New Developments in Mathematical Control and Information for Fuzzy Systems
View this Special IssueResearch Article  Open Access
YeongHwa Chang, ChunLin Chen, WeiShou Chan, HungWei Lin, ChiaWen Chang, "Fuzzy Formation Control and Collision Avoidance for Multiagent Systems", Mathematical Problems in Engineering, vol. 2013, Article ID 908180, 18 pages, 2013. https://doi.org/10.1155/2013/908180
Fuzzy Formation Control and Collision Avoidance for Multiagent Systems
Abstract
This paper aims to investigate the formation control of leaderfollower multiagent systems, where the problem of collision avoidance is considered. Based on the graphtheoretic concepts and locally distributed information, a neural fuzzy formation controller is designed with the capability of online learning. The learning rules of controller parameters can be derived from the gradient descent method. To avoid collisions between neighboring agents, a fuzzy separation controller is proposed such that the local minimum problem can be solved. In order to highlight the advantages of this fuzzy logic based collisionfree formation control, both of the static and dynamic leaders are discussed for performance comparisons. Simulation results indicate that the proposed fuzzy formation and separation control can provide better formation responses compared to conventional consensus formation and potentialbased collisionavoidance algorithms.
1. Introduction
Recently, distributed multiagent coordination has attracted much attention in many fields, where only the information available locally from its neighbors is required for each agent [1–6]. There are many applications of multiagent systems, such as autonomous unmanned aerial vehicles [7], autonomous formation flight [8], congestioncontrolled communication network [9], wireless sensor network [10], and autonomous multivehicle formations [11]. Graph theory has been used to characterize network topologies for consensus studies. A general consensus problem solving is to find a distributed control strategy such that the states of agents converge to a common value. Averageconsensus problem was investigated for distributed networks with fixed and switching topologies [12]. Relying on graph theory, matrix theory, and control theory, the analysis of consensus protocols was thoroughly discussed. In [13], an impulsive control protocol was presented for multiagent linear dynamic systems with fixed topology based on the local information of agents. A fuzzy slidingmode controller was proposed to investigate the formation control problem in directed graphs [14]. Cai et al. [15] addressed the controllability improvement problem for two types of linear timeinvariant dynamic multiagent systems by adjusting the configuration of graphs. A general case of leaderfollowing consensus problems under fixed and switching topologies was discussed in [16]. The secondorder agents under switching topology were concerned in [17], where the condition of communication delays was determined for achieving consensus. The consensus problem for a group of highorder dynamic agents with switching topology and timevarying communication delays was discussed in [18]. In the work of [19], linear consensus protocol and saturated consensus protocol were presented for the consensus problem of heterogeneous multiagent system. The heterogeneous multiagent system consists of firstorder and secondorder integrator agents.
Lately, the fuzzy logic control which consists of linguistic control rules is a technique to design controllers based on human expert knowledge and experience. This technique is a good alternative to overcome the difficulties in the requirement of exact mathematics models for plants with unexpected complex dynamics and external disturbances. Although the method has been practically successful, it has proved extremely difficult to develop a general analysis and design theory for conventional fuzzy control systems. Recently, based on TakagiSugeno (TS) fuzzy techniques, there have appeared in the literature a great number of results concerning stability analysis and design. TS fuzzy techniques have been applied to many applications of interest [20–25]. However, the membership functions of the above method are through manual adjustment. Lately, neural fuzzy control, combining with the capability of fuzzy reasoning to handle uncertain information and the capability of artificial neural networks to learn from processes, has been popularly addressed. In [26], a robust fuzzy neural network control (FNN) scheme including a parameter tuning algorithm was designed for a linear Maglev rail system to achieve the objective of modelfree control. A neural networkbased selflearning control strategy including an FNN controller and a recurrent neural network identifier was proposed for electronic throttle valves [27]. Lin and Shen [28] proposed an adaptive fuzzy neural network control scheme for a fieldoriented control permanent magnet linear synchronous motor servodrive system to track periodic reference trajectories. In [29], an adaptive network based fuzzy inference system was presented for speed and position estimations of a permanentmagnet synchronous generator. In [30], an adaptive neurofuzzy controller was presented for the tracking control of dynamic systems with unknown nonlinearities. A recurrent fuzzy neural controller for the robust tracking of a robot manipulator with adaptive observers was addressed in [31]. A robust selforganizing neural fuzzy control scheme was proposed for a class of uncertain nonlinear MIMO systems [32]. In the study of [33], an adaptive neurofuzzy inference system was employed to identify hand motion commands based on surface electromyogram signals. In [34], a new approach was proposed for machine health condition prognosis with the integration of neurofuzzy system and Bayesian algorithms. Moreover, a TS fuzzyneural model was adopted for identification and robust adaptive control of an antilock braking system [35]. In addition, a hybrid evolutionary algorithm using fuzzy rules to adjust optimization parameters was proposed in [36].
In multiagent networks, the problem of collision avoidance is also an important and interesting topic that is worthy of being discussed. A cooperative control law was proposed for general nonlinear dynamic models to guarantee collisionfree conflict resolution [37]. In [38], a fuzzy logic was designed for potential functions to achieve the separation control with input constrains. In the work of [39], a flocking algorithm was presented for separation forces generated to avoid collisions with external obstacles. A modified avoidance function was proposed for nonlinear Lagrange systems to achieve collision avoidance with bounded disturbances [40]. In [41], a potential field method was discussed for mobile robots to solve the local minimum problem. In addition, Wang and Gu [42] presented a fuzzy potential force for the separating potential function in flocking control. However, only few of existing results have been presented to solve the problem of local minima in multiagent systems.
This paper aims to investigate the formation control of leaderfollower multiagent systems, where the problem of collision avoidance is also considered. The graph theory is used to model the communication topology between agents. To improve the control performance, a novel formation algorithm, neuralfuzzy formation controller, is proposed for multiagent systems in directed graphs. The neuralfuzzy control parameter consists of input Gaussian membership functions and output fuzzy singletons, where the parameters of input and output membership functions can be adaptively adjusted. The proposed neuralfuzzy formation controller has the capability of online learning, and the adaptive rules can be derived using the gradient descent method. Moreover, a fuzzy based separation control is presented for collision avoidance, and the local minimum problem of traditional potentialbased separation control can be solved. The fuzzy based separation control consists of triangular input membership functions and singleton output membership functions, where the control output provides an alternative moving direction for agents to achieve collisionfree tasks. Numerical simulations are provided to validate the collisionfree formation responses.
This paper is organized as follows. In Section 2, some essential graphtheoretic concepts and a network of singleintegrator agents are introduced. In Section 3, the framework of a neuralfuzzy formation controller is investigated, where the updating rules for controller parameters are derived. In Section 4, the conventional potentialbased collision avoidance is introduced. Moreover, a novel fuzzyoriented separation control is presented. In Section 5, simulation results are provided for performance validations. Some concluding remarks are given in Section 6.
2. Preliminaries
2.1. Graph Theory
Considering a multiagent system of agents, let be a directed graph (digraph), consisting of a vertex set and an edge set . The vertexes and represent the th and th agents, respectively. In digraphs, an edge of is an ordered pair of distinct nodes , in which and are the head and tail of the edge, respectively [43]. The weighted adjacency matrix of a digraph is denoted as where is the link weight; , if , and , if .
In this paper, a leaderfollower problem will be dealt with, where the multiagent system consists of agents, one leader, and followers. In notations, the agents indexed by are followers and the item is the leader. Assume that the leader agent has only transmitting capability, that is, the leader acquires no information from followers, . In this case, let the topology relationship of follower agents be denoted as , a subgraph of . Then, the associated adjacency matrix of is represented as Consequently, the connection relationship between the leader and followers can be described as , where , .
2.2. SingleIntegrator Multiagent System
In this paper, a singleintegrator network is considered as where is the position vector and is the control input vector of the th agent, . It is assumed that all agents have the same environment sensing capability. In addition to formation keeping, each agent is not allowed to collide with other agents during the whole moving process.
The geometric relationship between agents is shown in Figure 1, where is the sensing radius. The node is a neighboring agent of the th node if the Euclidean distance between two agents is less or equal to the sensing radius, . Let stand for the neighbor set of the th agent. Once the th agent moves inside the sensing radius of the th agent, the collisionavoidance mechanism starts to work. In Figure 1, the notation denotes the avoidance radius of which the minimum distance allowed between two agents is . In this case, it is reasonable that for preventing collisions.
(a)
(b)
3. Neural Fuzzy Formation Control
3.1. Structure of Neural Fuzzy Controller
In this section, a neural fuzzy control (NFC) is proposed to deal with the leaderfollowing formation problem, where the singleintegrator model of (3) is considered. First, let the and axis error functions be, respectively, defined as where , and are coordinate positions regarding a desired formation pattern in and axes. It is noticed that means that the th agent can send position information to the th agent, and means that the th agent can receive position information from the leader. In this study, the leader is maneuvered along a prespecified trajectory, and the design of formation controller is focused on followers. Let the controller inputs of the th follower agent be designated as follows where and are positive constants, .
The network structure of NFC is shown in Figure 2. The fuzzy rules are given in Table 1, where the input and output spaces are fuzzily partitioned into six fuzzy sets, Negative Big (NB), Negative Medium (NM), Negative Small (NS), Positive Small (PS), Positive Medium (PM), and Positive Big (PB). The input and output membership functions are depicted in Figure 3. The corresponding ifthen fuzzy rules for the th agent are expressed as where and are the fuzzy sets of the antecedent and consequence parts, respectively, . In Figure 3, the th nodes of the membership layer are Gaussian functions, where . In (8) and (9), and are means and standard deviations, respectively, . By using the centroid defuzzification technique, the NFC output can be calculated as follows: where are the values corresponding to singleton outputs, in which and .

(a)
(b)
3.2. ParameterLearning Algorithms
In this section, the idea of gradient descent will be adopted to derive online learning algorithms to update NFC parameters. First, energy functions and are defined as follows: where . Then, the update laws of layer parameters are described in the following.
(1) Output Layer. According to the gradient decent method [44], and are updated by the following rules: where is a learning rate, . From (11), it can be obtained that In (14), the error term to be propagated can be reformulated as However, the terms and in (16) cannot be analytically determined. To overcome this problem, the following adaptive laws are adopted [45]: where and are positive constants.
In summary, from (15) and (17), the parameters of the output layer can be adaptively updated.
(2) Membership Layer. The parameters and are updated by the following rules: where is a learning rate, , . From (8), (9), and (11), it can be obtained that Similarly, and the are updated by the following amounts: where is a learning rate, , . From (8), (9), and (11), it can be obtained that In summary, from (18)–(21), the parameters of the input layer can be adaptively updated.
4. Separation Control for Collision Avoidance
4.1. PotentialBased Separation Control
Let be the separation force between the th and th agents. Then, the integrated separation force from all its neighboring agents can be formulated as To derive a proper separation force between two connected agents, a smooth potential function is considered, such that In a twodimensional case, , the gradient computations of can be obtained as follows: where is the maximum allowable separation force. Integrating the formation and separation forces, the net control action to an agent can be obtained as where is the neural fuzzy formation action.
4.2. Fuzzy Separation Control
In a multiagent system, an agent could be standstill or move back and forth if the net force acting on this agent is balanced. This phenomenon is known as the local minimum problem. In Figure 4, the case of agent with two neighboring agents is considered, where and are neighboring agents, and the related target is located at . The notation is denoted as the attractive force to the target, and and are separation forces corresponding to neighboring agents. Then, the integrated separation force to the th agent is the vector sum of and , where is the related direction of to the axis. In addition, is the net force of the th agent, and is angle between the agent and its target. In case , the direction of is opposite to the attractive action . Moreover, if the magnitude of is less or equal to the magnitude of , the th agent will be stuck in the local minimum. To solve this problem, a fuzzy separation control method will be presented, and the key idea is depicted in Figure 5, where an extra angle is added to the original separation force, . In Figure 5, is the modified separation force, of which the magnitude keeps unchanged but the direction is changed because of . Consequently, the integrated net force of attractive force and separation force can be represented as Basically, can provide a new route to bypass the neighboring agents when a local minimum situation happens. It is noticed that the target of a follower is its temporary destination for next movement during the process of avoiding collision. For those follower agents, communicationconnected to the leader, their respective targets can be obtained according to the leader position and designated formation pattern; however, a substitute solution is required for other followers. Alternatively, from (4) and (5), targets can be equivalently viewed as the propagated errors from other followers, where follower is not communicated to the leader, , .
The design of fuzzy separation controller will be illustrated in the following. First, let the fuzzy inputs of the th agent be where is the position of the center gravity of neighboring agents, where is the cardinality of a set, that is, is the number of neighboring agents of the th agent.
The fuzzy rules are given in Table 2, where the input and output spaces are fuzzily partitioned into ten fuzzy sets, Negative Big (NB), Negative Small (NS), Zero (ZO), Positive Small (PS), Positive Big (PB), Very Close (VC), Close (C), Medium (M), Far (F), and Very Far (VF). The input and output membership functions are depicted in Figures 6 and 7, respectively. The corresponding ifthen fuzzy rules for the th agent are expressed as where and are the fuzzy sets of the antecedent part, and is the fuzzy set of the consequence part, , and . By using the centroid defuzzification technique, the defuzzified fuzzy output is calculated as where a minmax operation is performed over all rules mapping to the same output fuzzy set:

(a)
(b)
5. Simulation Results
In the following, all the agents are assumed to be homogeneous with the same specifications, (m) and (m). To verify the feasibility of proposed neurofuzzy formation controller and fuzzy separation controller, both the collision avoidance and formation preservation are considered.
5.1. Collision Avoidance with Static Target
One dynamical agent is initially placed at the point (m), and four fixed agents are located at , , , and (m), respectively. The target position is given as (m). It is desired that the dynamic agent can reach the designated target without colliding with fixed agents. Figures 8 and 9 illustrate the collisionavoidance responses corresponding to potentialbased separation control (PSC) and the proposed fuzzy separation control (FSC), where the formation control of each case is the conventional consensus algorithm [46]. In Figure 8, it can be seen that the agent lingers in front of neighboring agents with PSC. On the other hand, the agent can successfully bypass neighboring agents with the proposed FSC. The responses of position errors, shown in the bottom two subplots of Figure 8, indicate that the desired target can be asymptotically achieved with the proposed FSC. However, a localminimum behavior exists by using the PSC, and the related separation force keeps oscillation. The oscillation behavior of the separation force is coincided with the response of , shown in Figure 9(a). In Figure 9(b), the angle after 8.26 (sec) means that the dynamic agent successfully bypasses the fixed agents, and thus, there is no separation force.
(a)
(b)
(a)
(b)
5.2. Formation Control and Collision Avoidance with Static Leader
In leaderfollower formation control, the case of five agents, one static leader and four followers, is considered. The communication topology is shown in Figure 10, where the circles labelled 1 to 4 denote the follower agents and the circle represents the leader agent. From (2), the information exchanges between leader and followers can be modelled as The followers are initially placed at the points , , , and (m), respectively, and the position of the leader is (m). The formation pattern is designated as , , , , and (m). The parameters of the NFC are originally chosen as , , , , , , , and . Three unique distributed strategies, consensus algorithm with potentialbased separation control (CA + PSC), consensus algorithm with fuzzy separation control (CA + FSC), and neurofuzzy formation control with fuzzy separation control (NFC + FSC), are considered. Accordingly, simulation results are shown in Figures 11, 12, and 13, including the error trajectories of  and axis, and the relative distances between a pair of agents. Since the relative distance between any two agents is greater than , shown in Figures 11(b), 12(b), and 13(b), the desired collision avoidance can be accomplished using these three control strategies. In addition, the formation responses are shown in Figures 11(a), 12(a), and 13(a). It can be seen that there exist significant steadystate errors using CA + PSC. Conversely, the formation pattern can be asymptotically achieved with the use of NFC + FSC. The position errors, , are summarized in Table 3, where IAE is the integral absolute error, ISE is the integral square error, ITAE is the integral time absolute error, and ITAE stands for the integral time square error. In summary, the proposed NFC + FSC can provide better performance than the counterparts of the other two methods.

(a)
(b)
(a)
(b)
(a)
(b)
5.3. Formation Control and Collision Avoidance with Dynamic Leader
In the following, previous leaderfollower system is addressed, where all initial settings are the same except that one fixed agent is added at the position (m). This fixed agent can be viewed as a standstill obstacle. Besides, a timevarying leader is considered, where the velocity vector is (m/s). The formation responses are shown in Figures 14, 15, and 16, where the methods of CA + PSC, CA + FSC, and NFC + FSC are considered. Similar to the previously mentioned illustrations, the moving trajectories and the relative distances between two follower agents are depicted in Figures 14, 15, and 16. It can be seen that even the collision avoidance can be achieved with CA + PSC and CA + FSC; however, these two strategies eventually fail to preserve the desired formation pattern. On the other hand, from Figure 16, collisionfree formation can be obtained by using the proposed NFC + FSC control scheme. In particular, it can be seen that the third follower can successfully bypass the fixed agent and keep its way to form a designated pattern. The formation errors of different control strategies are summarized in Table 4. It can be concluded that the proposed neurofuzzy formation and fuzzy separation combined controller can provide better responses than the counterparts of conventional consensus algorithm with potentialbased separation control.

(a)
(b)
(a)
(b)
(a)
(b)
Remark 1. From simulation results, it can be seen that the collisionfree formation task can be achieved for singleintegrator agents. In particular, the local minimum problem can be overcome using the proposed fuzzy separation control. It is promising that the proposed works can be applied to some practical applications, such as multirobot systems and unmanned vehicle systems.
6. Conclusion
This paper presents a neural fuzzy formation controller for multiagent systems. The learning rules for controller parameters can be derived from the use of gradient decent method. In addition, a fuzzy separation control is proposed to achieve collision avoidance such that the problem of local minima can be solved. Simulation results are provided for the cases of static leader and dynamic leader. The collisionfree leaderfollower formation can be accomplished by the proposed fuzzy formation and separation control strategy. Performance comparisons indicate that the proposed fuzzybased control scheme has better formation responses compared to the counterparts of conventional consensus algorithm and potentialbased separation control. The current results are mainly limited in singleintegrator multiagent systems. It should be challenging and interesting to investigate the collision avoidance problem for highorder agents with nonlinear dynamics. For example, a fuzzy based separation control for kinematic agents will be undertaken in our future work.
Acknowledgments
This work is partially supported by the Taiwan National Science Council under the Grants NSC 1012221E182041MY2 and NSC 1002918I182002. The authors also want to appreciate the High Speed Intelligent Communication Research Center, Chang Gung University, Taiwan, for continuing support in the study of multiagent systems.
References
 Z. G. Hou, L. Cheng, and M. Tan, “Decentralized robust adaptive control for the multiagent system consensus problem using neural networks,” IEEE Transactions on Systems, Man, and Cybernetics, Part B, vol. 39, no. 3, pp. 636–647, 2009. View at: Publisher Site  Google Scholar
 H. Zhang and F. L. Lewis, “Adaptive cooperative tracking control of higherorder nonlinear systems with unknown dynamics,” Automatica, vol. 48, no. 7, pp. 1432–1439, 2012. View at: Publisher Site  Google Scholar  MathSciNet
 M. E. Valcher and P. Misra, “On the controllability and stabilizability of nonhomogeneous multiagent dynamical systems,” Systems & Control Letters, vol. 61, no. 7, pp. 780–787, 2012. View at: Publisher Site  Google Scholar  Zentralblatt MATH  MathSciNet
 B. RanjbarSahraei, F. Shabaninia, A. Nemati, and S.D. Stan, “A novel robust decentralized adaptive fuzzy control for swarm formation of multiagent systems,” IEEE Transactions on Industrial Electronics, vol. 59, no. 8, pp. 3124–3134, 2012. View at: Google Scholar
 R. Cui, B. Ren, and S. S. Ge, “Synchronised tracking control of multiagent system with highorder dynamics,” IET Control Theory & Applications, vol. 6, no. 5, pp. 603–614, 2012. View at: Publisher Site  Google Scholar  MathSciNet
 X. Luo, D. Liu, X. Guan, and S. Li, “Flocking in target pursuit for multiagent systems with partial informed agents,” IET Control Theory & Applications, vol. 6, no. 4, pp. 560–569, 2012. View at: Publisher Site  Google Scholar  MathSciNet
 D. J. Pack, P. DeLima, G. J. Toussaint, and G. York, “Cooperative control of UAVs for localization of intermittently emitting mobile targets,” IEEE Transactions on Systems, Man, and Cybernetics, Part B, vol. 16, no. 1, pp. 19–33, 2009. View at: Publisher Site  Google Scholar
 T. Keviczky, F. Borrelli, K. Fregene, D. Godbole, and G. J. Balas, “Decentralized receding horizon control and coordination of autonomous vehicle formations,” IEEE Transactions on Control Systems Technology, vol. 16, no. 1, pp. 19–33, 2008. View at: Publisher Site  Google Scholar
 K. S. Hwang, S. W. Tan, M. C. Hsiao, and C. S. Wu, “Cooperative multiagent congestion control for highspeed networks,” IEEE Transactions on Systems, Man, and Cybernetics, Part B, vol. 35, no. 2, pp. 255–268, 2005. View at: Publisher Site  Google Scholar
 L. Shi, A. Capponi, K. H. Johansson, and R. M. Murray, “Resource optimisation in a wireless sensor network with guaranteed estimator performance,” IET Control Theory and Applications, vol. 4, no. 5, pp. 710–723, 2010. View at: Publisher Site  Google Scholar  MathSciNet
 B. Fidan, C. Yu, and B. D. O. Anderson, “Acquiring and maintaining persistence of autonomous multivehicle formations,” IET Control Theory and Applications, vol. 1, no. 2, pp. 452–460, 2007. View at: Publisher Site  Google Scholar
 R. OlfatiSaber and R. M. Murray, “Consensus problems in networks of agents with switching topology and timedelays,” IEEE Transactions on Automatic Control, vol. 49, no. 9, pp. 1520–1533, 2004. View at: Publisher Site  Google Scholar  MathSciNet
 H. Jiang, J. Yu, and C. Zhou, “Consensus of multiagent linear dynamic systems via impulsive control protocols,” International Journal of Systems Science, vol. 42, no. 6, pp. 967–976, 2011. View at: Publisher Site  Google Scholar  Zentralblatt MATH  MathSciNet
 Y.H. Chang, C.W. Chang, C.L. Chen, and C.W. Tao, “Fuzzy slidingmode formation control for multirobot systems: design and implementation,” IEEE Transactions on Systems, Man and Cybernetics, Part B, vol. 42, no. 2, pp. 444–457, 2012. View at: Google Scholar
 N. Cai, J.X. Xi, Y.S. Zhong, and H.Y. Ma, “Controllability improvement for linear timeinvariant dynamic multiagent systems,” International Journal of Innovative Computing, Information and Control, vol. 8, no. 5, pp. 3315–3328, 2012. View at: Google Scholar
 W. Ni and D. Cheng, “Leaderfollowing consensus of multiagent systems under fixed and switching topologies,” Systems and Control Letters, vol. 59, no. 34, pp. 209–217, 2010. View at: Publisher Site  Google Scholar  Zentralblatt MATH  MathSciNet
 J. Qin, H. Gao, and W. X. Zheng, “Secondorder consensus for multiagent systems with switching topology and communication delay,” Systems and Control Letters, vol. 60, no. 6, pp. 390–397, 2011. View at: Publisher Site  Google Scholar  Zentralblatt MATH  MathSciNet
 F. Jiang, L. Wang, and G. Xie, “Consensus of highorder dynamic multiagent systems with switching topology and timevarying delays,” Journal of Control Theory and Applications, vol. 8, no. 1, pp. 52–60, 2010. View at: Publisher Site  Google Scholar  Zentralblatt MATH  MathSciNet
 Y. Zheng, Y. Zhu, and L. Wang, “Consensus of heterogeneous multiagent systems,” IET Control Theory & Applications, vol. 5, no. 16, pp. 1881–1888, 2011. View at: Publisher Site  Google Scholar  MathSciNet
 X. Su, P. Shi, L. Wu, and Y.D. Song, “A novel control design on discretetime TakagiSugeno fuzzy systems with timevarying delays,” IEEE Transactions on Fuzzy Systems, 2012. View at: Publisher Site  Google Scholar
 X. Su, P. Shi, L. Wu, and Y.D. Song, “A novel approach to filter design for TS fuzzy discretetime systems with timevarying delay,” IEEE Transactions on Fuzzy Systems, vol. 20, no. 6, pp. 1114–1129, 2012. View at: Google Scholar
 X. Su, P. Shi, L. Wu, and S. K. Nguang, “Induced ${\ell}_{2}$ filtering of fuzzy stochastic systems with timevarying delays,” IEEE Transactions on Systems, Man and Cybernetics, Part B, 2012. View at: Publisher Site  Google Scholar
 L. Wu, X. Su, P. Shi, and J. Qiu, “Model approximation for discretetime statedelay systems in the TS fuzzy framework,” IEEE Transactions on Fuzzy Systems, vol. 19, no. 2, pp. 366–378, 2011. View at: Publisher Site  Google Scholar
 L. Wu and W. X. Zheng, “${L}_{2}{L}_{\infty}$ control of nonlinear fuzzy itô stochastic delay systems via dynamic output feedback,” IEEE Transactions on Systems, Man, and Cybernetics, Part B, vol. 39, no. 5, pp. 1308–1315, 2009. View at: Publisher Site  Google Scholar
 L. Wu, X. Su, P. Shi, and J. Qiu, “A new approach to stability analysis and stabilization of discretetime TS fuzzy timevarying delay systems,” IEEE Transactions on Systems, Man, and Cybernetics, Part B, vol. 41, no. 1, pp. 273–286, 2011. View at: Publisher Site  Google Scholar
 R. J. Wai and J. D. Lee, “Robust levitation control for linear Maglev rail system using fuzzy neural network,” IEEE Transactions on Control Systems Technology, vol. 17, no. 1, pp. 4–14, 2009. View at: Publisher Site  Google Scholar
 X. Yuan, Y. Wang, L. Wu, X. Zhang, and W. Sun, “Neural network based selflearning control strategy for electronic throttle valve,” IEEE Transactions on Vehicular Technology, vol. 59, no. 8, pp. 3757–3765, 2010. View at: Publisher Site  Google Scholar
 F. J. Lin and P. H. Shen, “Adaptive fuzzyneuralnetwork control for a DSPbased permanent magnet linear synchronous motor servo drive,” IEEE Transactions on Fuzzy Systems, vol. 14, no. 4, pp. 481–495, 2006. View at: Publisher Site  Google Scholar
 M. Singh and A. Chandra, “Application of adaptive networkbased fuzzy inference system for sensorless control of PMSGbased wind turbine with nonlinearloadcompensation capabilities,” IEEE Transactions on Power Electronics, vol. 26, no. 1, pp. 165–175, 2011. View at: Publisher Site  Google Scholar
 S. P. Moustakidis, G. A. Rovithakis, and J. B. Theocharis, “An adaptive neurofuzzy tracking control for multiinput nonlinear dynamic systems,” Automatica, vol. 44, no. 5, pp. 1418–1425, 2008. View at: Publisher Site  Google Scholar  MathSciNet
 S. H. Park and S. I. Han, “Robusttracking control for robot manipulator with deadzone and friction using backstepping and RFNN controller,” IET Control Theory & Applications, vol. 5, no. 12, pp. 1397–1417, 2011. View at: Publisher Site  Google Scholar  MathSciNet
 C.S. Chen, “Robust selforganizing neuralfuzzy control with uncertainty observer for MIMO nonlinear systems,” IEEE Transactions on Fuzzy Systems, vol. 19, no. 4, pp. 694–706, 2011. View at: Google Scholar
 M. Khezri and M. Jahed, “A neurofuzzy inference system for sEMGbased identification of hand motion commands,” IEEE Transactions on Industrial Electronics, vol. 58, no. 5, pp. 1952–1960, 2011. View at: Publisher Site  Google Scholar
 C. Chen, B. Zhang, and G. Vachtsevanos, “Prediction of machine health condition using neurofuzzy and Bayesian algorithm,” IEEE Transactions on Instrumentation and Measurement, vol. 61, no. 2, pp. 297–306, 2012. View at: Google Scholar
 W.Y. Wang, M.C. Chen, and S.F. Su, “Hierarchical TS fuzzyneural control of antilock braking system and active suspension in a vehicle,” Automatica, vol. 48, no. 8, pp. 1698–1706, 2012. View at: Publisher Site  Google Scholar  MathSciNet
 T. Niknam, H. D. Mojarrad, and M. Nayeripour, “A new hybrid fuzzy adaptive particle swarm optimization for nonconvex economic dispatch,” International Journal of Innovative Computing, Information and Control, vol. 7, no. 1, pp. 189–202, 2011. View at: Google Scholar
 D. M. Stipanović, P. F. Hokayem, M. W. Spong, and D. D. Šiljak, “Cooperative avoidance control for multiagent systems,” Journal of Dynamic Systems, Measurement and Control, Transactions of the ASME, vol. 129, no. 5, pp. 699–707, 2007. View at: Publisher Site  Google Scholar
 D. Gu and H. Hu, “Using fuzzy logic to design separation function in flocking algorithms,” IEEE Transactions on Fuzzy Systems, vol. 16, no. 4, pp. 826–838, 2008. View at: Publisher Site  Google Scholar
 D. Gu and Z. Wang, “Leaderfollower flocking: algorithms and experiments,” IEEE Transactions on Control Systems Technology, vol. 17, no. 5, pp. 1211–1219, 2009. View at: Publisher Site  Google Scholar
 P. F. Hokayem, D. M. Stipanović, and M. W. Spong, “Coordination and collision avoidance for Lagrangian systems with disturbances,” Applied Mathematics and Computation, vol. 217, no. 3, pp. 1085–1094, 2010. View at: Publisher Site  Google Scholar  Zentralblatt MATH  MathSciNet
 S. S. Ge and Y. J. Cui, “Dynamic motion planning for mobile robots using potential field method,” Autonomous Robots, vol. 13, no. 3, pp. 207–222, 2002. View at: Publisher Site  Google Scholar
 Z. Wang and D. Gu, “Cooperative target tracking control of multiple robots,” IEEE Transactions on Industrial Electronics, vol. 59, no. 8, pp. 3232–3240, 2012. View at: Google Scholar
 M. Mesbahi and M. Egerstedt, Graph Theoretic Methods in Multiagent Networks, Princeton University Press, Princeton, NJ, USA, 2010. View at: MathSciNet
 L. Khan, S. Anjum, and R. Badar, “Standard fuzzy model identification using gradient methods,” World Applied Sciences Journal, vol. 8, no. 1, pp. 1–9, 2010. View at: Google Scholar
 F. J. Lin, R. J. Wai, and C. C. Lee, “Fuzzy neural network position controller for ultrasonic motor drive using pushpull DCDC converter,” IEE Proceedings: Control Theory and Applications, vol. 146, no. 1, pp. 99–107, 1999. View at: Publisher Site  Google Scholar
 W. Ren and R. Beard, Distributed Consensus in MultiVehicle Cooperative Control: Theory and Applications, Springer, 2007.
Copyright
Copyright © 2013 YeongHwa Chang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.