Motion Coordination and Other Distributed Control Problems for Large - - PowerPoint PPT Presentation

Prabir Barooah João P. Hespanha

Motion Coordination and Other Distributed Control Problems for Large Networks of Agents

45th CDC, San Diego Dec 12, 2006,

Center for Control, Dynamical-systems and Computation, University of California, Santa Barbara, USA

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) Ignore communication delay, switching between formations, etc. etc. Consider –

• stability,
• tracking error due to measurement noise,
• disturbance propagation

Tool : electrical analogy

Motion Coord. with Noisy Measurements

agent ’s velocity directs agent so as to decreases error with respect to immediate neighbors At steady state, desired position

• f relative to

measurement noise (covariance u,v = v,u) position of relative to

• “Measurement” graph

How to compute ? How large is ? How does it depend on ? Tracking error of node

(Generalized) Dirichlet Laplacian

The (generalized) Laplacian

(Generalized) Dirichlet Laplacian

The (generalized) Laplacian

• 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
• =
• is a symmetric positive definite matrix as long as the graph is connected.

Steady State Tracking Error Covariance :

Electrical Analogy - Effective Resistance

Steady State Tracking Error Variance = Effective Resistance* !

covariance of x‘s tracking error with as the leader. Effective resistance* between node and . Re Edge Covariance Gen. Resistance Re

*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 2) Disturbance propagation in 1-D platoons Larger effective resistances => Smaller gain margins Effective resistances provide bounds on the (Dirichlet) Laplacian eigenvalues Im Re Nyquist plot of P(s)K(s) should not intersect Depend on the smallest eigenvalue of the Dirichlet Laplacian

(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.

desired position

• f relative to

position of relative to

(Back to) Tracking Error

Tracking Error Variance of = Effective Resistance* of w.r.t. *matrix valued Why is the Electrical Analogy useful?

(Gen.) Rayleigh’s Monotonicity Law

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 “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 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

• f the graph?

Tracking Error – Scaling Laws?

Question – How does the tracking error of an agent vary with The size of the network? The structure of the network? leader variance Distance from leader

Effective Resistance in Lattices

1-D lattice 2-D lattice 3-D lattice

3 2 1

What about graphs that are not lattices ?

1 2

Embed a “nice looking” graph in the graph of interest, compute the effective resistance in the “nice” graph : upper bound Embed the graph of interest in a “nice looking” graph, compute the effective resistance in the “nice” graph. : lower bound Bound the effective resistance by embedding and Rayleigh’s Monotonicity Law

3

Outline of Scaling Law results

in 1-D in 2-D in 3-D “sparse” “dense”

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: 1. One cannot fit arbitrarily large balls between nodes 2. “Small Euclidean distance between nodes implies small graphical distance”

γ (node density) not dense in 2D

(small Euclidean distance does not mean small graphical distance)

dense in 2D

(small Euclidean distance means small graphical distance)

ρ> 0 (edge/cross-connection density)

= 2 = 2

γ

Embedding a Lattice in (sth like) the graph

• 2
• The lattice
• 2 cannot

be embedded in

• (2)

But the lattice can be embedded in a 2-fuzz of

• If a graph is dense in ,

the -dimensional lattice can be can be embedded in an -fuzz of

• fuzz of
• :=
• ()

if (,) , then

• () has an edge between and .

Doyle and Snell (‘84)

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 is dense

in 1-dimension

is dense

in 2-dimension

is dense

in 3-dimension Proof of upper bounds:

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) Sparse in 1-D (as well as 2-D) Not sparse in 1D (but sparse in 2-D)

= 2 = 1

Sparse Graphs – lower bound

If a graph is sparse in , can be embedded in a -fuzz of the -dimensional lattice is sparse

in 1-dimension

is sparse

in 2-dimension

is sparse

in 3-dimension Proof of lower bounds: Doyle & Snell ’84

node degree

Counterexamples to conventional wisdom

# of nodes or edges per unit area # of indep. paths

None of the following determines error scaling:

Steady State Tracking Error Scaling Laws

in 1-D in 2-D in 3-D sparse dense

Euclidean distance between and

• Deeper geometric structure

determines density/sparsity

• Graphs encountered in

cooperative control

• - dense or sparse in at least one

dimension ( natural drawing )

Scalability in Natural Swarms

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 Robustness with respect to measurement noise…

Function > Structure > Limitation

• bjective of

multi-agent team network structure suggests limits

Formation flying – V shape reduces drag

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 if the controller stabilizes each of the plants Im Re SISO : Nyquist plot of P(s)K(s) should not intersect Eigenvalues of the Dirichlet Laplacian Eigenvalues of the 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

– Seiler et. al. (04) – Sai Krishna et. al. (05) – Barooah et. al. (05) Closed Loop Platoon Dynamics

Stability, Disturbance Amplification

Tracking errors due to leader trajectory Im Re Platoon is stable if and only if the Nyquist plot of (s)(s) does not intersect

Result doesn’t change when more than two neighbors are allowed Sparse in 1-D: sparse in 1-D : effective resistance is linear with distance Independent of ()

Other Applications of Electrical Analogy

Distributed Estimation from (noisy) Relative Measurements

• 1. Localization in Sensor Networks
• 2. Time synchronization in ad-hoc networks
• 3. Direction estimation in motion coordination

Variance of the optimal estimation error of a node = Effective Resistance between the node and the reference More in “Graph Effective Resistance and Distributed Control: Spectral Properties and Applications”, CDC 2006, Thursday Session B14.4, 11:30-11:50 A.M. Convergence rate of distributed algorithm (Jacobi):

Summary

Electrical analogy is useful for examining scalability issues in motion-coordination problems

• Scaling laws of effective resistance for large graphs
• Dirichlet Laplacian and Laplacian eigenvalue bounds

More in “Graph Effective Resistance and Distributed Control: Spectral Properties and Applications”, CDC 2006, Thursday Session B14.4, 11:30-11:50 A.M.