Motion Coordination and Other Distributed Control Problems for Large Networks of Agents Prabir Barooah João P. Hespanha Center for Control, Dynamical-systems and Computation, University of California, Santa Barbara, USA 45 th CDC, San Diego Dec 12, 2006,
Scalability in Motion Coordination Formation Control � A number of agents move together � One* leader moves independently � Goal : keep a desired formation / meet at a point Issue : Scalability ( performance degradation with increasing number of agents) Consider – Ignore � � communication delay, stability, � � switching between tracking error due to measurement noise, formations, etc. etc. � disturbance propagation Tool : electrical analogy
Motion Coord. with Noisy Measurements position of � relative to � measurement noise (covariance � u,v = � v,u ) desired position agent � ’s of � relative to � velocity directs agent so as to decreases error with respect to immediate neighbors “Measurement” graph � Tracking error of node � � � How to compute ? At steady state, How large is ? How does it depend on � ?
(Generalized) Dirichlet Laplacian The (generalized) Laplacian � � � � � � � � �
(Generalized) Dirichlet Laplacian The (generalized) Laplacian � � � � � � � � � � o = The (generalized) Dirichlet Laplacian � o Remove the rows and columns of Laplacian that corresponds to the leader node. The submatrix that is left is the Dirichlet Laplacian � o Steady State Tracking Error Covariance : � o is a symmetric positive definite matrix as long as the graph is connected.
Electrical Analogy - Effective Resistance Edge Covariance � Gen. Resistance R e R e covariance of x � ‘s tracking Effective resistance* error with � as the leader. between node � and � . Steady State Tracking Error Variance = Effective Resistance* ! *matrix valued
Other Issues: Stability and Robustness 1) Stability:(Fax and Murray ’04) Formation of � vehicles stable if and only if the controller stabilizes each of the � plants Nyquist plot of P(s)K(s) should not intersect Im Re 0 Larger effective resistances => Smaller gain margins 2) Disturbance propagation in 1-D platoons Depend on the smallest eigenvalue of the Dirichlet Laplacian Effective resistances provide bounds on the (Dirichlet) Laplacian eigenvalues
(More) Applications… • Distributed Estimation in Sensor Networks (Localization, Time synchronization, …) – Scaling laws for the minimum possible error – Convergence Rate of distributed algorithms More in “ Graph Effective Resistance and Distributed Control: Spectral Properties and Applications ”, CDC 2006, Thursday Session B14.4, 11:30-11:50 A.M.
(Back to) Tracking Error position of � relative to � desired position of � relative to � Tracking Error Variance of � = Effective Resistance* of � w.r.t. � Why is the Electrical Analogy useful? *matrix valued
(Gen.) Rayleigh’s Monotonicity Law “If the resistance in any branch of an electric network is increased, the effective resistance between any two nodes can only increase, and vice versa” - Rayleigh �� 1 can be embedded in �� 2 � Effective Resistance in �� 1 is higher than in �� 2 (generalized) Rayleigh’s Monotonicity Law � � Doyle & Snell ’84 (scalar valued resistances) Also true for matrix-valued resistances � � �� 1 �� 2 If we introduce more agents, the tracking errors of the agents decrease !
The Story so far … Motion coordination - scalability A typical example - motion coordination with noisy measurements Electrical analogy effective resistance Scaling laws in large networks Other applications How does the tracking error in different graphs scale with the size of the graph?
Tracking Error – Scaling Laws? Distance from leader variance leader Question – How does the tracking error of an agent vary with � The size of the network? � The structure of the network?
Effective Resistance in Lattices 1-D lattice � 1 2-D lattice � 2 3-D lattice � 3
What about graphs that are not lattices ? Bound the effective resistance by embedding and Rayleigh’s Monotonicity Law Embed a “nice looking” graph in the graph of interest, compute the effective resistance in the “nice” graph : upper bound �� 1 � 2 Embed the graph of interest in a “nice looking” graph, compute the effective resistance in the “nice” graph. : lower bound �� 3
Outline of Scaling Law results “sparse” “dense” in 1-D in 2-D in 3-D Euclidean distance between � and �
Dense Graphs A graph is said to be dense in � -dimensions if it can be drawn in a � -dimensional space so that: γ 1. One cannot fit arbitrarily large balls between nodes γ �� � (node density) � = 2 2. “Small Euclidean distance between nodes implies small graphical distance” ρ> 0 (edge/cross-connection density) � = 2 dense in 2D not dense in 2D (small Euclidean distance (small Euclidean distance does not means small graphical distance) mean small graphical distance)
Embedding a Lattice in (sth like) the graph If a graph �� is dense in � � , the � -dimensional lattice can be can be embedded in an � -fuzz of � � -fuzz of � � := � � ( � ) � � � � Doyle and Snell (‘84) � ( � ) has an edge between � and � . if � � ( � , � ) � � , then � � � The lattice � � 2 cannot � � � (2) � � � � � � � be embedded in � � � � But the lattice can be embedded in a 2-fuzz of �� �� �� �� � 2 � � �
Dense Embedding Lemma A � � graph is dense in � - dimension ( � =1,2,..) � � � (1) The � -dimensional lattice � d can be embedded in an � -fuzz of � � , for � � some � . (2) Every node of the graph that is not mapped into a node of the lattice � d is at an uniformly bounded graphical distance from a node in the graph that is mapped into a node of the lattice. Fuzzing changes effective resistance only by a constant factor* *Doyle and Snell (’84) Also true for matrix-valued effective resistance networks
Dense Graphs – upper bound If a graph �� is dense in � � , the � -dimensional lattice can be can be embedded in an � -fuzz of � Proof of upper bounds: �� is dense in 1-dimension �� is dense in 2-dimension �� is dense in 3-dimension
Sparse Graphs* A graph is said to be sparse in � -dimensions if it can be drawn in a � -dimensional space so that: 1. The minimum distance between two nodes is non-zero 2. The maximum length of an edge is bounded. *Graphs that can be drawn in a civilized manner in d-dimensions (Doyle & Snell ‘84) � = 2 Not sparse in 1D Sparse in 1-D (as well as 2-D) (but sparse in 2-D) � = 1
Sparse Graphs – lower bound If a graph �� is sparse in � � , � can be embedded in a � -fuzz of the � -dimensional lattice Doyle & Snell ’84 Proof of lower bounds: �� is sparse in 1-dimension �� is sparse in 2-dimension �� is sparse in 3-dimension
Counterexamples to conventional wisdom None of the following determines error scaling: # of nodes or # of indep. node degree edges per unit paths area
Steady State Tracking Error Scaling Laws sparse dense in 1-D in 2-D in 3-D • Deeper geometric structure determines density/sparsity • Graphs encountered in cooperative control -- dense or sparse in at least one dimension ( natural drawing ) Euclidean distance between � and �
Scalability in Natural Swarms Robustness with respect to measurement noise… Dense and Sparse in 1-D tracking error variance � O( # agents) Dense and Sparse in 2-D tracking error variance � O( Log # agents Dense and Sparse in 3-D tracking error variance � constant
Function > Structure > Limitation Formation flying – V shape reduces drag suggests objective of network multi-agent structure team limits Network structure imposes limitation on control/estimation performance
Scalability in Motion Coordination tracking error, stability, disturbance propagation, … Electrical analogy - Effective Resistance - Scaling laws in large networks (ex: tracking error) Other Issues stability of multi-vehicle formations disturbance propagation (effect of leader trajectory) Smallest eigenvalue of the Dirichlet Laplacian
Ex 1 : Stability of Multi-Vehicle Formation Formation of � vehicles stable if and only SISO : Nyquist plot of if the controller stabilizes each of the � P(s)K(s) should not plants intersect Im Eigenvalues of the Laplacian Re Eigenvalues of the Dirichlet Laplacian
Ex 2 : Platoon – Stability / Robustness – Scenario : Platoon of �� vehicles moving in a straight line following a single vehicle (leader) – Objective : maintain constant inter- vehicular distance – Architecture : symmetric bi-directional • Literature Closed Loop – Seiler et. al. (04) Platoon – Sai Krishna et. al. (05) Dynamics – Barooah et. al. (05)
Stability, Disturbance Amplification Platoon is stable if and only if the Nyquist plot of � (s) � (s) does not intersect Im Re Tracking errors due to leader trajectory
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