Flocking of Discrete-time Multi-Agent Systems with Predictive Mechanisms

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1 Preprints of the 8th IFAC World Congress Milano (Italy) August 28 - September 2, 2 Flocing of Discrete-time Multi-Agent Systems Predictive Mechanisms Jingyuan Zhan, Xiang Li Adaptive Networs and Control Lab Electronic Engineering Department, Fudan University Shanghai 2433, China ( @fudan.edu.cn, lix@fudan.edu.cn) Abstract: For decades, scientists from various fields have been fascinated by flocing behavior for its wide engineering applications. Inspired by the predictive intelligence of bio-groups, in this paper we design a novel discrete-time model predictive control based flocing protocol, and investigate the role of predictive mechanisms in flocing of multi-agent systems by using mathematical analyses and numerical simulations. Compared flocing algorithm, the newly proposed model predictive control based flocing protocol yields several improvements on the convergence rate, -lattice regularity and the feasible range of the sampling period. Keywords: flocing, predictive intelligence, model predictive control, multi-agent system, convergence rate, -lattice.. INTRODUCTION Flocing, a form of global collective behaviour from distributed agent-to-agent interactions, has stimulated quantities of investigations by Reynolds (987), Vicse et al. (995), Jadbabaie et al. (23), Gazi and Passino (23), and Murray (23, 24), Tanner et al. (23a, 23b, 27), Moreau (25), Ren and Beard (25), Olfati- Saber (26), et al. (27), Gazi (28), and Su et al. (29a, 29b). Flocing behaviour is widespread in nature, for example, crowds of birds and fishes coordinate their movements to achieve coherent displacement for migration or group safety. It s a high-level self-organized behaviour of a group resulting from the interactions among individuals out a central authority governing the group. The self-organization and emergence features of flocs as reported in Couzin & Krause (23) have attracted a lot of attention for its extensive applications in engineering areas including massive distributed sensing using mobile sensor networs; self-assembly of connected mobile networs; automated parallel delivery of payloads; and combat using cooperative unmanned aerial vehicles (UAVs) which were studied for example in Zhao et al. (22), Rabbat et al. (25), Antoniou et al. (29), Kar and Moura (2), Patterson et al. (2), Pereira and Pagès-Zamora (2), Ribeiro et al. (2). To help explain flocing behaviour in nature, Reynolds (987) introduced a set of simple rules that led to creation of the first computer animation of flocing: ) Cohesion: attempt to stay close to nearby floc-mates; 2) Separation: avoid collisions nearby floc-mates; 3) Alignment: attempt to match velocity nearby floc mates. Afterwards, many algorithms to realize these three rules have been proposed. Among the first groups of scientists who studied flocing from a theoretical perspective were Vicse et al. (995) whose focus was on the emergence of alignment in selfdriven particle systems, followed by volumes of investigations on consensus (alignment) problem covering algorithms, stability and convergence rate analyses such as Jadbabaie et al. (23), and Murray (23, 24), Moreau (25), Ren and Beard (25), et al. (27). In more detail, Gazi and Passino (23) proposed an effective A/R swarm model to deal separation and cohesion rules. And Tanner et al. (23a, 23b, 27) used an artificial potential function together a consensus protocol to form flocing algorithms for both fixed and switched topology systems. (26) proposed three flocing algorithms a more physical artificial potential function. The first algorithm led to regular fragmentation for generic initial states, while the second one had an additional term represented by a virtual leader leading to flocing, and the third algorithm had obstacle avoidance capabilities. More close to the interest of this paper, Woods (959) and Montague et al. (995) have experimentally indicated for decades of years that natural bio-groups possess predictive intelligence, i.e. each individual can predict the future motion of itself and its neighbours according to their past trajectories. Motivated by the predictive intelligence, Zhang et al. (28) illustrated that predictive mechanisms play an important role in the emergence of collective consensus, and presented a discrete-time consensus model predictive controller for multiagent systems. Following the wor of Zhang et al., it s reasonable for us to conjecture that predictive mechanisms also play an important role in flocing control of multi-agent systems. On the basis of the available flocing control algorithms such as artificial potential functions and consensus protocols, we see in this article to propose a new flocing algorithm predictive mechanisms for multi-agent systems by using mathematical analyses and numerical simulations. The objective of this Copyright by the International Federation of Automatic Control (IFAC) 5669

2 Preprints of the 8th IFAC World Congress Milano (Italy) August 28 - September 2, 2 article is to design a Model Predictive Control () based flocing protocol in an attempt to obtain better and quicer formation of flocs. We approach the problem in two stages. Firstly, we design a discrete-time based flocing protocol for multi-agent systems. Secondly, we provide numerical simulation results as the verification of the newly proposed flocing protocol and focus on the comparison the second flocing algorithm, which reveals that the based flocing protocol yields several improvements concerning the convergence rate, -lattice regularity, and the feasible range of the sampling period. 2. PRELIMINARIES AND PROBLEM STATEMENTS To simplify the flocing protocol description, some basic concepts are stated first in this section. A graph is a pair that consists of a set of vertices { } and edges { }. The quantities and are the order and size of the graph, respectively. The graph is undirected if, and the problem addressed in this paper is concerned undirected graphs. An undirected graph is called connected if there exists a path, i.e., a sequence of distinct edges such that consecutive edges are joint, between any two vertices. Consider a discrete-time system of inematics of agent as agents the { () where ( corresponds to the dimension of space, e.g., ) denote the position, velocity and acceleration (or the control input) of agent,, respectively, and denotes the sampling period. Let, and. Then the multiagent system can be written in the vector form as { (2) Define as the interaction range between two agents. Then the set of neighbours of node is described by { } (3) where is the Euclidean norm in. Given the interaction range, the set of edges is defined by which depends on. { } (4) An -lattice type of structure proposed by (26) is used to model the geometry of desired conformation of agents in a floc. It can be described as the solutions of the following set of algebraic constraints (5) where is the desired distance of each pair of neighboured agents. To describe the conformation that are very close to an -lattice satisfying (5), (26) also presented the following set of inequalities (6) and referred to its solution as a quasi -lattice. In addition to the constraints for, there are velocity consensus that should be guaranteed to realize flocing the average velocity vector., (7) Therefore in this paper, we focus on the solution of the control input in accordance the protocol to achieve the -lattice flocing satisfying Eqs. (5) and (7). 3. DISCRETE-TIME BASED FLOCKING PROTOCOL In this section, we design a discrete-time based flocing protocol. As shown in Fig., a scheme illustrating the based flocing protocol is presented. The topology and state information of the networ are used to predict the future states in steps and to compute the future control inputs, which are optimized respect to the cost function. At Prediction Time Global info input Cost function () Reference Fig.. The illustration of model predictive control based flocing protocol. 567

3 Preprints of the 8th IFAC World Congress Milano (Italy) August 28 - September 2, 2 the next discrete-time instant, only the first control input is implemented, after which a new networ state prediction is taen and the whole process restarts again. Since the goal of flocing control is to obtain an -lattice type structure and velocity consensus, namely satisfying Eqs. (5) and (7), define the cost function at step as below ( ) denoting the vertical vector pointing from agent to, corresponding to the unit vertical vector pointing from to, and representing the desired distance of each pair of neighboured agents; and are the prediction horizon and the control horizon, respectively. Define the state variable ( ), and transform (2) into a first-order state equation based on as follows (8) (9) [ ], [ ]. Based on the state variable (8) as below the output prediction, we write the cost function () (), [ ], where is a weighting coefficient, and denotes the maximum number of all possible edges among agents. When, corresponds to the existing edges { ( ) when, corresponds to the deviation from the average velocity ( ( ) ). And the desired output ; consists of two parts: [ ] (2) when, corresponds to the fixed distance between each pair of neighboring agents { and when,, representing velocity consensus. Note that is determined by, and hence is a nonlinear function of. So here we mae a slight modification (3) available at each discrete-time instant when is unresolved. For simplicity, we rewrite the cost function () into a vector form { and (4) [ ( )] [ ( )] [ ] [ ] are compatible weighting matrices. We set,, and. ; (5) Using the dynamics (9), one can predict the future states of the networ in steps as follows [ (6) * ( ) ( ) +, ] To minimize the cost function (4), we compute according to Eqs. (5) and (6), and consequently, we obtain control input. (7) [ ] (8) 567

4 Preprints of the 8th IFAC World Congress Milano (Italy) August 28 - September 2, 2 can be written in a simplified form (9) where [ ] and [ ]. Substituting into the discrete-time dynamics (9) comes (2) distribution of eigenvalues cluster of ( in Fig. 3. ) as illustrated The circle in Fig. 3 denotes the unit circle in the complex plane, and the small dots represent the eigenvalues of ( ) at all discrete-time instants. For each discrete-time instant, there is only one eigenvalue equalling and all the others lie in the unit circle, hence, the convergence of the discrete-time based flocing protocol is visualized. 4. SIMULATION RESULTS AND ANALYSES We present numerical simulation results of the based flocing its convergence analysis, and we also give further analyses on the velocity consensus and -Lattice regularity in comparison the second flocing algorithm. 4. Numerical Verification Considering a multi-agent system 25 agents in a 2-D free space, they are initially scattered in the square [ ], forming an undirected connected graph, and their velocity coordinates are randomly chosen from the uniform distribution on [ ]. In all simulations,, the sampling period ranges between, the prediction horizon and the control horizon both range between, and a quasi -lattice structure is defined the set of inequalities. The system finally forms a quasi -lattice structure in all simulations as one example shown in Fig. 2. Fig The illustration of a quasi -lattice structure. Furthermore, we next provide the convergence analysis of the based flocing protocol and comparison results the second flocing algorithm on velocity consensus and -Lattice regularity. In the following analyses, though only one case for specific parameters is discussed figures, the results apply to other parameters as well. 4.2 Convergence Analysis According to the discrete-time dynamics (2), the convergence of the system can be explained in terms of the Fig. 3. Eigenvalue distribution of the coefficient matrix, and. 4.3 Velocity Consensus Analysis As illustrated in Section 4., a quasi -lattice structure is finally obtained and preserved in all simulations, and hence all the agents eventually achieve velocity consensus. To study the velocity convergence rate, a velocity disagreement index is defined as, and its time evolution is used to measure velocity consensus speed. Figure 4 shows the time evolution of two different exampled sampling period as.5 and.. The diamond marers and the plus ones denote in the second Olfati- Saber flocing algorithm and the based flocing protocol respectively. Figure 4(a) demonstrates that both the two flocing algorithms achieve velocity consensus as indicated by approaching to zero, and the steeper decline of the plus marers reveals higher velocity convergence rate of the based flocing protocol. In another case a larger, consistent result is shown in Fig. 4(b), while the inset, i.e., illustration of s evolvement in the later period, indicates that the final lattice structure stability of the second flocing algorithm is no more guaranteed due to the ripples of diamonds, compared the smooth trajectory of plus marers. 4.4 Quasi -Lattice Regularity Analysis To quantify the regularity of quasi -Lattice structure, define 5672

5 Γ() Γ() Γ() Γ() D() D() D() Preprints of the 8th IFAC World Congress Milano (Italy) August 28 - September 2, (a) (b) Fig. 4. Time evolution of comparison between the algorithm (, ) and the second algorithm. (a) (b) (a) Fig. 5. Quasi -Lattice regularity comparison between the algorithm (, ) and the second Olfati- Saber algorithm. (a) (b). (b) as the indicator of regularity deviation between quasi - Lattice and -Lattice, where is the total number of edges satisfying. When, the quasi - Lattice structure becomes -Lattice structure; the larger is, the less regular -Lattice achieved. Figure 5 shows the regularity deviation varying the running step in two different sampling period cases. Similarly, the diamond marers denote in the second flocing algorithm while the plus ones correspond to the based flocing protocol. The inset illustrates s evolvement starting from step 26. Figure 5(a) reveals that the based flocing protocol converges a higher rate and forms a more regular -Lattice structure as indicated by the smaller. Moreover, as shown in the inset of Fig. 5(b), there are ripples of diamond marers indicating that the is too large to guarantee the stable convergence to a regular lattice structure, while the plus marers eep unchanged, which is in consistence the counterpart in velocity consensus analysis. In summary, the proposed flocing protocol yields several improvements compared the second flocing algorithm as follows: ) The based flocing protocol improves the convergence rate of velocity consensus as well as - lattice structure. 2) The quasi -lattice structure formed by applying the based flocing protocol is a more regular -lattice structure. 3) The feasible range of the sampling period is remarably expanded in the based flocing protocol. 5. CONCLUSIONS To investigate the role of predictive mechanisms in flocing for multi-agent systems, we have proposed a novel discretetime based flocing protocol for multi-agent systems in this paper. We have presented the mathematical description of the based flocing protocol, and compared it the second flocing algorithm by using 5673

6 Preprints of the 8th IFAC World Congress Milano (Italy) August 28 - September 2, 2 numerical simulation and mathematical analyses, the result of which indicated that the based flocing protocol improved the convergence rate, -lattice regularity and the feasible range of sampling period. Due to these advantages, such predictive mechanisms in flocing of multi-agent systems have great potential to find the way into industrial applications, and motivated us to develop a distributed flocing protocol in Zhan and Li (2). ACKNOWLEDGEMENTS This wor was partly supported by the National Key Basic Research & Development Program (No. 2CB7343), the National Science Foundation of China (No ), and the NCET Program (No. 9-37). REFERENCES Antoniou, P., Pitsillides, A., Blacwell, T., & Engelbrecht, A. (29). Employing the Flocing Behavior of Birds for Controlling Congestion in Autonomous Decentralized Networs. IEEE Congress on Evolutionary Computation, Couzin, I. D., & Krause, J. (23). Self-Organization and Collective Behavior in Vertebrates. Advances in the Study of Behavior, 32, 75. Gazi, V., & Passino, K. M. (23). Stability analysis of swarms. IEEE Trans. Automat. Control, 48(4), Gazi, V. (28). Stability of a Discrete-Time Asynchronous Swarm Time-Dependent Communication Lins. IEEE Transactions on Systems, Man, and Cybernetics: Part B, 38(), Jadbabaie, A., Lin, J., & Morse, A. S. (23). Coordination of groups of mobile agents using nearest neighbor rules. IEEE Trans. Autom. Control, 48(6), 988. Kar, S., & Moura José, M. F. (2). Distributed Consensus Algorithms in Sensor Networs: Quantized Data and Random Lin Failures. IEEE Transaction on Signal Processing, 58(3), Montague, P. R., Dayan, P., Person, C., & Sejnowsi, T. J. (995). Bee foraging in uncertain environments using predictive hebbian learning. Nature, 377, Moreau, L. (25). Stability of multiagent systems timedependent communication lins. IEEE Trans. Autom. Control, 5(2), , R., & Murray, R. M. (23). Consensus protocols for networs of dynamic agents. Proc. Amer. Control Conf., 2, , R., & Murray, R. M. (24). Consensus problems in networs of agents switching topology and time-delays. IEEE Trans. Autom. Control, 49(9), , R. (26). Flocing for Multi-Agent Dynamic Systems: Algorithms and Theory. IEEE Transaction on Automatic Control, 5(3), 4 42., R., Fax, A., & Murray, R. M. (27). Consensus and cooperation in multi-agent networed systems. Proceedings of the IEEE, 95(), Patterson, S., Bamieh, B., & Abbadi, A. E. (2). Convergence Rates of Distributed Average Consensus Stochastic Lin Failures. IEEE Transaction on Automatic Control, 55(4), Pereira, S. S., & Pagès-Zamora, A. (2). Mean Square Convergence of Consensus Algorithms in Random WSNs. IEEE Transaction on Signal Processing, 58(5), Rabbat, M., Nowa, R., & Buclew, J. (25). Generalized consensus computation in networed systems erasure lins. Proc. IEEE 6th Worshop Signal Processing Advances Wireless Communications (SPAWC), Ren, W., & Beard, R. W. (25). Consensus seeing in multiagent systems under dynamically changing interaction topologies. IEEE Trans. Autom. Control, 5(5), Reynolds, C. W. (987). Flocs, herds, and schools: A distributed behavioral model. Comput. Graph. (ACM SIGGRAPH 87 Conf. Proc.), 2, Ribeiro, A., Schizas, I., Roumeliotis, S., & Giannais, G. (2). Kalman Filtering in Wireless Sensor Networs. IEEE Control Systems Magazine, 3(2), Su, H. S., Wang, X. F., & Chen, G. R. (29a). A connectivity-preserving flocing algorithm for multi-agent systems based only on position measurements. International Journal of Control, 82(7), Su, H. S., Wang, X. F., & Lin, Z. L. (29b). Flocing of Multi-Agents a Virtual Leader. IEEE Transaction on Automatic Control, 54(2), Tanner, H. G., Jadbabaie, A., & Pappas, G. J. (23a). Stable flocing of mobile agents, part I: Fixed topology. Proc. 42nd IEEE Conf. Decision Control, Tanner, H. G., Jadbabaie, A., & Pappas, G. J. (23b). Stable flocing of mobile agents, part II: Dynamic topology. Proc. 42nd IEEE Conf. Decision Control, Tanner, H. G., Jadbabaie, A., & Pappas, G. J. (27). Flocing in Fixed and Switching Networs. IEEE Transaction on Automatic Control, 52(2), Vicse, T., Cziroó, A., Ben-Jacob, E., Cohen, I., & Shochet, O. (995). Novel type of phase transition in a system of selfderiven particles. Phys. Rev. Lett., 75(6), Woods, E. F. (959). Electronic prediction of swarming in bees. Nature, 84, Zhao, F., Shin, J., & Reich, J. (22). Information-Driven Dynamic Sensor Collaboration. IEEE Signal Processing Magazine, 9(2), Zhang, H. T., Chen, Z. Q. Michael, Stan, G. B., Zhou, T., & Jan, M. M. (28). Collective behavior coordination predictive mechanisms. IEEE Circuits and Systems Magazine, 8(3), Zhan, J. Y., & Li, X. (2). Distributed Flocing Protocol of Multi-agent Systems Predictive Mechanisms, submitted. 5674