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Robust cluster consensus of general fractional-order nonlinear multi agent systems via adaptive sliding mode controller
Journal Pre-proof Robust cluster consensus of general fractional-order nonlinear multi agent systems via adaptive sliding mode controller Zahra Yaghoubi
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S0378-4754(20)30003-3 https://doi.org/10.1016/j.matcom.2020.01.002 MATCOM 4926
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Mathematics and Computers in Simulation
Received date : 1 September 2019 Revised date : 3 January 2020 Accepted date : 4 January 2020 Please cite this article as: Z. Yaghoubi, Robust cluster consensus of general fractional-order nonlinear multi agent systems via adaptive sliding mode controller, Mathematics and Computers in Simulation (2020), doi: https://doi.org/10.1016/j.matcom.2020.01.002. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. c 2020 Published by Elsevier B.V. on behalf of International Association for Mathematics and ⃝ Computers in Simulation (IMACS).
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Robust Cluster Consensus of General Fractional-Order Nonlinear Multi Agent Systems via Adaptive Sliding Mode Controller Zahra Yaghoubi1 Department of Electrical Engineering, Amirkabir University of Technology, Tehran, Iran. Abstract: In this paper robust cluster consensus is investigated for general fractional-order multi agent systems with nonlinear dynamics with dynamic uncertainty and external disturbances via adaptive sliding mode controller. First, robust cluster consensus for general fractional-order nonlinear multi agent systems is investigated with dynamic uncertainty and external disturbances which multi agent systems are weakly heterogeneous because they have identical nominal dynamics with different normbounded parameter uncertainties. Then, robust cluster consensus for the fractional-order nonlinear
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multi agent systems with general form dynamics is investigated by using adaptive sliding mode
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controller. Robust cluster consensus for general fractional-order nonlinear multi agent systems is achieved asymptotically without disturbance. It is shown that the errors between agents can converge
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to a small region in the presence of disturbances based on the linear matrix inequality (LMI) and Mittag-Leffler stability theory. Finally, simulation examples are presented for general form multi agent systems, i.e. a single-link flexible joint manipulator which demonstrates the efficiency of the proposed adaptive controller.
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Keywords: Robust cluster consensus; Fractional-order multi agent systems; Adaptive sliding mode controller; External disturbances; Dynamic uncertainty.
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1. Introduction
Application of multi agent systems is interesting topic such as formation [7],
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swarming, flocking and synchronization in physical and chemical systems and recently attracts more attention. Consensus is a one of these applications which means all the states/outputs of agents converge to the same value [32]. Consensus without leader [28] and
1
Corresponding author: Email: [email protected].
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consensus tracking with leader in which all agents follow the leader are two categories for consensus problem [31]. Recently, many researchers are studied about cluster consensus which means that agents are divided into a few clusters and these clusters may change in different situations and environments [33]. All the agents in the same cluster converge to a same attitude which is different from other clusters [41]. Multi agent systems with integer-order dynamic are studied in most literatures. Consensus problem for first-order multi agent systems is studied in [24], for second-order dynamics in [19], and for higher-order dynamics in [10]. Fractional-order dynamics can
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describe behaviors of agents in complex environments , such as traveling aircraft in rainy and dusty environment [3, 5, 17, 27] and the integer-order dynamics are parts of fractional-order dynamics [36]. In [5], control of fractional-order linear multi agent systems are investigated
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under the directed graph and the convergence speed is checked to achieve consensus. Consensus of fractional-order nonlinear multi agent systems is studied with leader and without leader under directed topologies in [11, 13]. The consensus for nonlinear fractional-
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order multi agent systems with an unknown leader is studied in [12]. In [39], The observer base consensus for fractional-order multi agent systems is investigated. Therefore, cluster
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consensus for fractional-order multi agent systems with high-order nonlinear dynamics are a new study which is investigated in this paper.
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One feature to achieve consensus is the smallest non-zero eigenvalue of the Laplacian matrix [22, 25]. Consensus via adaptive controller is used to overcome to this limitation, [4, 23]. Consensus for second-order nonlinear multi agent systems for both with and without leader via adaptive controller under undirected topologies is studied in [40]. In [38],
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consensus for nonlinear multi agent systems with uncertainty is studied via adaptive controller. The distributed consensus for second-order multi agent systems under directed topologies is investigated in [26]. One of inevitable part of many industries is influence of disturbances. For example, friction of robot arm and electromagnetic interference in multi-robot systems is defined as a external disturbances [15]. Disturbances have an effect in formation of satellite [34] and power systems [2]. The robust consensus of fractional-order multi-agent systems with exogenous disturbance is studied in [30]. Distributed robust consensus for multi agent systems which dynamics are unknown fractional-order is studied in [14]. Therefore, multi-
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agent systems with disturbance is one of the important subjects [9]. In real environment, there are unmodelled dynamics of agents but in most researches dynamics of agents are identical and this is invalid [20, 43].So study on uncertain multi-agent
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system is one of the important issues.
The main contribution of this paper is divided in fourfold:
The robust cluster consensus of fractional-order nonlinear system is presented for the first
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time. Note that the underlying dynamics are in general nonlinear form. The controller for multi agent system is designed by using adaptive sliding mode
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principles and σ modification method and the efficiency of which are compared with that of non-adaptive case.
General fractional-order multi agent system with dynamic uncertainty is investigated.
Fractional-order nonlinear system is presented with disturbances.
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The above discussion motivates that robust cluster consensus for fractional-order nonlinear multi agent systems can be investigated by using adaptive sliding mode controller
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with dynamic uncertainty and external disturbances. The Lyapunov stability theorem for fractional-order systems is used to prove the Mittag-Leffler stability and the convergence of consensus. The rest of the paper is organized as follows. In Section 2, graph theory is discussed. A brief explanation of fractional calculus is given in Section 3. In Section 4, problem formulation for fractional-order nonlinear multi agent systems is discussed. In Section 5, simulation examples are given. Finally, the conclusion is presented in Section 6. 2. Graph theory The information exchange between 𝑛 agents is described with a graph and shown by
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𝐺 = (𝑉, 𝐸, 𝐴), where 𝑉 = {𝜈1 , 𝜈2 , … , 𝜈𝑛 } is the node set and 𝐸 = {(𝜈𝑗 , 𝜈𝑖 ): 𝜈𝑗 , 𝜈𝑖 ∈ 𝑉, 𝜈𝑗 ≠ 𝜈𝑖 } ∈ 𝑉 × 𝑉 is the edge set of the graph. 𝒩𝑖 = {𝜈𝑗 ∈ 𝑉: (𝜈𝑗 , 𝜈𝑖 ) ∈ 𝐸} is the neighbors set of the
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𝑖 th agent which the 𝑖 th agent receives information from the 𝑗th agent. A directed spanning tree means there is at least a directed path from one node to all other nodes and there is in directed
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graph shown in Fig.1.
Figure. 1. A directed graph
Two matrices 𝐴 and 𝐿 describe the information exchange, where 𝐴 = [𝑎𝑖𝑗 ] ∈ ℝ𝑛×𝑛 is
a matrix with 𝑎𝑖𝑗 > 0 if (𝜈𝑗 , 𝜈𝑖 ) ∈ 𝐸, otherwise 𝑎𝑖𝑗 = 0 and 𝐿 is a positive semi-definite
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matrix which is defined as 𝐿 = 𝐷 − 𝐴 where 𝐷 = 𝑑𝑖𝑎𝑔{∑𝑛𝑗=1 𝑎𝑖𝑗 } ∈ ℝ𝑛×𝑛 . Some properties of the Kronecker product define in Lemma 1. Lemma 1. There are some properties of the Kronecker product ⊗ which is defined as follows [8]: (1) (𝜉𝐴) ⊗ B = A ⊗ (𝜉𝐵) (2) (𝐴 + 𝐵) ⊗ C = A ⊗ C + 𝐵 ⊗ C (3) (𝐴 ⊗ 𝐵)(C ⊗ D) = (AC) ⊗ (BD) (4) (A ⊗ B)T = AT ⊗ BT where A, B, C and D are matrices with appropriate dimensions and 𝜉 ∈ ℝ.
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3. Fractional-order calculus Some definitions and Lemmas of fractional-order calculus are expressed as follows. Definition 1. Caputo fractional derivative is used in this paper defined below [29]: 𝑥(𝑡) =
𝑡 1 ∫ (𝑡 − 𝜏)𝑚−𝛼−1 𝑥 (𝑚) (𝜏)𝑑𝜏, Γ(𝑚 − 𝛼) 0
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𝑐 𝛼 0𝒟𝑡
(1)
where 𝑚 − 1 < 𝛼 < 𝑚, 𝑚 ∈ ℕ+ and Γ(. ) is the gamma function.
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The notation 𝑥 𝛼 (𝑡) is used instead of 0𝑐𝒟𝑡𝛼 𝑥(𝑡) for simplicity. Definition 2. The Mittag-Leffler function is used to solve fractional-order systems like
∞
𝐸𝛼,𝛽 (𝑧) = ∑
𝑧𝑘 Γ(𝑘𝛼 + 𝛽)
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𝑘=0
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the exponential function for integer-order systems. This function with 𝛼, 𝛽 > 0 is defined as:
The stability of fractional-order systems is defined in the following Lemmas [1, 18, 42]. Lemma 2. For any time 𝑡 ≥ 𝑡0 , the following inequality is defined: 1 𝑐 𝛼 2 𝑐 𝛼 𝑡0 𝒟𝑡 𝑥 (𝑡) ≤ 𝑥(𝑡) 𝑡0 𝒟𝑡 𝑥(𝑡), 𝛼 ∈ (0,1) 2
(2)
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where 𝑥(𝑡) ∈ ℝ is a continuous and derivable function. Lemma 3. The following inequality is defined: 𝑐 𝛼 𝑡0 𝒟𝑡
(𝑥 𝑇 (𝑡)𝑃𝑥(𝑡)) ≤ 2𝑥 𝑇 (𝑡)𝑃 𝑡𝑐0 𝒟𝑡𝛼 𝑥(𝑡), 𝛼 ∈ (0,1)
(3)
where 𝑥(𝑡) ∈ ℝ𝑛 is continuous and derivable and 𝑃 ∈ ℝ𝑛×𝑛 is a positive definite symmetric matrix. The systems are proved to be asymptotically stable by using the Lyapunov direct method. The Lyapunov direct method for fractional order systems is derived from the Lyapunov direct method for integer-order systems and it leads to the Mittag-Leffler stability [21].
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Lemma 4. Mittag-Leffler stability for the fractional-order system in 𝑡0 = 0 at the equilibrium point 𝑥 = 0 is proved if the Lyapunov function 𝑉(𝑡, 𝑥(𝑡)) exists with the
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following condition: 𝛼1 ‖𝑥‖𝑐 ≤ 𝑉(𝑡, 𝑥(𝑡)) ≤ 𝛼2 ‖𝑥‖𝑐𝑑 , 𝑐 𝛼 0𝒟𝑡 𝑉(𝑡, 𝑥(𝑡))
(4)
≤ −𝜆 𝑉(𝑡, 𝑥(𝑡))
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where 𝑡 ∈ ℝ+ , 𝛼 ∈ (0,1], 𝛼1 , 𝛼2 , 𝜆, 𝑐 and 𝑑 are positive constants [37]. 4. Problem Formulation
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In this part, cluster consensus problem is explained, where the graph of agents is divided into two parts and each part reaches a different consistent value. Consider a graph
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with 𝑛 agents divided like 𝑙1 = {1,2, … , 𝑁1 }, 𝑙2 = {𝑁1 + 1, … , 𝑁1 + 𝑁2 } where 𝑛 = 𝑁1 + 𝑁2. Some assumptions for analyzing the cluster consensus problem are expressed in this section.
Assumption 1. The Lipschitz condition is defined as follow for function 𝑓(𝑥, 𝑡).
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|𝑓(𝑥2 , 𝑡) − 𝑓(𝑥1 , 𝑡)| ≤ 𝑙|𝑥2 − 𝑥1 |,
∀𝑥1 , 𝑥2 ∈ ℝ, ∀𝑡 ≥ 0
where 𝑙 is the Lipschitz constant. Assumption 2. (a) Each subnetwork or cluster has a directed spanning tree. (b) The information exchange between two clusters should be balanced. In other 𝑁1 +𝑁2 words, two clusters graph, satisfy the following conditions: ∑𝑗=𝑁 𝑎 = 0, for all 𝑖 ∈ 𝑙1, 1 +1 𝑖𝑗 1 ∑𝑁 𝑗=1 𝑎𝑖𝑗 = 0, for all 𝑖 ∈ 𝑙2 .
4.1.Robust cluster consensus of fractional-order nonlinear multi agent systems In this section, agents with general high-order nonlinear system are studied. There are
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𝑛 agents which the 𝑖 th agent, 𝑖 = 1, … , 𝑛 is modeled as follows [35]: 𝑥𝑖𝛼 = (𝐴 + ∆𝐴𝑖 )𝑥𝑖 + 𝐶 𝑓𝑖̂ (𝑡, 𝑥𝑖 ) + 𝐵𝑢𝑖 (𝑡) + 𝑆𝑤𝑖 (𝑡)
(5)
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where 𝑥𝑖 ∈ ℝ𝑝 are the states of 𝑖 th agent for 𝑖 = 1,2, … , 𝑛 with 𝑝 dimensions, 𝛼 ∈ (0,1], 𝑓𝑖̂ (𝑡, 𝑥𝑖 ) = (𝑓1 (𝑥𝑖 , 𝑡), 𝑓2 (𝑥𝑖 , 𝑡), … , 𝑓𝑝 (𝑥𝑖 , 𝑡))𝑇 is a nonlinear vector-valued function for each cluster which satisfies Lipschitz condition, 𝑢𝑖 (𝑡) is the control input, 𝑤𝑖 (𝑡) is the external
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disturbance, ∆𝐴𝑖 is the parameter uncertainty of agent 𝑖, 𝐴, 𝐵, 𝐶 and 𝑆 are constant matrices. The 𝑖 th leader is described by:
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𝛼 𝑥𝑟𝑖 = 𝐴𝑥𝑟𝑖 + 𝐶𝑓𝑖̂ (𝑡, 𝑥𝑟𝑖 ), 𝑖 = 1,2, … , 𝑛
(6)
The controller is designed as follows:
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𝑛
𝑢𝑖 (𝑡) = −𝑘𝜃 (∑ 𝑎𝑖𝑗 [𝑥𝑖 − 𝑥𝑗 ] + 𝑑𝑖 [𝑥𝑖 − 𝑥𝑟𝑖 ]) , 𝑖 = 1,2, … , 𝑛, 𝑖 ≠ 𝑗
(7)
𝑗=1
where 𝑘 > 0, 𝜃 ∈ ℝ1×𝑝 is the feedback gain matrix, 𝑎𝑖𝑗 , 𝑖, 𝑗 = 1,2, … , 𝑛 is the (𝑖, 𝑗)th element of the adjacency matrix 𝐴. 𝐷 = 𝑑𝑖𝑎𝑔( 𝑑1 , 𝑑2 , … , 𝑑𝑛 ) is the leader adjacency matrix.
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𝑑𝑖 is used for describing the leader information exchange to 𝑖 th agent. 𝑑𝑖 = 1, if information is transmitted from leader to 𝑖 th agent and otherwise 𝑑𝑖 = 0. For each cluster, 𝑑𝑖 = 1 for only
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one follower. Block diagram of the cluster consensus algorithm is shown in Fig. 2.
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Figure. 2. Block diagram of cluster consensus algorithm. Assumption 3.
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We can suppose that the parameter uncertainty satisfies ∆𝐴𝑖 = 𝐵 𝐸𝑖 , where 𝐸𝑖 is defined as follows with constant 𝜌 > 0:
𝐸𝑖 < 𝜌 𝐼𝑝 for all 𝑖 = 1,2, … , 𝑛 and 𝐸 = diag (𝐸1 , 𝐸2 , … , 𝐸𝑛 ) [43].
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Definition 3 ([6]). Robust cluster consensus is achieved for the multi agent system (6) if the error of agents satisfies:
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𝑙𝑖𝑚 ‖𝑥𝑖 − 𝑥𝑗 ‖ ≤ ℎ(𝛾),
𝑡→∞
𝑙𝑖𝑚 ‖𝑥𝑖 − 𝑥𝑗 ‖ ≤ ℎ(𝛾),
∀𝑖, 𝑗 ∈ 𝑙2
where ℎ(𝛾) is the monotonous increasing function and ℎ(0) = 0.
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𝑡→∞
∀𝑖, 𝑗 ∈ 𝑙1
Assumption 4. The external disturbances 𝑤𝑖 (𝑡) satisfy |𝑤𝑖 (𝑡)| < 𝛾 < +∞ for all 𝑖 = 1,2, … , 𝑛. By substituting the controller (8), into the system (6), we can write:
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𝑛
𝑥𝑖𝛼
= (𝐴 + ∆𝐴𝑖 )𝑥𝑖 + 𝐶 𝑓(𝑥𝑖 , 𝑡) + 𝑘𝐵𝜃 ∑ 𝑎𝑖𝑗 [𝑥𝑖 − 𝑥𝑗 ] + 𝑑𝑖 [𝑥𝑖 − 𝑥𝑟𝑖 ]
(8)
𝑗=1
+ 𝑆𝑤𝑖 (𝑡),
𝑖 = 1, … , 𝑛
Now system (9) can be written as: 𝑋 𝛼 = (𝐼𝑛 ⊗ 𝐴 + ∆𝐴) 𝑋 + (𝐼𝑛 ⊗ 𝐶) 𝐹(𝑋, 𝑡) − 𝑘(𝑀 ⊗ 𝐵𝜃)𝑋 (9)
+ (𝐼𝑛 ⊗ 𝑆)𝑊(𝑡) where
𝑋 = [𝑥1 , 𝑥2 , … , 𝑥𝑁1 , 𝑥𝑁1 +1 , … , 𝑥𝑁1 +𝑁2 ]𝑇 ,
∆𝐴 = diag(∆𝐴1 , ∆𝐴2 , … , ∆𝐴𝑛 ),
𝐹(𝑡, 𝑋) = [𝑓𝑖̂ (𝑡, 𝑥1 (𝑡)), 𝑓𝑖̂ (𝑡, 𝑥2 (𝑡)), … , 𝑓𝑖̂ (𝑡, 𝑥𝑁1 (𝑡)) , 𝑓𝑖̂ (𝑡, 𝑥𝑁1 +1 (𝑡)) , … , 𝑓𝑖̂ (𝑡, 𝑥𝑛 (𝑡))]𝑇 , 𝑇
𝑊(𝑡) = [𝑤1 , 𝑤2 , … , 𝑤𝑁1 , 𝑤𝑁1 +1 , … , 𝑤𝑁1 +𝑁2 ] , ⊗
is
Kronecker
product
and
𝜃=
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𝑑𝑖𝑎𝑔(𝜃1, , 𝜃2 , … , 𝜃𝑛 ). 𝑀 = 𝐻 + 𝐷, where 𝐻 is the Laplacian matrix of G described for two clusters as follows:
𝐿1 Ω21
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𝐻=[
Ω12 ] 𝐿2
where 𝐿𝑖 , 𝑖 = 1,2 are the Laplacian matrix of the subnetworks and 𝛺𝑖𝑗 , 𝑖, 𝑗 = 1,2 are the information exchange between the two subnetworks as: ⋯ 𝑎1(𝑁1 +𝑁2 ) ⋱ ⋮ ] ⋯ 𝑎𝑁1 (𝑁1 +𝑁2 )
Ω21
𝑎(𝑁1 +1)1 ⋮ = −[ 𝑎(𝑁1 +𝑁2 )1
⋯ 𝑎(𝑁1 +1)𝑁1 ⋱ ⋮ ] ⋯ 𝑎(𝑁1 +𝑁2 )𝑁1
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Ω12
𝑎1(𝑁1 +1) ⋮ = −[ 𝑎𝑁1 (𝑁1 +1)
In the undirected graph 𝐺, 𝑀 is always a symmetric positive definite matrix and in
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directed graph 𝐺, if information is sent and received between two nodes, 𝑀 is a symmetric positive definite matrix. Let 𝑥̃𝑖 = 𝑥𝑖 − 𝑥𝑟𝑖 , so we can write as follows:
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𝑋̃ 𝛼 (𝑡) = (𝐼𝑛 ⊗ 𝐴 + ∆𝐴)𝑋̃ + (𝐼𝑛 ⊗ 𝐶)(𝐹(𝑡, 𝑋) − 𝐹(𝑡, 𝑋𝑟 )) − 𝑘(𝑀 ⊗ 𝐵𝜃)𝑋̃ (10) + (𝐼𝑛 ⊗ 𝑆)𝑊(𝑡) where, 𝑋̃ = [𝑥̃1 , 𝑥̃2 , … , 𝑥̃𝑁1 , 𝑥̃𝑁1 +1 , … , 𝑥̃𝑛 ]𝑇 and 1𝑛 = [1 1 … 1]𝑇 . Theorem 1. Assume that Assumption1 and Assumption2 are satisfied. i.
The cluster consensus problem for system (6) is achieved with controller (8) if 𝑃 is a positive definite matrix satisfying:
𝐴𝑃 + 𝑃𝐴𝑇 + 𝑙 2 𝐶𝐶 𝑇 + (1 − 𝑘𝜆𝑚𝑖𝑛 (𝑀))𝐵𝐵 𝑇 + 𝑆𝑆 𝑇 + 𝛽𝑃 [ 𝑃
𝑃 ] −(𝜌 + 1)𝐼𝑝 2
(11)
1 2
ii.
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where 𝛽 > 0 and take 𝜃 = 𝐵 𝑇 𝑃−1.
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<0
The errors between two agents can achieve to the region 𝜇, When 𝑤𝑖 (𝑡) ≠ 0 and
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Assumption 3 and part i are established, where 𝜇 = {‖𝑋̃(𝑡)‖ ≤ √𝛽 𝜆
𝑛
𝑚𝑖𝑛 (𝐼𝑛 ⊗𝑃
−1 )
𝛾}.
Proof: The following Lyapunov function is considered: (12)
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𝑉 = 𝑋̃ 𝑇 (𝐼𝑛 ⊗ 𝑃−1 )𝑋̃
The fractional-order derivative of 𝑉 is given as follows:
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𝑉 𝛼 ≤ 𝑋̃ 𝑇 (𝐼𝑛 ⊗ 𝑃 −1 𝐴 + 𝐴𝑇 𝑃−1 )𝑋̃ + 𝑋̃ 𝑇 ∆𝐴𝑇 (𝐼𝑛 ⊗ 𝑃 −1 )𝑋̃ + 𝑋̃ 𝑇 (𝐼𝑛 ⊗ 𝑃 −1 )∆𝐴 𝑋̃ + 2𝑋̃ 𝑇 (𝐼𝑛 ⊗ 𝑃−1 𝐶)[𝐹(𝑡, 𝑋) − 𝐹(𝑡, 𝑋𝑟 )] − 2𝑘𝑋̃ 𝑇 (𝐼𝑛 ⊗ 𝑃−1 )(𝑀 ⊗ 𝐵𝜃)𝑋̃ + 2𝑋̃ 𝑇 (𝐼𝑛 ⊗ 𝑃−1 )(𝐼𝑛 ⊗ 𝑆) 𝑊(𝑡)
(13)
Consider Lemma 5 to continue the proof of Lyapunov stability. Lemma 5. The following inequality with vectors 𝑥, 𝑦 and matrices 𝑃, 𝐷 and 𝑆 is established:
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(14)
2𝑥 𝑇 𝐷𝑆𝑦 ≤ 𝑥 𝑇 𝐷𝑃𝐷𝑇 𝑥 + 𝑦 𝑇 𝑆 𝑇 𝑃−1 𝑆𝑦
By using Lemma 5 and Assumption 1and by getting ∆𝐴 = (𝐼𝑛 ⊗ 𝐵)𝐸 is given (16) as follows: 𝑉 𝛼 ≤ 𝑋̃ 𝑇 (𝐼𝑛 ⊗ 𝑃−1 𝐴 + 𝐴𝑇 𝑃−1 )𝑋̃ + 𝑋̃ 𝑇 𝐸 𝑇 (𝐼𝑛 ⊗ 𝐵 𝑇 𝑃−1 )𝑋̃ + 𝑋̃ 𝑇 (𝐼𝑛 ⊗ 𝑃−1 𝐵)𝐸 𝑋̃ + 𝑋̃ 𝑇 [𝐼𝑛 ⊗ (𝑙 2 𝑃−1 𝐶𝐶 𝑇 𝑃 −1 + 𝐼𝑝 )]𝑋̃
(15)
− 2𝑘𝑋̃ 𝑇 (𝑀 ⊗ 𝑃−1 𝐵𝜃)𝑋̃ + 2𝑋̃ 𝑇 (𝐼𝑛 ⊗ 𝑃−1 𝑆) 𝑊(𝑡) 1
Substituting 𝜃 = 𝐵 𝑇 𝑃−1, (16) is rewritten as follows: 2
V α (𝑡) ≤ 𝑋̃ 𝑇 [𝐼𝑛 ⊗ (𝑃−1 𝐴 + 𝐴𝑇 𝑃−1 )]𝑋̃ + 𝑋̃ 𝑇 𝐸 𝑇 𝐸𝑋̃
f
+ 𝑋̃ 𝑇 (𝐼𝑛 ⊗ 𝑃−1 𝐵𝐵 𝑇 𝑃−1 )𝑋̃
roo
+ 𝑋̃ 𝑇 [𝐼𝑛 ⊗ (𝑙 2 𝑃−1 𝐶𝐶 𝑇 𝑃−1 + 𝐼𝑝 )]𝑋̃
(16)
+ 𝑊𝑇 𝑊
rep
− 𝑘𝑋̃ 𝑇 (𝑀 ⊗ 𝑃−1 𝐵𝐵 𝑇 𝑃−1 )𝑋̃ + 𝑋̃ 𝑇 [𝐼𝑛 ⊗ (𝑃−1 𝑆𝑆 𝑇 𝑃 −1 )]𝑋̃
Using 𝐸𝑖𝑇 𝐸𝑖 < 𝜌2 𝐼𝑝 which implies 𝐸 𝑇 𝐸 < 𝜌2 𝐼𝑛𝑝 = (𝐼𝑛 ⊗ 𝜌2 𝐼𝑝 ) is given (18) as follows:
lP
V α (𝑡) ≤ 𝑋̃ 𝑇 [𝐼𝑛 ⊗ (𝑃−1 𝐴 + 𝐴𝑇 𝑃−1 + 𝑙 2 𝑃−1 𝐶𝐶 𝑇 𝑃 −1 + 𝐼𝑝 + 𝑃 −1 𝐵𝐵 𝑇 𝑃−1
rna
+ 𝑃−1 𝑆𝑆 𝑇 𝑃−1 ) + (𝐼𝑛 ⊗ 𝜌2 𝐼𝑝 )]𝑋̃ − 𝑘𝑋̃ 𝑇 (𝑀 ⊗ 𝑃−1 𝐵𝐵 𝑇 𝑃−1 )𝑋̃ + 𝑊 𝑇 𝑊
(17)
= 𝜀 𝑇 [𝐼𝑛 ⊗ (𝐴𝑃 + 𝑃𝐴𝑇 + 𝑙 2 𝐶𝐶 𝑇 + (𝜌2 + 1)𝑃𝑇 𝑃 + 𝐵𝐵 𝑇
Jo u
+ 𝑆𝑆 𝑇 )]𝜀 − 𝑘𝜀 𝑇 (𝑀 ⊗ 𝐵𝐵 𝑇 )𝜀 + 𝑊 𝑇 𝑊
where 𝜀 = (𝐼𝑛 ⊗ 𝑃−1 )𝑋̃ . V α (𝑡) ≤ 𝜀 𝑇 [𝐼𝑛 ⊗ (𝐴𝑃 + 𝑃𝐴𝑇 + 𝑙 2 𝐶𝐶 𝑇 + (𝜌2 + 1)𝑃𝑇 𝑃 + 𝐵𝐵 𝑇 + 𝑆𝑆 𝑇 )]𝜀 − 𝑘𝜆𝑚𝑖𝑛 (𝑀)𝜀 𝑇 (𝐼𝑛 ⊗ 𝐵𝐵 𝑇 )𝜀 + 𝑊 𝑇 𝑊
(18)
Journal Pre-proof
Using (12) and Schur complement lemma [16], (20) is rewritten as follows: V α (𝑡) ≤ −𝛽𝜀 𝑇 (𝐼𝑛 ⊗ 𝑃)𝜀 + ‖𝑊‖2 = −𝛽𝑋̃ 𝑇 (𝐼𝑛 ⊗ 𝑃−1 )𝑋̃ + ‖𝑊‖2
(19)
≤ −𝛽 𝑉(𝑡) + 𝑛𝛾 2 Let 𝑅(𝑡) = 𝑉(𝑡) −
𝑛𝛾2 𝛽
, so:
𝑅 𝛼 (𝑡) ≤ −𝛽 𝑉(𝑡)
(20)
By using function 𝑚(𝑡), (22) can be write as follows: 𝑅 𝛼 (𝑡) + 𝑚(𝑡) = −𝛽 𝑉(𝑡)
(21)
So the unique solution of (22) is as follows: 𝑅(𝑡) = 𝑅(0)𝐸𝛼 (−𝛽𝑡 𝛼 ) − 𝑚(𝑡) ∗ (𝑡 𝛼−1 𝐸𝛼 (−𝛽𝑡 𝛼 ))
f
𝑛𝛾 2 = 𝑅(𝑡) ≤ 𝑅(0)𝐸𝛼 (−𝛽𝑡 𝛼 ) 𝛽
roo
Therefore 𝑉(𝑡) −
(22)
(23)
2
rep
From (12), 𝑉(𝑡) ≥ 𝜆𝑚𝑖𝑛 (𝐼𝑛 ⊗ 𝑃 −1 )‖𝑋̃(𝑡)‖ and lim 𝑅(0)𝐸𝛼 (−𝛽𝑡 𝛼 ) → 0, so:
lP
𝑛 lim ‖𝑋̃(𝑡)‖ ≤ √ 𝛾 𝑡→∞ 𝛽 𝜆𝑚𝑖𝑛 (𝐼𝑛 ⊗ 𝑃 −1 )
𝑡→∞
(24)
When 𝑤𝑖 (𝑡) = 0, V α ≤ −𝛽 𝑉(𝑡) which we can conclude from Lemma 4, Mittag-
Jo u
Definition 2.
rna
Leffler stability results in (11). System (6) is achieved to robust cluster consensus by using
Journal Pre-proof
4.2.Robust cluster consensus of fractional-order nonlinear multi agent systems by adaptive law Consider a group of 𝑛 agents which the model of 𝑖 th agent, 𝑖 = 1, … , 𝑛 is described in general form as follows: 𝑥𝑖𝛼 = (𝐴 + ∆𝐴𝑖 )𝑥𝑖 + 𝐶 𝑓𝑖̂ (𝑡, 𝑥𝑖 ) + 𝐵𝑢𝑖 (𝑡) + 𝑆𝑤𝑖 (𝑡)
(25)
where 𝑥𝑖 ∈ ℝ𝑝 are the states of 𝑖 th agent for 𝑖 = 1,2, … , 𝑛 with 𝑝 dimensions, 𝛼 ∈ (0,1], 𝑓𝑖̂ (𝑡, 𝑥𝑖 ) = (𝑓1 (𝑥𝑖 , 𝑡), 𝑓2 (𝑥𝑖 , 𝑡), … , 𝑓𝑝 (𝑥𝑖 , 𝑡))𝑇 is a nonlinear vector-valued function for each cluster which satisfies Lipschitz condition, 𝑢𝑖 (𝑡) is the control input, 𝑤𝑖 (𝑡) is the external disturbance, ∆𝐴𝑖 is the parameter uncertainty of agent 𝑖, 𝐴, 𝐵, 𝐶 and 𝑆 are constant
The 𝑖 th leader is described by:
rep
𝛼 𝑥𝑟𝑖 = 𝐴𝑥𝑟𝑖 + 𝐶𝑓𝑖̂ (𝑡, 𝑥𝑟𝑖 ), 𝑖 = 1,2, … , 𝑛
roo
f
matrices.
(26)
The adaptive sliding mode controller and adaptive law are designed as follows: 𝑛
𝑢𝑖 (𝑡) = −𝜃𝑖 𝑘 (∑ 𝑎𝑖𝑗 [𝑥𝑖 − 𝑥𝑗 ] + 𝑑𝑖 [𝑥𝑖 − 𝑥𝑟𝑖 ]) 𝑗=1
lP
𝑛
(27)
− 𝜔. 𝑠𝑔𝑛 (∑ 𝑎𝑖𝑗 [𝑥𝑖 − 𝑥𝑗 ] + 𝑑𝑖 [𝑥𝑖 − 𝑥𝑟𝑖 ]) , 𝑗=1
𝑛
𝜃𝑖𝛼
rna
𝑖, 𝑗 = 1,2, … , 𝑛, 𝑖 ≠ 𝑗
𝑇
𝑛
= 𝛽𝑖 (𝑘 ∑ 𝑎𝑖𝑗 [𝑥𝑖 − 𝑥𝑗 ] + 𝑑𝑖 [𝑥𝑖 − 𝑥𝑟𝑖 ]) (𝑘 ∑ 𝑎𝑖𝑗 [𝑥𝑖 − 𝑥𝑗 ] 𝑗=1
𝑗=1
(28)
Jo u
+ 𝑑𝑖 [𝑥𝑖 − 𝑥𝑟𝑖 ])
where 𝑘 > 0, 𝑎𝑖𝑗 , 𝑖, 𝑗 = 1,2, … , 𝑛 is the (𝑖, 𝑗)th element of the adjacency matrix 𝐴. 𝐷 = 𝑑𝑖𝑎𝑔( 𝑑1 , 𝑑2 , … , 𝑑𝑛 ) is the leader adjacency matrix. 𝑑𝑖 is used for describing the leader
Journal Pre-proof
information exchange to 𝑖 th agent. 𝑑𝑖 = 1, if information is transmitted from leader to 𝑖 th agent and otherwise 𝑑𝑖 = 0. For each cluster, 𝑑𝑖 = 1 for only one follower. By substituting the controller (28), into the system (26), we can write: 𝑛
𝑥𝑖𝛼
= (𝐴 + ∆𝐴𝑖 )𝑥𝑖 + 𝐶 𝑓(𝑥𝑖 , 𝑡) − 𝑘𝐵𝜃𝑖 ∑ 𝑎𝑖𝑗 [𝑥𝑖 − 𝑥𝑗 ] + 𝑑𝑖 [𝑥𝑖 − 𝑥𝑟𝑖 ] 𝑗=1 𝑛
(29)
− 𝜔. 𝑠𝑔𝑛 (∑ 𝑎𝑖𝑗 [𝑥𝑖 − 𝑥𝑗 ] + 𝑑𝑖 [𝑥𝑖 − 𝑥𝑟𝑖 ]) + 𝑆𝑤𝑖 (𝑡), 𝑗=1
𝑖 = 1, … , 𝑛 where
𝑋 = [𝑥1 , 𝑥2 , … , 𝑥𝑁1 , 𝑥𝑁1 +1 , … , 𝑥𝑁1 +𝑁2 ]𝑇 ,
∆𝐴 = diag(∆𝐴1 , ∆𝐴2 , … , ∆𝐴𝑛 ),
is
roo
𝑊(𝑡) = [𝑤1 , 𝑤2 , … , 𝑤𝑁 , 𝑤𝑁+1 , … , 𝑤𝑁+𝑀 ]𝑇 , ⊗
f
𝐹(𝑡, 𝑋) = [𝑓𝑖̂ (𝑡, 𝑥1 (𝑡)), 𝑓𝑖̂ (𝑡, 𝑥2 (𝑡)), … , 𝑓𝑖̂ (𝑡, 𝑥𝑁1 (𝑡)) , 𝑓𝑖̂ (𝑡, 𝑥𝑁1 +1 (𝑡)) , … , 𝑓𝑖̂ (𝑡, 𝑥𝑛 (𝑡))]𝑇 ,
𝑑𝑖𝑎𝑔(𝜃1, , 𝜃2 , … , 𝜃𝑛 ).
Kronecker
product
and
𝜃=
rep
Let 𝑥̃𝑖 = 𝑥𝑖 − 𝑥𝑟𝑖 , so we can write as follows:
𝑋̃ 𝛼 (𝑡) = (𝐼𝑛 ⊗ 𝐴 + ∆𝐴)𝑋̃ + (𝐼𝑛 ⊗ 𝐶)(𝐹(𝑡, 𝑋) − 𝐹(𝑡, 𝑋𝑟 )) − (𝜃𝑀 ⊗ 𝐵𝑘)𝑋̃ (30)
− (𝜔𝐼𝑛 ⊗ 𝐵) 𝑠𝑔𝑛 ((𝑀 ⊗ 𝐼𝑛 )𝑋̃) + (𝐼𝑛 ⊗ 𝑆)𝑊(𝑡)
lP
where, 𝑋̃ = [𝑥̃1 , 𝑥̃2 , … , 𝑥̃𝑁 , 𝑥̃𝑁+1 , … , 𝑥̃𝑛 ]𝑇 , 1𝑛 = [1 1 … 1]𝑇 , 𝑀 = 𝐻 + 𝐷 , 𝐻 is the Laplacian matrix of G described for two clusters which is described in (10).
i.
rna
Theorem 2. Assume that Assumption1 and Assumption2 are satisfied. The cluster consensus problem for system (26) is achieved with controller (28) if 𝑃 is
Jo u
a positive definite matrix satisfying: 𝐴𝑃 + 𝑃𝐴𝑇 + 𝑙 2 𝐶𝐶 𝑇 + (1 − 2𝜃0 𝜆𝑚𝑖𝑛 (𝑀))𝐵𝐵 𝑇 + 𝑆𝑆 𝑇 + 𝛽𝑃 [ 𝑃 <0
where 𝛽 > 0 and take 𝑘 = 𝐵 𝑇 𝑃−1 .
𝑃 ] −(𝜌 + 1)𝐼𝑝 2
(31)
Journal Pre-proof
The errors between two agents can achieve to the region 𝜇, When 𝑤𝑖 (𝑡) ≠ 0 and
ii.
Assumption 3 and part i are established, where 𝜇 = {‖𝑋̃(𝑡)‖ ≤ √
𝑛
𝛽 𝜆𝑚𝑖𝑛 (𝑀⊗𝑃 −1 )
𝛾}.
Proof: The following Lyapunov function is considered: 𝑛
𝑉 = 𝑋̃ 𝑇 (𝑀 ⊗ 𝑃−1 )𝑋̃ + ∑ 𝑖=1
1 (𝜃 − 𝜃0 )2 𝛽𝑖 𝑖
(32)
The fractional-order derivative of 𝑉 is given as follows: 𝑉 𝛼 ≤ 𝑋̃ 𝑇 (𝑀 ⊗ 𝑃−1 𝐴 + 𝐴𝑇 𝑃−1 )𝑋̃ + 𝑋̃ 𝑇 ∆𝐴𝑇 (𝑀 ⊗ 𝑃−1 )𝑋̃ + 𝑋̃ 𝑇 (𝑀 ⊗ 𝑃−1 )∆𝐴 𝑋̃ + 2𝑋̃ 𝑇 (𝑀 ⊗ 𝑃−1 𝐶)[𝐹(𝑡, 𝑋) − 𝐹(𝑡, 𝑋𝑟 )]) − 𝑋̃ 𝑇 (𝑀 ⊗ 𝑃−1 )(𝜃𝑀 ⊗ 𝐵𝑘)𝑋̃ − 𝑋̃ 𝑇 (𝜃𝑀 ⊗ 𝑘 𝑇 𝐵 𝑇 )(𝑀 ⊗ 𝑃−1 )𝑋̃ − 2𝑋̃ 𝑇 𝜔(𝑀 ⊗ 𝑃−1 𝐵) 𝑠𝑔𝑛 ((𝑀 ⊗ 𝐼𝑛 )𝑋̃) 𝑛
𝑇
𝑛
𝑛
(33)
𝑖=1
roo
f
+2 ∑(𝜃𝑖 − 𝜃0 ) (𝑘 ∑ 𝑎𝑖𝑗 (𝑥̃𝑖 − 𝑥̃𝑗 ) + 𝑑𝑖 𝑥̃𝑖 ) (𝑘 ∑ 𝑎𝑖𝑗 (𝑥̃𝑖 − 𝑥̃𝑗 ) + 𝑑𝑖 𝑥̃𝑖 ) 𝑗=1
𝑗=1
+ 2𝑋̃ 𝑇 (𝑀 ⊗ 𝑃−1 )(𝐼𝑛 ⊗ 𝑆) 𝑊(𝑡)
rep
By using Lemma 5 and Assumption 1and by getting ∆𝐴 = (𝐼𝑛 ⊗ 𝐵)𝐸 is given (35) as follows:
𝑉 𝛼 ≤ 𝑋̃ 𝑇 (𝑀 ⊗ 𝑃−1 𝐴 + 𝐴𝑇 𝑃−1 )𝑋̃ + 𝑋̃ 𝑇 𝐸 𝑇 (𝑀 ⊗ 𝐵 𝑇 𝑃−1 )𝑋̃
lP
+ 𝑋̃ 𝑇 (𝑀 ⊗ 𝑃−1 𝐵)𝐸 𝑋̃ + 𝑋̃ 𝑇 [𝑀 ⊗ (𝑙 2 𝑃−1 𝐶𝐶 𝑇 𝑃 −1 + 𝐼𝑝 )]𝑋̃
𝑛
rna
− 2𝑋̃ 𝑇 (𝑀𝜃𝑀 ⊗ 𝑃−1 𝐵𝑘)𝑋̃ 𝑛
𝑇
(34) 𝑛
+2 ∑(𝜃𝑖 − 𝜃0 ) (𝑘 ∑ 𝑎𝑖𝑗 (𝑥̃𝑖 − 𝑥̃𝑗 ) + 𝑑𝑖 𝑥̃𝑖 ) (𝑘 ∑ 𝑎𝑖𝑗 (𝑥̃𝑖 − 𝑥̃𝑗 ) + 𝑑𝑖 𝑥̃𝑖 ) 𝑗=1
𝑗=1
+ 2𝑋̃ 𝑇 (𝑀 ⊗ 𝑃 −1 𝑆) 𝑊(𝑡)
Jo u
𝑖=1
Substituting 𝑘 = 𝐵 𝑇 𝑃−1, (35) is rewritten as follows:
Journal Pre-proof
V α (𝑡) ≤ 𝑋̃ 𝑇 [𝑀 ⊗ (𝑃−1 𝐴 + 𝐴𝑇 𝑃 −1 )]𝑋̃ + 𝑋̃ 𝑇 𝐸 𝑇 (𝑀 ⊗ 𝐼𝑝 )𝐸𝑋̃ + 𝑋̃ 𝑇 (𝑀 ⊗ 𝑃−1 𝐵𝐵 𝑇 𝑃−1 )𝑋̃ + 𝑋̃ 𝑇 [𝑀 ⊗ (𝑙 2 𝑃−1 𝐶𝐶 𝑇 𝑃 −1 + 𝐼𝑝 )]𝑋̃ − 2𝑋̃ 𝑇 (𝑀𝜃𝑀 ⊗ 𝑃−1 𝐵𝐵 𝑇 𝑃−1 )𝑋̃ 𝑛
𝑇
𝑛
(35) 𝑛
+2 ∑(𝜃𝑖 − 𝜃0 ) (𝑘 ∑ 𝑎𝑖𝑗 (𝑥̃𝑖 − 𝑥̃𝑗 ) + 𝑑𝑖 𝑥̃𝑖 ) (𝑘 ∑ 𝑎𝑖𝑗 (𝑥̃𝑖 − 𝑥̃𝑗 ) + 𝑑𝑖 𝑥̃𝑖 ) 𝑖=1
𝑗=1
𝑗=1
+ 𝑋̃ 𝑇 [𝑀 ⊗ (𝑃−1 𝑆𝑆 𝑇 𝑃−1 )]𝑋̃ + 𝑊 𝑇 𝑊 Using 𝐸𝑖𝑇 𝐸𝑖 < 𝜌2 𝐼𝑝 which implies 𝐸 𝑇 𝐸 < 𝜌2 𝐼𝑛𝑝 = (𝐼𝑛 ⊗ 𝜌2 𝐼𝑝 ) is given (37) as
f
follows:
roo
V α (𝑡) ≤ 𝑋̃ 𝑇 [𝑀 ⊗ (𝑃−1 𝐴 + 𝐴𝑇 𝑃−1 + 𝑙 2 𝑃−1 𝐶𝐶 𝑇 𝑃 −1 + 𝐼𝑝 + 𝑃 −1 𝐵𝐵 𝑇 𝑃−1 + 𝑃−1 𝑆𝑆 𝑇 𝑃−1 ) + (𝑀 ⊗ 𝜌2 𝐼𝑝 )]𝑋̃ − 2𝜃0 𝑋̃ 𝑇 (𝑀2
rep
⊗ 𝑃−1 𝐵𝐵 𝑇 𝑃−1 )𝑋̃ + 𝑊 𝑇 𝑊
(36)
= 𝜀 𝑇 [𝑀 ⊗ (𝐴𝑃 + 𝑃𝐴𝑇 + 𝑙 2 𝐶𝐶 𝑇 + (𝜌2 + 1)𝑃𝑇 𝑃 + 𝐵𝐵 𝑇 + 𝑆𝑆 𝑇 − 2𝜃0 𝜆𝑚𝑖𝑛 (𝑀)𝐵𝐵 𝑇 )]𝜀 + 𝑊 𝑇 𝑊
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where 𝜀 = (𝐼𝑛 ⊗ 𝑃−1 )𝑋̃ .
V α (𝑡) ≤ 𝜀 𝑇 [𝑀 ⊗ (𝐴𝑃 + 𝑃𝐴𝑇 + 𝑙 2 𝐶𝐶 𝑇 + (𝜌2 + 1)𝑃𝑇 𝑃 + (1 (37)
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− 2𝜃0 𝜆𝑚𝑖𝑛 (𝑀))𝐵𝐵 𝑇 + 𝑆𝑆 𝑇 )]𝜀 + 𝑊 𝑇 𝑊 Using (32) and Schur complement lemma [16], (38) is rewritten as follows:
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V α (𝑡) ≤ −𝛽𝜀 𝑇 (𝑀 ⊗ 𝑃)𝜀 + ‖𝑊‖2 = −𝛽𝑋̃ 𝑇 (𝑀 ⊗ 𝑃−1 )𝑋̃ + ‖𝑊‖2 ≤ −𝛽 𝑉(𝑡) + 𝑛𝛾
Let 𝑅(𝑡) = 𝑉(𝑡) −
𝑛𝛾2 𝛽
2
, so:
(38)
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𝑅 𝛼 (𝑡) ≤ −𝛽 𝑉(𝑡)
(39)
By using function 𝑚(𝑡), (40) can be write as follows: 𝑅 𝛼 (𝑡) + 𝑚(𝑡) = −𝛽 𝑉(𝑡)
(40)
So the unique solution of (41) is as follows: 𝑅(𝑡) = 𝑅(0)𝐸𝛼 (−𝛽𝑡 𝛼 ) − 𝑚(𝑡) ∗ (𝑡 𝛼−1 𝐸𝛼 (−𝛽𝑡 𝛼 ))
(41)
Therefore 𝑛𝛾 2 𝑉(𝑡) − = 𝑅(𝑡) ≤ 𝑅(0)𝐸𝛼 (−𝛽𝑡 𝛼 ) 𝛽
(42) 2
From (32), 𝑉(𝑡) ≥ 𝜆𝑚𝑖𝑛 (𝑀 ⊗ 𝑃−1 )‖𝑋̃(𝑡)‖ and lim 𝑅(0)𝐸𝛼 (−𝛽𝑡 𝛼 ) → 0, so:
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𝑛 lim ‖𝑋̃(𝑡)‖ ≤ √ 𝛾 𝑡→∞ 𝛽 𝜆𝑚𝑖𝑛 (𝑀 ⊗ 𝑃 −1 )
f
𝑡→∞
(43)
When 𝑤𝑖 (𝑡) = 0, from (39) V α ≤ −𝛽 𝑉(𝑡) which we can conclude from Lemma 4,
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Mittag-Leffler stability results in (31). System (26) is achieved to robust cluster consensus by using Definition 2.
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4.3.Robust cluster consensus of fractional-order nonlinear multi agent systems by σ modification adaptive law
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Consider a group of 𝑛 agents and leaders which are modeled by (26) and (27). The adaptive sliding mode controller and adaptive law are designed as follows: 𝑛
𝑢𝑖 (𝑡) = −𝜃𝑖 𝑘 (∑ 𝑎𝑖𝑗 [𝑥𝑖 − 𝑥𝑗 ] + 𝑑𝑖 [𝑥𝑖 − 𝑥𝑟𝑖 ])
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𝑗=1
𝑛
− 𝜔. 𝑠𝑔𝑛 (∑ 𝑎𝑖𝑗 [𝑥𝑖 − 𝑥𝑗 ] + 𝑑𝑖 [𝑥𝑖 − 𝑥𝑟𝑖 ]) , 𝑗=1
𝑖, 𝑗 = 1,2, … , 𝑛, 𝑖 ≠ 𝑗
(44)
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𝑇
𝑛
𝜃𝑖𝛼
𝑛
= 𝛽𝑖 (𝑘 ∑ 𝑎𝑖𝑗 [𝑥𝑖 − 𝑥𝑗 ] + 𝑑𝑖 [𝑥𝑖 − 𝑥𝑟𝑖 ]) (𝑘 ∑ 𝑎𝑖𝑗 [𝑥𝑖 − 𝑥𝑗 ] 𝑗=1
𝑗=1
(45)
+ 𝑑𝑖 [𝑥𝑖 − 𝑥𝑟𝑖 ]) − 𝜎(𝜃𝑖 − 𝜃0 ) Like section 4.2, let 𝑥̃𝑖 = 𝑥𝑖 − 𝑥𝑟𝑖 , so we can write as follows: 𝑋̃ 𝛼 (𝑡) = (𝐼𝑛 ⊗ 𝐴 + ∆𝐴)𝑋̃ + (𝐼𝑛 ⊗ 𝐶)(𝐹(𝑡, 𝑋) − 𝐹(𝑡, 𝑋𝑟 )) − (𝜃𝑀 ⊗ 𝐵𝑘)𝑋̃ − (𝜔𝐼𝑛 ⊗ 𝐵) 𝑠𝑔𝑛 ((𝑀 ⊗ 𝐼𝑛 )𝑋̃) + (𝐼𝑛 ⊗ 𝑆)𝑊(𝑡)
(46)
Robust cluster consensus is achieved if the conditions of Theorem 2 are held. For proof it, the Lyapunov function (33) is considered. Like section 4.2, the fractional-order
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derivative of 𝑉 is given as follows:
V α (𝑡) ≤ 𝑋̃ 𝑇 [𝑀 ⊗ (𝑃−1 𝐴 + 𝐴𝑇 𝑃−1 + 𝑙 2 𝑃−1 𝐶𝐶 𝑇 𝑃 −1 + 𝐼𝑝 + 𝑃 −1 𝐵𝐵 𝑇 𝑃−1 + 𝑃−1 𝑆𝑆 𝑇 𝑃−1 ) + (𝑀 ⊗ 𝜌2 𝐼𝑝 )]𝑋̃ − 2𝜃0 𝑋̃ 𝑇 (𝑀2
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𝑛
⊗ 𝑃 𝐵𝐵 𝑃 )𝑋̃ − 2𝜎 ∑(𝜃𝑖 − 𝜃0 )2 + 𝑊 𝑇 𝑊 −1
𝑇 −1
(47)
𝑖=1
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≤ 𝜀 𝑇 [𝑀 ⊗ (𝐴𝑃 + 𝑃𝐴𝑇 + 𝑙 2 𝐶𝐶 𝑇 + (𝜌2 + 1)𝑃𝑇 𝑃 + 𝐵𝐵 𝑇 + 𝑆𝑆 𝑇 − 2𝜃0 𝜆𝑚𝑖𝑛 (𝑀)𝐵𝐵 𝑇 )]𝜀 + ‖𝑊‖2
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This proof is continued like previous section and robust cluster consensus is achieved. 5. Simulation results
In this section, three numerical simulation examples are given to evaluate the
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theoretical results.
Example 1: In this example, each agent is a single-link flexible-joint manipulator shown in Fig. 3. The revolving joints are actuated by a DC motor and a linear spring is used to
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model the elasticity of the joint. The state-space model which the states of the system are position and velocity of motor and link is as follows: 𝛼 𝜃𝑚 = 𝜔𝑚 𝑘 𝑘 𝐾𝑇 𝛼 𝜔𝑚 = (𝜃𝑙 − 𝜃𝑚 ) − 𝜔𝑚 + 𝑢 𝐽𝑚 𝐽𝑚 𝐽𝑚 𝜃𝑙𝛼 = 𝜔𝑙 𝑘 𝑚𝑔ℎ 𝜔𝑙𝛼 = − (𝜃𝑙 − 𝜃𝑚 ) − 𝑠𝑖𝑛(𝜃𝑙 ) 𝐽𝑙 𝐽𝑙
where 𝐽𝑚 and 𝐽𝑙 are the inertia of the motor and link, 𝜃𝑚 and 𝜃𝑙 are the angular rotation of the motor and the angular position of the link, respectively; and 𝜔𝑚 and 𝜔𝑙 are the angular velocity of the motor and link [44].
parameters are defined as follows:
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The manipulator is described by the form of (6) with α = 0.95 and the numerical
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0 1 0 −48.6 −1.26 48.6 𝐴=[ 0 0 0 1.95 0 −1.95
0 0 0 21.6 ],𝐵 = [ ] , 𝐶 = 𝐼4 10 0 0 0
𝑇
𝑥𝑖 (𝑡) = [𝜃𝑚 𝑖 , 𝜔𝑚 𝑖 , 𝜃𝑙 𝑖 , 𝜔𝑙 𝑖 ] = [𝑥𝑖1 (𝑡), 𝑥𝑖2 (𝑡), 𝑥𝑖3 (𝑡), 𝑥𝑖4 (𝑡)]𝑇 ,
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𝑥𝑖3 (𝑡) 𝑇 𝑓1 (𝑡, 𝑥𝑖 (𝑡)) = (0,0,0, 𝑠𝑖𝑛 ( ) ) , 10
𝑥𝑖3 (𝑡) 𝑇 𝑓2 (𝑡, 𝑥𝑖 (𝑡)) = (0,0,0, 2𝑠𝑖𝑛 ( ) ) , 𝐸𝑖 = [0 0 𝛾𝑖 0], 10
𝑙 = 0.333,
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𝑤1 (𝑡) = (0,1.1 sin(8𝑡 − 1) , 0,0)𝑇 , 𝑤2 (𝑡) = (0, − cos(2t) , 0,0)𝑇 , 𝑤3 (𝑡) = (0,0.8 sin(𝑡) + 0.7 cos(𝑡) , 0,0)𝑇 , 𝑤4 (𝑡) = (0,0.2 cos(11𝑡 − 4) , 0,0)𝑇 ,
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𝑤5 (𝑡) = (0,1.5 sin(7𝑡 − 1) , 0,0)𝑇 , 𝑤6 (𝑡) = (0, cos(3𝑡) , 0,0)𝑇 , 𝑤7 (𝑡) = (0, −0.6 sin(𝑡) − 0.9 cos(𝑡) , 0,0)𝑇
Cluster consensus for the manipulator is achieved for graph in Fig. 4 with 𝑙 = 0.333, 𝛽 = 0.7, 𝜌 = 1 and 𝛾𝑖 ∈ [−1,1] is the uncertainty which is randomly chose. 𝑘 = 2.11 and 𝑃 is achieved by LMI (12) as follows:
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0.9438 −1.0792 −1.0792 12.2318 𝑃=[ 0.8298 −0.404 −0.1177 −0.1063 Simulation results are shown in Fig. 5.
0.8298 −0.404 0.9114 −0.08
−0.1177 −0.1063 ] −0.08 0.028
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Figure. 3. single-link manipulator with a flexible joint
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Figure. 4. The topology interaction of the agents.
(a)
(b)
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(c)
(d)
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Figure 5. Cluster consensus for (a) 𝐱 𝐢𝟏 (𝐭), (b) 𝐱 𝐢𝟐 (𝐭), (c) 𝐱 𝐢𝟑 (𝐭), (d) 𝐱 𝐢𝟒 (𝐭), 𝐢 = 𝟏, … , 𝟕. The state trajectories of agents are shown in Fig. 5. In Fig. 5(a), first state of agents
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x11 to x31 converge to one consensus value which is time varying and x41 to x71 converge to another consensus value at time 6.5 second. In Fig. 5(b), second state of agents x12 to x32 converge to one consensus value which is time-varying and x42 to x72 converge to another
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consensus value at time 6.5 second. In Fig. 5(c), third state of agents x13 to x33 converge to one consensus value which is time-varying and x43 to x73 converge to another consensus
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value at time 4 second. In Fig. 5(d), fourth state of agents x14 to x34 converge to one consensus value which is time varying and x44 to x74 converge to another consensus value at
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time 4 second. So cluster consensus is achieved for all states of agents. Example 2: A directed graph with seven followers and two leaders which communicated with first and fourth agents is shown in Fig. 4. In this example, each agent is a single-link flexible-joint manipulator and this is described like Example 1 with 𝛼 = 0.95, 𝜔 =
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𝑥𝑖3 (𝑡)
𝑥𝑖3 (𝑡)
10
10
1, 𝑓1 (𝑡, 𝑥𝑖 (𝑡)) = (0,0,0, 𝑠𝑖𝑛 (
) )𝑇 , 𝑓2 (𝑡, 𝑥𝑖 (𝑡)) = (0,0,0, 2𝑠𝑖𝑛 (
) )𝑇 , 𝑙 = 0.333,
𝛽 = 0.7, 𝜃0 = 105.8, 𝜌 = 1 and 𝛾𝑖 ∈ [−1,1] is the uncertainty which is randomly chose. 𝑃 is achieved by LMI (32) as follows: 0.5349 −0.9707 𝑃=[ 0.4700 −0.0828
−0.9707 107.0421 −0.0183 0.0009
0.4700 −0.0183 0.5195 −0.0484
−0.0828 0.0009 ] −0.0484 0.0196
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Simulation results are shown in Fig. 6.
(b)
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(a)
(c)
(d)
Figure 6. Cluster consensus for (a) 𝐱 𝐢𝟏 (𝐭), (b) 𝐱 𝐢𝟐 (𝐭), (c) 𝐱 𝐢𝟑 (𝐭), (d) 𝐱 𝐢𝟒 (𝐭), 𝐢 = 𝟏, … , 𝟕.
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Fig. 6 shows the position trajectories of agents for each state. In Fig. 6(a), first state of agents x11 to x31 follow the first leader and x41 to x71 follow the second leader after 5 second. In Fig. 6(b), second state of agents x12 to x32 follow the first leader and x42 to x72 follow the second leader after 1 second. In Fig. 6(c), third state of agents x13 to x33 follow the first leader and x43 to x73 follow the second leader after 0.5 second. In Fig. 6(d), fourth state of agents x14 to x34 follow the first leader and x44 to x74 follow the second leader after 1 second. By comparison Fig. 3 and Fig. 6, the convergence speed increases by using adaptive sliding mode controller. The control signals of each agent are shown in Fig. 7 which they are limited to specified amplitudes. Cluster consensus with α = 1 is simulated for this system and it is
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rep
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f
shown in Fig.8, therefore the convergence speed is lower than fractional order system.
Figure 7. Control signals of agents
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(a)
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rep
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f
(b)
(c)
(d)
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Figure 8. Cluster consensus with 𝜶 = 𝟏 for (a) 𝐱 𝐢𝟏 (𝐭), (b) 𝐱 𝐢𝟐 (𝐭), (c) 𝐱 𝐢𝟑 (𝐭), (d) 𝐱 𝐢𝟒 (𝐭), 𝐢 = 𝟏, … , 𝟕. Example 3: Agents and parameters are defined same as example 2 with 𝜎 = 1.
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Simulation results are shown in figure 9.
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(b)
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rep
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f
(a)
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(c)
(d)
Figure 9. Cluster consensus for (a) 𝐱 𝐢𝟏 (𝐭), (b) 𝐱 𝐢𝟐 (𝐭), (c) 𝐱 𝐢𝟑 (𝐭), (d) 𝐱 𝐢𝟒 (𝐭), 𝐢 = 𝟏, … , 𝟕.
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Fig. 9 shows the position trajectories of agents for each state and the adaptive parameters are shown in Fig.10. The comparison of these examples is shown in Table.1. So, it demonstrates the efficiency of the proposed adaptive controller and the convergence speed increases by using adaptive sliding mode controller with σ modification method.
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Figure 10. trajectories of 𝜽𝒊
7 agents
Adaptive sliding mode controller
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General form system General form system General form system
Graph
7 agents
7 agents
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System
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Table.1. Comparison of examples σ modification
Average consensus time of the states 5.25 second 1.87 second
0.975 second
6. Conclusions
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In this paper, robust cluster consensus for the general fractional-order nonlinear multi agent systems has been studied. General form nonlinear systems are considered in the presence of uncertainty and external disturbance. Robust cluster consensus is achieved by an adaptive sliding mode controller and the sufficient conditions are given. In the simulations,
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the effectiveness of the proposed results is shown and compared with integer-order systems. Future work will concentrate on robust cluster consensus of the fractional-order nonlinear multi agent systems under switching communication topology in the presence of uncertainty and external disturbance. References [1] N. Aguila-Camacho, M.A. Duarte-Mermoud, J.A. Gallegos, Lyapunov functions for fractional order systems, Communications in Nonlinear Science and Numerical Simulation, 19 (2014) 2951-2957. [2] J. Ansari, A. Gholami, A. Kazemi, Multi-agent systems for reactive power control in smart grids, International Journal of Electrical Power & Energy Systems, 83 (2016) 411-425.
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[3] G. Arthi, J.H. Park, K. Suganya, Controllability of fractional order damped dynamical systems with distributed delays, Mathematics and Computers in Simulation, 165 (2019) 7491. [4] J. Bai, G. Wen, A. Rahmani , Y. Yu, Distributed consensus tracking for the fractionalorder multi-agent systems based on the sliding mode control method, Neurocomputing, 235 (2017) 210-216.
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[8] L. Ding, P. Yu, Z.-W. Liu, Z.-H. Guan, Consensus and performance optimisation of multiagent systems with position-only information via impulsive control, IET Control Theory & Applications, 7 (2013) 16-24. [9] H. Du, S. Li, P. Shi, Robust consensus algorithm for second-order multi-agent systems with external disturbances, International Journal of Control, 85 (2012) 1913-1928.
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[10] S. El-Ferik, A. Qureshi, F.L. Lewis, Neuro-adaptive cooperative tracking control of unknown higher-order affine nonlinear systems, Automatica, 50 (2014) 798-808. [11] P. Gong, Distributed consensus of non-linear fractional-order multi-agent systems with directed topologies, IET Control Theory & Applications, 10 (2016) 2515-2525. [12] P. Gong, Distributed tracking of heterogeneous nonlinear fractional-order multi-agent systems with an unknown leader, Journal of the Franklin Institute, 354 (2017) 2226-2244.
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[19] Y. Li, Y. Huang, P. Lin, W. Ren, Distributed rotating consensus of second-order multiagent systems with nonuniform delays, Systems & Control Letters, 117 (2018) 18-22. [20] Z. Li, Z. Duan, G. Chen, Global synchronised regions of linearly coupled Lur'e systems, International Journal of Control, 84 (2011) 216-227.
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[23] Z. Li, W. Ren, X. Liu, M. Fu, Consensus of multi-agent systems with general linear and Lipschitz nonlinear dynamics using distributed adaptive protocols, IEEE Transactions on Automatic Control, 58 (2013) 1786-1791. [24] J. Liang-Hao, L. Xiao-Feng, Consensus problems of first-order dynamic multi-agent systems with multiple time delays, Chinese Physics B, 22 (2013) 040203.
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[26] J. Mei, W. Ren, J. Chen, Distributed consensus of second-order multi-agent systems with heterogeneous unknown inertias and control gains under a directed graph, IEEE Transactions on Automatic Control, 61 (2016) 2019-2034. [27] A.J. Muñoz-Vázquez, M.B. Ortiz-Moctezuma, A. Sánchez-Orta, V. Parra-Vega, Adaptive robust control of fractional-order systems with matched and mismatched disturbances, Mathematics and Computers in Simulation, 162 (2019) 85-96. [28] R. Olfati-Saber, R.M. Murray, Consensus problems in networks of agents with switching topology and time-delays, IEEE Transactions on automatic control, 49 (2004) 1520-1533.
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[35] G. Wen, Z. Duan, G. Chen, W. Yu, Consensus tracking of multi-agent systems with Lipschitz-type node dynamics and switching topologies, IEEE Transactions on Circuits and Systems I: Regular Papers, 61 (2014) 499-511.
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[36] Z. Yaghoubi, H.A. Talebi, Consensus tracking for nonlinear fractional-order multi agent systems using adaptive sliding mode controller, Mechatronic Systems and Control, 47 (2019) 194-200. [37] X. Yang, C. Li, T. Huang, Q. Song, Mittag–Leffler stability analysis of nonlinear fractional-order systems with impulses, Applied Mathematics and Computation, 293 (2017) 416-422.
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[38] S.J. Yoo, Distributed adaptive containment control of uncertain nonlinear multi-agent systems in strict-feedback form, Automatica, 49 (2013) 2145-2153. [39] W. Yu, Y. Li, G. Wen, X. Yu, J. Cao, Observer design for tracking consensus in secondorder multi-agent systems: Fractional order less than two, IEEE Transactions on Automatic Control, 62 (2017) 894-900.
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[40] W. Yu, W. Ren, W.X. Zheng, G. Chen, J. Lü, Distributed control gains design for consensus in multi-agent systems with second-order nonlinear dynamics, Automatica, 49 (2013) 2107-2115.
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[41] J. Zhan, X. Li, Cluster consensus in networks of agents with weighted cooperative– competitive interactions, IEEE Transactions on Circuits & Systems II: Express Briefs, 65 (2018) 241-245. [42] S. Zhang, Y. Yu, J. Yu, LMI conditions for global stability of fractional-order neural networks, IEEE transactions on neural networks and learning systems, 28 (2017) 2423-2433. [43] Y. Zhao, Z. Duan, G. Wen, G. Chen, Robust consensus tracking of multi-agent systems with uncertain Lur’e-type non-linear dynamics, IET Control Theory & Applications, 7 (2013) 1249-1260.
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[44] F. Zhu, Z. Han, A note on observers for Lipschitz nonlinear systems, IEEE Transactions on automatic control, 47 (2002) 1751-1754.
PII: DOI: Reference:
S0378-4754(20)30003-3 https://doi.org/10.1016/j.matcom.2020.01.002 MATCOM 4926
To appear in:
Mathematics and Computers in Simulation
Received date : 1 September 2019 Revised date : 3 January 2020 Accepted date : 4 January 2020 Please cite this article as: Z. Yaghoubi, Robust cluster consensus of general fractional-order nonlinear multi agent systems via adaptive sliding mode controller, Mathematics and Computers in Simulation (2020), doi: https://doi.org/10.1016/j.matcom.2020.01.002. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. c 2020 Published by Elsevier B.V. on behalf of International Association for Mathematics and ⃝ Computers in Simulation (IMACS).
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Manuscript
Click here to view linked References
Robust Cluster Consensus of General Fractional-Order Nonlinear Multi Agent Systems via Adaptive Sliding Mode Controller Zahra Yaghoubi1 Department of Electrical Engineering, Amirkabir University of Technology, Tehran, Iran. Abstract: In this paper robust cluster consensus is investigated for general fractional-order multi agent systems with nonlinear dynamics with dynamic uncertainty and external disturbances via adaptive sliding mode controller. First, robust cluster consensus for general fractional-order nonlinear multi agent systems is investigated with dynamic uncertainty and external disturbances which multi agent systems are weakly heterogeneous because they have identical nominal dynamics with different normbounded parameter uncertainties. Then, robust cluster consensus for the fractional-order nonlinear
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multi agent systems with general form dynamics is investigated by using adaptive sliding mode
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controller. Robust cluster consensus for general fractional-order nonlinear multi agent systems is achieved asymptotically without disturbance. It is shown that the errors between agents can converge
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to a small region in the presence of disturbances based on the linear matrix inequality (LMI) and Mittag-Leffler stability theory. Finally, simulation examples are presented for general form multi agent systems, i.e. a single-link flexible joint manipulator which demonstrates the efficiency of the proposed adaptive controller.
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Keywords: Robust cluster consensus; Fractional-order multi agent systems; Adaptive sliding mode controller; External disturbances; Dynamic uncertainty.
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1. Introduction
Application of multi agent systems is interesting topic such as formation [7],
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swarming, flocking and synchronization in physical and chemical systems and recently attracts more attention. Consensus is a one of these applications which means all the states/outputs of agents converge to the same value [32]. Consensus without leader [28] and
1
Corresponding author: Email: [email protected].
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consensus tracking with leader in which all agents follow the leader are two categories for consensus problem [31]. Recently, many researchers are studied about cluster consensus which means that agents are divided into a few clusters and these clusters may change in different situations and environments [33]. All the agents in the same cluster converge to a same attitude which is different from other clusters [41]. Multi agent systems with integer-order dynamic are studied in most literatures. Consensus problem for first-order multi agent systems is studied in [24], for second-order dynamics in [19], and for higher-order dynamics in [10]. Fractional-order dynamics can
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describe behaviors of agents in complex environments , such as traveling aircraft in rainy and dusty environment [3, 5, 17, 27] and the integer-order dynamics are parts of fractional-order dynamics [36]. In [5], control of fractional-order linear multi agent systems are investigated
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under the directed graph and the convergence speed is checked to achieve consensus. Consensus of fractional-order nonlinear multi agent systems is studied with leader and without leader under directed topologies in [11, 13]. The consensus for nonlinear fractional-
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order multi agent systems with an unknown leader is studied in [12]. In [39], The observer base consensus for fractional-order multi agent systems is investigated. Therefore, cluster
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consensus for fractional-order multi agent systems with high-order nonlinear dynamics are a new study which is investigated in this paper.
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One feature to achieve consensus is the smallest non-zero eigenvalue of the Laplacian matrix [22, 25]. Consensus via adaptive controller is used to overcome to this limitation, [4, 23]. Consensus for second-order nonlinear multi agent systems for both with and without leader via adaptive controller under undirected topologies is studied in [40]. In [38],
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consensus for nonlinear multi agent systems with uncertainty is studied via adaptive controller. The distributed consensus for second-order multi agent systems under directed topologies is investigated in [26]. One of inevitable part of many industries is influence of disturbances. For example, friction of robot arm and electromagnetic interference in multi-robot systems is defined as a external disturbances [15]. Disturbances have an effect in formation of satellite [34] and power systems [2]. The robust consensus of fractional-order multi-agent systems with exogenous disturbance is studied in [30]. Distributed robust consensus for multi agent systems which dynamics are unknown fractional-order is studied in [14]. Therefore, multi-
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agent systems with disturbance is one of the important subjects [9]. In real environment, there are unmodelled dynamics of agents but in most researches dynamics of agents are identical and this is invalid [20, 43].So study on uncertain multi-agent
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system is one of the important issues.
The main contribution of this paper is divided in fourfold:
The robust cluster consensus of fractional-order nonlinear system is presented for the first
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time. Note that the underlying dynamics are in general nonlinear form. The controller for multi agent system is designed by using adaptive sliding mode
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principles and σ modification method and the efficiency of which are compared with that of non-adaptive case.
General fractional-order multi agent system with dynamic uncertainty is investigated.
Fractional-order nonlinear system is presented with disturbances.
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The above discussion motivates that robust cluster consensus for fractional-order nonlinear multi agent systems can be investigated by using adaptive sliding mode controller
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with dynamic uncertainty and external disturbances. The Lyapunov stability theorem for fractional-order systems is used to prove the Mittag-Leffler stability and the convergence of consensus. The rest of the paper is organized as follows. In Section 2, graph theory is discussed. A brief explanation of fractional calculus is given in Section 3. In Section 4, problem formulation for fractional-order nonlinear multi agent systems is discussed. In Section 5, simulation examples are given. Finally, the conclusion is presented in Section 6. 2. Graph theory The information exchange between 𝑛 agents is described with a graph and shown by
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𝐺 = (𝑉, 𝐸, 𝐴), where 𝑉 = {𝜈1 , 𝜈2 , … , 𝜈𝑛 } is the node set and 𝐸 = {(𝜈𝑗 , 𝜈𝑖 ): 𝜈𝑗 , 𝜈𝑖 ∈ 𝑉, 𝜈𝑗 ≠ 𝜈𝑖 } ∈ 𝑉 × 𝑉 is the edge set of the graph. 𝒩𝑖 = {𝜈𝑗 ∈ 𝑉: (𝜈𝑗 , 𝜈𝑖 ) ∈ 𝐸} is the neighbors set of the
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𝑖 th agent which the 𝑖 th agent receives information from the 𝑗th agent. A directed spanning tree means there is at least a directed path from one node to all other nodes and there is in directed
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graph shown in Fig.1.
Figure. 1. A directed graph
Two matrices 𝐴 and 𝐿 describe the information exchange, where 𝐴 = [𝑎𝑖𝑗 ] ∈ ℝ𝑛×𝑛 is
a matrix with 𝑎𝑖𝑗 > 0 if (𝜈𝑗 , 𝜈𝑖 ) ∈ 𝐸, otherwise 𝑎𝑖𝑗 = 0 and 𝐿 is a positive semi-definite
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matrix which is defined as 𝐿 = 𝐷 − 𝐴 where 𝐷 = 𝑑𝑖𝑎𝑔{∑𝑛𝑗=1 𝑎𝑖𝑗 } ∈ ℝ𝑛×𝑛 . Some properties of the Kronecker product define in Lemma 1. Lemma 1. There are some properties of the Kronecker product ⊗ which is defined as follows [8]: (1) (𝜉𝐴) ⊗ B = A ⊗ (𝜉𝐵) (2) (𝐴 + 𝐵) ⊗ C = A ⊗ C + 𝐵 ⊗ C (3) (𝐴 ⊗ 𝐵)(C ⊗ D) = (AC) ⊗ (BD) (4) (A ⊗ B)T = AT ⊗ BT where A, B, C and D are matrices with appropriate dimensions and 𝜉 ∈ ℝ.
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3. Fractional-order calculus Some definitions and Lemmas of fractional-order calculus are expressed as follows. Definition 1. Caputo fractional derivative is used in this paper defined below [29]: 𝑥(𝑡) =
𝑡 1 ∫ (𝑡 − 𝜏)𝑚−𝛼−1 𝑥 (𝑚) (𝜏)𝑑𝜏, Γ(𝑚 − 𝛼) 0
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𝑐 𝛼 0𝒟𝑡
(1)
where 𝑚 − 1 < 𝛼 < 𝑚, 𝑚 ∈ ℕ+ and Γ(. ) is the gamma function.
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The notation 𝑥 𝛼 (𝑡) is used instead of 0𝑐𝒟𝑡𝛼 𝑥(𝑡) for simplicity. Definition 2. The Mittag-Leffler function is used to solve fractional-order systems like
∞
𝐸𝛼,𝛽 (𝑧) = ∑
𝑧𝑘 Γ(𝑘𝛼 + 𝛽)
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𝑘=0
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the exponential function for integer-order systems. This function with 𝛼, 𝛽 > 0 is defined as:
The stability of fractional-order systems is defined in the following Lemmas [1, 18, 42]. Lemma 2. For any time 𝑡 ≥ 𝑡0 , the following inequality is defined: 1 𝑐 𝛼 2 𝑐 𝛼 𝑡0 𝒟𝑡 𝑥 (𝑡) ≤ 𝑥(𝑡) 𝑡0 𝒟𝑡 𝑥(𝑡), 𝛼 ∈ (0,1) 2
(2)
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where 𝑥(𝑡) ∈ ℝ is a continuous and derivable function. Lemma 3. The following inequality is defined: 𝑐 𝛼 𝑡0 𝒟𝑡
(𝑥 𝑇 (𝑡)𝑃𝑥(𝑡)) ≤ 2𝑥 𝑇 (𝑡)𝑃 𝑡𝑐0 𝒟𝑡𝛼 𝑥(𝑡), 𝛼 ∈ (0,1)
(3)
where 𝑥(𝑡) ∈ ℝ𝑛 is continuous and derivable and 𝑃 ∈ ℝ𝑛×𝑛 is a positive definite symmetric matrix. The systems are proved to be asymptotically stable by using the Lyapunov direct method. The Lyapunov direct method for fractional order systems is derived from the Lyapunov direct method for integer-order systems and it leads to the Mittag-Leffler stability [21].
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Lemma 4. Mittag-Leffler stability for the fractional-order system in 𝑡0 = 0 at the equilibrium point 𝑥 = 0 is proved if the Lyapunov function 𝑉(𝑡, 𝑥(𝑡)) exists with the
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following condition: 𝛼1 ‖𝑥‖𝑐 ≤ 𝑉(𝑡, 𝑥(𝑡)) ≤ 𝛼2 ‖𝑥‖𝑐𝑑 , 𝑐 𝛼 0𝒟𝑡 𝑉(𝑡, 𝑥(𝑡))
(4)
≤ −𝜆 𝑉(𝑡, 𝑥(𝑡))
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where 𝑡 ∈ ℝ+ , 𝛼 ∈ (0,1], 𝛼1 , 𝛼2 , 𝜆, 𝑐 and 𝑑 are positive constants [37]. 4. Problem Formulation
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In this part, cluster consensus problem is explained, where the graph of agents is divided into two parts and each part reaches a different consistent value. Consider a graph
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with 𝑛 agents divided like 𝑙1 = {1,2, … , 𝑁1 }, 𝑙2 = {𝑁1 + 1, … , 𝑁1 + 𝑁2 } where 𝑛 = 𝑁1 + 𝑁2. Some assumptions for analyzing the cluster consensus problem are expressed in this section.
Assumption 1. The Lipschitz condition is defined as follow for function 𝑓(𝑥, 𝑡).
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|𝑓(𝑥2 , 𝑡) − 𝑓(𝑥1 , 𝑡)| ≤ 𝑙|𝑥2 − 𝑥1 |,
∀𝑥1 , 𝑥2 ∈ ℝ, ∀𝑡 ≥ 0
where 𝑙 is the Lipschitz constant. Assumption 2. (a) Each subnetwork or cluster has a directed spanning tree. (b) The information exchange between two clusters should be balanced. In other 𝑁1 +𝑁2 words, two clusters graph, satisfy the following conditions: ∑𝑗=𝑁 𝑎 = 0, for all 𝑖 ∈ 𝑙1, 1 +1 𝑖𝑗 1 ∑𝑁 𝑗=1 𝑎𝑖𝑗 = 0, for all 𝑖 ∈ 𝑙2 .
4.1.Robust cluster consensus of fractional-order nonlinear multi agent systems In this section, agents with general high-order nonlinear system are studied. There are
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𝑛 agents which the 𝑖 th agent, 𝑖 = 1, … , 𝑛 is modeled as follows [35]: 𝑥𝑖𝛼 = (𝐴 + ∆𝐴𝑖 )𝑥𝑖 + 𝐶 𝑓𝑖̂ (𝑡, 𝑥𝑖 ) + 𝐵𝑢𝑖 (𝑡) + 𝑆𝑤𝑖 (𝑡)
(5)
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where 𝑥𝑖 ∈ ℝ𝑝 are the states of 𝑖 th agent for 𝑖 = 1,2, … , 𝑛 with 𝑝 dimensions, 𝛼 ∈ (0,1], 𝑓𝑖̂ (𝑡, 𝑥𝑖 ) = (𝑓1 (𝑥𝑖 , 𝑡), 𝑓2 (𝑥𝑖 , 𝑡), … , 𝑓𝑝 (𝑥𝑖 , 𝑡))𝑇 is a nonlinear vector-valued function for each cluster which satisfies Lipschitz condition, 𝑢𝑖 (𝑡) is the control input, 𝑤𝑖 (𝑡) is the external
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disturbance, ∆𝐴𝑖 is the parameter uncertainty of agent 𝑖, 𝐴, 𝐵, 𝐶 and 𝑆 are constant matrices. The 𝑖 th leader is described by:
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𝛼 𝑥𝑟𝑖 = 𝐴𝑥𝑟𝑖 + 𝐶𝑓𝑖̂ (𝑡, 𝑥𝑟𝑖 ), 𝑖 = 1,2, … , 𝑛
(6)
The controller is designed as follows:
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𝑛
𝑢𝑖 (𝑡) = −𝑘𝜃 (∑ 𝑎𝑖𝑗 [𝑥𝑖 − 𝑥𝑗 ] + 𝑑𝑖 [𝑥𝑖 − 𝑥𝑟𝑖 ]) , 𝑖 = 1,2, … , 𝑛, 𝑖 ≠ 𝑗
(7)
𝑗=1
where 𝑘 > 0, 𝜃 ∈ ℝ1×𝑝 is the feedback gain matrix, 𝑎𝑖𝑗 , 𝑖, 𝑗 = 1,2, … , 𝑛 is the (𝑖, 𝑗)th element of the adjacency matrix 𝐴. 𝐷 = 𝑑𝑖𝑎𝑔( 𝑑1 , 𝑑2 , … , 𝑑𝑛 ) is the leader adjacency matrix.
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𝑑𝑖 is used for describing the leader information exchange to 𝑖 th agent. 𝑑𝑖 = 1, if information is transmitted from leader to 𝑖 th agent and otherwise 𝑑𝑖 = 0. For each cluster, 𝑑𝑖 = 1 for only
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one follower. Block diagram of the cluster consensus algorithm is shown in Fig. 2.
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Figure. 2. Block diagram of cluster consensus algorithm. Assumption 3.
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We can suppose that the parameter uncertainty satisfies ∆𝐴𝑖 = 𝐵 𝐸𝑖 , where 𝐸𝑖 is defined as follows with constant 𝜌 > 0:
𝐸𝑖 < 𝜌 𝐼𝑝 for all 𝑖 = 1,2, … , 𝑛 and 𝐸 = diag (𝐸1 , 𝐸2 , … , 𝐸𝑛 ) [43].
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Definition 3 ([6]). Robust cluster consensus is achieved for the multi agent system (6) if the error of agents satisfies:
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𝑙𝑖𝑚 ‖𝑥𝑖 − 𝑥𝑗 ‖ ≤ ℎ(𝛾),
𝑡→∞
𝑙𝑖𝑚 ‖𝑥𝑖 − 𝑥𝑗 ‖ ≤ ℎ(𝛾),
∀𝑖, 𝑗 ∈ 𝑙2
where ℎ(𝛾) is the monotonous increasing function and ℎ(0) = 0.
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𝑡→∞
∀𝑖, 𝑗 ∈ 𝑙1
Assumption 4. The external disturbances 𝑤𝑖 (𝑡) satisfy |𝑤𝑖 (𝑡)| < 𝛾 < +∞ for all 𝑖 = 1,2, … , 𝑛. By substituting the controller (8), into the system (6), we can write:
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𝑛
𝑥𝑖𝛼
= (𝐴 + ∆𝐴𝑖 )𝑥𝑖 + 𝐶 𝑓(𝑥𝑖 , 𝑡) + 𝑘𝐵𝜃 ∑ 𝑎𝑖𝑗 [𝑥𝑖 − 𝑥𝑗 ] + 𝑑𝑖 [𝑥𝑖 − 𝑥𝑟𝑖 ]
(8)
𝑗=1
+ 𝑆𝑤𝑖 (𝑡),
𝑖 = 1, … , 𝑛
Now system (9) can be written as: 𝑋 𝛼 = (𝐼𝑛 ⊗ 𝐴 + ∆𝐴) 𝑋 + (𝐼𝑛 ⊗ 𝐶) 𝐹(𝑋, 𝑡) − 𝑘(𝑀 ⊗ 𝐵𝜃)𝑋 (9)
+ (𝐼𝑛 ⊗ 𝑆)𝑊(𝑡) where
𝑋 = [𝑥1 , 𝑥2 , … , 𝑥𝑁1 , 𝑥𝑁1 +1 , … , 𝑥𝑁1 +𝑁2 ]𝑇 ,
∆𝐴 = diag(∆𝐴1 , ∆𝐴2 , … , ∆𝐴𝑛 ),
𝐹(𝑡, 𝑋) = [𝑓𝑖̂ (𝑡, 𝑥1 (𝑡)), 𝑓𝑖̂ (𝑡, 𝑥2 (𝑡)), … , 𝑓𝑖̂ (𝑡, 𝑥𝑁1 (𝑡)) , 𝑓𝑖̂ (𝑡, 𝑥𝑁1 +1 (𝑡)) , … , 𝑓𝑖̂ (𝑡, 𝑥𝑛 (𝑡))]𝑇 , 𝑇
𝑊(𝑡) = [𝑤1 , 𝑤2 , … , 𝑤𝑁1 , 𝑤𝑁1 +1 , … , 𝑤𝑁1 +𝑁2 ] , ⊗
is
Kronecker
product
and
𝜃=
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𝑑𝑖𝑎𝑔(𝜃1, , 𝜃2 , … , 𝜃𝑛 ). 𝑀 = 𝐻 + 𝐷, where 𝐻 is the Laplacian matrix of G described for two clusters as follows:
𝐿1 Ω21
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𝐻=[
Ω12 ] 𝐿2
where 𝐿𝑖 , 𝑖 = 1,2 are the Laplacian matrix of the subnetworks and 𝛺𝑖𝑗 , 𝑖, 𝑗 = 1,2 are the information exchange between the two subnetworks as: ⋯ 𝑎1(𝑁1 +𝑁2 ) ⋱ ⋮ ] ⋯ 𝑎𝑁1 (𝑁1 +𝑁2 )
Ω21
𝑎(𝑁1 +1)1 ⋮ = −[ 𝑎(𝑁1 +𝑁2 )1
⋯ 𝑎(𝑁1 +1)𝑁1 ⋱ ⋮ ] ⋯ 𝑎(𝑁1 +𝑁2 )𝑁1
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Ω12
𝑎1(𝑁1 +1) ⋮ = −[ 𝑎𝑁1 (𝑁1 +1)
In the undirected graph 𝐺, 𝑀 is always a symmetric positive definite matrix and in
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directed graph 𝐺, if information is sent and received between two nodes, 𝑀 is a symmetric positive definite matrix. Let 𝑥̃𝑖 = 𝑥𝑖 − 𝑥𝑟𝑖 , so we can write as follows:
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𝑋̃ 𝛼 (𝑡) = (𝐼𝑛 ⊗ 𝐴 + ∆𝐴)𝑋̃ + (𝐼𝑛 ⊗ 𝐶)(𝐹(𝑡, 𝑋) − 𝐹(𝑡, 𝑋𝑟 )) − 𝑘(𝑀 ⊗ 𝐵𝜃)𝑋̃ (10) + (𝐼𝑛 ⊗ 𝑆)𝑊(𝑡) where, 𝑋̃ = [𝑥̃1 , 𝑥̃2 , … , 𝑥̃𝑁1 , 𝑥̃𝑁1 +1 , … , 𝑥̃𝑛 ]𝑇 and 1𝑛 = [1 1 … 1]𝑇 . Theorem 1. Assume that Assumption1 and Assumption2 are satisfied. i.
The cluster consensus problem for system (6) is achieved with controller (8) if 𝑃 is a positive definite matrix satisfying:
𝐴𝑃 + 𝑃𝐴𝑇 + 𝑙 2 𝐶𝐶 𝑇 + (1 − 𝑘𝜆𝑚𝑖𝑛 (𝑀))𝐵𝐵 𝑇 + 𝑆𝑆 𝑇 + 𝛽𝑃 [ 𝑃
𝑃 ] −(𝜌 + 1)𝐼𝑝 2
(11)
1 2
ii.
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where 𝛽 > 0 and take 𝜃 = 𝐵 𝑇 𝑃−1.
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<0
The errors between two agents can achieve to the region 𝜇, When 𝑤𝑖 (𝑡) ≠ 0 and
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Assumption 3 and part i are established, where 𝜇 = {‖𝑋̃(𝑡)‖ ≤ √𝛽 𝜆
𝑛
𝑚𝑖𝑛 (𝐼𝑛 ⊗𝑃
−1 )
𝛾}.
Proof: The following Lyapunov function is considered: (12)
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𝑉 = 𝑋̃ 𝑇 (𝐼𝑛 ⊗ 𝑃−1 )𝑋̃
The fractional-order derivative of 𝑉 is given as follows:
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𝑉 𝛼 ≤ 𝑋̃ 𝑇 (𝐼𝑛 ⊗ 𝑃 −1 𝐴 + 𝐴𝑇 𝑃−1 )𝑋̃ + 𝑋̃ 𝑇 ∆𝐴𝑇 (𝐼𝑛 ⊗ 𝑃 −1 )𝑋̃ + 𝑋̃ 𝑇 (𝐼𝑛 ⊗ 𝑃 −1 )∆𝐴 𝑋̃ + 2𝑋̃ 𝑇 (𝐼𝑛 ⊗ 𝑃−1 𝐶)[𝐹(𝑡, 𝑋) − 𝐹(𝑡, 𝑋𝑟 )] − 2𝑘𝑋̃ 𝑇 (𝐼𝑛 ⊗ 𝑃−1 )(𝑀 ⊗ 𝐵𝜃)𝑋̃ + 2𝑋̃ 𝑇 (𝐼𝑛 ⊗ 𝑃−1 )(𝐼𝑛 ⊗ 𝑆) 𝑊(𝑡)
(13)
Consider Lemma 5 to continue the proof of Lyapunov stability. Lemma 5. The following inequality with vectors 𝑥, 𝑦 and matrices 𝑃, 𝐷 and 𝑆 is established:
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(14)
2𝑥 𝑇 𝐷𝑆𝑦 ≤ 𝑥 𝑇 𝐷𝑃𝐷𝑇 𝑥 + 𝑦 𝑇 𝑆 𝑇 𝑃−1 𝑆𝑦
By using Lemma 5 and Assumption 1and by getting ∆𝐴 = (𝐼𝑛 ⊗ 𝐵)𝐸 is given (16) as follows: 𝑉 𝛼 ≤ 𝑋̃ 𝑇 (𝐼𝑛 ⊗ 𝑃−1 𝐴 + 𝐴𝑇 𝑃−1 )𝑋̃ + 𝑋̃ 𝑇 𝐸 𝑇 (𝐼𝑛 ⊗ 𝐵 𝑇 𝑃−1 )𝑋̃ + 𝑋̃ 𝑇 (𝐼𝑛 ⊗ 𝑃−1 𝐵)𝐸 𝑋̃ + 𝑋̃ 𝑇 [𝐼𝑛 ⊗ (𝑙 2 𝑃−1 𝐶𝐶 𝑇 𝑃 −1 + 𝐼𝑝 )]𝑋̃
(15)
− 2𝑘𝑋̃ 𝑇 (𝑀 ⊗ 𝑃−1 𝐵𝜃)𝑋̃ + 2𝑋̃ 𝑇 (𝐼𝑛 ⊗ 𝑃−1 𝑆) 𝑊(𝑡) 1
Substituting 𝜃 = 𝐵 𝑇 𝑃−1, (16) is rewritten as follows: 2
V α (𝑡) ≤ 𝑋̃ 𝑇 [𝐼𝑛 ⊗ (𝑃−1 𝐴 + 𝐴𝑇 𝑃−1 )]𝑋̃ + 𝑋̃ 𝑇 𝐸 𝑇 𝐸𝑋̃
f
+ 𝑋̃ 𝑇 (𝐼𝑛 ⊗ 𝑃−1 𝐵𝐵 𝑇 𝑃−1 )𝑋̃
roo
+ 𝑋̃ 𝑇 [𝐼𝑛 ⊗ (𝑙 2 𝑃−1 𝐶𝐶 𝑇 𝑃−1 + 𝐼𝑝 )]𝑋̃
(16)
+ 𝑊𝑇 𝑊
rep
− 𝑘𝑋̃ 𝑇 (𝑀 ⊗ 𝑃−1 𝐵𝐵 𝑇 𝑃−1 )𝑋̃ + 𝑋̃ 𝑇 [𝐼𝑛 ⊗ (𝑃−1 𝑆𝑆 𝑇 𝑃 −1 )]𝑋̃
Using 𝐸𝑖𝑇 𝐸𝑖 < 𝜌2 𝐼𝑝 which implies 𝐸 𝑇 𝐸 < 𝜌2 𝐼𝑛𝑝 = (𝐼𝑛 ⊗ 𝜌2 𝐼𝑝 ) is given (18) as follows:
lP
V α (𝑡) ≤ 𝑋̃ 𝑇 [𝐼𝑛 ⊗ (𝑃−1 𝐴 + 𝐴𝑇 𝑃−1 + 𝑙 2 𝑃−1 𝐶𝐶 𝑇 𝑃 −1 + 𝐼𝑝 + 𝑃 −1 𝐵𝐵 𝑇 𝑃−1
rna
+ 𝑃−1 𝑆𝑆 𝑇 𝑃−1 ) + (𝐼𝑛 ⊗ 𝜌2 𝐼𝑝 )]𝑋̃ − 𝑘𝑋̃ 𝑇 (𝑀 ⊗ 𝑃−1 𝐵𝐵 𝑇 𝑃−1 )𝑋̃ + 𝑊 𝑇 𝑊
(17)
= 𝜀 𝑇 [𝐼𝑛 ⊗ (𝐴𝑃 + 𝑃𝐴𝑇 + 𝑙 2 𝐶𝐶 𝑇 + (𝜌2 + 1)𝑃𝑇 𝑃 + 𝐵𝐵 𝑇
Jo u
+ 𝑆𝑆 𝑇 )]𝜀 − 𝑘𝜀 𝑇 (𝑀 ⊗ 𝐵𝐵 𝑇 )𝜀 + 𝑊 𝑇 𝑊
where 𝜀 = (𝐼𝑛 ⊗ 𝑃−1 )𝑋̃ . V α (𝑡) ≤ 𝜀 𝑇 [𝐼𝑛 ⊗ (𝐴𝑃 + 𝑃𝐴𝑇 + 𝑙 2 𝐶𝐶 𝑇 + (𝜌2 + 1)𝑃𝑇 𝑃 + 𝐵𝐵 𝑇 + 𝑆𝑆 𝑇 )]𝜀 − 𝑘𝜆𝑚𝑖𝑛 (𝑀)𝜀 𝑇 (𝐼𝑛 ⊗ 𝐵𝐵 𝑇 )𝜀 + 𝑊 𝑇 𝑊
(18)
Journal Pre-proof
Using (12) and Schur complement lemma [16], (20) is rewritten as follows: V α (𝑡) ≤ −𝛽𝜀 𝑇 (𝐼𝑛 ⊗ 𝑃)𝜀 + ‖𝑊‖2 = −𝛽𝑋̃ 𝑇 (𝐼𝑛 ⊗ 𝑃−1 )𝑋̃ + ‖𝑊‖2
(19)
≤ −𝛽 𝑉(𝑡) + 𝑛𝛾 2 Let 𝑅(𝑡) = 𝑉(𝑡) −
𝑛𝛾2 𝛽
, so:
𝑅 𝛼 (𝑡) ≤ −𝛽 𝑉(𝑡)
(20)
By using function 𝑚(𝑡), (22) can be write as follows: 𝑅 𝛼 (𝑡) + 𝑚(𝑡) = −𝛽 𝑉(𝑡)
(21)
So the unique solution of (22) is as follows: 𝑅(𝑡) = 𝑅(0)𝐸𝛼 (−𝛽𝑡 𝛼 ) − 𝑚(𝑡) ∗ (𝑡 𝛼−1 𝐸𝛼 (−𝛽𝑡 𝛼 ))
f
𝑛𝛾 2 = 𝑅(𝑡) ≤ 𝑅(0)𝐸𝛼 (−𝛽𝑡 𝛼 ) 𝛽
roo
Therefore 𝑉(𝑡) −
(22)
(23)
2
rep
From (12), 𝑉(𝑡) ≥ 𝜆𝑚𝑖𝑛 (𝐼𝑛 ⊗ 𝑃 −1 )‖𝑋̃(𝑡)‖ and lim 𝑅(0)𝐸𝛼 (−𝛽𝑡 𝛼 ) → 0, so:
lP
𝑛 lim ‖𝑋̃(𝑡)‖ ≤ √ 𝛾 𝑡→∞ 𝛽 𝜆𝑚𝑖𝑛 (𝐼𝑛 ⊗ 𝑃 −1 )
𝑡→∞
(24)
When 𝑤𝑖 (𝑡) = 0, V α ≤ −𝛽 𝑉(𝑡) which we can conclude from Lemma 4, Mittag-
Jo u
Definition 2.
rna
Leffler stability results in (11). System (6) is achieved to robust cluster consensus by using
Journal Pre-proof
4.2.Robust cluster consensus of fractional-order nonlinear multi agent systems by adaptive law Consider a group of 𝑛 agents which the model of 𝑖 th agent, 𝑖 = 1, … , 𝑛 is described in general form as follows: 𝑥𝑖𝛼 = (𝐴 + ∆𝐴𝑖 )𝑥𝑖 + 𝐶 𝑓𝑖̂ (𝑡, 𝑥𝑖 ) + 𝐵𝑢𝑖 (𝑡) + 𝑆𝑤𝑖 (𝑡)
(25)
where 𝑥𝑖 ∈ ℝ𝑝 are the states of 𝑖 th agent for 𝑖 = 1,2, … , 𝑛 with 𝑝 dimensions, 𝛼 ∈ (0,1], 𝑓𝑖̂ (𝑡, 𝑥𝑖 ) = (𝑓1 (𝑥𝑖 , 𝑡), 𝑓2 (𝑥𝑖 , 𝑡), … , 𝑓𝑝 (𝑥𝑖 , 𝑡))𝑇 is a nonlinear vector-valued function for each cluster which satisfies Lipschitz condition, 𝑢𝑖 (𝑡) is the control input, 𝑤𝑖 (𝑡) is the external disturbance, ∆𝐴𝑖 is the parameter uncertainty of agent 𝑖, 𝐴, 𝐵, 𝐶 and 𝑆 are constant
The 𝑖 th leader is described by:
rep
𝛼 𝑥𝑟𝑖 = 𝐴𝑥𝑟𝑖 + 𝐶𝑓𝑖̂ (𝑡, 𝑥𝑟𝑖 ), 𝑖 = 1,2, … , 𝑛
roo
f
matrices.
(26)
The adaptive sliding mode controller and adaptive law are designed as follows: 𝑛
𝑢𝑖 (𝑡) = −𝜃𝑖 𝑘 (∑ 𝑎𝑖𝑗 [𝑥𝑖 − 𝑥𝑗 ] + 𝑑𝑖 [𝑥𝑖 − 𝑥𝑟𝑖 ]) 𝑗=1
lP
𝑛
(27)
− 𝜔. 𝑠𝑔𝑛 (∑ 𝑎𝑖𝑗 [𝑥𝑖 − 𝑥𝑗 ] + 𝑑𝑖 [𝑥𝑖 − 𝑥𝑟𝑖 ]) , 𝑗=1
𝑛
𝜃𝑖𝛼
rna
𝑖, 𝑗 = 1,2, … , 𝑛, 𝑖 ≠ 𝑗
𝑇
𝑛
= 𝛽𝑖 (𝑘 ∑ 𝑎𝑖𝑗 [𝑥𝑖 − 𝑥𝑗 ] + 𝑑𝑖 [𝑥𝑖 − 𝑥𝑟𝑖 ]) (𝑘 ∑ 𝑎𝑖𝑗 [𝑥𝑖 − 𝑥𝑗 ] 𝑗=1
𝑗=1
(28)
Jo u
+ 𝑑𝑖 [𝑥𝑖 − 𝑥𝑟𝑖 ])
where 𝑘 > 0, 𝑎𝑖𝑗 , 𝑖, 𝑗 = 1,2, … , 𝑛 is the (𝑖, 𝑗)th element of the adjacency matrix 𝐴. 𝐷 = 𝑑𝑖𝑎𝑔( 𝑑1 , 𝑑2 , … , 𝑑𝑛 ) is the leader adjacency matrix. 𝑑𝑖 is used for describing the leader
Journal Pre-proof
information exchange to 𝑖 th agent. 𝑑𝑖 = 1, if information is transmitted from leader to 𝑖 th agent and otherwise 𝑑𝑖 = 0. For each cluster, 𝑑𝑖 = 1 for only one follower. By substituting the controller (28), into the system (26), we can write: 𝑛
𝑥𝑖𝛼
= (𝐴 + ∆𝐴𝑖 )𝑥𝑖 + 𝐶 𝑓(𝑥𝑖 , 𝑡) − 𝑘𝐵𝜃𝑖 ∑ 𝑎𝑖𝑗 [𝑥𝑖 − 𝑥𝑗 ] + 𝑑𝑖 [𝑥𝑖 − 𝑥𝑟𝑖 ] 𝑗=1 𝑛
(29)
− 𝜔. 𝑠𝑔𝑛 (∑ 𝑎𝑖𝑗 [𝑥𝑖 − 𝑥𝑗 ] + 𝑑𝑖 [𝑥𝑖 − 𝑥𝑟𝑖 ]) + 𝑆𝑤𝑖 (𝑡), 𝑗=1
𝑖 = 1, … , 𝑛 where
𝑋 = [𝑥1 , 𝑥2 , … , 𝑥𝑁1 , 𝑥𝑁1 +1 , … , 𝑥𝑁1 +𝑁2 ]𝑇 ,
∆𝐴 = diag(∆𝐴1 , ∆𝐴2 , … , ∆𝐴𝑛 ),
is
roo
𝑊(𝑡) = [𝑤1 , 𝑤2 , … , 𝑤𝑁 , 𝑤𝑁+1 , … , 𝑤𝑁+𝑀 ]𝑇 , ⊗
f
𝐹(𝑡, 𝑋) = [𝑓𝑖̂ (𝑡, 𝑥1 (𝑡)), 𝑓𝑖̂ (𝑡, 𝑥2 (𝑡)), … , 𝑓𝑖̂ (𝑡, 𝑥𝑁1 (𝑡)) , 𝑓𝑖̂ (𝑡, 𝑥𝑁1 +1 (𝑡)) , … , 𝑓𝑖̂ (𝑡, 𝑥𝑛 (𝑡))]𝑇 ,
𝑑𝑖𝑎𝑔(𝜃1, , 𝜃2 , … , 𝜃𝑛 ).
Kronecker
product
and
𝜃=
rep
Let 𝑥̃𝑖 = 𝑥𝑖 − 𝑥𝑟𝑖 , so we can write as follows:
𝑋̃ 𝛼 (𝑡) = (𝐼𝑛 ⊗ 𝐴 + ∆𝐴)𝑋̃ + (𝐼𝑛 ⊗ 𝐶)(𝐹(𝑡, 𝑋) − 𝐹(𝑡, 𝑋𝑟 )) − (𝜃𝑀 ⊗ 𝐵𝑘)𝑋̃ (30)
− (𝜔𝐼𝑛 ⊗ 𝐵) 𝑠𝑔𝑛 ((𝑀 ⊗ 𝐼𝑛 )𝑋̃) + (𝐼𝑛 ⊗ 𝑆)𝑊(𝑡)
lP
where, 𝑋̃ = [𝑥̃1 , 𝑥̃2 , … , 𝑥̃𝑁 , 𝑥̃𝑁+1 , … , 𝑥̃𝑛 ]𝑇 , 1𝑛 = [1 1 … 1]𝑇 , 𝑀 = 𝐻 + 𝐷 , 𝐻 is the Laplacian matrix of G described for two clusters which is described in (10).
i.
rna
Theorem 2. Assume that Assumption1 and Assumption2 are satisfied. The cluster consensus problem for system (26) is achieved with controller (28) if 𝑃 is
Jo u
a positive definite matrix satisfying: 𝐴𝑃 + 𝑃𝐴𝑇 + 𝑙 2 𝐶𝐶 𝑇 + (1 − 2𝜃0 𝜆𝑚𝑖𝑛 (𝑀))𝐵𝐵 𝑇 + 𝑆𝑆 𝑇 + 𝛽𝑃 [ 𝑃 <0
where 𝛽 > 0 and take 𝑘 = 𝐵 𝑇 𝑃−1 .
𝑃 ] −(𝜌 + 1)𝐼𝑝 2
(31)
Journal Pre-proof
The errors between two agents can achieve to the region 𝜇, When 𝑤𝑖 (𝑡) ≠ 0 and
ii.
Assumption 3 and part i are established, where 𝜇 = {‖𝑋̃(𝑡)‖ ≤ √
𝑛
𝛽 𝜆𝑚𝑖𝑛 (𝑀⊗𝑃 −1 )
𝛾}.
Proof: The following Lyapunov function is considered: 𝑛
𝑉 = 𝑋̃ 𝑇 (𝑀 ⊗ 𝑃−1 )𝑋̃ + ∑ 𝑖=1
1 (𝜃 − 𝜃0 )2 𝛽𝑖 𝑖
(32)
The fractional-order derivative of 𝑉 is given as follows: 𝑉 𝛼 ≤ 𝑋̃ 𝑇 (𝑀 ⊗ 𝑃−1 𝐴 + 𝐴𝑇 𝑃−1 )𝑋̃ + 𝑋̃ 𝑇 ∆𝐴𝑇 (𝑀 ⊗ 𝑃−1 )𝑋̃ + 𝑋̃ 𝑇 (𝑀 ⊗ 𝑃−1 )∆𝐴 𝑋̃ + 2𝑋̃ 𝑇 (𝑀 ⊗ 𝑃−1 𝐶)[𝐹(𝑡, 𝑋) − 𝐹(𝑡, 𝑋𝑟 )]) − 𝑋̃ 𝑇 (𝑀 ⊗ 𝑃−1 )(𝜃𝑀 ⊗ 𝐵𝑘)𝑋̃ − 𝑋̃ 𝑇 (𝜃𝑀 ⊗ 𝑘 𝑇 𝐵 𝑇 )(𝑀 ⊗ 𝑃−1 )𝑋̃ − 2𝑋̃ 𝑇 𝜔(𝑀 ⊗ 𝑃−1 𝐵) 𝑠𝑔𝑛 ((𝑀 ⊗ 𝐼𝑛 )𝑋̃) 𝑛
𝑇
𝑛
𝑛
(33)
𝑖=1
roo
f
+2 ∑(𝜃𝑖 − 𝜃0 ) (𝑘 ∑ 𝑎𝑖𝑗 (𝑥̃𝑖 − 𝑥̃𝑗 ) + 𝑑𝑖 𝑥̃𝑖 ) (𝑘 ∑ 𝑎𝑖𝑗 (𝑥̃𝑖 − 𝑥̃𝑗 ) + 𝑑𝑖 𝑥̃𝑖 ) 𝑗=1
𝑗=1
+ 2𝑋̃ 𝑇 (𝑀 ⊗ 𝑃−1 )(𝐼𝑛 ⊗ 𝑆) 𝑊(𝑡)
rep
By using Lemma 5 and Assumption 1and by getting ∆𝐴 = (𝐼𝑛 ⊗ 𝐵)𝐸 is given (35) as follows:
𝑉 𝛼 ≤ 𝑋̃ 𝑇 (𝑀 ⊗ 𝑃−1 𝐴 + 𝐴𝑇 𝑃−1 )𝑋̃ + 𝑋̃ 𝑇 𝐸 𝑇 (𝑀 ⊗ 𝐵 𝑇 𝑃−1 )𝑋̃
lP
+ 𝑋̃ 𝑇 (𝑀 ⊗ 𝑃−1 𝐵)𝐸 𝑋̃ + 𝑋̃ 𝑇 [𝑀 ⊗ (𝑙 2 𝑃−1 𝐶𝐶 𝑇 𝑃 −1 + 𝐼𝑝 )]𝑋̃
𝑛
rna
− 2𝑋̃ 𝑇 (𝑀𝜃𝑀 ⊗ 𝑃−1 𝐵𝑘)𝑋̃ 𝑛
𝑇
(34) 𝑛
+2 ∑(𝜃𝑖 − 𝜃0 ) (𝑘 ∑ 𝑎𝑖𝑗 (𝑥̃𝑖 − 𝑥̃𝑗 ) + 𝑑𝑖 𝑥̃𝑖 ) (𝑘 ∑ 𝑎𝑖𝑗 (𝑥̃𝑖 − 𝑥̃𝑗 ) + 𝑑𝑖 𝑥̃𝑖 ) 𝑗=1
𝑗=1
+ 2𝑋̃ 𝑇 (𝑀 ⊗ 𝑃 −1 𝑆) 𝑊(𝑡)
Jo u
𝑖=1
Substituting 𝑘 = 𝐵 𝑇 𝑃−1, (35) is rewritten as follows:
Journal Pre-proof
V α (𝑡) ≤ 𝑋̃ 𝑇 [𝑀 ⊗ (𝑃−1 𝐴 + 𝐴𝑇 𝑃 −1 )]𝑋̃ + 𝑋̃ 𝑇 𝐸 𝑇 (𝑀 ⊗ 𝐼𝑝 )𝐸𝑋̃ + 𝑋̃ 𝑇 (𝑀 ⊗ 𝑃−1 𝐵𝐵 𝑇 𝑃−1 )𝑋̃ + 𝑋̃ 𝑇 [𝑀 ⊗ (𝑙 2 𝑃−1 𝐶𝐶 𝑇 𝑃 −1 + 𝐼𝑝 )]𝑋̃ − 2𝑋̃ 𝑇 (𝑀𝜃𝑀 ⊗ 𝑃−1 𝐵𝐵 𝑇 𝑃−1 )𝑋̃ 𝑛
𝑇
𝑛
(35) 𝑛
+2 ∑(𝜃𝑖 − 𝜃0 ) (𝑘 ∑ 𝑎𝑖𝑗 (𝑥̃𝑖 − 𝑥̃𝑗 ) + 𝑑𝑖 𝑥̃𝑖 ) (𝑘 ∑ 𝑎𝑖𝑗 (𝑥̃𝑖 − 𝑥̃𝑗 ) + 𝑑𝑖 𝑥̃𝑖 ) 𝑖=1
𝑗=1
𝑗=1
+ 𝑋̃ 𝑇 [𝑀 ⊗ (𝑃−1 𝑆𝑆 𝑇 𝑃−1 )]𝑋̃ + 𝑊 𝑇 𝑊 Using 𝐸𝑖𝑇 𝐸𝑖 < 𝜌2 𝐼𝑝 which implies 𝐸 𝑇 𝐸 < 𝜌2 𝐼𝑛𝑝 = (𝐼𝑛 ⊗ 𝜌2 𝐼𝑝 ) is given (37) as
f
follows:
roo
V α (𝑡) ≤ 𝑋̃ 𝑇 [𝑀 ⊗ (𝑃−1 𝐴 + 𝐴𝑇 𝑃−1 + 𝑙 2 𝑃−1 𝐶𝐶 𝑇 𝑃 −1 + 𝐼𝑝 + 𝑃 −1 𝐵𝐵 𝑇 𝑃−1 + 𝑃−1 𝑆𝑆 𝑇 𝑃−1 ) + (𝑀 ⊗ 𝜌2 𝐼𝑝 )]𝑋̃ − 2𝜃0 𝑋̃ 𝑇 (𝑀2
rep
⊗ 𝑃−1 𝐵𝐵 𝑇 𝑃−1 )𝑋̃ + 𝑊 𝑇 𝑊
(36)
= 𝜀 𝑇 [𝑀 ⊗ (𝐴𝑃 + 𝑃𝐴𝑇 + 𝑙 2 𝐶𝐶 𝑇 + (𝜌2 + 1)𝑃𝑇 𝑃 + 𝐵𝐵 𝑇 + 𝑆𝑆 𝑇 − 2𝜃0 𝜆𝑚𝑖𝑛 (𝑀)𝐵𝐵 𝑇 )]𝜀 + 𝑊 𝑇 𝑊
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where 𝜀 = (𝐼𝑛 ⊗ 𝑃−1 )𝑋̃ .
V α (𝑡) ≤ 𝜀 𝑇 [𝑀 ⊗ (𝐴𝑃 + 𝑃𝐴𝑇 + 𝑙 2 𝐶𝐶 𝑇 + (𝜌2 + 1)𝑃𝑇 𝑃 + (1 (37)
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− 2𝜃0 𝜆𝑚𝑖𝑛 (𝑀))𝐵𝐵 𝑇 + 𝑆𝑆 𝑇 )]𝜀 + 𝑊 𝑇 𝑊 Using (32) and Schur complement lemma [16], (38) is rewritten as follows:
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V α (𝑡) ≤ −𝛽𝜀 𝑇 (𝑀 ⊗ 𝑃)𝜀 + ‖𝑊‖2 = −𝛽𝑋̃ 𝑇 (𝑀 ⊗ 𝑃−1 )𝑋̃ + ‖𝑊‖2 ≤ −𝛽 𝑉(𝑡) + 𝑛𝛾
Let 𝑅(𝑡) = 𝑉(𝑡) −
𝑛𝛾2 𝛽
2
, so:
(38)
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𝑅 𝛼 (𝑡) ≤ −𝛽 𝑉(𝑡)
(39)
By using function 𝑚(𝑡), (40) can be write as follows: 𝑅 𝛼 (𝑡) + 𝑚(𝑡) = −𝛽 𝑉(𝑡)
(40)
So the unique solution of (41) is as follows: 𝑅(𝑡) = 𝑅(0)𝐸𝛼 (−𝛽𝑡 𝛼 ) − 𝑚(𝑡) ∗ (𝑡 𝛼−1 𝐸𝛼 (−𝛽𝑡 𝛼 ))
(41)
Therefore 𝑛𝛾 2 𝑉(𝑡) − = 𝑅(𝑡) ≤ 𝑅(0)𝐸𝛼 (−𝛽𝑡 𝛼 ) 𝛽
(42) 2
From (32), 𝑉(𝑡) ≥ 𝜆𝑚𝑖𝑛 (𝑀 ⊗ 𝑃−1 )‖𝑋̃(𝑡)‖ and lim 𝑅(0)𝐸𝛼 (−𝛽𝑡 𝛼 ) → 0, so:
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𝑛 lim ‖𝑋̃(𝑡)‖ ≤ √ 𝛾 𝑡→∞ 𝛽 𝜆𝑚𝑖𝑛 (𝑀 ⊗ 𝑃 −1 )
f
𝑡→∞
(43)
When 𝑤𝑖 (𝑡) = 0, from (39) V α ≤ −𝛽 𝑉(𝑡) which we can conclude from Lemma 4,
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Mittag-Leffler stability results in (31). System (26) is achieved to robust cluster consensus by using Definition 2.
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4.3.Robust cluster consensus of fractional-order nonlinear multi agent systems by σ modification adaptive law
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Consider a group of 𝑛 agents and leaders which are modeled by (26) and (27). The adaptive sliding mode controller and adaptive law are designed as follows: 𝑛
𝑢𝑖 (𝑡) = −𝜃𝑖 𝑘 (∑ 𝑎𝑖𝑗 [𝑥𝑖 − 𝑥𝑗 ] + 𝑑𝑖 [𝑥𝑖 − 𝑥𝑟𝑖 ])
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𝑗=1
𝑛
− 𝜔. 𝑠𝑔𝑛 (∑ 𝑎𝑖𝑗 [𝑥𝑖 − 𝑥𝑗 ] + 𝑑𝑖 [𝑥𝑖 − 𝑥𝑟𝑖 ]) , 𝑗=1
𝑖, 𝑗 = 1,2, … , 𝑛, 𝑖 ≠ 𝑗
(44)
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𝑇
𝑛
𝜃𝑖𝛼
𝑛
= 𝛽𝑖 (𝑘 ∑ 𝑎𝑖𝑗 [𝑥𝑖 − 𝑥𝑗 ] + 𝑑𝑖 [𝑥𝑖 − 𝑥𝑟𝑖 ]) (𝑘 ∑ 𝑎𝑖𝑗 [𝑥𝑖 − 𝑥𝑗 ] 𝑗=1
𝑗=1
(45)
+ 𝑑𝑖 [𝑥𝑖 − 𝑥𝑟𝑖 ]) − 𝜎(𝜃𝑖 − 𝜃0 ) Like section 4.2, let 𝑥̃𝑖 = 𝑥𝑖 − 𝑥𝑟𝑖 , so we can write as follows: 𝑋̃ 𝛼 (𝑡) = (𝐼𝑛 ⊗ 𝐴 + ∆𝐴)𝑋̃ + (𝐼𝑛 ⊗ 𝐶)(𝐹(𝑡, 𝑋) − 𝐹(𝑡, 𝑋𝑟 )) − (𝜃𝑀 ⊗ 𝐵𝑘)𝑋̃ − (𝜔𝐼𝑛 ⊗ 𝐵) 𝑠𝑔𝑛 ((𝑀 ⊗ 𝐼𝑛 )𝑋̃) + (𝐼𝑛 ⊗ 𝑆)𝑊(𝑡)
(46)
Robust cluster consensus is achieved if the conditions of Theorem 2 are held. For proof it, the Lyapunov function (33) is considered. Like section 4.2, the fractional-order
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derivative of 𝑉 is given as follows:
V α (𝑡) ≤ 𝑋̃ 𝑇 [𝑀 ⊗ (𝑃−1 𝐴 + 𝐴𝑇 𝑃−1 + 𝑙 2 𝑃−1 𝐶𝐶 𝑇 𝑃 −1 + 𝐼𝑝 + 𝑃 −1 𝐵𝐵 𝑇 𝑃−1 + 𝑃−1 𝑆𝑆 𝑇 𝑃−1 ) + (𝑀 ⊗ 𝜌2 𝐼𝑝 )]𝑋̃ − 2𝜃0 𝑋̃ 𝑇 (𝑀2
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𝑛
⊗ 𝑃 𝐵𝐵 𝑃 )𝑋̃ − 2𝜎 ∑(𝜃𝑖 − 𝜃0 )2 + 𝑊 𝑇 𝑊 −1
𝑇 −1
(47)
𝑖=1
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≤ 𝜀 𝑇 [𝑀 ⊗ (𝐴𝑃 + 𝑃𝐴𝑇 + 𝑙 2 𝐶𝐶 𝑇 + (𝜌2 + 1)𝑃𝑇 𝑃 + 𝐵𝐵 𝑇 + 𝑆𝑆 𝑇 − 2𝜃0 𝜆𝑚𝑖𝑛 (𝑀)𝐵𝐵 𝑇 )]𝜀 + ‖𝑊‖2
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This proof is continued like previous section and robust cluster consensus is achieved. 5. Simulation results
In this section, three numerical simulation examples are given to evaluate the
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theoretical results.
Example 1: In this example, each agent is a single-link flexible-joint manipulator shown in Fig. 3. The revolving joints are actuated by a DC motor and a linear spring is used to
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model the elasticity of the joint. The state-space model which the states of the system are position and velocity of motor and link is as follows: 𝛼 𝜃𝑚 = 𝜔𝑚 𝑘 𝑘 𝐾𝑇 𝛼 𝜔𝑚 = (𝜃𝑙 − 𝜃𝑚 ) − 𝜔𝑚 + 𝑢 𝐽𝑚 𝐽𝑚 𝐽𝑚 𝜃𝑙𝛼 = 𝜔𝑙 𝑘 𝑚𝑔ℎ 𝜔𝑙𝛼 = − (𝜃𝑙 − 𝜃𝑚 ) − 𝑠𝑖𝑛(𝜃𝑙 ) 𝐽𝑙 𝐽𝑙
where 𝐽𝑚 and 𝐽𝑙 are the inertia of the motor and link, 𝜃𝑚 and 𝜃𝑙 are the angular rotation of the motor and the angular position of the link, respectively; and 𝜔𝑚 and 𝜔𝑙 are the angular velocity of the motor and link [44].
parameters are defined as follows:
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The manipulator is described by the form of (6) with α = 0.95 and the numerical
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0 1 0 −48.6 −1.26 48.6 𝐴=[ 0 0 0 1.95 0 −1.95
0 0 0 21.6 ],𝐵 = [ ] , 𝐶 = 𝐼4 10 0 0 0
𝑇
𝑥𝑖 (𝑡) = [𝜃𝑚 𝑖 , 𝜔𝑚 𝑖 , 𝜃𝑙 𝑖 , 𝜔𝑙 𝑖 ] = [𝑥𝑖1 (𝑡), 𝑥𝑖2 (𝑡), 𝑥𝑖3 (𝑡), 𝑥𝑖4 (𝑡)]𝑇 ,
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𝑥𝑖3 (𝑡) 𝑇 𝑓1 (𝑡, 𝑥𝑖 (𝑡)) = (0,0,0, 𝑠𝑖𝑛 ( ) ) , 10
𝑥𝑖3 (𝑡) 𝑇 𝑓2 (𝑡, 𝑥𝑖 (𝑡)) = (0,0,0, 2𝑠𝑖𝑛 ( ) ) , 𝐸𝑖 = [0 0 𝛾𝑖 0], 10
𝑙 = 0.333,
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𝑤1 (𝑡) = (0,1.1 sin(8𝑡 − 1) , 0,0)𝑇 , 𝑤2 (𝑡) = (0, − cos(2t) , 0,0)𝑇 , 𝑤3 (𝑡) = (0,0.8 sin(𝑡) + 0.7 cos(𝑡) , 0,0)𝑇 , 𝑤4 (𝑡) = (0,0.2 cos(11𝑡 − 4) , 0,0)𝑇 ,
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𝑤5 (𝑡) = (0,1.5 sin(7𝑡 − 1) , 0,0)𝑇 , 𝑤6 (𝑡) = (0, cos(3𝑡) , 0,0)𝑇 , 𝑤7 (𝑡) = (0, −0.6 sin(𝑡) − 0.9 cos(𝑡) , 0,0)𝑇
Cluster consensus for the manipulator is achieved for graph in Fig. 4 with 𝑙 = 0.333, 𝛽 = 0.7, 𝜌 = 1 and 𝛾𝑖 ∈ [−1,1] is the uncertainty which is randomly chose. 𝑘 = 2.11 and 𝑃 is achieved by LMI (12) as follows:
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0.9438 −1.0792 −1.0792 12.2318 𝑃=[ 0.8298 −0.404 −0.1177 −0.1063 Simulation results are shown in Fig. 5.
0.8298 −0.404 0.9114 −0.08
−0.1177 −0.1063 ] −0.08 0.028
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Figure. 3. single-link manipulator with a flexible joint
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Figure. 4. The topology interaction of the agents.
(a)
(b)
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(c)
(d)
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Figure 5. Cluster consensus for (a) 𝐱 𝐢𝟏 (𝐭), (b) 𝐱 𝐢𝟐 (𝐭), (c) 𝐱 𝐢𝟑 (𝐭), (d) 𝐱 𝐢𝟒 (𝐭), 𝐢 = 𝟏, … , 𝟕. The state trajectories of agents are shown in Fig. 5. In Fig. 5(a), first state of agents
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x11 to x31 converge to one consensus value which is time varying and x41 to x71 converge to another consensus value at time 6.5 second. In Fig. 5(b), second state of agents x12 to x32 converge to one consensus value which is time-varying and x42 to x72 converge to another
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consensus value at time 6.5 second. In Fig. 5(c), third state of agents x13 to x33 converge to one consensus value which is time-varying and x43 to x73 converge to another consensus
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value at time 4 second. In Fig. 5(d), fourth state of agents x14 to x34 converge to one consensus value which is time varying and x44 to x74 converge to another consensus value at
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time 4 second. So cluster consensus is achieved for all states of agents. Example 2: A directed graph with seven followers and two leaders which communicated with first and fourth agents is shown in Fig. 4. In this example, each agent is a single-link flexible-joint manipulator and this is described like Example 1 with 𝛼 = 0.95, 𝜔 =
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𝑥𝑖3 (𝑡)
𝑥𝑖3 (𝑡)
10
10
1, 𝑓1 (𝑡, 𝑥𝑖 (𝑡)) = (0,0,0, 𝑠𝑖𝑛 (
) )𝑇 , 𝑓2 (𝑡, 𝑥𝑖 (𝑡)) = (0,0,0, 2𝑠𝑖𝑛 (
) )𝑇 , 𝑙 = 0.333,
𝛽 = 0.7, 𝜃0 = 105.8, 𝜌 = 1 and 𝛾𝑖 ∈ [−1,1] is the uncertainty which is randomly chose. 𝑃 is achieved by LMI (32) as follows: 0.5349 −0.9707 𝑃=[ 0.4700 −0.0828
−0.9707 107.0421 −0.0183 0.0009
0.4700 −0.0183 0.5195 −0.0484
−0.0828 0.0009 ] −0.0484 0.0196
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Simulation results are shown in Fig. 6.
(b)
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(a)
(c)
(d)
Figure 6. Cluster consensus for (a) 𝐱 𝐢𝟏 (𝐭), (b) 𝐱 𝐢𝟐 (𝐭), (c) 𝐱 𝐢𝟑 (𝐭), (d) 𝐱 𝐢𝟒 (𝐭), 𝐢 = 𝟏, … , 𝟕.
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Fig. 6 shows the position trajectories of agents for each state. In Fig. 6(a), first state of agents x11 to x31 follow the first leader and x41 to x71 follow the second leader after 5 second. In Fig. 6(b), second state of agents x12 to x32 follow the first leader and x42 to x72 follow the second leader after 1 second. In Fig. 6(c), third state of agents x13 to x33 follow the first leader and x43 to x73 follow the second leader after 0.5 second. In Fig. 6(d), fourth state of agents x14 to x34 follow the first leader and x44 to x74 follow the second leader after 1 second. By comparison Fig. 3 and Fig. 6, the convergence speed increases by using adaptive sliding mode controller. The control signals of each agent are shown in Fig. 7 which they are limited to specified amplitudes. Cluster consensus with α = 1 is simulated for this system and it is
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rep
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f
shown in Fig.8, therefore the convergence speed is lower than fractional order system.
Figure 7. Control signals of agents
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(a)
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rep
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f
(b)
(c)
(d)
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Figure 8. Cluster consensus with 𝜶 = 𝟏 for (a) 𝐱 𝐢𝟏 (𝐭), (b) 𝐱 𝐢𝟐 (𝐭), (c) 𝐱 𝐢𝟑 (𝐭), (d) 𝐱 𝐢𝟒 (𝐭), 𝐢 = 𝟏, … , 𝟕. Example 3: Agents and parameters are defined same as example 2 with 𝜎 = 1.
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Simulation results are shown in figure 9.
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(b)
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rep
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f
(a)
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(c)
(d)
Figure 9. Cluster consensus for (a) 𝐱 𝐢𝟏 (𝐭), (b) 𝐱 𝐢𝟐 (𝐭), (c) 𝐱 𝐢𝟑 (𝐭), (d) 𝐱 𝐢𝟒 (𝐭), 𝐢 = 𝟏, … , 𝟕.
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Fig. 9 shows the position trajectories of agents for each state and the adaptive parameters are shown in Fig.10. The comparison of these examples is shown in Table.1. So, it demonstrates the efficiency of the proposed adaptive controller and the convergence speed increases by using adaptive sliding mode controller with σ modification method.
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Figure 10. trajectories of 𝜽𝒊
7 agents
Adaptive sliding mode controller
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General form system General form system General form system
Graph
7 agents
7 agents
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System
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Table.1. Comparison of examples σ modification
Average consensus time of the states 5.25 second 1.87 second
0.975 second
6. Conclusions
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In this paper, robust cluster consensus for the general fractional-order nonlinear multi agent systems has been studied. General form nonlinear systems are considered in the presence of uncertainty and external disturbance. Robust cluster consensus is achieved by an adaptive sliding mode controller and the sufficient conditions are given. In the simulations,
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the effectiveness of the proposed results is shown and compared with integer-order systems. Future work will concentrate on robust cluster consensus of the fractional-order nonlinear multi agent systems under switching communication topology in the presence of uncertainty and external disturbance. References [1] N. Aguila-Camacho, M.A. Duarte-Mermoud, J.A. Gallegos, Lyapunov functions for fractional order systems, Communications in Nonlinear Science and Numerical Simulation, 19 (2014) 2951-2957. [2] J. Ansari, A. Gholami, A. Kazemi, Multi-agent systems for reactive power control in smart grids, International Journal of Electrical Power & Energy Systems, 83 (2016) 411-425.
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[3] G. Arthi, J.H. Park, K. Suganya, Controllability of fractional order damped dynamical systems with distributed delays, Mathematics and Computers in Simulation, 165 (2019) 7491. [4] J. Bai, G. Wen, A. Rahmani , Y. Yu, Distributed consensus tracking for the fractionalorder multi-agent systems based on the sliding mode control method, Neurocomputing, 235 (2017) 210-216.
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[8] L. Ding, P. Yu, Z.-W. Liu, Z.-H. Guan, Consensus and performance optimisation of multiagent systems with position-only information via impulsive control, IET Control Theory & Applications, 7 (2013) 16-24. [9] H. Du, S. Li, P. Shi, Robust consensus algorithm for second-order multi-agent systems with external disturbances, International Journal of Control, 85 (2012) 1913-1928.
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[10] S. El-Ferik, A. Qureshi, F.L. Lewis, Neuro-adaptive cooperative tracking control of unknown higher-order affine nonlinear systems, Automatica, 50 (2014) 798-808. [11] P. Gong, Distributed consensus of non-linear fractional-order multi-agent systems with directed topologies, IET Control Theory & Applications, 10 (2016) 2515-2525. [12] P. Gong, Distributed tracking of heterogeneous nonlinear fractional-order multi-agent systems with an unknown leader, Journal of the Franklin Institute, 354 (2017) 2226-2244.
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[19] Y. Li, Y. Huang, P. Lin, W. Ren, Distributed rotating consensus of second-order multiagent systems with nonuniform delays, Systems & Control Letters, 117 (2018) 18-22. [20] Z. Li, Z. Duan, G. Chen, Global synchronised regions of linearly coupled Lur'e systems, International Journal of Control, 84 (2011) 216-227.
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[23] Z. Li, W. Ren, X. Liu, M. Fu, Consensus of multi-agent systems with general linear and Lipschitz nonlinear dynamics using distributed adaptive protocols, IEEE Transactions on Automatic Control, 58 (2013) 1786-1791. [24] J. Liang-Hao, L. Xiao-Feng, Consensus problems of first-order dynamic multi-agent systems with multiple time delays, Chinese Physics B, 22 (2013) 040203.
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[26] J. Mei, W. Ren, J. Chen, Distributed consensus of second-order multi-agent systems with heterogeneous unknown inertias and control gains under a directed graph, IEEE Transactions on Automatic Control, 61 (2016) 2019-2034. [27] A.J. Muñoz-Vázquez, M.B. Ortiz-Moctezuma, A. Sánchez-Orta, V. Parra-Vega, Adaptive robust control of fractional-order systems with matched and mismatched disturbances, Mathematics and Computers in Simulation, 162 (2019) 85-96. [28] R. Olfati-Saber, R.M. Murray, Consensus problems in networks of agents with switching topology and time-delays, IEEE Transactions on automatic control, 49 (2004) 1520-1533.
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[35] G. Wen, Z. Duan, G. Chen, W. Yu, Consensus tracking of multi-agent systems with Lipschitz-type node dynamics and switching topologies, IEEE Transactions on Circuits and Systems I: Regular Papers, 61 (2014) 499-511.
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[36] Z. Yaghoubi, H.A. Talebi, Consensus tracking for nonlinear fractional-order multi agent systems using adaptive sliding mode controller, Mechatronic Systems and Control, 47 (2019) 194-200. [37] X. Yang, C. Li, T. Huang, Q. Song, Mittag–Leffler stability analysis of nonlinear fractional-order systems with impulses, Applied Mathematics and Computation, 293 (2017) 416-422.
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[38] S.J. Yoo, Distributed adaptive containment control of uncertain nonlinear multi-agent systems in strict-feedback form, Automatica, 49 (2013) 2145-2153. [39] W. Yu, Y. Li, G. Wen, X. Yu, J. Cao, Observer design for tracking consensus in secondorder multi-agent systems: Fractional order less than two, IEEE Transactions on Automatic Control, 62 (2017) 894-900.
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