Transcript Document

Hybrid Systems and Hybrid Automata
Lecture 18 (03/21/02)
7/17/2015
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What is a Hybrid System?
Dynamic systems that require more than one modeling
language to characterize their dynamics
Provide a mathematical framework for analyzing systems with
interacting discrete and continuous dynamics
Capture the coupling between digital computations and analog
physical plant and environment
(another way: interaction between time-driven signals
(synchronous) and event-driven signals (asynchronous)
Continuous Dynamics: mechanical, fluid, thermal systems, linear
circuits, chemical reactions
Discrete dynamics: collisions, switches in circuits, valves and
pumps
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Hybrid Models of Physical Systems
Why Hybrid Models?
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Proliferation of Embedded Systems
Simplify Behavior analysis of complex non linear systems
+
-
signal domain
D/A
control
algorithm
actuators
A/D
embedded controller
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energy domain
plant
Tsample
sensors
physical system
Motivation(s): Need to accurately describe dynamic behavior
Monitoring & Diagnosis
Design, Control
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Practical Applications
Manufacturing systems:
part processed by machine only after its arrival at the machine
(triggered, event-driven process)
 Process within machine can be described by time-driven
dynamics
In the past, handled separately: event-driven: automata or Petri
nets; time-driven: differential or difference equations
But this is not good for high performance analysis and
optimization – becomes even more difficult when tight
coupling exists between the two forms

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Other Practical Applications
Automotive control – electronic fuel ignition system
with embedded controller; performance parameters:
reduce gas consumption, reduce emissions while
maintaining performance – requires tight integration of
continuous and discrete time processes
(in the past discrete time behavior, e.g., 4 stroke cycle
was reduced to continuous time behavior: models less
accurate and computationally more complex)
other examples: anti-lock system of brakes, etc.
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Third Practical Example
Interaction of discrete planning algorithms and
continuous processes or plant, e.g., spacecraft systems
on deep space missions: require them to be
autonomous, intelligent, highly reliable
Task: design sequential supervisory controllers for
continuous systems
Hierarchical organization may help maintain autonomy
Initially deal with simpler examples: bouncing ball,
thermostat systems, diode circuits
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Modeling Physical Systems
Continuous behavior governed by
- Conservation of energy
- Continuity of power
Discontinuous changes artifacts of
- Embedded control
- Simplification of complex nonlinear system behavior
Have to handle
- Discrete changes in model topology
- Initial value problem
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Supervisory Controller
Decision
Maker
Environment
Switching
Signal
Controller 1
Controller 2
••
•
Controller m
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u1
u2
y
u
Plant
um
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Example: Bouncing Ball
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ball position – x1 ball velocity – x2 acceleration – g
coefficient of restitution – c  [0,1]
x1 > 0  continuous flow governed by differential equation
when transition condition satisfied  discrete jump occurs
Behavior is zeno, i.e., infinite number of bounces occur in finite
time interval
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Example: Thermostat
Thermostat (controller) turns on radiator between 68 & 70 degrees and
turns off the radiator between 80 and 82 degrees.
Result: non deterministic system – for a given initial condition
there are a whole family of different executions.
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Example: Automatic Gear Box Control
Lateral position x1 and velocity x2
Control signals (i) gear -- v  {1,2,3,4}, (ii) throttle -- u  [-1,1]
Question: Optimal Control Strategy in going from point a to b.
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(given gear efficiencies)
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Classification of Hybrid Behavior
Continuous systems with phased operation
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Bouncing balls
Diode circuits
Walking robots
Continuous systems controlled by discrete inputs
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Thermostats
Circuits with switches
Processes with valves and pumps
Control modes

Aircraft autopilot modes – maintain altitude, maintain airspeed, maintain
angle of attack, take-off, landing
Coordination processes
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Multi-agent systems
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Hybrid Automata
Hybrid automata is a 6-tuple
H = (V, X, f, Init, Inv, Jump)
V  I – set of discrete modes
X  Rn – real-valued variables, often the state vector
f : V x Rn  Rn -- vector field
Init  V x Rn -- defines initial state of H (v,Z)
Inv  V x Rn -- invariant condition
Jump: V x Rn  P(V x Rn) – jump condition; what transitions from
one discrete mode to another are possible, and what value should
be assigned to state vector after the jump (reset condition)
(v,z)  V x Rn (z is an evaluation of x) state of H
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Hybrid Automata: Graphical
Representation
H  directed graph (V,E), vertices V, and edges E
E = {(v,v’)  V x V :  z,z’  Rn , (v’ ,z’)  Jump(v,z)]
Init(v) = {z  Rn : (v,z)  Init}
Inv(v) = {z  Rn : (v,z)  Inv}
G(e) = {z  Rn : z’  Rn (v,z),
(v’ ,z’)  Jump(v,z)}
J(e,z) = {z’  Rn : (v’ ,z’)  Jump(v,z)}
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Hybrid Time Trajectory
Interval Point Paradigm
x
x
k
x+
k
k
< ,t s >
1
x+
x+
k

2
x
m
m
[ts]
x+
x+
m
m
1
2
n
x+
n
<t s , >
Two key issues:
1. When do jumps occur, and how to model the transition ?
2. When system re-enters a continuous mode, what is the
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initial state vector ?
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Dynamic Physical Systems
Inherently continuous
Discontinuities attributed to modeling abstractions
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parameter abstraction
time scale abstraction
Implement discontinuities as transitions in
continuous behavior
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systematic principles
compositional modeling
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Abstraction Semantics
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Parameter Abstraction
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abstracts away complex non linear behaviors
intermediate modes mythical
switching model uses a posteriori state values
Time Scale Abstraction
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collapses behavior in small intervals to point in time
(pinnacle)
switching model uses a priori state values
Our Goal: systematic model building to facilitate building
Hybrid Automata for real-time analysis
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Example: Diode-Inductor Circuit
Diode-Inductor Circuit
R
I
Sw
D:
ON
V
D
L
V
ID
+
I th
+
VD < -Vdi ode
OFF
Mode Switching
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V
I
V
R
Sw
D
L
I
00
p
V
R
01
Sw
D
L
0
V
I
-V
R
Sw
D
L
?
Switch closed: inductor charges
Switch open: IL=0
No parasitic capacitance or resistance
Diode comes on
Sequence of instantaneous changes
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Simulation Result
Freewheeling Diode
I
V
V
Sw
D
R
L
I
p
R
V
Sw
D
0
L
R
V
a
0
0
a
1
0
I
-V
Sw
D
?
L
a0
1
0.04
0.03
IL
0.02
0.01
0.00
50
100
50
100
150
200
250
3 00
-0.01
3 50
400
t
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VL
10
5
0
0
150
200
250
3 00
350
t
400
Sw
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real mode 00
mythical mode
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01
00
t
00
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Parameter Abstractions
Principle of Invariance of State
Lemma: Any vector that represents the state of a linear physical system is
invariant across mode changes.
Proof: based on converting any state vector to particular state vector
involving energy variables. (Mosterman, Biswas, and Sztipanovits,
“A hybrid modeling and verification paradigm for embedded control systems,”
Control Engineering Practice, vol. 2, pp. 127-142, 1998.)
Conjecture: This may be extended to nonlinear systems provided an
inverse mapping can be computed uniquely.
Switching transition for parameter abstraction
depends on a posteriori state vector value
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Time Scale Abstraction
Perfect Elastic Collision
elasticity effects
condensed to a point
in time
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conservation of state
conservation of energy
Collision Chain
energy state
changes
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free
free
im pact
free
fr ee
impact
1
2
1
2
1
2
v
m1
m2
a0
0
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m3
m1
m2
m3
m1
m2
m3
v
v
a
1
0
a
0
1
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Time Scale Abstraction
State Vector change governed by the principle of
Conservation of State.
E.g., colliding bodies
Newton’s Collision rule: v2+ - v1+ =  (v2 - v1 ) ;  - coefficient of restitution
and equate Forces: m1 (v1+ - v1 ) = m2 (v2+ - v2 )
Mode change from interval to point to interval. Point is
called a pinnacle.
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Hybrid Systems: Issues and Challenges
Building Hybrid Models of Complex Systems
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Systematic introduction of abstraction phenomena
Composing hybrid automata
Design of Hybrid Systems
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Verification and Validation of Hybrid Trajectories
Monitoring and Control of Hybrid Systems
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Issues of switching transients
Fault Detection and Isolation
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Combining discrete-event and continuous paradigms
Fault Adaptive Control
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Fault detection, isolation, study of consequences, controller selection,
transient management
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