Abstract
A stochastic pinning approach for multiagent systems is developed, which guarantees such systems being almost surely stable. It is seen that the pinning is closely related to being a Bernoulli variable. It has been proved for the first time that a series of systems can be stabilized by a Brownian noise perturbation in terms of a pinning scheme. A new terminology named “stochastic pinning control” is introduced to describe the given pinning algorithm. Additionally, two general cases that the expectation of the Bernoulli variable with bounded uncertainty or being unknown are studied. Finally, two simulation examples are provided to demonstrate the effectiveness of the proposed methods.
1. Introduction
Due to the broad applications in cooperative control of unmanned air vehicles, formation control of mobile robots, sensor networks, and cooperative surveillance, multiagent problems have drawn a lot of attention. In particular, multiagent coordination with multiple leaders becomes more and more important, which forces a group of agents into a specific target region. Because of the spatial distribution of actuators and sensors, it is of high cost or even impossible in practice to implement a centralized controller. Instead, distributed control emerged to be a promising tool for coordination of multiagent systems, which is usually to design a controller to every subsystem. During the past years, many important results have been reported in [1–6]. Many natural and man-made systems, such as ecosystems, internet, Word Wide Web, social networks, and power grids, are described by it. In recent years, the analysis and control of complex behaviors in complex networks have become a hot topic across many fields such as in [7–11]. Especially, synchronization related to being the most important collective behavior of complex networks, such as ER random, small-world, and scale-free complex networks [12–14], has been extensively studied. Via introducing a Bernoulli stochastic variable describing the random switching of controllers, the distributed synchronization of complex networks was studied in [15, 16].
Due to the complexity of complex networks, it is usually not easy to control a complex network by adding controllers to all nodes. Instead, pinning control only uses a small number of controllers. In this sense, it is said that pinning control is a promising method, which can efficiently reduce the number of controllers. The pinning control strategy for linear coupled networks was investigated in [17, 18], in which two different pinning strategies, random pinning and special pinning methods, were theoretically and numerically compared. During the past decades, a lot of results on synchronization of various complex networks by pinning control have emerged, for example, [19–25]. By searching such references on pinning control, it is concluded that all the pinning methods are realized by a kind of regular controllers in terms of in the drift section of a system. However, it is possible to design a controller to stabilize a stochastic system almost surely which is unable to be stabilized in mean-square sense. Based on these facts, it is asked that can we design a pinning controller referred to be a Brownian noise perturbation to stabilize multiagent systems? To the best of authors’ knowledge, the control problem of multiagent systems via a pinning controller only in the diffusion part has not yet been investigated, which motivates the current research.
In this paper, the control problem of multiagent systems is firstly considered by a stochastic pinning viewpoint. In contrast to the existing results of pinning control methods, the main contributions of this paper are as follows. The control of multiagent systems is firstly realized by a pinning control method in terms of a Brownian noise perturbation. In order to achieve this goal, new pinning control in terms of stochastic pinning control (SPC) is developed, in which the Bernoulli variable plays an important role in SPC. More general cases such as the expectation of Bernoulli variable with uncertainty and being unknown are considered respectively. The relationship among the expectation, the pinning fraction, and the pinning control gain for both random and special pinning control is demonstrated in detail.
Notation. denotes the dimensional Euclidean space; is the set of all real matrices. denotes the Euclidean norm. is the expectation operator with respect to some probability measure. is the Kronecker product. In symmetric block matrices, we use “” as an ellipsis for the terms induced by symmetry, for a block-diagonal matrix. , where and .
2. Problem Formulation
Consider a multiagent system consisting of agents, such as in [6], the model of agent, , is described as where is the state vector, is the inherent nonlinear dynamic. However, it can be also be dealt with by the fuzzy method [26, 27]. In this paper, the nonlinear term is treated directly, and an assumption is needed here.
Assumption 1. Nonlinear function is assumed to satisfy the following condition: where is a given constant.
In this paper, a new pinning control method for system (1) in terms of stochastic pinning control is proposed as respectively, where is the coupling strength, is the coupling matrix with , , , which is an irreducible matrix, and is the control gain. Bernoulli variable in (4) is described as whose expectation is .
In this paper, such controllers will be used as a pinning controller in terms of Brownian noise perturbation. As a result, we have the closed-loop system as where is a -dimensional Brownian motion or Wiener process. Let , , and ; one has and , respectively. Without of loss generality, in this paper it is assumed that , for all . Based on these notations, we have where . Operation mode takes values in set and is described as whose probabilities are and , respectively.
Remark 2. It is worth pointing out that the proposed pinning control method is different from the existing pinning methods. Firstly, the pinning problem of this paper is realized by a Brownian noise perturbation, which cannot be solved by the usual analysis methods. Secondly, the introduced Bernoulli variable plays an important role in achieving the pinning control in terms of Brownian noise control. Such differences embody the property of stochastic pinning control. That is, when , only controller (6) works and is a pinning controller. On the contrary, if , the desired controller becomes a distributed controller, which is not pinning control in fact. In this sense, it is said that the developed pinning control is a stochastic control method.
Remark 3. It should be remarked that this framework is necessary to achieve pinning control through Brownian noise perturbations. If there is no in (7), the underlying systems become (6) with and (1) with , which can be obtained by applying the usual pinning methods directly. Unfortunately, it is concluded that this pinning framework is very hard to realize the desired object. The reason will be explained later. On the other hand, when , we have (6) only, which is seen as a distributed controller instead of a pinning controller. Thus, it is claimed that the given pinning control algorithm bridges the traditional pinning control and distributed control. In addition, can also choose other forms, such as , where both and are dependent or independent Bernoulli variables.
Definition 4. The equilibrium of system (8) is said to be almost surely exponentially stable if for any
3. Main Results
Theorem 5. For given scalars and , the equilibrium of system (8) is almost surely exponentially stable, if there exists such that where , .
Proof. For any given initial condition , it is known that will never reach zero with probability one, and by Itô formula, it is obtained that Then, it is obtained that where is a continuous martingale vanishing at . Taking into account (9), it is seen that On the other hand, it is concluded that for any finite given in , there exists a positive scalar that the quadratic variation of is Applying the strong law of the large numbers to , one has Based on (14) and (16), we conclude that which is guaranteed by (11). This completes the proof.
Remark 6. By Theorem 5, it is known that if condition (11) holds, one can stabilize system (1) by a pinning control tactic in terms of Brownian noise perturbation. However, condition (11) is impossible or hard to be satisfied if controller (7) is with . If , condition (11) becomes , where pinning controller is same as the traditional pinning controller. Unfortunately, it is said that with is impossible or hard to be satisfied because of . That is, due to the property of in (3) or (4), we have , and . It usually results in . In this sense, when , it is difficult to realize the pinning control goal of this paper.
From Theorem 5, it is seen that the expectation plays an important role in SPC which should be given exactly. In some applications, it is very hard or of high cost to obtain exactly. Instead, only its estimation is available. Then, it is natural and important to study how to realize SPC when is uncertain. If there exists an uncertainty in , we will use its estimation . It is described as where admissible uncertainty with . Then, we have the following theorem.
Theorem 7. For given scalars and , the equilibrium of system (8) is robust almost surely exponentially stable for any admissible uncertainty (18), if there exist and such that
Proof. Based on the proof of Theorem 5, it is obtained that the change of only takes place in (14); that is For with any , it is seen that Taking into account (19) and (21), one has (11). That completes the proof.
It is seen that the conditions of Theorems 5 and 7 are not LMIs, which are not solved directly. In the following, another condition in terms of LMIs with equation constraints is proposed, which could be solved easily.
Theorem 8. For given scalars and , the equilibrium of system (8) is almost surely exponentially stable, if there exist , , , , , and , such that the following LMIs hold for all : either or where .
Proof. Based on (11), it is known that if there are and such that hold, which are guaranteed by or respectively. Based on these, it is obvious that (28) can be obtained by (23), (26), or (27). Then, we have (11) which is insured by where and should be determined. Because of nonlinear term in (31), it cannot be solved by LMI tool box directly. By introducing a variable satisfying , it is concluded that it is equivalent to . By Schur complement and condition (25), one has (22)–(25) implying (11). This completes the proof.
If the expectation is unknown, how to achieve a useful SPC is another general case. For this case, we have the following theorem.
Theorem 9. For a given scalar , the equilibrium of system (8) is almost surely exponentially stable, if there exists such that In this case, the expectation can be unknown, but it should be satisfied
Proof. Based on the proof of Theorem 5, one can easily have the equilibrium of system (8) almost surely exponentially stable, if there exists such that which is equivalent to If is unknown but satisfies (33), one could always choose a sufficiently large scalar such that (35) holds. This completes the proof.
Corollary 10. For a given scalar , the equilibrium of system (8) is almost surely exponentially stable, if there exists such that and (32) hold. In this case, the expectation can be unknown, but it should be satisfied .
Proof. Similar to the proof of Theorem 9, (34) can be rewritten to be which could be guaranteed by Since and , it is obtained that there is always a sufficiently large constant such that (37) holds. On the other hand, under the conditions of and , we have (33) implying (38) directly. This completes the proof.
It is claimed that the key idea of SPC described by (3) and (4) can be used to construct a pinning controller in the drift section. That is Let , , and ; one has It is rewritten to be where . Without of loss generality, in matrix , it is also assumed that , for all . Since is a Bernoulli variable, it is known that .
Theorem 11. For given scalars and , the equilibrium of system (40) is exponentially mean-square stable, if there exists such that
Proof. Choose the following Lyapunov function: Based on (42), we have Then, there is always a sufficient small scalar such that By Dynkin’s formula, it is obtained that for Applying the Gronwall-Bellman lemma to (46), one gets This completes the proof.
Remark 12. It should be pointed out that the pinning controller in the drift section is also different from the existing methods such as [17, 18, 20, 21, 28]. It is said that the pinning method of this paper is a stochastic algorithm, where the expectation plays an important role. Compared with the traditional pinning methods, the desired pinning controller is not necessary implemented online, which is added to some nodes in terms of probability . The correlation among the expectation , the pinning fraction , and the control gain is firstly illustrated in Theorem 11, which is also shown by numerical examples.
Similarly, when the expectation is uncertain and satisfies (18), we have the following result.
Theorem 13. For given scalars and , the equilibrium of system (40) is robust exponentially mean-square stable for any admissible uncertainty (18), if there exist and such that where .
Proof. Based on the proof of Theorem 11 and taking in (48), it is obtained that (44) is rewritten as where is defined in (41). Obviously, it is known that (48) implies (49). This completes the proof.
When is unknown, we have the following theorem.
Theorem 14. For a given scalar , the equilibrium of system (40) is exponentially mean-square stable, if there exists such that In this case, there is an SPC such that (40) is exponentially mean-square stable with unknown .
Proof. Based on Theorem 11, the equilibrium of system (40) is exponentially mean-square stable, if there exists such that which is equivalent to By the definition of , it is known that , . Because of and , it is concluded that (52) is guaranteed by If (50) satisfies, one could always choose a sufficiently large scalar such that (53) holds. This completes the proof.
4. Numerical Examples
In this section, two numerical examples are used to demonstrate the utility of the proposed method.
Example 1. Without loss of generality, consider a multiagent system with agents, whose agent is described as
In this example, such multiagent system will be stabilized by a stochastic pinning controller whose connection of agents in terms of scale-free network is simulated in Figure 1. Then, the coupling matrix can be obtained from Figure 1 directly.
From system (54), it is obtained that can be chosen as . The correlations among the expectation , the pinning fraction , and the pinning control gain are given in Theorem 5, which are demonstrated in Figures 2 and 3, respectively. In this paper, the special pinning control means (3) takes place in the nodes with more degrees. From Figure 2, it is seen that for given , larger expectation results in smaller control gain . When is small, such as , the curve of along with changes sharply, while the other section is gentle. On the other hand, the change of along with under given is shown in Figure 3. By this simulation, one knows that larger results in larger . Different from Figure 2, it is seen that the whole curve is gentle. If , , by Theorem 5, we have with . Let initial condition of system (54) be ; the state response of the closed-loop system is given in Figure 4, which is stable and demonstrates that the desired SPC is effective.
When the stochastic pinning controller is realized by random pinning control in terms of (3) taking place in any nodes, one has the following results, which are given in Figures 5 and 6, respectively. Considering Figures 2 and 5, it is concluded that control gain in both of them becomes larger when takes larger values. Especially, from Figure 5, it is further obtained that in some cases with larger pinning fraction , smaller results in no solution to . On the other hand, in Figure 6, one has that for some given values of such as , and , there is no solution to control gain in terms of random pinning control when satisfies and should be an integer number. On the contrary, even if , we also have the control gain of special pinning controller. In this sense, it is said that special pinning control is better. For a given , when pinning fraction becomes larger, larger control gain is needed no matter which pinning control algorithm is selected. That means if one wants to pin a multiagent system by exploiting SPC described by (3) and (4) in terms of more agents controlled directly, he should provide a larger control gain . Moreover, there is an interesting phenomenon in Figure 2 with and Figure 5 with . That is, for the same , the gain of random pinning controller is smaller than one of special pinning controller. This phenomenon can be explained if the two pinning methods are effective, because of special pinning control pinning more nodes due to the controlled nodes more “important”, it needs its control gain larger.
Example 2. Consider a dynamical node of complex network is a Chua’s chaotic circuit described by where . When the parameters are , , , and , Chua’s system has a chaotic attractor shown in Figure 7. By computation, one has in view of Assumption 1. Suppose an undirected network consisting of nodes in terms of small word network, where the connection is given in Figure 8. Similarly, its coupling matrix is easily obtained from Figure 8.
By Theorem 11 with coupling strength , one has the relationship among the expectation , the pinning fraction , and the pinning control gain in terms of special pinning control, which are given Figures 9 and 10, respectively. From such simulations, it is seen that larger expectation results in smaller control gain with given pinning fraction , while larger pinning fraction also results in smaller with given expectation . This property is same as that in Example 1. Let initial condition of system (55) be and , , one has with by Theorem 11. The state curve of the closed-loop system is given in Figure 11. From Figure 11, it is said that the desired pinning controller is useful. If the desired pinning controller is realized by random pinning control, we also have the corresponding simulations of correlation among , , and . Such relationships are demonstrated in Figures 12 and 13, respectively, which are quite different to the above cases. That is, the array of curves , , and in Figure 12 is different from those in Figures 9, 2, and 5, though there is also a consistency that larger leads to smaller . Accordingly, a phenomenon different from Figures 3, 6, and 10 is shown in Figure 13. For a given , it is seen that the value change of is not in accordance with . Such differences come from the properties of complex network and random pinning control.
5. Conclusions
In this paper, a new pinning method with a stochastic pinning viewpoint is proposed to investigate the control problem of multiagent systems. It has been shown that a fraction of controllers added to nodes in terms of Brownian noise perturbations can stabilize the underlying systems, whose control method is defined as “stochastic pinning control." It is also seen that the Bernoulli variable plays an essential role in realizing SPC. Based on the given method, new sufficient conditions of the expectation with uncertainty and being unknown are also established. Finally, the utility of the developed theory is illustrated by numerical examples. In this paper, there is no delay in the underlying system. When there is time delay in the controller such as [29], one may design a similar stochastic pinning controller with time delay, which will be our further topics.
Conflict of Interests
The author declares that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
This work was supported by the National Natural Science Foundation of China under Grants 61104066 and 61374043, the China Postdoctoral Science Foundation funded Project under Grant 2012M521086, and the Program for Liaoning Excellent Talents in University under Grant LJQ2013040.