FEL3330/EL2910 Networked and Multi-Agent Control Systems. Spring 2013

Transcription

1 FEL3330/EL2910 Networked and Multi-Agent Control Systems Spring 2013 Automatic Control School of Electrical Engineering Royal Institute of Technology Lecture 1 1 May 6, 2013

2 FEL3330/EL2910 Networked and Multi-Agent Control Systems Disposition 7.5 credits, 24h lectures PhD Course with MS alternative code. Instructors Dimos Dimarogonas, lecturer and course responsible, Guodong Shi, lecturer, Lecture 1 2 May 6, 2013

3 Homework Three homework assignments First homework given after Lecture 3 One week deadline per HW Up to teams of two, but no more Lecture 1 3 May 6, 2013

4 Course goal After the course, you should be able to Know the essential theoretical tools to cope with Networked and Multi-Agent Systems Know the established problems and results in the area Apply the theoretical tools to problems in the area Contribute to the research frontier in the area Lecture 1 4 May 6, 2013

5 FEL3330/EL2910 Networked and Multi-Agent Control Systems Lecture 1: Introduction Practical information Motivating applications What is Multi-agent and Networked? Course outline Some Lyapunov theory background tools Poll: Take home exam vs. Research Projects? Lecture 1 5 May 6, 2013

6 Today s lecture What is the motivation and main theme of the course? Lyapunov theory background essentials Lecture 1 6 May 6, 2013

7 All info available at Course Information Lecture 1 7 May 6, 2013

8 Material Textbook: No textbook, but papers/notes related to each lecture Lecture slides: Online after each lecture a Blackboard: During the lecture Poll: Take home exam vs. Research Projects? a whenever the lecture is not only blackboard based Lecture 1 8 May 6, 2013

9 Motivation How to understand and achieve global behaviors from local behaviors Multi-robot/vehicle coordination Sensor networks Social networks Power networks Bio-inspired coordination Lecture 1 9 May 6, 2013

10 Design issues Scalability Limited information consideration Control objectives Lecture 1 10 May 6, 2013

11 Some nice figures B C A D E Lecture 1 11 May 6, 2013

12 What is in the course-why I should take it? Decentralized controllers at a high level of abstraction (Simple dynamics/sensing, complicated networks) Graph based models of networks with which the first step in control design can be held Disclaimer: the selection of topics may be biased towards the instructors interests Why I should take the course? 1. New tools for a large class of control problems 2. HOT area for the last 10 years and still growing strong! Lecture 1 12 May 6, 2013

13 3. Lots of really interesting unsolved problems Lecture 1 13 May 6, 2013

14 What is NOT in the course Cool computer graphics Behavioral robotics How to build devices Sensing/perception algorithms Communication protocols Lecture 1 14 May 6, 2013

15 Q: Why multi-agent? Course title A: Agents represent the different entities in each application Q: Why networked? A: Need to model the limited information on the rest of the group due to sensing and communication limitations Agents are the vertices in the graph that represents the network! Pairs of agents that can exchange info are the edges! Lecture 1 15 May 6, 2013

16 Limited Sensing and Communication aspects Limited Sensing: Vision based sensors, range sensors (sonars, laser scanners,...) Limited Communication: communication channel, bandwith, coding,... Lecture 1 16 May 6, 2013

17 Graph theoretic approach Limitations in communication/sensing do now allow each agent to communicate with everyone else Modelling of limitations through graphs G =(V,E) 5 4 Agents are the vertices V = {1,...,N} Edges E V V are pairs of agents that can communicate Lecture 1 17 May 6, 2013

18 Simple graphs Undirected graphs Weighted graphs Some graphs of interest Lecture 1 18 May 6, 2013

19 Relating graphs to networks Static networks Random networks State dependent/dynamic networks Lecture 1 19 May 6, 2013

20 Lyapunov stability Let x =0be an equilibrium point of ẋ = f(x). It is called Stable, if for all ɛ>0, thereexistsδ = δ(ɛ) > 0 such that x(0) <δimplies x(t) <ɛ, t 0 Asymptotically stable, if stable and there exists δ>0 such that x(0) <δimplies lim t x(t) =0. Lecture 1 20 May 6, 2013

21 Lyapunov s Second Method Let x =0be an equilibrium point of ẋ = f(x). Ifthere exists a C 1 function V : R n R such that V (0) = 0 V (x) > 0, x 0 V (x) 0, x R n, then x is stable. If V (x) < 0, forallx 0,thenx is asymptotically stable. Lecture 1 21 May 6, 2013

22 Lyapunov Function for Linear System Real λ i (A) < 0 for all i if and only if for every positive definite Q = Q T there exists a positive definite P = P T such that PA+ A T P = Q A Lyapunov function for a linear system ẋ = Ax is given by In particular, V (x) =x T Px V (x) =x T P ẋ +ẋ T Px = x T (PA+ A T P )x = x T Qx < 0 Lecture 1 22 May 6, 2013

23 LaSalle s Invariance Principle Let Ω be a compact set that is positively invariant with respect to ẋ = f(x). LetV be a C 1 function with V 0 in Ω. LetE be the set of all points in Ω where V =0.LetM be the largest invariant set in E. Then, every solution starting in Ω approaches M as t. Lecture 1 23 May 6, 2013

24 Switched systems Suppose x =0is an equilibrium of each mode q =1,...,m of the switched system ẋ = f q (x), x Ω q If there exist functions V 1,...,V m such that V q (0) = 0, V q (x) > 0, x R n \{0} V q (x(t)) 0, whenever x(t) Ω q and the sequences {V q (x(τ iq ))}, q =1,...,m are non-increasing, where τ iq are the time instances when mode q becomes active, then x is stable. Lecture 1 24 May 6, 2013

25 Example Let the origin be a stable equilibrium point for ẋ = f q (x), x Ω q, q =1, 2 Below, V 1 (x(t)) and V 2 (x(t)) are shown. The active parts are solid. The sequences {V q (x(τ iq ))}, q =1, 2, are indicated V q (x(t)) t Lecture 1 25 May 6, 2013

26 Research Projects Up to teams of four. Pick up one of the course topics and get back to me by this coming Friday. I assign four papers per topic. Purpose of the project: write a report based on these papers, and additionally identify other papers of interest within the topic, and propose new research directions. Short presentation in the final lecture. Lecture 1 26 May 6, 2013

27 Next Lecture Graphs and Matrices Graph theory essentials Relating graphs to matrices (algebraic graph theory) Lecture 1 27 May 6, 2013