Koopman Operator Approach for Instability Detection and Mitigation - - PowerPoint PPT Presentation

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Koopman Operator Approach for Instability Detection and Mitigation - - PowerPoint PPT Presentation

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Koopman Operator Approach for Instability Detection and Mitigation IEEE International Conference on Intelligent Transportation Systems Esther Ling*, Lillian Ratliff**, Samuel Coogan* * Georgia Institute of Technology ** University of

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Koopman Operator Approach for Instability Detection and Mitigation

1 Esther Ling*, Lillian Ratliff**, Samuel Coogan* * Georgia Institute of Technology ** University of Washington IEEE International Conference on Intelligent Transportation Systems

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The Traffic Control Loop

§ Can we automate detection of imminent traffic congestion? § Can we make data- driven models to “predict” effect of the control strategy?

2 November 2018

Image source: Sensys Networks

Control Devices Traffic Network Sensors Surveillance Control Strategy

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Outline

§ Koopman Operator review § Two Applications:

  • Early detection of congestion
  • Capturing effect of signal timings in queue model

3 November 2018

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Koopman Operator

§ Given a nonlinear discrete-time system § Koopman Operator

  • Linear
  • Infinite-dimensional

4 November 2018

Evolution of States Evolution of Functions on States (Observables)

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Approximating an Infinite-Dimensional Operator using Data

5 November 2018

Suppose the sensor measurements are realizations of the observables

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Dynamic Mode Decomposition

DMD: “approximate using proxy matrix ! by learning a locally-linear model”

6 November 2018

If ! is large, high compute cost to perform eigen-decomposition:

  • Use rank truncation in SVD (#̃ ≤ #)
  • Use projection !

' = )*!)

Abrupt decay in singular values

Singular Values Percentage

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Koopman Operator Applications

7 November 2018

Instability Analysis § Eigenvalues

Indicates unstable dynamics

Spatio-temporal Information § Modes

Provides relative spatio-temporal information

Prediction

Learn dynamics to predict future traffic

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. . . .

The Traffic Control Loop

§ Can we automate detection of imminent traffic congestion? § Can we make data- driven models to “predict” effect of the control strategy?

8 November 2018

Image source: Sensys Networks

Control Devices Traffic Network Sensors Surveillance Control Strategy

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. . . .

Outline

§ Koopman Operator review § Two Applications:

  • Early detection of congestion
  • Capturing effect of signal timings in queue model

9 November 2018

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Instability Detection

§ Local instability analysis to detect congestion § How local? Specify the range of data to include, N § Learn dynamics (!) in a rolling window § Keep track of consecutive unstable eigenvalues

10 November 2018

(Xk)

N

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Instability Detection

§ Local instability analysis to detect congestion § How local? Specify the range of data to include, N § Learn dynamics (!) in a rolling window § Keep track of consecutive unstable eigenvalues

11 November 2018

(Xk)

N

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Instability Detection

§ Local instability analysis to detect congestion § How local? Specify the range of data to include, N § Learn dynamics (!) in a rolling window § Keep track of consecutive unstable eigenvalues

12 November 2018

(Xk)

N

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Instability Detection

§ Local instability analysis to detect congestion § How local? Specify the range of data to include, N § Learn dynamics (!) in a rolling window § Keep track of consecutive unstable eigenvalues

13 November 2018

(Xk)

N

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Instability Detection

14 November 2018

Normal Day Accident Day

Indicates growth for 33 minutes (10s step between rolling window )

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Outline

§ Koopman Operator review § Two Applications:

  • Early detection of congestion
  • Capturing effect of signal timings in queue model

16 November 2018

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Effect of signal timings in queue model

§ Notice that the queue starts to clear at 3.30pm § Scheduled change in timing plan at 3.30pm § Did the extended green time for congested leg play a role?

17 November 2018

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Effect of signal timings in queue model

Learn + and , using

  • riginal xk and uk

§ Do + and , learn a good model? Reconstruct {x2,…, xN} using initial condition x1 and {u1,…, uN} § What is the effect of a modified phase-split? Reconstruct {x2,…, xN} using initial condition x1 and modified {u1,…, uN}

18 November 2018

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Effect of signal timings in queue model

§ Longer green times for congested leg ⇒ faster queue mitigation § Can visualize effect on all 4 legs with one model

19 November 2018

Queue Plots for Congested Legs (longer green time) Queue Plots for Congested Legs (shorter green time)

Legend Blue = Original queues Red = Reconstructed queues using original signal phases Green = Reconstructed queues using modified signal phases

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Summary and Q&A

§ Koopman Operator framework for data-driven modeling § Applications:

  • Automated early detection of traffic congestion
  • Modeling queue dynamics with signal phases to

anticipate effect of modified phase-splits

§ Future Directions:

  • What is an adequate amount of green time extension?
  • Model is currently intersection-level. Can this be

extended to include a network-level model?

20 November 2018