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 Washington . . . 1 .
The Traffic Control Loop Image source: Sensys Networks Control Traffic Sensors Devices Network § Can we automate detection of imminent traffic congestion? Control Surveillance § Can we make data- Strategy driven models to “predict” effect of the control strategy? . . . . November 2018 2
Outline § Koopman Operator review § Two Applications: • Early detection of congestion • Capturing effect of signal timings in queue model . . . . November 2018 3
Koopman Operator § Given a nonlinear discrete-time system Evolution of States § Koopman Operator • Linear Evolution of Functions on States • Infinite-dimensional (Observables) . . . . November 2018 4
Approximating an Infinite-Dimensional Operator using Data Suppose the sensor measurements are realizations of the observables . . . . November 2018 5
Dynamic Mode Decomposition DMD: “approximate using proxy matrix ! by learning a locally-linear model” Singular Values Percentage Abrupt decay in singular values If ! is large, high compute cost to perform eigen-decomposition: Use rank truncation in SVD ( #̃ ≤ #) - ' = ) * !) Use projection ! - . . . . November 2018 6
Koopman Operator Applications Instability Analysis Spatio-temporal Information § Eigenvalues § Modes Indicates unstable dynamics Provides relative spatio-temporal information Prediction Learn dynamics to predict future traffic . . . . November 2018 7
The Traffic Control Loop Image source: Sensys Networks Control Traffic Sensors Devices Network § Can we automate detection of imminent traffic congestion? Control Surveillance § Can we make data- Strategy driven models to “predict” effect of the control strategy? . . . . November 2018 8
Outline § Koopman Operator review § Two Applications: • Early detection of congestion • Capturing effect of signal timings in queue model . . . . November 2018 9
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 N (X k ) . . . . November 2018 10
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 N (X k ) . . . . November 2018 11
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 N (X k ) . . . . November 2018 12
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 N (X k ) . . . . November 2018 13
Instability Detection Normal Day Accident Day Indicates growth for 33 minutes (10s step between rolling window ) . . . . November 2018 14
Outline § Koopman Operator review § Two Applications: • Early detection of congestion • Capturing effect of signal timings in queue model . . . . November 2018 16
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? . . . . November 2018 17
Effect of signal timings in queue model Learn + and , using original x k and u k § Do + and , learn a good model? Reconstruct {x 2 ,…, x N } using initial condition x 1 and {u 1 ,…, u N } § What is the effect of a modified phase-split? Reconstruct {x 2 ,…, x N } using initial condition x 1 and modified { u 1 , …, u N } . . . . November 2018 18
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 Queue Plots for Congested Legs Queue Plots for Congested Legs (shorter green time) (longer green time) Legend Blue = Original queues Red = Reconstructed queues using original signal phases Green = Reconstructed queues using modified signal phases . . . . November 2018 19
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? . . . . November 2018 20
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