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Feedback Control and Multi-Agent Systems: Ubiquitous and Increasingly Interdependent Prof. Bill Dunbar Autonomous Systems Group Computer Engineering What are Systems? … ANYTHING in Engineering, usually with Dynamics. Some familiar examples: QuickTimeª and a TIFF (LZW) decompressor are needed to see this picture. How do we describe systems? … with math! Math: Describing Diverse Engineering Systems in a Common Way Internet backbone CA power grid In these Examples: San Fran ATC Control Systems are Hidden Engineering Systems “A Control System is a device in which a sensed quantity is used to modify the behavior of a system through computation and actuation.” • My (and Potentially Your) Research Robotics – Exploration – Toys – Competition (soccer) • Automated Freeways • Supply Chain Management QuickTimeª and a TIFF (LZW) decompressor are needed to see this picture. QuickTimeª and a TIFF (LZW) decompressor are needed to see this picture. Eventually…A Fully Autonomous Vehicle Off-Road Dessert Race QuickTimeª and a TIFF (Uncompressed) decompressor are needed to see this picture. The Potential is Enormous • Researchers at Caltech are working toward the math model of the “fruit fly system,” with the ultimate objective of making a micromechanical fly! QuickTimeª and a TIFF (LZW) decompressor are needed to see this picture. Distributed Optimization-Based Control of Multiagent Systems Ass. Prof. Bill Dunbar Autonomous Systems Group Computer Engineering Multiagent Systems are Everywhere • The Internet • Air traffic control • The Power Grid • Autonomous Formations Control Problems with: • Subsystem dynamics • Shared resources (constraints) • Communications topology • Shared objectives Multiagent Systems: Inherently Distributed and Cooperative Multiagent System: • autonomous agents • communication network Distributed: local decisions based on local information. Cooperative: agents agree on roles & dynamically coordinate. Agent sensor input Environment output action A Relevant Decision Method: Receding Horizon Control (RHC) RHC uses optimization to find feasible/optimal plans for near future. objective Minimize (distance to pump & fuel) s.t. Car model (dynamics) Without hitting wall (constraint) To mitigate uncertainty, plan is revised after a short time. actual computed X Mathematics of RHC is Finite Horizon Optimal Control objective Minimize (distance to pump& fuel) s.t. Car dynamics Without hitting wall (constraint) Convergence of RHC Requires Appropriate Planning Horizon and Terminal Penalty Theoretical conditions sufficient & in absence of explicit uncertainty. *[Mayne et al., 2000] RHC Compared to Other Control Techniques • Gives planning & feedback with builtin contingency plans. • Only technique that handles state and control constraints explicitly. • Tradeoff: computationally intensive. z(t0) state t0 t0+ z*(;t0) T zk() time RHC Successful in Applications: Process to Flight Control Caltech flight control experiment: Tracking ramp input of 16 meters in horizontal, step input of 1m in altitude. RHC updates at 10 Hz, trajectories generated by NTG software package. Movie RHC Admits Cooperation ok follow 3 ok 2 Get 1 to pump, 2 follow 1 & 3 follow 2. Decoupled dynamics Avoid collision follow 1 RHC of Multiagent Systems: What’s Missing? Enables autonomy of single agent. Amenable to cooperation for multiple agents. Missing?…Distributed Implementation* Why not Centralized?…Local decision require Global information Parallelization**?…If you can, but sometimes not applicable. *[Krogh et al, 2000, 2001] **[Bertsekas & Tsitsiklis, 1997] My Contribution: A Distributed Implementation of RHC Distributed: local decisions based on local information. Decoupled subsystem dynamics/constraints, Coupled cost L Decomposition Solution of Sub-problems requires Assumed Plan for Neighbors Agent 3 What 3 does z3(t0) state What 2 assumes t0 t0+ z3 (;t0) * z3 () k time Compatibility of Actual and Assumed Plans via Constraint Compatibility constraint Assumed plan Bounds discrepancy z3(tk) state tk tk+ time Distributed Implementation Requires Synchrony & Common Horizon T Conditions for Theory are General Convergence Conditions: Same as Centralized plus Bound on Update Period *[Dunbar & Murray, Accepted to Automatica, June, 2004] Venue: Multi-Vehicle Fingertip Formation 4 2 qref d31 Decomposition of Coupled Cost Simulation Parameters 4 2 : Reference signal : Actual COM of {1,2,3} Centralized RHC: Benchmark for Comparison QuickTimeª and a Microsoft Video 1 decompressor are needed to see this picture. Centralized RHC Simulation Distributed RHC is Comparable to Centralized RHC QuickTimeª and a Microsoft Video 1 decompressor are needed to see this picture. Distributed RHC Simulation Naïve Approach Produces Less Desirable Performance QuickTimeª and a Microsoft Video 1 decompressor are needed to see this picture. Naïve Approach: Bad Overshoot Summary of Contribution Distributed implementation of RHC is provable convergent, performs well, and is applicable to a class of Multiagent Systems: • Distributed & cooperative structure: ● Local decisions based on local information ● Decomposition and incorporation of compatibility constraint ● Coordination via sharing feasible plans • Applicable for: ● Heterogeneous nonlinear dynamics ● Generic objective function (need not be quadratic) ● Coupling constraints Conclusions • Theory conservative; useful as guideline for implementation. • Scalable: computational complexity independent of Na ; communication complexity independent of Na but dependent on Ni (size of neighborhood). • Communicating trajectories: more intensive than traditional decentralized control, but not too bad, given smoothness properties. • Less communication than required by parallelization. Tradeoff: not recovering centralized solution to original problem, but that of a modified problem. Current and Ongoing Work Theoretically: ● Locally synchronous and asynchronous versions ● DONE: Coupled subsystem dynamics. Potential applications: ● Process control ● Supply chain management ● Reduced order contingency plans ● Connection with rollout algorithms in MDPs Current and Ongoing Work Applications: ● Coordinated UAVs ● Mobile Sensor Networks ● Robots coordinating for toxin detection ● Intelligent Transportation Systems ● Automated freeways ● Semi-automated Air Traffic Control ● Interdisciplinary examples: ● Supply chain management (Business) ● Power/Water resource management