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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
Workshop on Hybrid and Embedded Systems 2006
What Are Hybrid Systems?
Dynamical systems with interacting continuousand discretedynamics
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
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
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
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]
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)
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
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
Workshop on Hybrid and Embedded Systems 2006
Automated Platoons on I-15
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
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
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
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
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
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
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
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
Workshop on Hybrid and Embedded Systems 2006
TCP/IPTCP/IP
Wireless LAN
GROUNDSTATIONVIRTUAL COCKPIT
GRAPHICALEMMULATION
WIRELESSHUB
UAV Laboratory Configuration
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
Workshop on Hybrid and Embedded Systems 2006
Hierarchical Hybrid Systems
•• Envelope Protecting ModeEnvelope Protecting Mode
•• Normal Flight ModeNormal Flight Mode
SafetyInvariant��
LivenessReachability
Tactical Planner
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
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
Workshop on Hybrid and Embedded Systems 2006
Newer Platforms 2003-04
Workshop on Hybrid and Embedded Systems 2006
Waypoint Navigation
Helicopter Mode transition
ForwardFlight
Take-off
Land
Hover
Ascend/Descend
PirouetteBank-to-turn
Sideslip
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
Workshop on Hybrid and Embedded Systems 2006
Waypoint Navigation using VCL (Jan 29, 2001)
1m
1m
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
Workshop on Hybrid and Embedded Systems 2006
Experimental Results: Pursuit Evasion Games with 4UGVs and 1 UAV (Spring’ 01)
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
Workshop on Hybrid and Embedded Systems 2006
Advanced Maneuver Experiments
Workshop on Hybrid and Embedded Systems 2006
Video tape of Maneuvers
Workshop on Hybrid and Embedded Systems 2006
Formation Flying Experiment: 2 real UAVs and 7 virtual UAVs Nov 2002
Workshop on Hybrid and Embedded Systems 2006
Formation Flying Visualized
Animation by A. Pant and X. Xiao
Withoutleaderinfo
Withleaderinfo
Workshop on Hybrid and Embedded Systems 2006
Tracking Pitching Target (Nov 02)
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
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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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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Workshop on Hybrid and Embedded Systems 2006
Controller Synthesis: Example
• 2D conflict resolution
• Ensure aircraft remain more than 5nmi from each other
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
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
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
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
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
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
Workshop on Hybrid and Embedded Systems 2006
Continuous Reachable Set
x
y
[Mitchell, Bayen, Tomlin 2001][Tomlin, Lygeros, Sastry 2000]
Workshop on Hybrid and Embedded Systems 2006
Fast Wavefront Approximation Methods (Tomlin, Mitchell)
Workshop on Hybrid and Embedded Systems 2006
Visualization of Unsafe Set:Mitchell-Tomlin
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
Workshop on Hybrid and Embedded Systems 2006
Transition Systems
• Transition System
• Define for
• Given equivalence relation define
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• 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?
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Workshop on Hybrid and Embedded Systems 2006
Bisimulation Algorithm
initialize :
while such that
define
refine
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• If algorithm terminates, we obtain a finite bisimulation
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
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• 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
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
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Semilinear sets
Semialgebraic sets
Bounded Subanalytic sets
Exponential flows
Spirals ?
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
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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
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• 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
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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
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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.
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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
Workshop on Hybrid and Embedded Systems 2006
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.
Workshop on Hybrid and Embedded Systems 2006
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
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)