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WORKSHOP ON HYBRID AND EMBEDDED SYSTEMS Hybrid Systems Modeling, Analysis, Control Datta Godbole, John Lygeros, Claire Tomlin, Gerardo Lafferiere, George Pappas, John Koo Jianghai Hu, Rene Vidal, Yi Ma, Shawn Shaffert, Jun Zhang, Slobodan Simic, Kalle Johansson, Maria Prandini David Shim, Jin Kim, Peter Ray, Omid Shakernia, Cedric Ma, Hoam Chung, Travis Pynn and Perry Kavros (with the interference of) Shankar Sastry

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WORKSHOP ON HYBRID AND EMBEDDED SYSTEMS

Hybrid Systems Modeling, Analysis, Control

Datta Godbole, John Lygeros, Claire Tomlin, Gerardo Lafferiere, George Pappas, John Koo

Jianghai Hu, Rene Vidal, Yi Ma, Shawn Shaffert, Jun Zhang,

Slobodan Simic, Kalle Johansson, Maria PrandiniDavid Shim, Jin Kim, Peter Ray, Omid Shakernia, Cedric

Ma, Hoam Chung, Travis Pynn and Perry Kavros (with the interference of) Shankar Sastry

Page 2: Hybrid Systems Modeling, Analysis, Control - mae.cuhk… … · WORKSHOP ON HYBRID AND EMBEDDED SYSTEMS Hybrid Systems Modeling, Analysis, Control Datta Godbole, John Lygeros, Claire

Workshop on Hybrid and Embedded Systems 2006

What Are Hybrid Systems?

Dynamical systems with interacting continuousand discretedynamics

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Workshop on Hybrid and Embedded Systems 2006

Why Hybrid Systems?

• Modeling abstraction of– Continuous systems with phased operation (e.g. walking robots,

mechanical systems with collisions, circuits with diodes)– Continuous systems controlled by discrete inputs (e.g. switches,

valves, digital computers)– Coordinating processes (multi-agent systems)

• Important in applications– Hardware verification/CAD, real time software– Manufacturing, chemical process control,– communication networks, multimedia

• Large scale, multi-agent systems– Automated Highway Systems (AHS)– Air Traffic Management Systems (ATM)– Uninhabited Aerial Vehicles (UAV), Power Networks

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Workshop on Hybrid and Embedded Systems 2006

Control Challenges

• Large number of semiautonomous agents

• Coordinate to– Make efficient use of common resource

– Achieve a common goal

• Individual agents have various modes of operation

• Agents optimize locally, coordinate to resolve conflicts

• System architecture is hierarchical and distributed

• Safety critical systems

Challenge: Develop models, analysis, and synthesis tools for designing and verifying the safety of multi-agent systems

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Workshop on Hybrid and Embedded Systems 2006

Proposed Framework

Control TheoryControl of individual agentsContinuous modelsDifferential equations

Computer ScienceModels of computationCommunication modelsDiscrete event systems

Hybrid Systems

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Workshop on Hybrid and Embedded Systems 2006

Different Approaches

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Workshop on Hybrid and Embedded Systems 2006

Research Issues

• Modeling & Simulation– Control: classify discrete phenomena, existence and uniquenessof execution, Zeno

[Branicky, Brockett, van der Schaft, Astrom, Sastry-Egerstedt-Johansson-Lygeros-Zhang, Ames]

– Computer Science: composition and abstraction operations [Sztipanovits, Karsai, Alur-Henzinger, Lynch, Sifakis,Varaiya]

• Analysis & Verification– Control: stability, Lyapunov techniques [Antsaklis, Branicky, Michel, Sastry-Simic], LMI

techniques [Johansson-Rantzer], optimal control [Branicky, Sussmann, Caines]– Computer Science:Algorithmic [Alur-Henzinger, Sifakis, Pappas-Lafferrier-Sastry] or

deductive methods [Lynch, Manna, Pnuelli]

• Controller Synthesis– Control: optimal control [Branicky-Mitter, Bensoussan-Menaldi], hierarchical control

[Caines, Pappas-Sastry], supervisory control [Lemmon-Antsaklis], model predictive techniques [Morari Bemporad], safety specifications[Lygeros-Tomlin-Sastry]

– Computer Science: algorithmic synthesis [Maler, Pnueli, Asarin, Wong-Toi]

• Observability and Diagnosability– Control: observers[Bemporad, Koutsoukos, Vidal, Ma]– Computer Science[ Biswas, Karsai, Zhao]

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Workshop on Hybrid and Embedded Systems 2006

Automated Highway Systems

• Goal– Increase highway throughput

– Same highway infrastructure

– Same level of safety

– Same level of passenger comfort

• Introduce automation– Partial: driver assistance, intelligent cruise control, warning system

– Full: individual vehicles, mixed traffic, platooning

• Complex problem– Technological issues (is it possible with current technology)

– Social/Political issues (insurance and legal issues, equality)

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Workshop on Hybrid and Embedded Systems 2006

Safety-Throughput Tradeoff

• Contradictory demands– Safety: vehicles far and moving slowly

– Throughput: vehicles close and moving fast

• Proposed compromise– Allow low relative velocity collisions

– In emergency situations

• Two possible safe arrangements– Large spacing (leadermode)

– Small spacing (follower mode)

• Platooning concept

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Workshop on Hybrid and Embedded Systems 2006

Control Hierarchy

• Implementation requires automatic control

• Control hierarchy proposed in [Varaiya 93]– Regulation layer: braking, acceleration and steering

– Coordination layer: maneuvers implemented by communication protocols

– Link layer: flow control, lane assignment

– Network layer: routing

• Hybrid phenomena appear throughout– Switching controllers for regulation

– Switching between maneuvers

– Lane and maneuver assignment

– Degraded modes of operation

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Workshop on Hybrid and Embedded Systems 2006

Automated Platoons on I-15

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Workshop on Hybrid and Embedded Systems 2006

Air Traffic Management Systems

• Studied by NEXTOR funded by NASA/FAA• Increased demandfor air travel

– Higher aircraft density/operator workload– Severe degradation in adverse conditions– High business volume

• Technological advances: Guidance, Navigation & Control– GPS, advanced avionics, on-board electronics– Communication capabilities– Air Traffic Controller (ATC) computation capabilities

• Greater demand and possibilities for automation– Operator assistance– Decentralization– Free / Flexible flight

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Workshop on Hybrid and Embedded Systems 2006

Current ATM System

TRACON

TRACON

CENTER ACENTER B

20 Centers, 185 TRACONs, 400 Airport TowersSize of TRACON: 30-50 miles radius, 11,000ftCenters/TRACONs are subdivided to sectorsApproximately 1200 fixed VOR nodesSeparation Standards

Inside TRACON : 3 miles, 1,000 ftBelow 29,000 ft : 5 miles, 1,000ftAbove 29,000 ft : 5 miles, 2,000ft

VOR

SUA

GATES

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Workshop on Hybrid and Embedded Systems 2006

Current ATM System Limitations

• Inefficient Airspace Utilization– Nondirect, wind independent, nonoptimal routes

• Centralized System Architecture– Increased controller workload resulting in holding patterns

• Obsolete Technology and Communications– Frequent computer and display failures

• Limitations amplified in oceanic airspace– Separation standards in oceanic airspace are very conservative

In the presence of the predicted soaring demand for airtravel, the above problems will be greatly amplified leading to both safety and performance degradation in the future

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Workshop on Hybrid and Embedded Systems 2006

A Future ATM Concept

TRACON

TRACON

CENTER ACENTER B

• Free Flightfrom TRACON to TRACON– Increases airspace utilization

• Tools for optimizing TRACON capacity– Increases terminal area capacity and throughput

• Decentralized Conflict Prediction & Resolution– Reduces controller workload and increases safety

PROTECTED ZONE

ALERT ZONE

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Workshop on Hybrid and Embedded Systems 2006

Hybrid Systems in ATM

• Automation requires interaction between – Hardware(aircraft, communication devices, sensors, computers)

– Software(communication protocols, autopilots)

– Operators(pilots, air traffic controllers, airline dispatchers)

• Interaction is hybrid– Mode switching at the autopilot level

– Coordination for conflict resolution

– Scheduling at the ATC level

– Degraded operation

• Requirement for formal design and analysis techniques– Safety critical system

– Large scale system

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Workshop on Hybrid and Embedded Systems 2006

Control Hierarchy

• Flight Management System (FMS)– Regulation & trajectory tracking

– Trajectory planning

– Tactical planning

• Strategic planning– Decentralized conflict detection

and resolution

– Coordination, through communication protocols

• Air Traffic Control– Scheduling

– Global conflict detection and resolution

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Workshop on Hybrid and Embedded Systems 2006

Hybrid Research Issues

• Hierarchy design

• FMS level– Mode switching

– Aerodynamic envelope protection

• Strategic level– Design of conflict resolution maneuvers

– Implementation by communication protocols

• ATC level– Scheduling algorithms (e.g. for take-offs and landings)

– Global conflict resolution algorithms

• Software verification

• Probabilistic analysis and degraded modes of operation

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Workshop on Hybrid and Embedded Systems 2006

Other Applications

• Uninhabited Aerial Vehicles (UAV)– Automated aerial vehicles (airplanes and/or helicopters)

– Coordinate for search and rescue, or seek and destroymissions

– Control hierarchy similar to ATM

– Mode switching, discrete coordination, flight envelope protection

• Power Electronic Building Blocks (PEBB)– Power electronics, with sensing, control, communication

– Improve power network efficiency and reliability for utilities, hybrid electric vehicle, universal power ships

– Control hierarchy: load balancing/shedding, network stabilization, pulse width modulation

– Hybrid phenomena: modulation, input characteristic switching, scheduling

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Workshop on Hybrid and Embedded Systems 2006

TCP/IPTCP/IP

Wireless LAN

GROUNDSTATIONVIRTUAL COCKPIT

GRAPHICALEMMULATION

WIRELESSHUB

UAV Laboratory Configuration

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Workshop on Hybrid and Embedded Systems 2006

Motivation

•• GoalGoal

–– Design a multiDesign a multi--agent multiagent multi--modal control system for modal control system for Unmanned Aerial Vehicles (Unmanned Aerial Vehicles (UAVsUAVs))

•• Intelligent coordination among agentsIntelligent coordination among agents

•• Rapid adaptation to changing environmentsRapid adaptation to changing environments

•• Interaction of models of operationInteraction of models of operation

–– Guarantee Guarantee

•• Safety Safety

•• PerformancePerformance

•• Fault toleranceFault tolerance

•• Mission completionMission completion

Conflict ResolutionConflict ResolutionCollision AvoidanceCollision AvoidanceEnvelope ProtectionEnvelope ProtectionTracking ErrorTracking Error

Fuel ConsumptionFuel ConsumptionResponse TimeResponse TimeSensor FailureSensor Failure

Actuator FailureActuator FailurePath FollowingPath FollowingObject SearchingObject SearchingPursuitPursuit--EvasionEvasion

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Workshop on Hybrid and Embedded Systems 2006

Hierarchical Hybrid Systems

•• Envelope Protecting ModeEnvelope Protecting Mode

•• Normal Flight ModeNormal Flight Mode

SafetyInvariant��

LivenessReachability

Tactical Planner

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Workshop on Hybrid and Embedded Systems 2006

The Legacy of Success in UAV Research at BErkeleyAeRobotics

• Architecture for multi-level rotorcraft UAVs 1996- to date• Pursuit-evasion games 2000- to date• Autonomous landing using vision on pitching decks 2001- to date• Multi-target tracking 2001- to date• Formation flying and formation change 2002

Page 24: Hybrid Systems Modeling, Analysis, Control - mae.cuhk… … · WORKSHOP ON HYBRID AND EMBEDDED SYSTEMS Hybrid Systems Modeling, Analysis, Control Datta Godbole, John Lygeros, Claire

Workshop on Hybrid and Embedded Systems 2006

Berkeley BEAR Team Fleet Line-up

Ursa Minor 3 (1999~March 2000)Basic navigation&control system, algorithm, software development and test platform

Ursa Magna 1,2 (June 1999~present)Advanced navigation&control algorithm development platformMulti-agent scenarios, formation flight,Vision-based landing

Ursa Maxima 2 (July 2000~present)High-payload platform for Multi-agent scenarios, formation flight

Ursa Major 1 (Nov. 2002 ~)Low-cost, high-payload platformAggressive Maneuver, Vision-based landingMulti-agent scenarios, Model-predictive control

Page 25: Hybrid Systems Modeling, Analysis, Control - mae.cuhk… … · WORKSHOP ON HYBRID AND EMBEDDED SYSTEMS Hybrid Systems Modeling, Analysis, Control Datta Godbole, John Lygeros, Claire

Workshop on Hybrid and Embedded Systems 2006

Newer Platforms 2003-04

Page 26: Hybrid Systems Modeling, Analysis, Control - mae.cuhk… … · WORKSHOP ON HYBRID AND EMBEDDED SYSTEMS Hybrid Systems Modeling, Analysis, Control Datta Godbole, John Lygeros, Claire

Workshop on Hybrid and Embedded Systems 2006

Waypoint Navigation

Helicopter Mode transition

ForwardFlight

Take-off

Land

Hover

Ascend/Descend

PirouetteBank-to-turn

Sideslip

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Workshop on Hybrid and Embedded Systems 2006

Vehicle Control Language

Objective

Develop an abstract and unified UAV mission control language environment

Features

• Mission-independent • Executed as batch or interactive mode• Seamlessly integrated with existing hierarchy• Can be integrated with graphic interface via automatic

code generator

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Workshop on Hybrid and Embedded Systems 2006

Waypoint Navigation using VCL (Jan 29, 2001)

1m

1m

Page 29: Hybrid Systems Modeling, Analysis, Control - mae.cuhk… … · WORKSHOP ON HYBRID AND EMBEDDED SYSTEMS Hybrid Systems Modeling, Analysis, Control Datta Godbole, John Lygeros, Claire

Workshop on Hybrid and Embedded Systems 2006

Pursuit-Evasion Game Experiment June 2001

PEG with four UGVs

• Global-Max pursuit policy• Simulated camera view (radius 7.5m with 50degree conic view)

• Pursuer=0.3m/s Evader=0.5m/s MAX

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Workshop on Hybrid and Embedded Systems 2006

Experimental Results: Pursuit Evasion Games with 4UGVs and 1 UAV (Spring’ 01)

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Workshop on Hybrid and Embedded Systems 2006

pirouette

maneuver2maneuver1maneuver3

Nose-inDuring circling

Heading kept the same

•Any variation of the following maneuvers in x-y direction •Any combination of the following maneuvers

Σετ οφ Μανυεϖερσ φορ δεµονστρατινγ ΡΛ

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Workshop on Hybrid and Embedded Systems 2006

Advanced Maneuver Experiments

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Workshop on Hybrid and Embedded Systems 2006

Video tape of Maneuvers

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Workshop on Hybrid and Embedded Systems 2006

Formation Flying Experiment: 2 real UAVs and 7 virtual UAVs Nov 2002

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Workshop on Hybrid and Embedded Systems 2006

Formation Flying Visualized

Animation by A. Pant and X. Xiao

Withoutleaderinfo

Withleaderinfo

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Workshop on Hybrid and Embedded Systems 2006

Tracking Pitching Target (Nov 02)

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Workshop on Hybrid and Embedded Systems 2006

Talk Outline

• Motivating Applications– Automated Highway Systems– Air Traffic Management Systems– Unmanned Aerial Vehicles

• Modeling– Basic formalism– Existence uniqueness

• Controller synthesis– Safety specifications– Applications to ATM and AHS

• Analysis– Bisimulations of transition systems– O-minimal and linear hybrid systems

• Conclusions & Future Research

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Workshop on Hybrid and Embedded Systems 2006

Hybrid Automata

• Hybrid Automaton

– State space

– Input space

– Initial states

– Vector field

– Invariant set

– Transition relation

• Remarks:– countable,

– State

– Can add outputs, etc. (not needed here)

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Page 39: Hybrid Systems Modeling, Analysis, Control - mae.cuhk… … · WORKSHOP ON HYBRID AND EMBEDDED SYSTEMS Hybrid Systems Modeling, Analysis, Control Datta Godbole, John Lygeros, Claire

Workshop on Hybrid and Embedded Systems 2006

Executions

• Hybrid time trajectory, , finite or infinite with

• Execution with and

– Initial Condition:

– Discrete Evolution:

– Continuous Evolution: over , continuous, piecewise

continuous, and

• Remarks:– x, v not function, multiple transitions possible

– q constant along continuous evolution

– Can study existence uniqueness

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Page 40: Hybrid Systems Modeling, Analysis, Control - mae.cuhk… … · WORKSHOP ON HYBRID AND EMBEDDED SYSTEMS Hybrid Systems Modeling, Analysis, Control Datta Godbole, John Lygeros, Claire

Workshop on Hybrid and Embedded Systems 2006

Controller Synthesis: Example

• 2D conflict resolution

• Ensure aircraft remain more than 5nmi from each other

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Workshop on Hybrid and Embedded Systems 2006

Hybrid Automaton Specification

• Discrete input variable determines maneuver initiation

• Safety specification

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Workshop on Hybrid and Embedded Systems 2006

More Abstractly ...

• Consider plant hybrid automaton, inputs partitioned to:– Controls, U

– Disturbances, D

• Controls specified by “us”

• Disturbances specified by the “environment”– Unmodeled dynamics

– Noise, reference signals

– Actions of other agents

• Memoryless controlleris a map

• The closed loop executionsare� � ��

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Workshop on Hybrid and Embedded Systems 2006

Controller Synthesis Problem

• Given H and find g such that

• A set is controlled invariantif there exists a controller such that all executions starting in remain in

Proposition:The synthesis problem can be solved iff there exists a unique maximal controlled invariant set with

• Seek maximal controlled invariant sets & (least restrictive) controllers that render them invariant

• Proposed solution: treat the synthesis problem as a non-cooperative gamebetween the controland the disturbance

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Workshop on Hybrid and Embedded Systems 2006

Gaming Synthesis Procedure

• Discrete Systems: games on graphs, Bellman equation• Continuous Systems: pursuit-evasion games, Isaacs PDE• Hybrid Systems: for define

– states that can be forced to jump to for some– states that may jump out of for some – states that whatever does can be

continuously driven to avoiding by – Initialization:

while do

end

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Workshop on Hybrid and Embedded Systems 2006

Proposition: If the algorithm terminates, the fixed point isthe maximal controlled invariant subset of F

Proposition: If the algorithm terminates, the fixed point isthe maximal controlled invariant subset of F

Algorithm Interpretation

X

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Workshop on Hybrid and Embedded Systems 2006

Computation

• One needs to compute , and

• Computation of the Pre is straight forward (conceptually!): invert the transition relation

• Computation of Reach through a pair of coupled Hamilton-Jacobi partial differential equations

• Semi-decidable if Pre, Reach are computable

• Decidableif hybrid automata are rectangular, initialized.

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Workshop on Hybrid and Embedded Systems 2006

Application: Control of Automated Highway Systems

• Design of vehicle controllers & performance estimation

• Two concepts– platooning & individual vehicles

Network

Link

Coordination

Regulation •Lane keeping•Vehicle following

•Maneuver selection•inter-vehicle comm

•Dynamic routing

•Flow optimization Entry

Exit

LaneChange

PlatoonFollowing

Join

Split

Speed,vehiclefollowing

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Workshop on Hybrid and Embedded Systems 2006

Vehicle Following & Lane Changing

• Control actions: (vehicle i)-- braking, lane change

• Disturbances: (generated by neighboring vehicles)-- deceleration of the preceding vehicle

-- preceding vehicle colliding with the vehicle ahead of it

-- lane change resulting in a different preceding vehicles

-- appearance of an obstacle in front

• Operational conditions:– state of vehicle i with respect to traffic

i

j

i-1 i-2

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Workshop on Hybrid and Embedded Systems 2006

Game Theoretic Formulation

• Requirements– Safety (no collision)

– Passenger Comfort

– Efficiency• trajectory tracking (depends on the maneuver)

• Safe controller (J1): Solve a two-person zero-sum game– saddle solution (u1*,d1*) given by

• Both vehicles i and i-1 applying maximum braking

• Both collisions occur at T=0 and with maximum impact

J x u d x t J Ct

10

03 1 1 0( , , ) i n f ( ) ;= − ≤ =

J x u d u t J C m st

20

02 2

32 5( , , ) s u p | ( ) | ; .= ≤ =≥

∀ ∈ ∀ ∈ ≤ ≤u U d D J x u d J x u d J x u d, , ( , , ) ( , , ) ( , , )* * * * * 10

1 10

1 1 10

1

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Workshop on Hybrid and Embedded Systems 2006

Safe Vehicle Following Controller

• Partition the state space into safe & unsafe sets

0min,3

04

02

01 ),,(: xxxxS a

Design comfortable andefficient controllers inthe interior•IEEE TVT 11/94

Safe set characterizationalso provides sufficientconditions for lane change•CDC 97, CDC98

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Workshop on Hybrid and Embedded Systems 2006

Automated Highway System Safety

• Theorem 1: (Individual vehicle based AHS) – An individual vehicle based AHS can be designed to produce no inter-

vehicle collisions, – moreover disturbances attenuate along the vehicle string.

• Theorem 2: (Platoon based AHS)– Assuming that platoon follower operation does not result in any

collisions even with a possible inter-platoon collision during join/split, a platoon based AHS can be safe under low relative velocity collision criterion.

• References– Lygeros, Godbole, Sastry, IEEE TAC, April 1998– Godbole, Lygeros, IEEE TVT, Nov. 1994

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Workshop on Hybrid and Embedded Systems 2006

Example: Aircraft Collision Avoidance

Two identical aircraft at fixed altitude & speed:

−−

++−==

ud

uxv

uyvv

duy

x

dt

d ψψ

ψsin

cos

),,(xf

‘evader’ (control) ‘pursuer’ (disturbance)

x

y

uv

d

v

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Workshop on Hybrid and Embedded Systems 2006

Continuous Reachable Set

x

y

[Mitchell, Bayen, Tomlin 2001][Tomlin, Lygeros, Sastry 2000]

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Workshop on Hybrid and Embedded Systems 2006

Fast Wavefront Approximation Methods (Tomlin, Mitchell)

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Workshop on Hybrid and Embedded Systems 2006

Visualization of Unsafe Set:Mitchell-Tomlin

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Workshop on Hybrid and Embedded Systems 2006

Talk Outline

• Motivating Applications– Automated Highway Systems– Air Traffic Management Systems– Unmanned Aerial Vehicles

• Modeling– Basic formalism– Existence uniqueness

• Controller synthesis– Safety specifications– Applications to ATM and AHS

• Analysis and Computability– Bisimulations of transition systems– O-minimal and linear hybrid systems

• Conclusions & Future Research

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Workshop on Hybrid and Embedded Systems 2006

Transition Systems

• Transition System

• Define for

• Given equivalence relation define

$ � �%����� %&� %��

' � �� � � %

�� '��� � �� � % � �( � � "�� ��' (

�� % �%

$) �� �%) ������� %&) �� %�) ��

• A ~ block is a union of equivalence classes

% &

% �

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Workshop on Hybrid and Embedded Systems 2006

Bisimulations of Transition Systems

A partition ~ is a bisimulation iff– are ~ blocks

– For all and all ~ blocks is a ~ block

A partition ~ is a bisimulation iff– are ~ blocks

– For all and all ~ blocks is a ~ block

• Why are bisimulations important?

%&�%�

' � � �� �� '���

• Alternatively, for ��� �� � %) �� �� � �� '���� � � *� ��

% &

% �

�� '�%��

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Workshop on Hybrid and Embedded Systems 2006

Bisimulation Algorithm

initialize :

while such that

define

refine

%) �� �%&� %�� % � �%& �%�

���� �� � %) �� ' � �

����� � �� '���� ������ � �� � �� '����� �� � �� � �� '����

%) �� �%) � � ���� � ���� ��

% &

%�

�� '�%��

• If algorithm terminates, we obtain a finite bisimulation

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Workshop on Hybrid and Embedded Systems 2006

Transitions of Hybrid Systems

• Transitions of hybrid systems are concatenations of– Discrete transitions

– Continuous transitions

• Because of initialized transitions

• If invariants, guards, resets are ~ blocks, then no refinement is necessary due to discrete transitions

��� �� � ��� ��

��� �� �� ��� ��

�� ��� �� ��+� � ��

�� � �� � � �� ��� � ���

� � ��� � � � ��� � � �

+� � �� �

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Workshop on Hybrid and Embedded Systems 2006

Bisimulation Algorithm

Initialize

for each

while such that

define

refine

end while; end for

) �� �� ��

���� �� � �� ����� � �� ����� ������ � �� � �� ������ �� � �� � �� �����

�� � ��� � ���� � ���� ��

• Algorithm must terminate for each discrete location

• Refinement process is therefore decoupled

• Consider for each discrete state the finite collection of sets

• Let be a partition compatible with

�� � ������ �&��� ���� � �+� �� �� �� � ,

�� ��

� � �

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Workshop on Hybrid and Embedded Systems 2006

• Decidability requires the bisimulation algorithm to – Terminate in finite number of steps and

– Be computable

• For the bisimulation algorithm to be computable we need to– Represent sets symbollically,

– Perform boolean combinations on sets

– Check emptyness of a set,

– Compute Pre(P) of a set P

• Class of sets and vector fields must be topologically simple– Set operations must not produce pathological sets

– Sets must have desirable finiteness properties

Computability & Finitiness

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Workshop on Hybrid and Embedded Systems 2006

Mathematical Logic

• Every theory of the reals has an associated language

• Decidable theories– Every formula is equivalent to a quantifier free formula

– Quantifier free formulas can be decided

• Quanitifier elimination

• Computational tools (REDLOG, QEPCAD)

��� -�� � � ���� � -�� �� � � ��

��� -�� �� � �� � ��

�. �.� � �. � / �

�� �� � � � � � � � � � ��

�� �� � � � � � � � � � �� �

�. �.� � �. � / � � �� � ��/ �

�� �� � � � � � � � � � � � � � � � �

�� �� � � � � � � � � � �� � - � � �

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Workshop on Hybrid and Embedded Systems 2006

O-Minimal Theories

• A definable set is

A theory of the reals is called o-minimal if every definable subset of the reals is a finite union of points and intervals

• Example: for polynomial

• Recent o-minimal theories

����� � � �� ��� � �� � 0���� � � �� ���

��� -�� � � ���� � -�� �� � � ��

��� -�� �� � �� � ����� -�� �� � ���� � ��

��� -�� �� � �� ���� � ��

�� � � � (��� � (���

Semilinear sets

Semialgebraic sets

Bounded Subanalytic sets

Exponential flows

Spirals ?

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Workshop on Hybrid and Embedded Systems 2006

O-Minimal Hybrid Systems

A hybrid system H is said to be o-minimal if• the continuous state lives in

• For each discrete state, the flow of the vector field is complete

• For each discrete state, all relevant sets and the flow of the vector field are definable in the same o-minimal theory

Main Theorem Main Theorem

Every oEvery o--minimal hybrid system admits a minimal hybrid system admits a finitefinite bisimulationbisimulation. .

• Bisimulation alg. terminates for o-minimal hybrid systems

• Various corollaries for each o-minimal theory

��

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Workshop on Hybrid and Embedded Systems 2006

O-Minimal Hybrid Systems

Consider hybrid systems where– All relevant sets are polyhedral

– All vector fields have linear flows

Then the bisimulation algorithm terminates

��� -�� � � ��

Consider hybrid systems where– All relevant sets are semialgebraic

– All vector fields have polynomial flows

Then the bisimulation algorithm terminates

�� � -�� �� � � ��

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Workshop on Hybrid and Embedded Systems 2006

O-Minimal Hybrid Systems

Consider hybrid systems where– All relevant sets are subanalytic

– Vector fields are linear with purely imaginary eigenvalues

Then the bisimulation algorithm terminates

Consider hybrid systems where– All relevant sets are semialgebraic

– Vector fields are linear with real eigenvalues

Then the bisimulation algorithm terminates

��� -�� �� � ���� � ��

��� -�� �� � �� � ��

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Workshop on Hybrid and Embedded Systems 2006

O-Minimal Hybrid Systems

Consider hybrid systems where– All relevant sets are subanalytic

– Vector fields are linear with real or purely imaginary eigenvalues

Then the bisimulation algorithm terminates

��� -�� �� � �� ���� � ��

• New o-minimal theories result in new finiteness results

• Can we find constructive subclasses?– Must remain within decidable theory

– Sets must be semialgebraic

– Need to perform reachability computations

• Reals with exp. does not have quantifier elimination

�� � -�� �� � � ��

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Workshop on Hybrid and Embedded Systems 2006

��

Linear Hybrid Systems

A hybrid system H is said to be linear if• the continuous state lives in

• For each discrete state, all relevant sets are semialgebraic

• For each discrete state, the vector field is of the form

where matrix has rational entries

• Let . Then we can express

• Focus on the subformula

�� � 1� 1

2 � �� � ����

�� ��2� � ��� �� ���� � � � � � � ��1� �

�� � � � �� �� ��1�

�� � � � � � ��1�

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Workshop on Hybrid and Embedded Systems 2006

Decidable Linear Hybrid Systems

Let H be a linear hybrid system H where for each discrete

location the vector field is of the form F(x)=Ax where

• A is rational and nilpotent

• A is rational, diagonalizable, with rational eigenvalues

• A is rational, diagonalizable, with purely imaginary, rational eigenvalues

Then the reachability problem for H is decidable.

�� � �� � ��� � � � ��� � � �

� � ��� � � � � � �� ��

� � � ��

� � �

� � � ��

�� � �� � �

�� ����

�� ��� �� � ��

�� ����

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Workshop on Hybrid and Embedded Systems 2006

Linear Hybrid Systems with Inputs

Let H be a linear hybrid system H where for each discrete

location, the dynamics are where A,B are

rational matrices and one of the following holds:

• A is nilpotent, and

• A is diagonalizable with rational eigenvalues, and

• A is diagonalizable with purely imaginary eigenvalues and

Then the reachability problem for H is decidable.

���� ��

" �

���� ��

" 3 �

���� ��

" ����4 ��

3 �� 5( !�1�

64 �� 5( !�1�

�� �1��7�

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Workshop on Hybrid and Embedded Systems 2006

Linear DTS (compare with Morari Bemporad)

• X = ℜn, U = {u|Eu≤η}, D = {d|Gd≤γ}, f = { Ax+Bu+Cd},

F = {x|Mx≤β}.

• Pre(Wl) = {x | ϕl(x)}

ϕl(x) = ∃u ∀d | [Mlx≤βl]c[Eu≤η]∧[(Gd>γ)∨(MlAx+MlBu+MlCd ≤βl)]

• Implementation

– Quantifier Elimination on d: Linear Programming

– Quantifier Elimination on u: Linear Algebra

– Emptiness: Linear Programming

– Redundancy: Linear Programming

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Decidability Results for Algorithm

The controlled invariant set calculation problem is

• Semi-decidablein general.

• Decidablewhen F is a rectangle, and A,b is in controllable canonical form for single input single disturbance.

Extensions:

Hybrid systems with continuous state evolving according to discrete time dynamics: difficulties arise because sets may not be convex or connected.

There are other classes of decidable systems which need to be identified.

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Workshop on Hybrid and Embedded Systems 2006

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Workshop on Hybrid and Embedded Systems 2006

Summary

• Methodology– Modeling Framework

– Game theoretic approach to controller synthesis

– Linear hybrid systems and computability

• Applications– Synthesis of safe conflict resolution maneuvers

– Safe controllers for automated highways

– Verification of avionic software (CTAS, TCAS)

– Flight Envelope Protection

– Flight Mode Switching

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Workshop on Hybrid and Embedded Systems 2006

Newer Research

• Modeling – Robustness, Zeno (Zhang, Simic, Johansson, Ames)– Simulation, on-line event detection (Johannson)– Hybrid systems toolbox (Sprinkle, Mitchell, Eklund)

• Control– Extension to more general properties (liveness, stability) (Koo)– Links to viability theory and viscosity solutions (Lygeros, Tomlin,

Mitchell, Bayen)– Numerical solution of PDEs (Tomlin, Mitchell)– Rapproachment between model predictive control and HJ (Eklund, Abate)

• Analysis– Develop (exact/approximate) reachability tools (Vidal, Shaffert)– Complexity analysis (Pappas, Shakernia)

• Probabilistic Hybrid Systems (Hu, Wu, Mitchell)• Observability of Hybrid Systems (Vidal, Ma)