the - uta talks/2011 chinese academy- design for consensus.pdfstandard local voting protocol i iijji...

Automation & Robotics Research Institute (ARRI)The University of Texas at Arlington

F.L. LewisMoncrief-O’Donnell Endowed Chair

Head, Controls & Sensors Group

Cooperative Control Design for Multi‐Agent Systems Supported by AFOSR, NSF, ARO

The way that can be told is not the Constant WayThe name that can be named is not the Constant Name

For nameless is the true wayBeyond the myriad experiences of the world

To experience without intention is to sense the world

All experience is an archwherethrough gleams that untravelled landwhose margins fade forever as we move

Dao ke dao feichang daoMing ke ming feichang ming

Lao Tze

He who exerts his mind to the utmost knows nature’s pattern.

The way of learning is none other than finding the lost mind.

Meng Tze

J.J. Finnigan, Complex science for a complex world

The Internet

ecosystem ProfessionalCollaboration network

Barcelona rail network

Structure of Natural andManmade Systems

Local nature of Physical LawsPeer-to-Peer Relationships

in networked systems

Clusters of galaxies

Motions of Biological Groups

Fishschool

Birdsflock

Locustsswarm

Firefliessynchronize

Local / Peer-to-Peer Relationships in socio-biological systems

Outline

Duality for cooperative systems

Distributed observer and dynamic regulator

Distributed adaptive control

Finite-time consensus and simplified protocol

Control Design Methodsfor Multi‐Agent Systems

Acks. to:Guanrong Chen – Pinning controlLihua Xie - Local nbhd. tracking errorZhihua Qu - Lyapunov eq. for di-graphs

Communication Graph1

2

3

4

56

N nodes

[ ]ijA a

0 ( , )ij j i

i

a if v v E

if j N

oN1

Noi ji

jd a

Out-neighbors of node iCol sum= out-degree

42a

Adjacency matrix

0 0 1 0 0 01 0 0 0 0 11 1 0 0 0 00 1 0 0 0 00 0 1 0 0 00 0 0 1 1 0

A

iN1

N

i ijj

d a

In-neighbors of node iRow sum= in-degreei

(V,E)

i

Social Standing

Standard Control Protocol with Linear Integrator SystemEach node has an associated state i ix uStandard local voting protocol ( )

i

i ij j ij N

u a x x

( )u Dx Ax D A x Lx L=D‐A = graph Laplacian matrix

x Lx If x is an n‐vector then ( )nx L I x

1

N

uu

u

1

N

dD

d

Closed‐loop dynamics

i

j

1

N

xx

x

L has e‐val at zero, simple if exists a spanning tree

Type I system

Modal decomposition 2 1 22 2 1 1 2 2

1( ) (0) (0) (0) 1 (0)

Nt t tT T T

j jj

x t v e w x v e w x v e w x x

2 determines the rate of convergence ‐ Fiedler e‐value

1 1 2Tw determines the consensus value in terms of the initial conditions

No freedom to determine the consensus value or Convergence Rate

We call this the Cooperative Regulator Problem

Standard Control Protocol with Linear Integrator SystemEach node has an associated state i ix u

Standard local voting protocol ( )i

i ij j ij N

u a x x

1

1i i

i i ij ij j i i i iNj N j N

N

xu x a a x d x a a

x

( )u Dx Ax D A x Lx L=D‐A = graph Laplacian matrix

x Lx

If x is an n‐vector then ( )nx L I x

x

1

N

uu

u

1

N

dD

d

Closed‐loop dynamics

i

j

1 1

( ) (0) (0) (0)i i

N Nt tLt T T

i i i ij j

x t e x w e v x w x e v

Convergence Value and Rate

x Lx Closed‐loop system with local voting protocol

Modal decomposition

Let be simple. Then for large t1 0

2 1 22 2 1 1 2 2

1( ) (0) (0) (0) 1 (0)

Nt t tT T T

j jj

x t v e w x v e w x v e w x x

2 determines the rate of convergence ‐ Fiedler e‐value

1 1 2Tw determines the consensus value in terms of the initial conditions

Depends on Communication Graph TopologyNo freedom to determine the consensus value

L has e‐val at zero

We call this the Cooperative Regulator Problem

We want Design Freedom that overcomes graph topology constraints

Duality

Observers

Dynamic Regulators

Coop. Adaptive Control

Simplified Protocol for Finite-Time Consensus

Decouple Control Design from Graph Topology constraints

For Directed graphs

i i ix Ax Bu

A. State Feedback Design for Cooperative Systems on Graphs

Cooperative Regulator vs. Cooperative Tracker problemN nodes with dynamics

Synchronization Tracker design problem 0( ) ( ),ix t x t i

0 0x AxControl node or Command generator (Exosystem)

0( ) ( )i

i ij j i i ij N

e x x g x x

0n ne L G I x x L G I

1 2 ,TT T T nN

Ne R 0 0 ,nNx Ix R 1 nN nnI I R

0nNx x R

Local neighborhood tracking error

Overall error vector

Consensus or synchronization error

where

= Local quantity

= Global quantity

i

j

, ,n mi ix R u R x0(t)

0n ne L G I x x L G I

Local quantity Global quantity

Local control objectives imply global performance

Local Neighborhood Tracking Error

0( ) ( )i

i i ij j i i ij N

u cK cK e x x g x x

0( ) ( )i

i i i i ij j i i ij N

x Ax Bu Ax cBK e x x g x x

0( ) ( ) ( )Nx I A c L G BK x c L G BK x

0Gr Grc cx A x B x

( ) ( ) GN cI A c L G BK A

Local SVFB

Closed loop system

Overall c.l. dynamics

Global synch. error dynamics

Fax and Murray 2004

1 2 ,TT T T nN

Nx x x x R Overall state

Graph structure Control structure

Lewis and Syrmos1995

Emre Tuna 2008 paper online

OPTIMAL Design at each node gives global guaranteed performance on any strongly connected communication graph

OPTIMALDesign at Each node

Cui-Qin Ma and Ji-Feng Zhang, Necessary and Sufficient Conditions for Consensusability of Linear Multi-Agent Systems,” IEEE TAC, vol. 55, no. 5, May 2010

0T T

T

A P PA PBB P IK B P

Example: Unbounded Region of Consensus for Optimal Feedback Gains.

2 1 1,

2 1 0A B

0.5 0.5K

b. Unbounded Consensus Region forOptimal SVFB Gain

a. Bounded Consensus Region forArbitrarily Chosen Stabilizing SVFB Gain

Q=I, R=1

1.544 1.8901K

Example from [Li, Duan, Chen 2009]

Im{ }

Re{ }

Im{ }

Re{ }

A c BK E-vals of (L+G)

1. Cooperative State Feedback, Observers, Duality, and Optimal Design for Synchronization

i i ix Ax Bu

0( ) ( )i

i i ij j i i ij N

u cK cK e x x g x x

0( ) ( ) ( )Nx I A c L G BK x c L G BK x

Thm 1. Design of SVFB Gain for Coop. Tracking Stability

1 10 ,T T TA P PA Q PBR B P K R B P

1min Re( ( ))ii

cL G

Unbounded Region of Consensus for Optimal Gains if coupling gain is

i iy Cx

0 ˆ( ) ( ) ,i

oi ij j i i i i i i i

j Ne y y g y y y y y

ˆ ˆ oi i i ix Ax Bu cF

ˆ ˆ( ) ( ) ( ) ( )N Nx I A c L G FC x c L G F y I B u

Thm 2. Design of Observer Gain for Coop. Estimation

1 10 ,T T TAP PA Q PC R CP F PC R

Cooperative system node dynamics

State feedback with local neighborhood tracking error

Overall Cooperative Team Dynamics

Output measurements at each node

Local nbhd. estimation error

Use local coop. observer dynamics at each node

Overall Team Observer/Sensor Fusion Dynamics

Use OPTIMAL feedback gain

Then synchronization is achieved for ANY strongly connected digraph

Use OPTIMAL observer gain

Then estimates converge achieved for ANY strongly connected digraph

CONTROL DISTRIBUTED ESTIMATION & SENSOR FUSION

Results: Distributed dynamic regulator for synchronization of teams using only output measurementsDuality structure theory extended to networked cooperative feedback systems on graphsOptimal Design yields synchronization on ANY strongly connected Communication di‐graph

Pinning control node

Duality

B. Observer Design for Cooperative Systems on Graphs

,i i ix Ax Bu i iy Cx

ˆ ˆ,i i i i i ix x x y y y

ˆ ( ) ,nix t R ˆ ˆ( ) ( ) p

i iy t Cx t R

0 0 ,x Ax 0 0y Cx

Cooperative Observer design problem

N nodes with dynamics

State and output estimates

State and output estimation errors

Control node or Command generator dynamics

( ) 0,ix t i ˆ ( ) ( ),i ix t x t i or

0( ) ( )i

oi ij j i i i

j Ne y y g y y

Local neighborhood estimation error

0o

n ne L G I y y L G I y L G C x Ren; Beard, Kingston 2008

Overall estimation error

i

j

Local node observersˆ ˆ o

i i i ix Ax Bu cF

0ˆ ˆ ( ) ( )i

i i i ij j i i ij N

x Ax Bu cF e y y g y y

0ˆ ˆ ( ) ( )i

i i i ij j i i ij N

x Ax Bu cFC e x x g x x

ˆ ˆ( ) ( ) ( ) ( )N Nx I A c L G FC x c L G F y I B u

1 2ˆ ˆ ˆ ˆTT T T nN

Nx x x x R

ˆ ˆ ( )Gr Gro o Nx A x F y I B u

( ) ( ) GrN ox I A c L G FC x A x

Overall observer dynamics

where

Overall estimation error dynamics

Li, Duan, Ron Chen 2009

Ren and Beard

c.f. Finsler Lemma design in Li, Duan, Ron Chen 2009

Emre Tuna 2008 paper online(without observer design)

OPTIMALDesign at Each node

OPTIMAL Design at Each node gives guaranteed performanceOn any strongly connected communication di-graph topology

C. Control/Observer Duality on Graphs

Converse, reverse, or transpose graph = reverse edge arrows

0( ) ( ) ,i

i i ij j i i ij N

u cK cK e x x g x x

0ˆ ˆ ( ) ( )

i

i i i ij j i i ij N

x Ax Bu cFC e x x g x x

Must use local nbhd. Tracking error and local nbhd. Estimation erroror duality does not work

SVFB Observer

D. Dynamic Tracker for Synchronization of Cooperative Systems Using Output Feedback

,i i ix Ax Bu i iy Cx

0 0 ,x Ax 0 0y Cx

ˆ ˆ oi i i ix Ax Bu cF

0ˆ ˆ ( ) ( )i

i i i ij j i i ij N


0ˆ ˆ ˆ ˆ( ) ( )i

i i ij j i i ij N

u cK cK e x x g x x

0ˆ ˆ ˆ( ) ( )i

i i ij j i i ij N

x Ax cBK e x x g x x

0ˆ( ) ( ) ( )Nx I A x c L G BKx c L G BK x

0

ˆ ˆ( ) ( ) ( )

ˆ( ) ( )Nx I A c L G FC x c L G F y

c L G BKx c L G BK x

N nodes with dynamics

Command generator dynamics (exosystem)

Observers at each node

Estimated SVFB

Closed‐loop systems

Overall system/observer dynamics

Must use local nbhd. Tracking error and local nbhd. Estimation erroror it is not nice.

OPTIMAL Design at Each node gives guaranteed performanceOn any strongly connected communication di-graph topology

Three Regulator Designs

0ˆ ˆ ( ) ( )i

i i i ij j i i ij N


0ˆ ˆ ˆ ˆ( ) ( )i

i i ij j i i ij N

u cK cK e x x g x x

Nbhd Observers

Nbhd Controls

1. Neighborhood Controller and Neighborhood Observer

2. Neighborhood Controller and Local Observer

0ˆ ˆ ˆ ˆ( ) ( )i

i i ij j i i ij N

u cK cK e x x g x x

Local Observers

Nbhd Controls

ˆ ˆ .i i i ix Ax Bu cF y

3. Local Controller and Neighborhood Observer

ˆi iu Kx

0ˆ ˆ ( ) ( )i

i i i ij j i i ij N


Nbhd Observers

Local Controls

i

j

i

j

i

j

i

j

ij

ij

E. Local vs. Global Variables

0nNx x R

ˆ ˆ,nN pNx x x R y y y R

ne L G I

one L G I y L G C x

Local nbhd. tracking error

Global synchronization error

Global estimation errors

Local nbhd. estimation error

MULTIPLY GLOBAL VARIABLES BY (L+G) TO GETIMPLEMENTABLE LOCAL VARIABLES FOR CONTROL

1i i ix k Ax k Bu k

0 01x k Ax k

0i

i ij j i i ij N

e x x g x x

11i i i iu c d g K

kBKgdckAxkx iiiii 111

11 c Nk A k I A c I D G L G BK k

GLGDI 1

, 1,k K N

Discrete-Time Optimal Design for Synchronization

Distributed systems

Command generator

Local Nbhd Tracking Error

Local closed-loop dynamics

Local cooperative SVFB - normalized

0 ( )k x k x k Global disagreement error dynamics

Normalized Graph Matrix

Normalized graph eigenvalues

Work with Kristian Movric, Lihua Xie, Keyou You

1

r

c0

r0

Covering circle of graph eigenvalues

Synchronization region contains this circle

Single-Input case with Real Graph Eigenvalues

1/21/2 1 1/20max

0

( ( ) )T T Tr r Q A PB B PB B PAQc

0 max min

0 max min

.rc

If graph eigenvalues are real

u

u Ar

1For SI systems, for proper choice of Q

Mahler measure

2log uii

A intrinsic entropy rate = minimum data rate in a networked control system that enables stabilization of an unstable system

min max/ Eigen-ratio = ‘condition number’ of the communication graph

condition

Lihua Xie and You Keyou

Distributed Systems

( )i i i i ix f x u d

( )x f x u d

1 1 1 1

2 2 2 2

( )( )

, , ( )

( )

N N N

N N N N

x u f xx u f x

x R u R f x R

x u f x

0 0( , )x f x t

x0(t)

Distributed Adaptive Control for Networked Dynamical Systems

Node dynamics

Overall network dynamics

Command generator orControl node dynamics

ib

All Nonlinearities can be differentderivation is for General Di‐graphsNonlinearities and disturbances are unknown

Work of Abhijit Das

unknown

exosystem

unknown

Cooperative TrackerProblem

0i

i ij j i i ij N

e a x x b x x

0 01e L B x x L B x x L B

/ ( )e L B

Local nbhd. tracking error‐ Lihua Xie

0x x Synch. error

e(t)=0 implies synchronization

Overall Local nbhd tracking error

1

0 1 0( )i i

i ij j ij i i i i i i i iN ij N j N

N

xe a x a x b x x d b x a a b x

x

Lemma‐

0 0 01 ( ) 1 ( ) 1e Dx Bx Ax B x B D A x B x B L x B x

Synchronization or Tracker control problem‐

Design local control protocols so that local neighborhood erroris bounded to a small residual set.

( )e t

Every node has to go to x0(t)

Then ( ) ( )i jx t x t and 0( ) ( )ix t x t are small

Cooperative UUBClose enough synchronizationRobust or Practical synchronization

0 0( ) ( , ) ( )e L B x x L B f x f x t u d t

ˆ ( )i i i iu v f x

( )i i i i ix f x u d

ˆ ( )u v f x

0ˆ( ) ( ) ( , )e L B f x f x f x t v d

Local Control Inputs and Error Dynamics

Node dynamics

Node control protocols

Overall node control protocols

Error dynamics

Closed‐loop error dynamics

( )x f x u d Overall network dynamics

0i

i ij j i i ij N

e a x x b x x

Local nbhd. tracking error

0 01e L B x x L B x x L B

Overall Local nbhd tracking error

( )Ti i i i i if x W x

ˆ ˆ ( )Ti i i i if x W x

1 1 111 1

2 2 222 2

( )

( )( )

( )( )

( ) ( )

( )( )

T

T

T

N N TN N NN

x

xWf x

xWf x

f x W x

f xxW

1 1 1

2 22

( )

ˆ ( )ˆ ( )

ˆ ˆ( ) ( )

( )ˆ

T

T

T

TN NN

x

W xxW

f x W x

xW

ˆ( ) ( ) ( ) ( )Tf x f x f x W x

0( ) ( ) ( , )Te L B W x v d t f x t

Neural Network Approximation of Unknown Node Nonlinearities

Assume

Parameterized approximation

Overall network

Approx. error

Error Dynamics

Each node keeps a small NN to approximate its own nonlinearities and compensate

diagonal

11 2 1T

Nq q q q L B

diag diag 1/i iP p q

0TQ P L B L B P

Lemma‐ Zihua Qu

L B

Fact:

is irreducible diagonally dominant, hence nonsingular with all e‐vals in ORHP

Define

Then

diagonal

Background Facts

Digraph Lyapunov EquationAllows one to do design for Directed graphs

Let di‐graph be strongly connected with at least one pinned node. Then

0ˆ ˆ( ) ( )

i

Ti i i i ij j i i i i i i

j Nu ce f x c a x x cb x x W x

ˆ ( )Tu ce W x

ˆ ˆ( )Ti i i i i i i i iW F e p d b FW

. 01.2

. ( ) M

i c

ii c Q

iii c Q N P A

Theorem‐ Distributed Adaptive Control for Synchronization

Take local control protocols

Tune NN by local tuning law

Take control gains c big enough. Select

Let graph be strongly connected.Then e(t) is coop UUB and all nodes synchronize to the control node x0(t)

Main Result

Control gains Pinning gains

Local NN tuning law

Local control protocol

Abhijit Das

Left e-vector elementsBut Fi > 0 is arbitrary

11 12 2

T TV e Pe tr W F W

1T TV e Pe tr W F W

10( ) ( , )T T TV e P L B W x ce d f x t tr W F W

10( , )T T T TV ce P L B e e P L B d f x t tr W F W e P D B A

01 ( ) ( , ) ( )2

T T T T TV ce Qe e P L B d t f x t tr W W W tr W e PA

221 ( )

2 M M M M MF F FV c Q e e P L B d F W W W W e P A

2 12

M M

M

B P L B We

c Q P A

2 12

M M

M

B P L B WW

c Q P A

Proof and Error Bounds

This is negative if

UUB Error Bounds

1. Use Local nbhd error e(t)

3. Frobenius norm only cares about diagonal terms

2. P is diagonal

For general digraphs

4. Zhihua Qu Lyapunov eq. allows design for DI‐graphs

ˆ ˆ( )Ti i i i i i i n i iW F e p d b I FW

0( ) ( , ) ( )ne L B I f x f x t u d t

Vector Node States

( ) , ni i i i i ix f x u d x R

Node dynamics

Modifications needed:

( ), ( ), ( )A L B P

i

i ijj N

d a

oi ji

jd a

1( ) ( )

( ) max( ) max( )

N

ii

oi ii i

A d vol G

A d d

( ) max( )iiA N d

1( ) ( )

( ) max( ) max( 2 )

No

i i ii

oi i i i ii i

L B b d d

L B b d d b d

( ) max( )oi i ii

L B N b d d

1max( )o

i i iiL B b d d

Singular Value Bounds and Network Structure

Error bounds

So we like

In‐degree

Therefore

Representative bounds ‐Column sum= Degree sum

to be small

Out‐degree

1 1 12 13

21 2 2 23

31 32 3 3

d b a aa d b a

L B D B Aa a d b

2 12

M M

M

B P L B We

c Q P A

0TQ P L B L B P

diag diag 1/i iP p q

Dependence of Convergence Speed and Synchronization Error on Graph Structural Properties

Let NN estimation errors be zero. Then‐

12

TV e Pe

212 ( ) ( ) ( ) MV c Q e e P L P B

This is negative if ( ) ( )2( )M

P L Be Bc Q

When e(t) is large one has

12 ,TV e Pe

212 ( )V c Q e

V V so( )( )QcP

with

So convergence rate is /2( ) (0) te t e e

For undirected graphs, ( ) / ( )Q P is replaced by ( )L B

For general digraphs

0TQ P L B L B P

1 , 1 , , 1

k k k

i i j i ji i j i j

i j i j

i ij ijL G L g L g g L g g g Li ij ij

F. Selection of Pinned Nodes

Pin into nodes with LARGE OUT DEGREE

c.f. Xiao Fan Wang and Ron Chen

11 1 1

1

1

11 1 1 11 1

1 1

1 1

0

0

i N

i i ii

N Ni NN

i N N

i ii i i

N Ni NN N NN

ii i

L L LL L

L G

L L L

L L L L LL L L

L L L L L

L L

1

2

3

4

5

L

31 1 1 1

22 2 2 2

43 3 3 3

4 4 4 45

5 5 5 5

x x u d

x x u d

x x u d

x x u d

x x u d

0

ˆ ( )ˆ ( )

i

i i i i

Tij j i i i i i i

j N

u ce f x

c a x x cb x x W x

1a. Pinning Control

Communication Graph

Control gain c=500

Node dynamics

Simulations‐ Synchronization on Graphs

0 0.05 0.1 0.15 0.2 0.25 0.30

1

2

3

Time(t)St

ates

xi∀

i

State dynamics and input

0 0.05 0.1 0.15 0.2 0.25 0.3−300

−200

−100

0

100

200

300

Time(t)

Inpu

tu

i∀i

No synchronizationUnstable

1b. Pinning Control Control gain c=1500

0 0.05 0.1 0.15 0.2 0.25 0.30

1

2

3

Time(t)

Stat

esx

i∀i


0 0.05 0.1 0.15 0.2 0.25 0.3−300

−200

−100

0

100

200

300

Time(t)

Inpu

tu

i∀i

0 0.05 0.1 0.15 0.2 0.25 0.3−2

−1.5

−1

−0.5

0

0.5

1Error Plot

Time(t)δ i∀i

Synchronization error 5%

2. Distributed Adaptive Control Control gain c=300

0ˆ ˆ( ) ( )

i

Ti i i i ij j i i i i i i

j Nu ce f x c a x x cb x x W x

0 0.1 0.2 0.3 0.4 0.50

1

2

3

Time(t)

Stat

esx

i∀i


0 0.1 0.2 0.3 0.4 0.5−300

−200

−100

0

100

200

300

Time(t)

Inpu

tu

i∀i

0 0.1 0.2 0.3 0.4 0.5−2

−1

0

1Error Plot

Time(t)

δ i∀i

0 0.1 0.2 0.3 0.4 0.5−200

0

200

400

600

Time(t)

f i(x

)-f̂ i

(x)∀

i

0 0.1 0.2 0.3 0.4 0.5−150

−100

−50

0

50

100

150State Dynamics

Time(t)

f i(x

)∀i

0 0.1 0.2 0.3 0.4 0.5−150

−100

−50

0

50

100

150Estimated State Dynamics

Time(t)

f̂ i(x

)∀i

nonlinearities

estimated nonlinearities

3. Second‐Order Node Dynamics Control gain c=300

1 2 1

12 2 2 1sin( )i i

i i i

i

ri i i i i

q q u

q J u B q M gl q

Node Dynamics Target Dynamics

0 0 0 0 0 0 ( )om q d q k q u t

Standard Second-Order Consensus1 2

2i i

i i i

x x

x u w

1 1 1 1 10

i

i ij j i i ij N

e a x x b x x

2 2 2 2 20

i

i ij j i i ij N

e a x x b x x

1 20 020 0

x x

x u

1 1 2 2i i i i iu k e k e

1 1

1 22 2

0( ) ( )

Ix xK L B K L Bx x

For undirected graphs- stable for any positive gainsFor digraphs- the stabilizing gains depend on the graph topology

Node dynamics

Target dynamics

Local tracking errors

Standard approach

Closed-loop dynamics

Second-Order Consensus Using Sliding Variable

2 1i i i ir e e

1 1 1 1 10

i

i ij j i i ij N

e a x x b x x

2 2 2 2 20

i

i ij j i i ij N

e a x x b x x


Sliding variable

1 2

2 ( )i i

i i i i i

x x

x f x u w

1 20 020 0 0( , )

x x

x f x t

Node dynamics

Target dynamics

Decouple control design fromGraph topology

- any positive gain

Results for Digraphs

Work by Abhijit Das

position

velocity

( ) ( , ) ( ) ( )i i i i i i i i i i i i iM q q C q q q H q g q

0 0 0 0 0 0 0 0 0 0 0 0( ) ( , ) ( ) ( )M q q C q q q H q g q

Synchronization of Unknown Lagrangian SystemsGang Chen, College of Automation Chongqing University

Node dynamics

Target dynamics

All nonlinearities and disturbances unknown

Second-Order Consensus Using Sliding Variable

2 1i i i ir e e

2 2 20 0 0

20 0

1 ( , ) 1

1 ( , )

r L B x f x t L B x x

L B f x u w L B f x t e

2ˆ ( )T ii i i i i i

i i

u cr W x ed b

ˆ ˆ( )Ti i i i i i i i iW F r p d b FW

D BP A

1 1 1 1 10

i

i ij j i i ij N

e a x x b x x

2 2 2 2 20

i

i ij j i i ij N

e a x x b x x


Sliding variable

Error dynamics

Adaptive Control protocol

Key design parameter

1 2

2 ( )i i

i i i i i

x x

x f x u w

1 20 020 0 0( , )

x x

x f x t

Node dynamics

Target dynamics

Decouple control design fromGraph topology

- any positive gain

Results for Digraphs

Work by Abhijit Das

position

velocity

( ) ( , ) ( ) ( )i i i i i i i i i i i i iM q q C q q q H q g q

0 0 0 0 0 0 0 0 0 0 0 0( ) ( , ) ( ) ( )M q q C q q q H q g q

Synchronization of Unknown Lagrangian SystemsGang Chen, College of Automation Chongqing University

Node dynamics

Target dynamics

101

( ) ( )ni ij j i i ij

e a q q b q q

2

01( ) ( )n

i ij j i i ije a q q b q q

2 1i i i is e e


Sliding variable

All nonlinearities and disturbances unknown

position

velocity

Higher-Order Consensus Hongwei Zhang


Sliding variable

ChainedIntegratorNode dynamics

Target dynamics

Adaptive Control protocol

Position errors

Velocity errors

( 1) ( 1)

1 12 3

0 1 0 00 0 1 0

0 0 0 1

M M

M

R

1 1T P IP

0TQ P L B L B P

Proof:

Lyapunov equation for sliding gains

Lyapunov equation for Pinned Graph - Z. Qu

Decouple Control design from Graph topology

Control design Lyapunov eq.

Graph topology Lyapunov eq.

Hongwei Zhang idea

1. Natural biological groups do not measure synchronization errors !

Agents sense rough relative positions or motions

and adjust their motion accordingly

2. We want FINITE-TIME synchronization

0sgn( ) sgn( )i

i ij j i i ij N

u a x x b x x

Propose a Signum protocol

Binary, 1-bit quantization, reduced information needed

Discontinuous- makes a big mess.

Gang Chen, College of Automation Chongqing University

11

1

2 node consensus example in Wassim Haddad paper

sgn( )x uu x

u

t

1

-1

x

t

Linear state evolution

( ) ( ( ))x t f x t nx R f(x) Lebesgue measurable and locally essentially bounded

Filippov set-valued map0 ( ) 0

[ ]( ) { ( ( ) \ )}S

K f x co f B x S

Filippov solution on

An absolutely continuous function that satisfies the differential inclusion

0 1[ , ]t t

0 1: [ , ] nx t t R

( ) [ ]( )x t K f x for almost all 0 1[ , ]t t t

A Filippov solution is maximal if it cannot be extended forward in time

Set is weakly (resp. strongly) invariant if it contains a (resp. all) maximal solutions0x M nM R

Same as lim ( ) : :i i i fiK f x co f x x x y N N

,

Cortes 2008

Bacciotti & Ceragioli, 1999, Shevitz & Paden, 1994

Wassim Haddad 2008

Paden &Sastry 1987

Differential Equations with Discontinuous Right-Hand Sides

NOT necessarily Lipschitz

Caratheodory solutions work, in fact. But we need Filippov set-valued map on the next page.

: nV R R a locally Lipschitz function

the set of measure zero where gradient does not existVN

( ) {lim ( ) : , } [ ]( )i i i ViV x co V x x x x N N K V x

Clarke generalized gradient

Set-valued Lie Derivative [ ]( ) { | [ ]( ) s.t. , ( )}TfL V x a R K f x a V x

Cortes 2008


Wassim Haddad 2008Paden &Sastry 1987

Lyapunov Analysis for Discontinuous Dynamical Systems

Lyapunov function second derivative is strictly negative implies finite-time consensus

, 1, 2, ,i ix u i n

1 2 0( ) ( ) ( ) ,nx t x t x t x t T

0sgn( ) sgn( )i

i ij j i i ij N

u a x x b x x

1

-1

(

)

0sgn( ) sgn( )i

i ij j i i ij N

x a x x b x x

0i ie x x

sgn( ) sgn( )i

i ij j i i ij N

e a e e b e

Error dynamics

( ) ( )i

i ij j i i ij N

e a SGN e e b SGN e

Paden and Sastry 1987- calculus for Filippov map

sgn(.)

1

-1 SGN(.)

Node dynamics

Finite-time Synchronization in time T

Signum protocol

Closed-loop system

Synch. error

Binary, 1-bit quantization, reduced information needed

Cortes 2006, 2008


Wassim Haddad 2008

Filippov solutions (Caratheodory works)Fillipov set-valued map

11 1 1 1

1sgn( )2

n n n nT

ij i j i ij j ii j i j

a x x x a x x

Lemma 1. For undirected network one has

Theorem 1. Let the communication graph be undirected and connected. If there is at least one pinned node,

then the signum protocol solves the controlled consensus problem asymptotically.

Proof. Choose the candidate Lyapunov function

12

TV e e

1 2[ , , , ]T T T Tne e e e

[ ] [ ]TfL V K e e

1( sgn( ) sgn( ))

i

nTi ij j i i i

i j NK e a e e b e

1. Undirected Graphs Gang Chen, College of Automation Chongqing University

Interesting fact: the Lyapunov function and derivative are continuous for undirected graphs.

Proof: check out the second derivative of Lyapunov function

20[ ( )]L V t

12

TV e e

Convergence time bounded by

Depends on the initial synchronization errors

Must use Clarke generalized gradient and set-valued Lie derivative

Undirected Graphs – Finite Time Convergence

max{ }i inb d max din = max in-degree


2[ ( )]( ) 0f fL L V x Cortes 2006

2. Directed Graphs

, ,i ij j jip a p a i j

Detail-balanced

There exist such that0ip

Sum over j. Then 1 2 Np p p p is a left e-vector of L for zero e-val


11

12 22

0LF

F F

Let there exist a spanning tree.

Frobenius form of graph Laplacian L

11L

22F

1S

2S

3S

4S

2,max{ }ind

1,max{ }ind3

4,jll S

a some j S

Proof: first show that the leaders reach consensus in finite time, then show followers do so also.2

1 0 1, 1[ ( )] iL V t p Convergence times bounded like

leaders

followers

Must use Clarke generalized gradient and set-valued Lie derivative

Finite Time Tracking Control

0 0( , )x g t xCommand generator node (exosystem)

0 01 sgn( ) , sgn( )

i

i

i ij j j i i ij Nij i

j N

u a u x x b g t x x xa b

Idea from Lihua Xie and Liu Shuai -allows tracking a moving leader with FIRST-ORDER dynamics

i ix u

0 0 01 sgn ( ) ( ) ( ( , ) sgn(( ) ( )))

i

i

i ij j j j i i i i ij Nij i

j N


Finite Time Formation Control

Idea from Lihua Xie and Liu Shuai -formation position offsets


Node dynamics

Control protocol

Finite Time Tracking Control

0 0( , )x g t xCommand generator node (exosystem)

0 01 sgn( ) , sgn( )

i

i

i ij j j i i ij Nij i

j N


i ix u

0 0 01 sgn ( ) ( ) ( ( , ) sgn(( ) ( )))

i

i

i ij j j j i i i i ij Nij i

j N


Finite Time Formation Control


Node dynamics

Control protocol

What does Finite-Time Consensus Look Like? - Simulation

leaders

followers

Finite Time Consensus!

Communication Topology


Leaders reach consensus

the - uta talks/2011 chinese academy- design for consensus.pdfstandard local voting protocol i iijji...

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