1. 集体行为的 “ 软控制 ” 韩 靖 数学与系统科学研究院 系统与控制重点实验室 、复杂系统研究中心 合作者：郭 雷 、 黎 明

2. 集体行为 (Collective Behavior) 集体行为 (Collective Behavior) 目前复杂系统研究中的热点： network dynamics, spin glasses, self-organized criticality… 目前复杂系统研究中的热点： network dynamics, spin glasses, self-organized criticality… 特点： 特点：  大量相似个体  局部相互作用：吸引、抗拒、对齐 …  集体特性不同于单个个体  集体行为类型：相变、模式形成、群组运动、同步 … 典型例子： “ 磁化 ” 现象：个体行为  邻居动作的平均方向 鸟群、人群恐慌、同步掌声、 Google 引擎、组合优化问题的相变 … 典型例子： “ 磁化 ” 现象：个体行为  邻居动作的平均方向 鸟群、人群恐慌、同步掌声、 Google 引擎、组合优化问题的相变 …

3. Three Categories of Research on Collective Behavior 1. Given the local rules of the agent, what is the collective behavior of the overall system? Spin Glasses, panic model, network dynamics, 2. Given the desired collective behavior, what are the local rules for agents? Swarm Intelligence, BA model for internet 3. Given the local rule of the agent, how we control the collective behavior? Soft Control

4. A Group of Birds Filmed by Jing HAN. 2004. Sept.

5. How we drive them away? Can we guide their flight if we know how birds fly?Can we guide their flight if we know how birds fly? How we ‘control’ the collective behavior of a group of autonomous agents?How we ‘control’ the collective behavior of a group of autonomous agents?

6. Outline The Model for a Group of Birds The Model for a Group of Birds The Key Points of ‘Soft Control’ The Key Points of ‘Soft Control’ Computer Simulations Computer Simulations A Case Study A Case Study Soft Control of Complex Systems Soft Control of Complex Systems

7. r A Simple Model* ( 1995, Vicsek etc. ) http://angel.elte.hu/~vicsek/ A bird’s Neighborhood Alignment: steer towards the average heading of neighbors ii xixi 0 BasicShow *It is a simplified model of BOID (1986) Every bird only knows the local information, and follows a local rule to make decisions. n Agents: (X,  ) x i (t), i=1,…,n n Agents: (X,  ) x i (t), i=1,…,n  i (t): heading (moving direction) of the i-th agent.  i (t): heading (moving direction) of the i-th agent. v: the constant speed of birds v: the constant speed of birds r: radius of neighborhood, so N i (t)={j | ||x j -x i ||  r}, n i (t)=|N i (t)| r: radius of neighborhood, so N i (t)={j | ||x j -x i ||  r}, n i (t)=|N i (t)|  i (t+1)= r, where  i (t+1)= r, where = arc tan [ r / r ] = arc tan [ r / r ] x i (t+1)=x i (t) + V(v,  i (t+1)) x i (t+1)=x i (t) + V(v,  i (t+1))

8. Key Points of ‘Soft Control’ The system : The system :  Many agents more is different  Each agent follows the local rules Autonomous, distributed  Agents are connected, no one is isolated (most of the time) The local effect will affect the whole The “Control”: The “Control”:  Not to change the local rule of existing agents Can not implement by changing adjustable global parameters Can not implement by changing adjustable global parameters  Put a few spies to guide (seduce) Spy: is controlled by us, not following the local rules, Spy: is controlled by us, not following the local rules, is treated as an ordinary agent by other ordinary agents. is treated as an ordinary agent by other ordinary agents. y(t) U(t)  (t+1)=(  (t)+  (t)+  ’(t))/3 x(t+1)=x(t)+v·  (t+1)  0 (t+1)= controlled x 0 (t+1)= x 0 (t)+v·  (t+1) or controlled v would be variable Spy is not leader, not leader-follower type.Spy is not leader, not leader-follower type. The power of spy seems limited The ‘control’ is soft and seems weak.The power of spy seems limited The ‘control’ is soft and seems weak. Different from distributed control approach. Intervention to the distributed systemDifferent from distributed control approach. Intervention to the distributed system

9. There Are Lots of Questions … What is the purpose/task of control here? What is the purpose/task of control here?  Synchronization  Group connected / Dissolve a group  Turning (Minimal Circling)  Lead to a destination (in a shortest time)  Avoid hitting an object  Tracking  … In what degree we can control the spy? (heading, position, speed, …) In what degree we can control the spy? (heading, position, speed, …) How much information the spy can observe ? (positions, headings, …) How much information the spy can observe ? (positions, headings, …) …

10. Soft Control: A New Problem? The current control theory can not be applied directly, because The current control theory can not be applied directly, because  It is a many-body self-organized system. The purpose of control aims to collective behavior. Not allowed to change the local rules of the existing agents; Not allowed to change the local rules of the existing agents;  Formation Control: special task of formation, distributed control  Pinning Control: Networked system, imposed controllers on selected nodes  No one appear to directly study this problem yet. There are some preliminary results about synchronization for the Vicsek’s model There are some preliminary results about synchronization for the Vicsek’s model  Some a priori connectivity conditions  Leader Following (do not tell how to do)  And the model itself has problems

11. Computer Simulations Synchronization Put a few constant-heading spies  k (t)=  k (0), x k (t+1)=x k (t)+v·  k (t+1) Put a few constant-heading spies  k (t)=  k (0), x k (t+1)=x k (t)+v·  k (t+1) SynchronizationSynchronization, Group or Dissolve, Turning: Group or DissolveTurning SynchronizationGroup or DissolveTurning Put a powerful constant-heading spy x k (t) is controlled by us – a spy can move to anywhere in one step, with deceptive heading Put a powerful constant-heading spy x k (t) is controlled by us – a spy can move to anywhere in one step, with deceptive heading Put a powerful inconstant-heading spy both  k (t) and x k (t) are controlled by us Put a powerful inconstant-heading spy both  k (t) and x k (t) are controlled by us Put an inconstant-heading spy  k (t) is controlled by us v is variable: has an accelerator Put an inconstant-heading spy  k (t) is controlled by us v is variable: has an accelerator aFewFixHeadingSpy aPowerfulFixHeadingSpy

12. A Case Study Problem: Problem:  A group of n agents with initial headings  i (0)  [0,  )  Goal: all agents move to the direction of  eventually.  What is the control law? Assumptions: Assumptions:   The local rule about the ordinary agents is known  T  The position x 0 (t) and heading  0 (t) of the spy can be controlled at any time step t   The state information (headings and positions) of all ordinary agents are observable at any time step

13. The Control Law u  Theorem 1: For any initial headings and positions  i (0)  [0,  ), x i (0)  R 2, 1  i  n, the update rule and the control law u  will lead to the asymptotic synchronization of the group, i.e.,, where  (t) =  -  s(t) (t). Demo

14. Soon Other Questions … What if considers the energy of spy? What if considers the energy of spy? What if the spy can only see locally? What if the spy can only see locally? What if considers keeping the group connected? What if considers keeping the group connected? …

15. Further Questions … How about if we don’t know the local rule of birds, how does the spy learn and lead the group? (Learning and Adaptation) How about if we don’t know the local rule of birds, how does the spy learn and lead the group? (Learning and Adaptation) How about if we can remove an agent? How about if we can remove an agent? A game question: how if two spies competing for the group? A game question: how if two spies competing for the group? …

16. Afterthought Can we solve the bird problem for the airport? Can we solve the bird problem for the airport? This kind of models (multi-agent, many-particles) (Many autonomous agents with local interactions) This kind of models (multi-agent, many-particles) (Many autonomous agents with local interactions) catch the property of many real world systems: catch the property of many real world systems:  Magnetization, Spin glasses, Spintronics  Crowd Panic  Combinatorial Problems  Food web  Protein Interaction Network  Fluidization in Chemical Engineering  Language diffusion  Economics: Multi-agent games  Ant/bee colony ………… They are part of the study of Complex Systems!

17. Soft Control of Complex Systems Claytronics Claytronics Evolution of Language Evolution of Language Multi-player Game Multi-player Game Escape Panic of Crowd (Nature, 2000, Vol407, p.487-490) Escape Panic of Crowd (Nature, 2000, Vol407, p.487-490) … Intervention to complex systems ? We need a theory for Soft Control !

18. This picture is cut from the poster of movie This picture is cut from the poster of movie Thank You ! It is just the beginning! http://complex.amss.ac.cn/hanjing/softcontrol/