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University of Pennsylvania

Demonstrations

University of Pennsylvania

Demonstrations

Alur, Kumar, Lee, Pappas

Rafael Fierro

Yerang Hur

Franjo Ivancic

PK Mishra

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CHARONToolset

Analysis ToolsAnalysis Tools

AbstractionExploit hierarchy, modularitySimplified model for controller design or analysis

Reachability computationsToolboxes for reachability computationsAccurate event detection for hybrid systems

Symbolic toolbox for reachability analysis of nilpotent systems

Accurate event detection for power train and V2V controller

Abstractions and approximations for the power train

Hierarchy for the power train and V2V controller

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Powertrain and V2V OEPsPowertrain and V2V OEPs

V2V Abstraction The input variable u(t) is the desired acceleration sent by the V2V

controller The actual throttle position and brake commands are computed

using backstepping from u(t)

Powertrain Approximation Engine has physical limitation and the vehicle cannot change its state (velocity and acceleration) instantaneously. Acceleration mode

The engine, transmission and throttle systems are approximated by a linear system in series with a saturation element

Deceleration mode The engine, transmission and braking systems are approximated by a

saturation element in series with a first-order system

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Approximate Hybrid ModelApproximate Hybrid Model

on brake_sw if )(

offbrake_sw if )(

1

1

sin1

max

max

2

/

bU

tU

bbb

b

ttt

t

airrr

dragrrBt

usat

usat

u

uku

uku

MgvcFuM

v

FFFFMa

b

t

P P'

Approximate model for Engine and Transmission

1 sk

t

t tu

Tttt kuu maxmin

2 ,)(ˆmin

1 sk

b

b bu

Tbbb kuu maxmin

2 ,)(ˆmin

5

mode Acceleration mode Brake

= 90 = 10

= 0

= 10

= 40

= 20

= 10

Comparison: Approximate Model and OEP ModelComparison: Approximate Model and OEP Model

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CHARON Visual EditorCHARON Visual Editor

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CHARON Visual EditorCHARON Visual Editor

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CHARON Visual EditorCHARON Visual Editor

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CHARON Visual EditorCHARON Visual Editor

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CHARON Text EditorCHARON Text Editor

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Two Keys to AnalysisTwo Keys to Analysis

ModularityEfficient and accurate Integration of sub modes, agents at different time scales

Detection of Events Accurate detection of constraint violations or transitions

Applications Reachability Analysis Simulation

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Event DetectionEvent Detection

xfx dynamics

0xgoutput

input hdk

dt

Given:

We re-parameterize time by controlling the integration step size:

dk

dtgL

dk

dt

dt

dgf

output dynamics

Using feedback linearization we select our “speed” (step-size) along the integral curves to converge to the event surface

Event !x(t)

g(x)

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RequiemRequiem

Reachability of the system

A is nilpotent g(x, u) can be any nonlinear function

Functions Forward and Backward reachable sets for continuous and discrete

systems. Forward and Backward reachable sets for timed continuous systems. Forward and Backward reachable sets continuous systems under

invariants. Forward and Backward reachable sets for discrete systems with guards. Parametric forward and backward reachable set for continuous

systems.http://www.seas.upenn.edu/hybrid/requiem.html

0,

uxg

BuAxx