Optimization-based Formation Reconfiguration Planning For
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Transcript Optimization-based Formation Reconfiguration Planning For
EECE 396-1
Hybrid and Embedded Systems: Computation
T. John Koo
Institute for Software Integrated Systems
Department of Electrical Engineering and Computer Science
Vanderbilt University
300 Featheringill Hall
January 14, 2004
[email protected]
http://www.vuse.vanderbilt.edu/~kootj
Hybrid Systems
UC Berkeley
Stanford University
Spring 2002 by T. John Koo, S. Shankar Sastry
http://robotics.eecs.berkeley.edu/~koo/Sp02/
Spring 2001 by T. John Koo, S. Shankar Sastry
http://robotics.eecs.berkeley.edu/~koo/Sp01/
Spring 2000 by Karl. H. Johansson, Luca de Alfaro, Thomas A. Henzinger
http://www.s3.kth.se/~kallej/eecs291e/
Spring 1999 by John Lygeros, S. Shankar Sastry
http://robotics.eecs.berkeley.edu/~lygeros/Teaching/ee291E.html
Spring 1998 by Thomas A. Henzinger, S. Shankar Sastry
Spring 2002 by Claire Tomlin
http://www.stanford.edu/class/aa278a/
University of Pennsylvania
Fall 2000 by Rajeev Alur, George J. Pappas
http://www.seas.upenn.edu/~pappasg/EE601/
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Hybrid System
A system built from atomic discrete
components and continuous
components by parallel and serial
composition, arbitrarily nested.
q1
The behaviors and interactions of
components are governed by models
of computation (MOCs).
Discrete Components
u
q2
q3
x
xç = f (x) + g(x)u
Finite State Machine (FSM)
Discrete Event (DE)
Synchronous Data Flow (SDF)
Continuous Components
Ordinary Differential Equation (ODE)
Partial Differential Equation (PDE)
3
Hybrid System
Continuous systems with phased
operations
Bouncing ball
Circuits with diodes
Switching circuits
Continuous systems controlled by
discrete inputs
Thermostat
Water tank
Engine control systems
Multi-modal systems
Embedded control systems
q1
u
q2
q3
x
xç = f (x) + g(x)u
4
The Heterogeneity of Systems
engine
E State
H
Finite
Machine
C
power train
I
Continuous Time
fuel
air
Discrete Event
embedded controller
sensors
An Engine Control System
5
Models of Computation
Finite State Machine
• states
engine
• transitions
E
H
C
I
fuel
air
power train
Continuous Time
• continuous functions
• continuous time
• continuous signals
Discrete Event
• operations on events
embedded controller
• continuous time
• discrete events
sensors
6
The Hierarchical View of
Systems
controller
car model
engine
power
train
7
Embedded Systems
Embedded systems
composed of hardware and
software components are
designed to interact with a
physical environment in
real-time in order to fulfill
control objectives and
design specifications.
Embedded Software
Operating System
Board Support Packages
Embedded Hardware
Environment
8
Embedded Systems
Embedded software refers
to application software to
process information to and
fro between the information
and physical worlds.
q1
D/A
u
q2
Embedded Software
Operating System
q3
Board Support Packages
Embedded Hardware
A/D
x
Environment
xç = f (x) + g(x)u
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High-Confidence
Embedded Software
Embedded Computer
Embedded
Software
From Design to
Implementation
u[k]
q1
u(t )
q2
q3
x (t )
Servos
q1
q2
q3
x [k]
GPS Card
INS
How?
xç = f (x) + g(x)u
u(t )
x (t )
1. Guaranteed closed-loop performance
2. Interaction between asynchronous and
synchronous components
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High-Confidence
Embedded Software
10Hz
Nav Data to
Vision computer
@10Hz
PERIODIC
VCOMM
ULREAD
4±1Hz
Ultrasonic
sensors@4±1Hz
APERIODIC
Nav data
Control output
at 50Hz
Relative Altitude
PERIODIC
100Hz
DQICONT
INS Update Boeing DQI-NP
RX values
Yamaha Receiver
(using HW INT & proxy)
DGPS measurement
Ground
Station
ANYTIME
Ground computer
Win 98
Processes
running on QNX
PRTK@ 5Hz
PXY@1Hz
PERIODIC
DQIGPS
GPS Update
RS-232
Shared Memory
Radio link
NovAtel GPS RT-2
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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, 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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Different Approaches
13
Research Directions
14
What Are Hybrid Systems?
Dynamical systems with interacting continuous and
discrete dynamics
15
Proposed Framework
Control Theory
Computer Science
Models of computation
Communication models
Discrete event systems
Control of individual agents
Continuous models
Differential equations
Hybrid Systems
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Power Electronics
Power electronics found in:
DC-DC converters
Power supplies
Electric machine drives
Circuits can be defined as networks of:
Voltage and current sources (DC or AC)
Linear elements (R, L, C)
Semiconductors used as switches (diodes, transistors)
ENNA GmbH
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Power Electronics
Discrete dynamics
N switches, (up to) 2N discrete states
Only discrete inputs (switching): some
discrete transitions under control, others
not
Continuous dynamics
Linear or affine dynamics at each discrete
state
+
ENNA GmbH
+
23=8 possible configurations
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Power Electronics : DC-DC
Converters
iL
+
Vin
-
L
sw1
2
sw2
C
R
1
2
+
Vout
-
Vout
Have a DC supply (e.g. battery), but need
a different DC voltage
Different configurations depending on
whether Vin<Vout or Vin>Vout
Control switching to maintain Vout with
changes in load (R), and Vin
iL
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Two Output DC-DC Converter
sw3
iL
C3
+
Vin
sw1
1
iL
VoutA
2
sw2
L
3 1
2
C2
R2
R3
+
+
VoutB
VoutA
-
-
3
Want two DC output voltages
Inductors are big and heavy, so
only want to use one
Similar to “two tank” problem
VoutB
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Circuit Operation
One and only one switch
closed at any time
Each switch state has a
continuous dynamics
sw1: iL, VoutA, VoutB
sw2: iL , VoutA , VoutB
sw3: iL , VoutA, VoutB
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Design Objective
iL , VoutA, VoutB
iL, VoutA, VoutB
iL , VoutA , VoutB
Objective: Regulate two output voltages and limit current by
switching between three discrete states with continuous dynamics.
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Typical Circuit
Analysis/Control
T
Governing equations
T
Time domain, steady state
Energy balance
System dynamics
i1
Discretization in time
(1- )T
Switched quantity only sampled at
discrete instants
Assumes a fixed clock
i0
match!
Averaging
Switched quantity approximated
by a moving average
Assumes switching is much faster
than system time constants
i2
iL(t)
iL(t)
Control
Linearize with duty () as input
Use classical control techniques
iL[k]
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Outline
Background on Power Electronics
Hybrid Modeling of DC-DC Converters
Controlled Invariant Balls
Conclusions
iL
+
Vin
-
L
sw1
sw2
C
R
+
Vout
-
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Problem Formulation
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Problem Formulation
Parallel Composition of Hybrid
H1
Automata
q1
û = û1
x 2 G12
x 2 G21
x2X
q2
û = û2
Given a collection of Modes
and Edges, design Guards
H2
q1
xç(t) = f q1(x(t))
x(t) 2 I q1
û2 Î
û = û2
û = û1
q2
xç(t) = f q2(x(t))
x(t) 2 I q2
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Research Issues
Modeling & Simulation
Control: classify discrete phenomena, existence and uniqueness of
execution, Zeno [Branicky, Brockett, van der Schaft, Astrom]
Computer Science: composition and abstraction operations [AlurHenzinger, Lynch, Sifakis, Varaiya]
Analysis & Verification
Control: stability, Lyapunov techniques [Branicky, Michel], LMI
techniques [Johansson-Rantzer]
Computer Science: Algorithmic [Alur-Henzinger, Sifakis, PappasLafferrier-Sastry] or deductive methods [Lynch, Manna, Pnuelli],
Abstraction [Pappas-Tabuada, Koo-Sastry]
Controller Synthesis
Control: optimal control [Branicky-Mitter, Bensoussan-Menaldi],
hierarchical control [Caines, Pappas-Sastry], supervisory control
[Lemmon-Antsaklis], safety specifications [Lygeros-Sastry, TomlinLygeros-Sastry], control mode switching [Koo-Pappas-Sastry]
Computer Science: algorithmic synthesis [Maler et.al., Wong-Toi],
synthesis based on HJB [Mitchell-Tomlin]
27
Hybrid Systems
28
Hybrid Systems
Hybrid Automata (Lygeros-Tomlin-Sastry, 2001)
Ref: J. Lygeros, C. Tomlin, and S. Sastry,
The Art of Hybrid Systems, July 2001.
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Hybrid Systems
Enabled Discrete Evolution
Guard AB
Q
Reset AB
Invariant set A
X
Invariant set B
30
Hybrid Systems
Forced Discrete Evolution
Guard AB
Q
Reset AB
Invariant set A
X
Invariant set B
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Hybrid Systems
32
Thermostat
Non-deterministic Hybrid
Automaton
t
33
Motivating Examples:Two
Tanks
34
Zeno—infinitely many jumps
in finite time
If
Water Tank Automaton
35
Motivating Examples:
Bouncing Ball
Zeno Hybrid Autamaton
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Computational Tools
Simulation
Ptolemy II: ptolemy.eecs.berkeley.edu
Modelica: www.modelica.org
SHIFT: www.path.berkeley.edu/shift
Dymola: www.dynasim.se
OmSim: www.control.lth.se/~cace/omsim.html
ABACUSS: yoric.mit.edu/abacuss/abacuss.html
Stateflow: www.mathworks.com/products/stateflow
CHARON: http://www.cis.upenn.edu/mobies/charon/
Masaccio:
http://www-cad.eecs.berkeley.edu/~tah/Publications/masaccio.html
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Computational Tools
Simulation
Masaccio
CHARON
Ptolemy II
Dymola
Modelica
StateFlow/Simulink
System
Complexity
ABACUSS
SHIFT
OmSim
Models of Computation
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Verification
Deductive Methods
Theorem-Proving techniques [Lynch, Manna, Pnuelli]
Model Checking
State-space exploration [Alur-Henzinger, Sifakis, Pappas-LafferrierSastry]
Reachability Problem
Check if Post (X S) \ X F = ; ?
XF
XS
Post (X S)
Forward Reachable Set
Post (P) = f x 2 X j 9x 0 2 P 9t õ 0 s:t: x = þ(t; r i ; x 0)g
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Computational Tools – Hybrid
Systems
Reach Sets Computation
Finite
Automata
COSPAN
SMV
VIS
…
Timed
Automata
Linear
Automata
xç = 1
Axç ô b
xç = Ax
HYTECH
Requiem
Timed COSPAN
KRONOS
Timed HSIS
VERITI
UPPAAL
Linear
Hybrid Systems
Si (r i )
Nonlinear
Hybrid Systems
xç = f (x )
d/dt
CheckMate
Sj (r j )
Pr ei (Sj (r j ); r i )
40
Research Directions
Development of formal methods for the design of
high-confidence embedded software based on
hybrid system theory with applications to
distributed, network-centric, embedded systems
such as sensor networks, power electronics
circuits, and cooperative UAV systems
Hybrid Systems
Embedded Software
High-Confidence Embedded Systems
Network-Centric Distributed Systems
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Research Collaboration
Institutions
Center for Hybrid and Embedded Systems and Software (CHESS),
University of California at Berkeley
GRASP Laboratory, University of Pennsylvania
Hybrid Systems Laboratory, Stanford University
Control Group, Cambridge University
INRIA, France
KTH, Sweden
Honeywell Laboratories
Cadence Berkeley Laboratory
Conferences
Workshop on Hybrid Systems: Computation and Control (HSCC)
Workshop on Embedded Software (EMSOFT)
IEEE Conference on Decision and Control (CDC)
IEEE Conference on Robotics and Automation (ICRA)
…
42
International Workshop on
Hybrid Systems: Computation and Control
University of Pennsylvania
March, 2004
http://www.seas.upenn.edu/hybrid/HSCC04/
End
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