Queues and Network Control for Urban Traffic Systems Workshop on - - PowerPoint PPT Presentation

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Queues and Network Control for Urban Traffic Systems Workshop on - - PowerPoint PPT Presentation

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Queues and Network Control for Urban Traffic Systems Workshop on Control for Networked Transportation Systems July 8 2019 Ketan Savla ( ksavla@usc.edu ) University of Southern California Ketan Savla (USC) CNTS Workshop July 8 2019 1 / 11

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SLIDE 1

Queues and Network Control for Urban Traffic Systems

Workshop on Control for Networked Transportation Systems July 8 2019

Ketan Savla

(ksavla@usc.edu) University of Southern California

Ketan Savla (USC) CNTS Workshop July 8 2019 1 / 11

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SLIDE 2

Queues and Network Control for Urban Traffic Systems

Workshop on Control for Networked Transportation Systems July 8 2019

Ketan Savla

(ksavla@usc.edu) University of Southern California Thanks to NSF EPCN and DCSD, CALTRANS

Ketan Savla (USC) CNTS Workshop July 8 2019 1 / 11

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SLIDE 3

Overview

Transportation Science Operations Research Control Theory

Ketan Savla (USC) CNTS Workshop July 8 2019 2 / 11

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SLIDE 4

Overview

Transportation Science Operations Research Control Theory Queues Network Control

Symbiosis between transportation and systems sciences

Ketan Savla (USC) CNTS Workshop July 8 2019 2 / 11

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SLIDE 5

Overview

Transportation Science Operations Research Control Theory Queues Network Control DATA

Symbiosis between transportation and systems sciences Tight integration essential for efficient use of data

Ketan Savla (USC) CNTS Workshop July 8 2019 2 / 11

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SLIDE 6

Transportation Queues jobs server

Ketan Savla (USC) CNTS Workshop July 8 2019 3 / 11

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SLIDE 7

Transportation Queues jobs server

mobility on demand

jobs: pickup/delivery requests server: vehicle fleet

signalized intersection

jobs: vehicles server: intersection

freeway (w/ CAVs)

jobs: vehicles server: freeway infrastructure Ketan Savla (USC) CNTS Workshop July 8 2019 3 / 11

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SLIDE 8

Transportation Queues jobs server

mobility on demand

jobs: pickup/delivery requests server: vehicle fleet

signalized intersection

jobs: vehicles server: intersection

freeway (w/ CAVs)

jobs: vehicles server: freeway infrastructure

Service paradigms determined by automation and control

Ketan Savla (USC) CNTS Workshop July 8 2019 3 / 11

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SLIDE 9

Performance Evaluation

λ server

capacity? wait time?

Ketan Savla (USC) CNTS Workshop July 8 2019 4 / 11

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SLIDE 10

Performance Evaluation

λ server

capacity? wait time?

Constant Service Rate λ − c

  • service rate

= queue growth rate

Ketan Savla (USC) CNTS Workshop July 8 2019 4 / 11

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SLIDE 11

Performance Evaluation

λ server

capacity? wait time?

Constant Service Rate λ − c

  • service rate

= queue growth rate capacity = c

Ketan Savla (USC) CNTS Workshop July 8 2019 4 / 11

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SLIDE 12

Performance Evaluation

λ server

capacity? wait time?

Constant Service Rate λ − c

  • service rate

= queue growth rate capacity = c Example: M/M/1

Ketan Savla (USC) CNTS Workshop July 8 2019 4 / 11

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SLIDE 13

Performance Evaluation

λ server

capacity? wait time?

Constant Service Rate λ − c

  • service rate

= queue growth rate capacity = c Example: M/M/1

c ≡ c(queue length)

Ketan Savla (USC) CNTS Workshop July 8 2019 4 / 11

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SLIDE 14

State Dependent Transportation Queues

wait time

λ

capacity

  • Ketan Savla (USC)

CNTS Workshop July 8 2019 5 / 11

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SLIDE 15

State Dependent Transportation Queues

wait time

λ

capacity

  • Ketan Savla (USC)

CNTS Workshop July 8 2019 5 / 11

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SLIDE 16

State Dependent Transportation Queues

wait time

λ

capacity

  • Ketan Savla (USC)

CNTS Workshop July 8 2019 5 / 11

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SLIDE 17

State Dependent Transportation Queues

wait time

λ

capacity

  • verestimate
  • Ketan Savla (USC)

CNTS Workshop July 8 2019 5 / 11

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SLIDE 18

State Dependent Transportation Queues

wait time

λ

capacity

  • Ketan Savla (USC)

CNTS Workshop July 8 2019 5 / 11

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SLIDE 19

State Dependent Transportation Queues

wait time

λ

capacity underestimate

  • Ketan Savla (USC)

CNTS Workshop July 8 2019 5 / 11

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SLIDE 20

State Dependent Transportation Queues

wait time

λ

capacity

spatial queue

Θ(λ/m2)

vs

Θ(λ2/m3)

  • Ketan Savla (USC)

CNTS Workshop July 8 2019 5 / 11

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SLIDE 21

State Dependent Transportation Queues

wait time

λ

capacity

spatial queue

Θ(λ/m2)

vs

Θ(λ2/m3)

vacation queue

  • Link 1

Link 2

100 200 300 400 Time 2 4 6 8 10 Average Queue Length Vacation Queue Webster Model Vacation Queue (Time Average) Uninterrupted Model Akcelik Model

Ketan Savla (USC) CNTS Workshop July 8 2019 5 / 11

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SLIDE 22

State Dependent Transportation Queues

wait time

λ

capacity

spatial queue

Θ(λ/m2)

vs

Θ(λ2/m3)

vacation queue

  • Link 1

Link 2

100 200 300 400 Time 2 4 6 8 10 Average Queue Length Vacation Queue Webster Model Vacation Queue (Time Average) Uninterrupted Model Akcelik Model

processor sharing queue

vs Ketan Savla (USC) CNTS Workshop July 8 2019 5 / 11

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SLIDE 23

Current Notions of Capacity

Traffic Capacity [Highway Capacity Manual]

“. . . maximum number of vehicles that can pass a given point . . . (assuming) no influence from downstream traffic operation . . .”

Ketan Savla (USC) CNTS Workshop July 8 2019 6 / 11

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SLIDE 24

Current Notions of Capacity

Traffic Capacity [Highway Capacity Manual]

“. . . maximum number of vehicles that can pass a given point . . . (assuming) no influence from downstream traffic operation . . .” “. . . rate at which . . . vehicles can traverse an intersection approach . . . assuming that the green signal is available at all times . . .”

Ketan Savla (USC) CNTS Workshop July 8 2019 6 / 11

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SLIDE 25

Current Notions of Capacity

Traffic Capacity [Highway Capacity Manual]

“. . . maximum number of vehicles that can pass a given point . . . (assuming) no influence from downstream traffic operation . . .” “. . . rate at which . . . vehicles can traverse an intersection approach . . . assuming that the green signal is available at all times . . .”

f ≤ c

Ketan Savla (USC) CNTS Workshop July 8 2019 6 / 11

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SLIDE 26

Current Notions of Capacity

Traffic Capacity [Highway Capacity Manual]

“. . . maximum number of vehicles that can pass a given point . . . (assuming) no influence from downstream traffic operation . . .” “. . . rate at which . . . vehicles can traverse an intersection approach . . . assuming that the green signal is available at all times . . .”

f ≤ c

c − f : local robustness

Ketan Savla (USC) CNTS Workshop July 8 2019 6 / 11

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SLIDE 27

Current Notions of Capacity

Traffic Capacity [Highway Capacity Manual]

“. . . maximum number of vehicles that can pass a given point . . . (assuming) no influence from downstream traffic operation . . .” “. . . rate at which . . . vehicles can traverse an intersection approach . . . assuming that the green signal is available at all times . . .”

f ≤ c

c − f : local robustness

Current capacity notions are local

Ketan Savla (USC) CNTS Workshop July 8 2019 6 / 11

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SLIDE 28

Towards Network Capacity

network capacity : ({ci}, physical constraints, control)

Ketan Savla (USC) CNTS Workshop July 8 2019 7 / 11

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SLIDE 29

Dynamical Network Flow λ λout(t)

fi xi

Mass Conservation ˙ x = λ + RT (x)f(x, u)

  • inflow

− f(x, u)

  • utflow

xi : queue on link i R(x) : routing matrix

Ketan Savla (USC) CNTS Workshop July 8 2019 8 / 11

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SLIDE 30

Dynamical Network Flow λ λout(t)

fi xi

Mass Conservation ˙ x = λ + RT (x)f(x, u)

  • inflow

− f(x, u)

  • utflow

xi : queue on link i R(x) : routing matrix

equilibrium x∗: λout(t) = λ existence, stability, and robustness of x∗

Ketan Savla (USC) CNTS Workshop July 8 2019 8 / 11

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SLIDE 31

Distributed Feedback Control

min

u

T J(x(t), u(t)) dt

  • subj. to

˙ x = traffic flow dynamics u ≡ ramp metering, variable speed limit, routing

Ketan Savla (USC) CNTS Workshop July 8 2019 9 / 11

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SLIDE 32

Distributed Feedback Control

min

u

T J(x(t), u(t)) dt

  • subj. to

˙ x = traffic flow dynamics u ≡ ramp metering, variable speed limit, routing

  • pen-loop: u(t)

exact convex relaxation distributed computation

Ketan Savla (USC) CNTS Workshop July 8 2019 9 / 11

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SLIDE 33

Distributed Feedback Control

min

u

T J(x(t), u(t)) dt

  • subj. to

˙ x = traffic flow dynamics u ≡ ramp metering, variable speed limit, routing

  • pen-loop: u(t)

exact convex relaxation distributed computation

feedback: u(x) [ThC02.3]

principled distributed control global computation of u(.)

Ketan Savla (USC) CNTS Workshop July 8 2019 9 / 11

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SLIDE 34

Distributed Feedback Control

min

u

T J(x(t), u(t)) dt

  • subj. to

˙ x = traffic flow dynamics u ≡ ramp metering, variable speed limit, routing

  • pen-loop: u(t)

exact convex relaxation distributed computation

feedback: u(x) [ThC02.3]

principled distributed control global computation of u(.)

Ketan Savla (USC) CNTS Workshop July 8 2019 9 / 11

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SLIDE 35

From State to Output Feedback Control

𝑕2[𝑙] 𝜐1 𝜐2 𝜐3 𝑕 ̃2[𝑙] = ∑ 𝜐𝑗

3 𝑗=1

𝑕2[𝑙] 𝜐1 𝜐2 𝜐3 𝑕 ̃2[𝑙] = ∑ 𝜐𝑗

4 𝑗=1

𝜐4 𝑕2[𝑙] 𝜐1 𝜐2 𝜐3 𝑕 ̃2[𝑙] = ∑ 𝜐𝑗

3 𝑗=1

𝑕2[𝑙] 𝜐1 𝜐2 𝜐3 𝑕 ̃2[𝑙] = ∑ 𝜐𝑗

4 𝑗=1

𝜐4

direct access to x not available y: detector measurement

Ketan Savla (USC) CNTS Workshop July 8 2019 10 / 11

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SLIDE 36

From State to Output Feedback Control

𝑕2[𝑙] 𝜐1 𝜐2 𝜐3 𝑕 ̃2[𝑙] = ∑ 𝜐𝑗

3 𝑗=1

𝑕2[𝑙] 𝜐1 𝜐2 𝜐3 𝑕 ̃2[𝑙] = ∑ 𝜐𝑗

4 𝑗=1

𝜐4 𝑕2[𝑙] 𝜐1 𝜐2 𝜐3 𝑕 ̃2[𝑙] = ∑ 𝜐𝑗

3 𝑗=1

𝑕2[𝑙] 𝜐1 𝜐2 𝜐3 𝑕 ̃2[𝑙] = ∑ 𝜐𝑗

4 𝑗=1

𝜐4

direct access to x not available y: detector measurement “estimator” approach: y → ˆ x → u(ˆ x)

Ketan Savla (USC) CNTS Workshop July 8 2019 10 / 11

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SLIDE 37

From State to Output Feedback Control

𝑕2[𝑙] 𝜐1 𝜐2 𝜐3 𝑕 ̃2[𝑙] = ∑ 𝜐𝑗

3 𝑗=1

𝑕2[𝑙] 𝜐1 𝜐2 𝜐3 𝑕 ̃2[𝑙] = ∑ 𝜐𝑗

4 𝑗=1

𝜐4 𝑕2[𝑙] 𝜐1 𝜐2 𝜐3 𝑕 ̃2[𝑙] = ∑ 𝜐𝑗

3 𝑗=1

𝑕2[𝑙] 𝜐1 𝜐2 𝜐3 𝑕 ̃2[𝑙] = ∑ 𝜐𝑗

4 𝑗=1

𝜐4

direct access to x not available y: detector measurement “estimator” approach: y → ˆ x → u(ˆ x)

  • utput feedback: u(y)

Ketan Savla (USC) CNTS Workshop July 8 2019 10 / 11

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SLIDE 38

From State to Output Feedback Control

𝑕2[𝑙] 𝜐1 𝜐2 𝜐3 𝑕 ̃2[𝑙] = ∑ 𝜐𝑗

3 𝑗=1

𝑕2[𝑙] 𝜐1 𝜐2 𝜐3 𝑕 ̃2[𝑙] = ∑ 𝜐𝑗

4 𝑗=1

𝜐4 𝑕2[𝑙] 𝜐1 𝜐2 𝜐3 𝑕 ̃2[𝑙] = ∑ 𝜐𝑗

3 𝑗=1

𝑕2[𝑙] 𝜐1 𝜐2 𝜐3 𝑕 ̃2[𝑙] = ∑ 𝜐𝑗

4 𝑗=1

𝜐4

direct access to x not available y: detector measurement “estimator” approach: y → ˆ x → u(ˆ x)

  • utput feedback: u(y)
  • ptimal output feedback traffic signal control

Ketan Savla (USC) CNTS Workshop July 8 2019 10 / 11

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SLIDE 39

From State to Output Feedback Control

𝑕2[𝑙] 𝜐1 𝜐2 𝜐3 𝑕 ̃2[𝑙] = ∑ 𝜐𝑗

3 𝑗=1

𝑕2[𝑙] 𝜐1 𝜐2 𝜐3 𝑕 ̃2[𝑙] = ∑ 𝜐𝑗

4 𝑗=1

𝜐4 𝑕2[𝑙] 𝜐1 𝜐2 𝜐3 𝑕 ̃2[𝑙] = ∑ 𝜐𝑗

3 𝑗=1

𝑕2[𝑙] 𝜐1 𝜐2 𝜐3 𝑕 ̃2[𝑙] = ∑ 𝜐𝑗

4 𝑗=1

𝜐4

direct access to x not available y: detector measurement “estimator” approach: y → ˆ x → u(ˆ x)

  • utput feedback: u(y)
  • ptimal output feedback traffic signal control

pilot test: ∼ 20% improvement w.r.t. incumbent

Ketan Savla (USC) CNTS Workshop July 8 2019 10 / 11

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SLIDE 40

From State to Output Feedback Control

𝑕2[𝑙] 𝜐1 𝜐2 𝜐3 𝑕 ̃2[𝑙] = ∑ 𝜐𝑗

3 𝑗=1

𝑕2[𝑙] 𝜐1 𝜐2 𝜐3 𝑕 ̃2[𝑙] = ∑ 𝜐𝑗

4 𝑗=1

𝜐4 𝑕2[𝑙] 𝜐1 𝜐2 𝜐3 𝑕 ̃2[𝑙] = ∑ 𝜐𝑗

3 𝑗=1

𝑕2[𝑙] 𝜐1 𝜐2 𝜐3 𝑕 ̃2[𝑙] = ∑ 𝜐𝑗

4 𝑗=1

𝜐4

direct access to x not available y: detector measurement “estimator” approach: y → ˆ x → u(ˆ x)

  • utput feedback: u(y)
  • ptimal output feedback traffic signal control

pilot test: ∼ 20% improvement w.r.t. incumbent

Ketan Savla (USC) CNTS Workshop July 8 2019 10 / 11

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SLIDE 41

Concluding Remarks

Transportation Science Operations Research Control Theory Queues Network Control DATA Ketan Savla (USC) CNTS Workshop July 8 2019 11 / 11

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SLIDE 42

Concluding Remarks

Transportation Science Operations Research Control Theory Queues Network Control DATA

state-dependent queues distributed/output feedback control for nonlinear systems . . .

Ketan Savla (USC) CNTS Workshop July 8 2019 11 / 11