Hybrid Systems Modeling, Analysis, Control

Datta Godbole, John Lygeros, Claire Tomlin, Gerardo Lafferiere, George Pappas, John Koo Jianghai Hu, Rene Vidal, Shawn Shaffert, Jun Zhang, Slobodan Simic, Kalle Johansson, Maria Prandini David Shim, Jin Kim, Omid Shakernia, Cedric Ma, Judy Liebmann and Ben Horowitz

(with the interference of) Shankar Sastry

What Are Hybrid Systems?

Dynamical systems with interacting continuous and discrete dynamics

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

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

Proposed Framework

Control Theory Control of individual agents Continuous models Differential equations Computer Science Models of computation Communication models Discrete event systems Hybrid Systems

Different Approaches

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)

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 ( leader mode) – Small spacing ( follower mode) • Platooning concept

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

Air Traffic Management Systems

• Studied by NEXTOR and NASA • Increased demand for 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 flight

Automated Platoons on I-15

CENTER A VOR

Current ATM System

CENTER B SUA TRACON TRACON GATES

20 Centers, 185 TRACONs, 400 Airport Towers Size of TRACON: 30-50 miles radius, 11,000ft Centers/TRACONs are subdivided to sectors Approximately 1200 fixed VOR nodes Separation Standards Inside TRACON : 3 miles, 1,000 ft Below 29,000 ft : 5 miles, 1,000ft Above 29,000 ft : 5 miles, 2,000ft Computable Hybrid Systems

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 air travel, the above problems will be greatly amplified leading to both safety and performance degradation in the future

Computable Hybrid Systems

A Future ATM Concept

CENTER B CENTER A ALERT ZONE TRACON PROTECTED ZONE TRACON

• • • Free Flight from 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 Computable Hybrid Systems

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

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

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

Other Applications

• Uninhabited Aerial Vehicles ( UAV ) – Automated aerial vehicles (airplanes and/or helicopters) – Coordinate for search and rescue , or seek and destroy missions – 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

UAV Laboratory Configuration

Wireless LAN TCP/IP WIRELESS HUB GROUNDSTATION VIRTUAL COCKPIT TCP/IP GRAPHICAL EMMULATION

Motivation

•

Goal

–

Design a multi-agent multi-modal control system for Unmanned Aerial Vehicles (UAVs)

• • •

Intelligent coordination among agents Rapid adaptation to changing environments Interaction of models of operation

–

Guarantee

•

Safety

• • •

Performance Fault tolerance Mission completion

Conflict Resolution Collision Avoidance Envelope Protection Fuel Consumption Object Searching Pursuit-Evasion

Hierarchical Hybrid Systems

• •

Envelope Protecting Mode Normal Flight Mode

Tactical Planner Safety Invariant  Liveness Reachability

The UAV Aerobot Club at Berkeley

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

Hybrid Automata

• Hybrid Automaton – State space – Input space – Initial states – Vector field – Invariant set – Transition relation • Remarks: – – State countable, – Can add outputs, etc. (not needed here)

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

Controller Synthesis: Example

• 2D conflict resolution • Ensure aircraft remain more than 5nmi from each other

Hybrid Automaton Specification

• Discrete input variable determines maneuver initiation • Safety specification

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 controller is a map • The closed loop executions are

Controller Synthesis Problem

• Given H and find g such that • A set is controlled invariant if 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 game between the control and the disturbance

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 – does can be continuously driven to states that whatever avoiding by –

Initialization

:

while do end

X

Algorithm Interpretation

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

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 Decidable if hybrid automata are rectangular, initialized.

Application: Control of Automated Highway Systems

• Design of vehicle controllers & performance estimation • Two concepts – platooning & individual vehicles Network •Flow optimization Link •Dynamic routing Coordination •Maneuver selection •inter-vehicle comm Regulation •Lane keeping •Vehicle following Lane Change Entry Speed, vehicle following Exit Join Split Platoon Following

Vehicle Following & Lane Changing

i-2 i i-1 j • 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 Computable Hybrid Systems

Game Theoretic Formulation

• Requirements – Safety (no collision) ( 0   – Passenger Comfort inf

t

 0 ( );

J

1 

C

1  0

J

2 (

x

0 – Efficiency  sup| ( )|;

t

 0

J

2 

C

2  • trajectory tracking (depends on the maneuver) .

ms

 3 • Safe controller (

J 1

): Solve a two-person zero-sum game – saddle solution (

u 1 *,d 1 *

) 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

1 * )  1 0 1 * )

Safe Vehicle Following Controller

• Partition the state space into safe & unsafe sets

S

: (

x

0 1 ,

x

2 0 ,

x

4 0 )  0

x

3 , min Design comfortable and efficient controllers in the interior •IEEE TVT 11/94 Safe set characterization also provides sufficient conditions for lane change •CDC 97, CDC98

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

Example: Aircraft Collision Avoidance

Two identical aircraft at fixed altitude & speed:

x d dt

 

y

   

f

(

x

,

u

,

d

)    

v



v

cos 

v

sin

d

 

ux



u



uy

 

y x u v v d

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

Continuous Reachable Set

x



y

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

Fast Wavefront Approximation Methods (Tomlin, Mitchell)

Visualization of Unsafe Set: Mitchell-Tomlin

Transition Systems

• Transition System • Define for • Given equivalence relation define • A ~ block is a union of equivalence classes

Bisimulations of Transition Systems

A partition ~ is a bisimulation iff

– –

are ~ blocks For all and all ~ blocks is a ~ block

• Alternatively, for • Why are bisimulations important?

Bisimulation Algorithm

initialize

:

while such that define refine

• If algorithm terminates, we obtain a finite bisimulation

Computability & Finitiness

• 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

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 )

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 Semilinear sets Semialgebraic sets Exponential flows Bounded Subanalytic sets Spirals ?

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 Every o-minimal hybrid system admits a finite bisimulation. • Bisimulation alg. terminates for o-minimal hybrid systems • Various corollaries for each o-minimal theory

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

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

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 perfrom reachability computations • Reals with exp. does not have quantifier elimination

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

Diagonalizable, Rational Eigenvalues

diagonalizable with rational eigenvalues. Then is definable in the decidable theory of reals

Example:

Diagonalizable, Imaginary Eigenvalues

• Procedure is conceptually similar if is diagonalizable with purely imaginary, rational eigenvalues • Equivalence is obtained by • Suffices to compute over a period

Let be a linear vector field, rational, diagonalizable with purely imaginary rational eigenvalues. Then is definable in the decidable theory of reals

• Composing all the constructive results together gives in…

Semidecidable 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 semidecidable .

•

Above result also holds if discrete transitions are not necessarily initialized but computable

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 .

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 .

Linear DTS (compare with Morari Bemporad)

•

X

= 

n

, U = {

u|Eu

 },

D

= {

d|Gd

 },

f =

{

Ax+Bu+Cd

},

F

= {

x|Mx

 }.

• Pre(

W l

) = {

x |



l

(

x

)} 

l

(

x

) = 

u



d

| [

M l x



l

]c[

Eu

 ]  [(

Gd

>  )  (

M l Ax+M l Bu+M l Cd



l

)] • Implementation – Quantifier Elimination on

d

: – Quantifier Elimination on

u

: – Emptiness: – Redundancy: Linear Programming Linear Algebra Linear Programming Linear Programming

Decidability Results for Algorithm

The controlled invariant set calculation problem is • Semi-decidable in general.

• Decidable when

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.

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

Newer Research

• Modeling – Robustness, Zeno (Zhang, Simic, Johansson) – Simulation, on-line event detection (Johannson, Ames) • 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) • Analysis – Develop (exact/approximate) reachability tools (Vidal, Shaffert) – Complexity analysis (Pappas, Kumar) • Probabilistic Hybrid Systems (Hu) • Observability of Hybrid Systems (Vidal)