Team-triggered coordination of networked systems Cameron Nowzari - - PowerPoint PPT Presentation

▶
team triggered coordination of networked systems
SMART_READER_LITE
LIVE PREVIEW

Team-triggered coordination of networked systems Cameron Nowzari - - PowerPoint PPT Presentation

Oct 10, 2022 316 likes •864 views

Team-triggered coordination of networked systems Cameron Nowzari Jorge Cort es Mechanical and Aerospace Engineering University of California, San Diego cnowzari@ucsd.edu American Control Conference Washington D.C. June 18, 2013

slide-1
SLIDE 1

Team-triggered coordination

  • f networked systems

Cameron Nowzari Jorge Cort´ es

Mechanical and Aerospace Engineering University of California, San Diego cnowzari@ucsd.edu American Control Conference Washington D.C. June 18, 2013

slide-2
SLIDE 2
  • Coordination of networked systems-

Each individual subsystem senses immediate environment communicates with others wirelessly processes gathered information takes action in response Multiple subsystems provide inherent robustness adaptive behavior enable tasks beyond individuals’ capabilities

Cameron Nowzari (UCSD) Team-triggered coordination June 18, 2013 2 / 23

slide-3
SLIDE 3

Review of real-time implementation strategies

Given desired task, information

design

− − − − → agent plans If continuous feedback and control is possible, agent plans

ideal execution

− − − − − − − − − → ideal performance

Cameron Nowzari (UCSD) Team-triggered coordination June 18, 2013 3 / 23

slide-4
SLIDE 4

Review of real-time implementation strategies

Given desired task, information

design

− − − − → agent plans If continuous feedback and control is possible, agent plans

ideal execution

− − − − − − − − − → ideal performance Instead, given desired performance, we can use

Cameron Nowzari (UCSD) Team-triggered coordination June 18, 2013 3 / 23

slide-5
SLIDE 5

Review of real-time implementation strategies

Given desired task, information

design

− − − − → agent plans If continuous feedback and control is possible, agent plans

ideal execution

− − − − − − − − − → ideal performance Instead, given desired performance, we can use Time-triggered control: information

periodic

− − − − − → agent plans

periodic execution

− − − − − − − − − − − → desired performance Find suitable period T Simple Wasteful

Cameron Nowzari (UCSD) Team-triggered coordination June 18, 2013 3 / 23

slide-6
SLIDE 6

Review of real-time implementation strategies

Given desired task, information

design

− − − − → agent plans If continuous feedback and control is possible, agent plans

ideal execution

− − − − − − − − − → ideal performance Instead, given desired performance, we can use Event-triggered control: information

continuous

− − − − − − − → agent plans

triggered execution

− − − − − − − − − − − − → desired performance Design suitable event-trigger Only necessary updates Requires continuous information

Cameron Nowzari (UCSD) Team-triggered coordination June 18, 2013 3 / 23

slide-7
SLIDE 7

Review of real-time implementation strategies

Given desired task, information

design

− − − − → agent plans If continuous feedback and control is possible, agent plans

ideal execution

− − − − − − − − − → ideal performance Instead, given desired performance, we can use Self-triggered control: information?

design

← − − − − agent plans?

execute

← − − − − − desired performance What is necessary information to achieve a desired level of performance? Implementable, asynchronous Conservative

Cameron Nowzari (UCSD) Team-triggered coordination June 18, 2013 3 / 23

slide-8
SLIDE 8
  • Team-triggered- control

Objective: Combine best properties of event- and self-triggered strategies into a unified, implementable approach How?

Cameron Nowzari (UCSD) Team-triggered coordination June 18, 2013 4 / 23

slide-9
SLIDE 9
  • Team-triggered- control

Objective: Combine best properties of event- and self-triggered strategies into a unified, implementable approach How? Agents make promises to neighbors about their future states Agents warn each other when promises need to be broken

Cameron Nowzari (UCSD) Team-triggered coordination June 18, 2013 4 / 23

slide-10
SLIDE 10

Outline

1 Motivation 2 Problem Formulation

problem statement review of existing strategies

3 Team-Triggered Coordination

promises control policies self-triggered updates broken promises (event-triggered updates)

4 Algorithm Certification

performance guarantees simulations

Cameron Nowzari (UCSD) Team-triggered coordination June 18, 2013 5 / 23

slide-11
SLIDE 11

Problem statement

Coordination task: Drive N agents with linear dynamics ˙ xi = Aixi + Biui, xi ∈ Xi, ui ∈ Ui, to a set D. Agents can communicate with other agents through a graph with a fixed topology

Cameron Nowzari (UCSD) Team-triggered coordination June 18, 2013 6 / 23

slide-12
SLIDE 12

Problem statement

Coordination task: Drive N agents with linear dynamics ˙ xi = Aixi + Biui, xi ∈ Xi, ui ∈ Ui, to a set D. Agents can communicate with other agents through a graph with a fixed topology Given a Lyapunov function V with a distributed gradient d dtV (x) =

N

  • i=1

∇iV (xi

N ) ˙

xi, and a distributed continuous control law u∗ that monotonically optimizes V while x is not in D, how can we design a real-time implementation?

Cameron Nowzari (UCSD) Team-triggered coordination June 18, 2013 6 / 23

slide-13
SLIDE 13

Time-triggered coordination

Communication and actuator updates happen at a fixed period T

t0 t3 t1 t2 T

T must be small enough such that

d dtV (x) < 0 at all times

Cameron Nowzari (UCSD) Team-triggered coordination June 18, 2013 7 / 23

slide-14
SLIDE 14

Time-triggered coordination

Communication and actuator updates happen at a fixed period T

t0 t3 t1 t2 T

T must be small enough such that

d dtV (x) < 0 at all times

Drawbacks: Requires global information to compute T T can be quite conservative Assumes all agents to be synchronized

Cameron Nowzari (UCSD) Team-triggered coordination June 18, 2013 7 / 23

slide-15
SLIDE 15

Event-triggered coordination

Given an update at time tlast, the next update time tnext is decided by the

  • ccurrence of an event

d dtV (x(tnext) = 0

Cameron Nowzari (UCSD) Team-triggered coordination June 18, 2013 8 / 23

slide-16
SLIDE 16

Event-triggered coordination

Given an update at time tlast, the next update time tnext is decided by the

  • ccurrence of an event

d dtV (x(tnext) =

N

  • i=1

∇iV (xi

N (tnext))(Aixi + Biu∗ i (xN i (tlast)) = 0

Cameron Nowzari (UCSD) Team-triggered coordination June 18, 2013 8 / 23

slide-17
SLIDE 17

Event-triggered coordination

Given an update at time tlast, the next update time tnext is decided by the

  • ccurrence of an event

d dtV (x(tnext) =

N

  • i=1

∇iV (xi

N (tnext))(Aixi + Biu∗ i (xN i (tlast)) = 0

Requires global information.

Cameron Nowzari (UCSD) Team-triggered coordination June 18, 2013 8 / 23

slide-18
SLIDE 18

Event-triggered coordination

Given an update at time tlast, the next update time tnext is decided by the

  • ccurrence of an event

d dtV (x(tnext) =

N

  • i=1

∇iV (xi

N (tnext))(Aixi + Biu∗ i (xN i (tlast)) = 0

Requires global information. Instead, use local trigger ∇iV (xi

N (ti next)

  • exact

)(Aixi(ti

next) + Biu∗ i (xN i (ti last)) = 0

Cameron Nowzari (UCSD) Team-triggered coordination June 18, 2013 8 / 23

slide-19
SLIDE 19

Event-triggered coordination

Given an update at time tlast, the next update time tnext is decided by the

  • ccurrence of an event

d dtV (x(tnext) =

N

  • i=1

∇iV (xi

N (tnext))(Aixi + Biu∗ i (xN i (tlast)) = 0

Requires global information. Instead, use local trigger ∇iV (xi

N (ti next)

  • exact

)(Aixi(ti

next) + Biu∗ i (xN i (ti last)) = 0

Drawback: Requires continuous information from neighbors to check trigger

Cameron Nowzari (UCSD) Team-triggered coordination June 18, 2013 8 / 23

slide-20
SLIDE 20

Self-triggered coordination

Agent i receives information at time ti

last, constructs guaranteed sets

Xi

j(t) = Rj(t − tlast, xj(tlast)) ⊂ Xj

Cameron Nowzari (UCSD) Team-triggered coordination June 18, 2013 9 / 23

slide-21
SLIDE 21

Self-triggered coordination

Agent i receives information at time ti

last, constructs guaranteed sets

Xi

j(t) = Rj(t − tlast, xj(tlast)) ⊂ Xj

Recall the local trigger ∇iV (xi

N (ti next))(Aixi(ti next) + Biu∗ i (xN i (ti last))

Cameron Nowzari (UCSD) Team-triggered coordination June 18, 2013 9 / 23

slide-22
SLIDE 22

Self-triggered coordination

Agent i receives information at time ti

last, constructs guaranteed sets

Xi

j(t) = Rj(t − tlast, xj(tlast)) ⊂ Xj

Recall the local trigger ∇iV (xi

N (ti next))(Aixi(ti next) + Biu∗ i (xN i (ti last))

≤ sup

yN ∈Xi

N (ti next)

∇iV (yN )(Aixi(ti

next) + Biu∗ i (xN i (ti last))

Cameron Nowzari (UCSD) Team-triggered coordination June 18, 2013 9 / 23

slide-23
SLIDE 23

Self-triggered coordination

Agent i receives information at time ti

last, constructs guaranteed sets

Xi

j(t) = Rj(t − tlast, xj(tlast)) ⊂ Xj

Recall the local trigger ∇iV (xi

N (ti next))(Aixi(ti next) + Biu∗ i (xN i (ti last))

≤ sup

yN ∈Xi

N (ti next)

∇iV (yN )(Aixi(ti

next) + Biu∗ i (xN i (ti last))

Drawback: Conservative updates

Cameron Nowzari (UCSD) Team-triggered coordination June 18, 2013 9 / 23

slide-24
SLIDE 24

Outline

1 Motivation 2 Problem Formulation

problem statement review of existing strategies

3 Team-Triggered Coordination

promises control policies self-triggered updates broken promises (event-triggered updates)

4 Algorithm Certification

performance guarantees simulations

Cameron Nowzari (UCSD) Team-triggered coordination June 18, 2013 10 / 23

slide-25
SLIDE 25

Promises

Agent j makes either a control promise to agent i at time tlast U i

j[tlast] ∈ C0([tlast, ∞); 2Uj),

  • r a state promise

Xi

j[tlast] ∈ C0([tlast, ∞); 2Xj),

Cameron Nowzari (UCSD) Team-triggered coordination June 18, 2013 11 / 23

slide-26
SLIDE 26

Promises

Agent j makes either a control promise to agent i at time tlast U i

j[tlast] ∈ C0([tlast, ∞); 2Uj),

  • r a state promise

Xi

j[tlast] ∈ C0([tlast, ∞); 2Xj),

Promises contain more information than guaranteed sets xj(t) ∈ Xi

j[tlast](t) ⊂ Xi j(t)

Cameron Nowzari (UCSD) Team-triggered coordination June 18, 2013 11 / 23

slide-27
SLIDE 27

Promises

Agent j makes either a control promise to agent i at time tlast U i

j[tlast] ∈ C0([tlast, ∞); 2Uj),

  • r a state promise

Xi

j[tlast] ∈ C0([tlast, ∞); 2Xj),

Promises contain more information than guaranteed sets xj(t) ∈ Xi

j[tlast](t) ⊂ Xi j(t)

Promises that agent i sends depends on the information it has

Cameron Nowzari (UCSD) Team-triggered coordination June 18, 2013 11 / 23

slide-28
SLIDE 28

Control policies

Since agents have set-valued information, want controllers that operate on sets rather than points u∗

i : j∈N (i)∪{i} Xj → Ui

Cameron Nowzari (UCSD) Team-triggered coordination June 18, 2013 12 / 23

slide-29
SLIDE 29

Control policies

Since agents have set-valued information, want controllers that operate on sets rather than points u∗

i : j∈N (i)∪{i} Xj → Ui

u∗∗

i

:

  • j∈N (i)∪{i}

2Xj → Ui

Cameron Nowzari (UCSD) Team-triggered coordination June 18, 2013 12 / 23

slide-30
SLIDE 30

Control policies

Since agents have set-valued information, want controllers that operate on sets rather than points u∗∗

i

:

  • j∈N (i)∪{i}

2Xj → Ui Many different ways to design u∗∗

i

Cameron Nowzari (UCSD) Team-triggered coordination June 18, 2013 12 / 23

slide-31
SLIDE 31

Control policies

Since agents have set-valued information, want controllers that operate on sets rather than points u∗∗

i

:

  • j∈N (i)∪{i}

2Xj → Ui Many different ways to design u∗∗

i

Should decrease Lyapunov function when perfect information is available Can be constructed from u∗

Cameron Nowzari (UCSD) Team-triggered coordination June 18, 2013 12 / 23

slide-32
SLIDE 32

Self-triggered updates

Recall the local trigger ∇iV (xi

N (ti next))(Aixi(ti next) + Biu∗∗ i ({xN i (ti last)})

≤ sup

yN ∈Xi

N (ti next)

∇iV (yN )(Aixi(ti

next) + Biu∗∗ i (Xi N (ti last))

Cameron Nowzari (UCSD) Team-triggered coordination June 18, 2013 13 / 23

slide-33
SLIDE 33

Self-triggered updates

Recall the local trigger ∇iV (xi

N (ti next))(Aixi(ti next) + Biu∗∗ i ({xN i (ti last)})

≤ sup

yN ∈Xi

N (ti next)

∇iV (yN )(Aixi(ti

next) + Biu∗∗ i (XN i (ti last))

≤ sup

yN ∈Xi

N (ti next)

∇iV (yN )(Aixi(ti

next) + Biu∗∗ i (Xi N (ti last))

Cameron Nowzari (UCSD) Team-triggered coordination June 18, 2013 13 / 23

slide-34
SLIDE 34

Self-triggered updates

Recall the local trigger ∇iV (xi

N (ti next))(Aixi(ti next) + Biu∗∗ i ({xN i (ti last)})

≤ sup

yN ∈Xi

N (ti next)

∇iV (yN )(Aixi(ti

next) + Biu∗∗ i (XN i (ti last))

≤ sup

yN ∈Xi

N (ti next)

∇iV (yN )(Aixi(ti

next) + Biu∗∗ i (Xi N (ti last))

Self-trigger information update At any time t agent i ∈ {1, . . . , N} receives new promise(s) Xi

j[t] from

neighbor(s) j ∈ N(i), agent i performs:

1: compute time tnext ≥ t 2: request information from neighbors at time tnext

Respond to information request At any time t a neighbor j ∈ N(i) requests information, agent i per- forms:

1: send new promise Xj

i [t] to agent j

Cameron Nowzari (UCSD) Team-triggered coordination June 18, 2013 13 / 23

slide-35
SLIDE 35

Event-triggered updates

Evolution of V is monotonic assuming promises are not broken Agents must warn each other when they break their promises

Cameron Nowzari (UCSD) Team-triggered coordination June 18, 2013 14 / 23

slide-36
SLIDE 36

Event-triggered updates

Evolution of V is monotonic assuming promises are not broken Agents must warn each other when they break their promises Event-trigger information update At all times t, agent i performs:

1: if there exists j ∈ N(i) such that xi(t) /

∈ Xj

i [·](t) then

2:

send new promise Xj

i [t] to agent j

3: end if

Cameron Nowzari (UCSD) Team-triggered coordination June 18, 2013 14 / 23

slide-37
SLIDE 37

Outline

1 Motivation 2 Problem Formulation

problem statement review of existing strategies

3 Team-Triggered Coordination

promises control policies self-triggered updates broken promises (event-triggered updates)

4 Algorithm Certification

performance guarantees simulations

Cameron Nowzari (UCSD) Team-triggered coordination June 18, 2013 15 / 23

slide-38
SLIDE 38

Convergence guarantees

Proposition

Given a networked system executing the team-triggered law, the Lyapunov function V is monotonically nonincreasing along the network evolution

Cameron Nowzari (UCSD) Team-triggered coordination June 18, 2013 16 / 23

slide-39
SLIDE 39

Convergence guarantees

Proposition

Given a networked system executing the team-triggered law, the Lyapunov function V is monotonically nonincreasing along the network evolution The team-triggered law is not amenable to standard discrete-time stability analysis because the agents’ memories evolve discontinuously

Cameron Nowzari (UCSD) Team-triggered coordination June 18, 2013 16 / 23

slide-40
SLIDE 40

Convergence guarantees

Proposition

Given a networked system executing the team-triggered law, the Lyapunov function V is monotonically nonincreasing along the network evolution The team-triggered law is not amenable to standard discrete-time stability analysis because the agents’ memories evolve discontinuously Memory of agent i is in Si = C0(R; Xi ×

  • j∈N (i)

2Xj)

Cameron Nowzari (UCSD) Team-triggered coordination June 18, 2013 16 / 23

slide-41
SLIDE 41

Convergence guarantees

Proposition

Given a networked system executing the team-triggered law, the Lyapunov function V is monotonically nonincreasing along the network evolution The team-triggered law is not amenable to standard discrete-time stability analysis because the agents’ memories evolve discontinuously Memory of agent i is in Si = C0(R; Xi ×

  • j∈N (i)

2Xj) Instead of analyzing the team-triggered law directly, we construct a discrete-time set-valued map T : S ⇒ S containing the trajectories of the team-triggered law that is “continuous” in an appropriate sense

Cameron Nowzari (UCSD) Team-triggered coordination June 18, 2013 16 / 23

slide-42
SLIDE 42

Convergence guarantees

Proposition

Given a networked system executing the team-triggered law, the Lyapunov function V is monotonically nonincreasing along the network evolution The team-triggered law is not amenable to standard discrete-time stability analysis because the agents’ memories evolve discontinuously Memory of agent i is in Si = C0(R; Xi ×

  • j∈N (i)

2Xj) Instead of analyzing the team-triggered law directly, we construct a discrete-time set-valued map T : S ⇒ S containing the trajectories of the team-triggered law that is “continuous” in an appropriate sense

Cameron Nowzari (UCSD) Team-triggered coordination June 18, 2013 16 / 23

slide-43
SLIDE 43

Convergence guarantees

The set-valued map T captures all possibilities of each agent i receiving no new information receiving information from some neighbor(s) j ∈ N(i)

Cameron Nowzari (UCSD) Team-triggered coordination June 18, 2013 17 / 23

slide-44
SLIDE 44

Convergence guarantees

The set-valued map T captures all possibilities of each agent i receiving no new information receiving information from some neighbor(s) j ∈ N(i)

Lemma

The set-valued map T is closed

Cameron Nowzari (UCSD) Team-triggered coordination June 18, 2013 17 / 23

slide-45
SLIDE 45

Convergence guarantees

The set-valued map T captures all possibilities of each agent i receiving no new information receiving information from some neighbor(s) j ∈ N(i)

Lemma

The set-valued map T is closed

Proposition

Given a networked system executing the team-triggered law, the network trajectories converge to the set D

Cameron Nowzari (UCSD) Team-triggered coordination June 18, 2013 17 / 23

slide-46
SLIDE 46

Simulations

Consider a 4 agent formation control problem with ˙ xi = ui, ui2 ≤ vmax = 50 Desired formation is a rectangle of side lengths 1 and 2

Cameron Nowzari (UCSD) Team-triggered coordination June 18, 2013 18 / 23

slide-47
SLIDE 47

Simulations

Consider a 4 agent formation control problem with ˙ xi = ui, ui2 ≤ vmax = 50 Desired formation is a rectangle of side lengths 1 and 2 Promises generated from static ball-radius rule: U i

j[t](t′) = B(uj(t), 2λvmax) ∩ Uj,

λ = 0.25 The promise Xi

j[t](t′) is then just the reachable set from xj(t) in t′ − t

seconds using controls in U i

j[t](t′)

Cameron Nowzari (UCSD) Team-triggered coordination June 18, 2013 18 / 23

slide-48
SLIDE 48

Simulations

6 7 8 9 10 11 12 13 14 3 4 5 6 7 8 9 10 11 12 13

  • netwothree1
  • netwothree2
  • netwothree3
  • netwothree4

0.5 1 1.5 10 20 30 40 50 60 70 80 90

  • netwothreefour1
  • netwothreefour2

V Time

Cameron Nowzari (UCSD) Team-triggered coordination June 18, 2013 19 / 23

slide-49
SLIDE 49

Simulations

0.5 1 1.5 5 10 15 20 25 30 35 40 45 50

  • netwo1
  • netwo2
  • netwo3
  • netwo4

NS Time

0.5 1 1.5 5 10 15 20 25

  • netwo1
  • netwo2
  • netwo3
  • netwo4

NS Time

0.5 1 1.5 5 10 15 20 25

  • netwo1
  • netwo2
  • netwo3
  • netwo4

NE Time

Cameron Nowzari (UCSD) Team-triggered coordination June 18, 2013 20 / 23

slide-50
SLIDE 50

Simulations

0.2 0.4 0.6 0.8 1 0.24 0.26 0.28 0.3 0.32 0.34 0.36 0.38 0.4 0.42

V (1.5) λ

0.2 0.4 0.6 0.8 1 100 200 300 400 500 600 700 800 900 1000

Ncomm λ

Cameron Nowzari (UCSD) Team-triggered coordination June 18, 2013 21 / 23

slide-51
SLIDE 51

Conclusions

Novel approach for real-time implementation of distributed controllers on networked systems correct, adaptive, distributed, asynchronous same convergence guarantees as synchronous algorithm with perfect information at all times robustness against packet drops and communication noise/delays extensions to adaptive promises

  • submitted to IEEE TAC

Cameron Nowzari (UCSD) Team-triggered coordination June 18, 2013 22 / 23

slide-52
SLIDE 52

Conclusions

Novel approach for real-time implementation of distributed controllers on networked systems correct, adaptive, distributed, asynchronous same convergence guarantees as synchronous algorithm with perfect information at all times robustness against packet drops and communication noise/delays extensions to adaptive promises

  • submitted to IEEE TAC

Future work: analytical characterization of trade-offs standard methods of developing controllers that

  • perate on sets

robustness against disturbances or sensor noise changing topologies

Cameron Nowzari (UCSD) Team-triggered coordination June 18, 2013 22 / 23

slide-53
SLIDE 53

Thank you!

Cameron Nowzari (UCSD) Team-triggered coordination June 18, 2013 23 / 23