
and
(40)
Further, (40) can be written as
(2 − qT )s˜(k ) > T sign(s˜(k )).
| s˜(k ) |η sign(s˜(k ))] − D (k ).
Comparing (47) and (49), we may get
s˜(k + 1 ) + s˜(k ) = [(1 − qT )s˜(k ) − T sign(s˜(k ))] + s˜(k ) > 0, (39) s˜(k + 1 ) + s˜(k ) = [(2 − qT )s˜(k ) − T sign(s˜(k ))] > 0.
u(k ) = −(csT γ B˜ )−1 [csT γ A˜ X (k ) + csT γ (−1N Ax0 (k )) − (1 − qT )s˜(k )
s˜(k + 1 ) = csT γ (X (k + 1 ) − 1N x0 (k + 1 )).
Further, (35) can be written as
−qT s˜(k ) < T sign(s˜(k )).
Theorem 2. For the given DMAS (5), the synchronization with a leader (6) is achieved using a DMAS consensus protocol given by
+ T
Substituting X (k + 1 ) and x(k + 1 ) defined in (5) and (6) respectively, we get
−1N Ax0 (k ))] − s˜(k ).
In this section, fixed time steps DMAS consensus protocol for the global DMAS is designed using modified Power rate reaching law of DSMC in the form of Theorem 2. The condition for global stability of the DMAS with the proposed DSMC law is also derived.
| s˜(k ) |η sign(s˜(k )).
(50)
Further, (50) can be expressed in terms of consensus protocol as
u(k ) = −(csT γ B˜ )−1 [csT γ A˜ X (k ) + csT γ (−1N Ax0 (k )) − (1 −qT )s˜(k ) (41)
From (41) select the parameter value of q and such that s˜(k + 1 ) + s˜(k ) > 0. From the above two cases, it is informed that by choosing appropriate values of tuning parameters q and , the stability of global DMAS is guaranteed.
+ T
| s˜(k ) |η sign(s˜(k ))] − D (k ).
(51)
The consensus of leader and follower is obtained using DMAS consensus protocol based on Power rate reaching law.sliding mode
Please cite this article as: K. Patel and A. Mehta, Discrete-time sliding mode protocols for leader-following consensus of discrete multiagent system with switching graph topology, European Journal of Control, https://doi.org/10.1016/j.ejcon.2019.06.011
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5
Fig. 1. 2-DOF helicopter system. Courtesy : Quanser Inc. Fig. 2. Free body diagram of 2-DOF helicopter system.
band in which system remains steadily is defined for ith agent from (48) as
i =
T 1−1ηi i
(52)
1 − qi T
Proper selection of η reduce the chattering effect and enhance the robustness. Stability Analysis: The global stability of leader following consensus using consensus protocol (51) is guaranteed if the close loop error dynamics in (11) for global DMAS in (5) drives towards the sliding surface (16) and maintain on it for any gain q, > 0, 0 < η < 1, 0 < 1 − qT < 1, and 1 − qT < , provided the following conditions hold true:
−s˜(k ) < 0,
(62)
and
s˜(k + 1 ) − s˜(k ) = −qT s˜(k ) − T
| s˜(k ) |η sign(s˜(k )) < 0.
Further, (63) can be written as
−qT s˜(k ) < T
| s˜(k ) |η sign(s˜(k )).
(64)
From (64), choose the proper value of q and such that s˜(k + 1 ) − s˜(k ) < 0 Case 2:. s˜(k + 1 ) + s˜(k ) > 0 Let us consider s˜(k + 1 ) define in (18) as
s˜(k + 1 ) + s˜(k ) = [csT γ (X (k + 1 ) − 1N x0 (k + 1 ))] + s˜(k ).
(i)
−qT s˜(k ) < T
|
s˜(k ) |η sign(s˜(k )),
s˜(k + 1 ) < s˜(k )
(53)
(63)
(65)
Using X (k + 1 ) and x(k + 1 ) defined in (5) and (6)
s˜(k + 1 ) + s˜(k ) = csT γ [(A˜ X (k ) + B˜(u(k ) + D (k )) − 1N Ax0 (k ))]
(ii)
(2 − qT )s˜(k ) > T | s˜(k ) |η sign(s˜(k )),
Proof. Consider sliding surface given in the (17)
s˜(k ) =
csT
+s˜(k ).
s˜(k + 1 ) > s˜(k ) (54)
(( In )(X (k ) − 1N x0 (k ))).
(55)
Vs (k ) = s˜2 (k ).
Now, substituting consensus protocol define in (51) into (66) we may get
s˜(k + 1 ) + s˜(k ) = [(1 −qT )s˜(k ) − T
Let us select the Lyapunov function as
(66)
| s˜(k ) |η sign(s˜(k ))] + s˜(k ) > 0, (67)
(56)
Writing the forward derivative function and substituting for (56)
Vs (k ) = v˜ (k + 1 ) − v˜ (k ),
(57)
and
s˜(k + 1 ) + s˜(k ) = [(2 − qT )s˜(k ) − T
| s˜(k ) |η sign(s˜(k ))] > 0. (68)
Vs (k ) = s˜ (k + 1 ) − s˜ (k ),
(58)
Vs (k ) = (s˜(k + 1 ) − s˜(k ))(s˜(k + 1 ) + s˜(k )).
(59)
2
2
For stability, it is required to ensure Vs (k ) < 0. For proving the same let us consider two cases: Case 1:. s˜(k + 1 ) − s˜(k ) < 0 Now substituting s˜(k + 1 ) using (18), we get
s˜(k + 1 ) − s˜(k ) = [csT γ (X (k + 1 ) − 1N x0 (k + 1 ))] − s˜(k ).
(60)
considering X (k + 1 ) and x(k + 1 ) using (5) and (6)
s˜(k + 1 ) − s˜(k ) = csT γ [(A˜ X (k ) + B˜(u(k ) + D (k )) −1N Ax0 (k ))] − s˜(k )
(61)
Now substituting consensus protocol define in (51) into (61) we may get
s˜(k + 1 ) − s˜(k ) = [(1 − qT )s˜(k ) − T
| s˜(k ) |η sign(s˜(k ))]
Further (68) can be written as
(2 − qT )s˜(k ) > T | s˜(k ) |η sign(s˜(k )).
(69)
From (69), select the parameter value of q and such that s(k + 1 ) + s ( k ) > 0. From the above two cases, we may conclude that by choosing appropriate values of tuning parameters q and the stability of global DMAS is guaranteed. 5. 2 DOF helicopter systems In order to validate the efficacy of the proposed consensus protocols in Sections 3 and 4, let us consider six number of 2-DOF helicopter systems as agents in a leader-following network shown in (Fig. 3). Out of which one helicopter act as a leader and other five act as follower agent. The 2-DOF helicopter is an important model from the control system engineering point of view due to its wide non-linear
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nose. The center of mass acts at l distance from the pitch axis along the helicopter body length. Where l is the distance between the center of mass and the intersection of the pitch and yaw axes. The notations used are given as under: θ pitch angle, deg ψ yaw angle, deg K pp thrust force constant of yaw motor/propeller, N.m/V Kyy thrust torque constant of yaw axis from yaw motor/propeller, N.m/V K py thrust torque constant acting on pitch axis from yaw motor/propeller, N.m/V Kyp thrust torque constant acting on yaw axis from pitch motor/propeller, N.m/V Bp viscous damping about pitch axis, N/V By viscous damping about yaw axis, N/V mh total moving mass of the helicopter (body, two propeller assemblies, etc.), kg l Center of mass length along helicopter body from pitch axis, meter(m) Jeq , p rotal moment of inertia about pitch axis, kg.m2 Jeq , y total moment of inertia about yaw axis, kg.m2 Vp , Vy respective voltages applied to the front and rear motors g gravitational constant, 9.8 m/s2 The 2-DOF helicopter modeling conventions are: Fig. 3. Switching graph topology.
characteristics, highly cross coupling effects and instability in open loop. The 2-DOF helicopter model (fixed base) with two propellers driven by DC motors is shown in Fig. 1 The elevation of the nose over the pitch axis is controlled by the front propeller and the rotational motion around the yaw axis is controlled by the back propeller. The voltages across the pitch and yaw motors are ± 24V and ± 15V, respectively [10]. The 2-DOF helicopter model dynamics are shown in Fig. 2 The thrust forces Fp and Fy are applied across the pitch and yaw axis, respectively. The torques act at a distance rp and ry from the respective axis. The gravitational force Fg pulls down the helicopter
• The helicopter is horizontal when the pitch angle equals θ = 0. • The pitch angle increases positively, θ˙ > 0, when the nose is moved upwards and the body rotates in the counter-clockwise (CCW) direction. • The yaw angle increases positively, ψ˙ > 0 when the body rotates in the clockwise (CW) direction. • Pitch increases, θ > 0, when the pitch thrust force is positive Fp > 0. • Yaw increases, ψ > 0, when the yaw thrust force is positive, Fy > 0. Consider the dynamics of 2-DOF helicopter acting as ith agent is written as:
Fig. 4. Pitch position and velocity consensus.
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Fig. 5. Yaw position and velocity consensus.
Fig. 6. Control effort (u) using Gao’s reaching law.
For the pitch axis angle θ , the dynamic equation is written as
1 θ¨ = [{K ppVp + K pyVy } − {B p θ˙ + mh glcos(θ ) Jeq,p + mh l 2 +mh l 2 sin(θ )cos(θ )ψ˙ 2 }].
(70)
Consider Jt p = Jeq,p + mh l 2 , we may write (70) as
θ¨ =
(72)
Consider Jty = Jeq,y + mh l 2 cos2 (θ ), we can obtain from the above equation
ψ¨ =
1 [{K ppVp + K pyVy } − {B p θ˙ + mh glcos(θ ) Jt p +mh l 2 sin(θ )cos(θ )ψ˙ 2 }].
+2mh l 2 sin(θ )cos(θ )ψ˙ ψ˙ 2 }].
1 [{KypVp + KyyVy } − {By ψ˙ + 2mh l 2 sin(θ )cos(θ )ψ˙ ψ˙ 2 }]. Jty (73)
(71)
Similarly, for the yaw axis it is defined as
The state space model for ith agent system is defined as
1 ψ¨ = [{KypVp + KyyVy } − {By ψ˙ Jeq,y + mh l 2 cos2 (θ )
xi (k + 1 ) = Axi (k ) + B(ui (k ) + Di (k ))
∀i ∈ N.
(74)
Please cite this article as: K. Patel and A. Mehta, Discrete-time sliding mode protocols for leader-following consensus of discrete multiagent system with switching graph topology, European Journal of Control, https://doi.org/10.1016/j.ejcon.2019.06.011
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Fig. 7. Control effort (u) using Power rate reaching law.
Fig. 8. Sliding surface of individual agent using Gao’s reaching law.
⎡
0 ⎢0
0 ⎢ 0
⎢ kpp ⎢J ⎣ Tp ky p JTy
1 0
A = ⎢0
0
−J
0
0
0
⎢ ⎣
Where,
⎡
0 0
⎤
Bp Tp
⎤
0 1 ⎥
⎥ ⎦
0 ⎥ B
−J y
Di ∈ Rm are state vector, input vector and matched disturbance respectively acting on ith follower agent. ∈ Rn×n
and
B=
Ty
0 0 ⎥ k py ⎥ ⎥∈ Rn×m are the system matrix input matrix reJTp kyy JTy
⎦
spectively. xi (k ) = [θi (k )ψi (k )θ˙ i (k )ψ˙ i (k )] ∈ Rn , ui (k) ∈ Rm and
6. Simulation and result discussion In this section, Pitch position(θ ) and its velocity (θ˙ ), Yaw position(ψ ) and its velocity (ψ˙ ) states of 2 DOF helicopter system are to be considered as a single agent parameters for consensus. As discussed, it is assumed here that the leader node has same dynamics as the follower node. The simulation is carried out using Matlab R15. The following notations are used for simulation results.
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Fig. 9. Sliding surface of individual agent using power rate reaching law.
⎡0
Discretizing the continuous-time system at T = 0.03 sampling period, we obtain the discrete-time system model as
∀i ∈ N.
xi (k + 1 ) = Axi (k ) + B(ui (k ) + Di (k ))
⎡
(75)
⎤
1 0 0.0262 0 0 0.0285⎥ ⎢0 1 Where, A = ⎣ ⎦, 0 0 0.7571 0 0 0 0 0.9004 ⎡ ⎤ 0.0010 0 ⎢0.0 0 01 0.0 0 03⎥ B=⎣ ⎦ 0.0620 0.0021 0.0069 0.0225 As discussed in Section 3, consider MAS with five follower node and one leader node as undirected switching graph topology. The leader node notation is given as 0 while follower node notation is given as 1,2,3,4,5 respectively. Their communication topology is shown in the Fig. 3. Initial states of each follower agent are generated randomly in the span[−1, 1] while the initial state of the leader node is [0.1, 0.5, 0.7, 0.8]T and Laplacian matrix of individual graph topology Gσ [p] are defined as
⎡0 ⎢−1 L1 = ⎢ 0 ⎣
0 1 0 0 0
0 0
⎡0 L3 =
⎢0 ⎢0 ⎣ 0 0
0 1 0 0 0
0 0 0 0 0
0 0 0 0 0
0 −1 1 0 0
0 0 −1 0 0
⎤
⎡
0 0 0⎥ ⎢0 0⎥, L2 = ⎢0 ⎦ ⎣ 0 0 0 0
⎤
0 0 0 0 0
0 0 1 0 0
0 0 −1 1 0
⎤
0 0⎥ 0⎥ ⎦ −1 0
0 0⎥ 0⎥. ⎦ 0 0
(76)
Further, pinning gains matrix are also defined as
⎡1 B1 =
⎢0 ⎢0 ⎣ 0 0
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
⎤
⎡
0 0 0⎥ ⎢0 0⎥, B2 = ⎢0 ⎦ ⎣ 0 0 0 0
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
⎤
0 0⎥ 0⎥ , ⎦ 0 1
B3 =
⎢0 ⎢0 ⎣ 0 0
0 0 0 0 0
0 0 0 0 0
0 0 0 1 0
⎤
0 0⎥ 0⎥. ⎦ 0 0
(77)
The sliding gain for surface define in (16) is calculated using the pole-placement method which comes out to be
csT
93.64 = 126.18
52.39 482.90
13.17 −1.5626
−0.0043 . 45.98
(78)
In order to check the robustness of the derived consensus protocol for the global system, a slow varying disturbance is applied to each DMAS with a magnitude Di (k ) = 0.002 ∗ cos(0.86k ) for individual graph topology. The gain for each follower agent i , qi are chosen as 2 and 1 respectively in the case of Gao’s reaching law and the gain value of η in the case of Power rate reaching law is taken as 0.6. Proper selection of the value η decrease the sliding mode band in Power rate reaching law. Fig. 4 shows the comparison results of the pitch position and velocity states consensus with leader position and velocity states using Gao’s and Power rate reaching law using switching graph topology for the graph G1 , G2 , G3 . Graph G1 has changed to G2 after 50 no. of steps. Simultaneously graph G2 has changed to G3 and remain stable for 49 no. of steps after that G3 graph topology steady for another 49 no. of steps. In this way, we switch the topology and find that consensus begins after 148 no. of steps. However, in this case, Power rate reaching law gives faster convergence speed compared to Gao’s reaching law. Fig. 5 shows the comparative study of yaw position and velocity states consensus of the follower with leader states using Gao’s and Power rate reaching law and found that all the followers synchronize with leader in fixed steps size using switching graph topology. Figs. 6 and 7 show the consensus effort(u) of the follower agents, which further applied to individual agent into the overall network of DMAS. Figs. 8 and 9 show the sliding surface (consider as consensus error trajectories) of each agent of DMAS. It crosses first time sliding surface band called as QSMB from any initial condition in fixed
Please cite this article as: K. Patel and A. Mehta, Discrete-time sliding mode protocols for leader-following consensus of discrete multiagent system with switching graph topology, European Journal of Control, https://doi.org/10.1016/j.ejcon.2019.06.011
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Fig. 10. Pitch and yaw position consensus tracking error.
Fig. 11. Pitch and yaw velocity consensus tracking error.
time steps and steady within this band and value of this band is vary from -0.03 to +0.03 (O(T)) calculated using (21) in the case of Gao’s reaching law and similarly in the case of Power rate reaching law sliding surface remains steadily in O(T2 ) band once it is on the sliding surface. Figs. 10 and 11 express the Pitch and yaw Position and velocity tracking consensus error using two different reaching law and observed that reached within fixed time steps. It may be noted that Power rate reaching law quickly reaches to the zero as compared to Gao’s reaching law.
7. Conclusion In this paper, two discrete-time sliding mode control protocols using Gao’s reaching law and Power rate reaching law are proposed for leader following consensus of discrete multi-agent system configured with a fixed, undirected switching graph topology. Both the consensus protocols synchronize leader with the follower agents under the influence of switching topology at different intervals. The condition for global stability is also derived using the Lyapunov function. Finally, both the consensus protocols
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are validated in simulation for the 2-DOF helicopter system. In future, the DSMC algorithms for consensus of DMAS with bipartite and event-triggered control shall be explored. Supplementary material Supplementary material associated with this article can be found, in the online version, at doi:10.1016/j.ejcon.2019.06.011. References [1] W. Gao, Y. Wang, A. Homaifa, Discrete-time variable structure control systems, IEEE Trans. Ind. Electron. 42 (2) (1995) 117–122, doi:10.1109/41.370376. [2] Q. Cao, Y. Song, J.M. Guerrero, S. Tian, Coordinated control for flywheel energy storage matrix systems for wind farm based on charging/discharging ratio consensus algorithms, IEEE Trans. Smart Grid 7 (3) (2016) 1259–1267, doi:10.1109/TSG.2015.2470543. [3] C. Ding, X. Dong, C. Shi, Y. Chen, Z. Liu, Leaderless output consensus of multiagent systems with distinct relative degrees under switching directed topologies, IET Control Theory Appl. 13 (3) (2019) 313–320, doi:10.1049/iet-cta.2018. 5140. [4] Y. Dong, J. Huang, A leader-following rendezvous problem of double integrator multi-agent systems, Automatica 49 (5) (2013) 1386–1391. https://doi.org/10. 1016/j.automatica.2013.02.024. [5] M.-C. Fan, H.-T. Zhang, M. Wang, Bipartite flocking for multi-agent systems, Commun. Nonlinear Sci. Numer. Simulat. 19 (9) (2014) 3313–3322. https://doi. org/10.1016/j.cnsns.2013.10.009. [6] K. Fang, F. Shu, Y. Deng, Lag consensus of second-order leader-following multiagent systems with communication delays and switching topologies, in: Proceedings of the Chinese Automation Congress (CAC), 2017, pp. 4061–4066, doi:10.1109/CAC.2017.8243491. [7] L. Guo, X. Nian, H. Pan, Leader-following consensus of multi-agent systems with switching topologies and time-delays, J. Control Theory Appl. 11 (2) (2013) 306–310, doi:10.1007/s11768- 013- 1191- 2. [8] J. Han, C. Wang, G. Yi, Cooperative control of uav based on multi-agent system, in: Proceedings of the IEEE 8th Conference on Industrial Electronics and Applications (ICIEA), 2013, pp. 96–101, doi:10.1109/ICIEA.2013.6566347. [9] M. Hu, L. Guo, A. Hu, Y. Yang, Leader-following consensus of linear multi-agent systems with randomly occurring nonlinearities and uncertainties and stochastic disturbances, Neurocomputing 149 (2015) 884–890, doi:10.1016/j.neucom. 2014.07.047. [10] Q. Inc, Quanser 2-DOF helicopter control user manual, Technical Report, 2015. [11] Y. Jiang, J. Liu, S. Wang, Robust integral sliding-mode consensus tracking for multi-agent systems with time-varying delay, Asian J. Control 18 (1) (2014) 224–235, doi:10.1002/asjc.1007. [12] T.-C. Lee, W. Xia, Y. Su, J. Huang, Exponential consensus of discrete-time systems based on a novel Krasovskii-LaSalle theorem under directed switching networks, Automatica 97 (2018) 189–199. https://doi.org/10.1016/j.automatica. 2018.07.022. [13] Y. Li, C. Tang, K. Li, X. He, S. Peeta, Y. Wang, Consensus-based cooperative control for multi-platoon under the connected vehicles environment, IEEE Trans. Intell. Transp. Syst. (2018) 1–10, doi:10.1109/TITS.2018.2865575. [14] M.S. Mahmoud, G.D. Khan, Leader-following discrete consensus control of multi-agent systems with fixed and switching topologies, J. Frankl. Inst. 352 (6) (2015) 2504–2525, doi:10.1016/j.jfranklin.2015.03.026. [15] T. Nguyen, Z. Miao, Y. Pan, N.H. Amini, V. Ugrinovskii, M.R. James, Pole placement approach to coherent passive reservoir engineering for storing quantum information, in: Proceedings of the American Control Conference (ACC), 2017, pp. 234–239, doi:10.23919/ACC.2017.7962959. [16] W. Ni, D. Cheng, Leader-following consensus of multi-agent systems under fixed and switching topologies, Syst. Control Lett. 59 (3–4) (2010) 209–217, doi:10.1016/j.sysconle.2010.01.006. [17] M.M. Rana, L. Li, S.W. Su, W. Xiang, Consensus-based smart grid state estimation algorithm, IEEE Trans. Ind. Inf. 14 (8) (2018) 3368–3375, doi:10.1109/TII. 2017.2782750.
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