Hybrid Systems and Hybrid Automata

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Hybrid Systems and Hybrid Automata

• Lecture 18 (03/21/02)

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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?
• Proliferation of Embedded Systems
• Simplify Behavior analysis of complex non linear
  systems
• Motivation(s) Need to accurately describe
  dynamic behavior
• Monitoring Diagnosis
• Design, Control

signal domain
energy domain
D/A
A/D
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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
Environment
Decision Maker
Switching Signal
u1
Controller 1
u2
Controller 2
y
u
Plant

um
Controller m
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Example Bouncing Ball
ball position x1 ball velocity x2
acceleration g coefficient of restitution c ?
0,1 x1 gt 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. (given gear efficiencies)
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Classification of Hybrid Behavior

• Continuous systems with phased operation
• Bouncing balls
• Diode circuits
• Walking robots
• Continuous systems controlled by discrete inputs
• 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
• 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

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
initial state vector ?
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Dynamic Physical Systems

• Inherently continuous
• Discontinuities attributed to modeling
  abstractions
• parameter abstraction
• time scale abstraction
• Implement discontinuities as transitions in
  continuous behavior
• systematic principles
• compositional modeling

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Abstraction Semantics

• Parameter Abstraction
• abstracts away complex non linear behaviors
• intermediate modes mythical
• switching model uses a posteriori state values
• Time Scale Abstraction
• 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

Mode Switching
Switch closed inductor charges Switch open
IL0 Diode comes on
No parasitic capacitance or resistance Sequence
of instantaneous changes
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Simulation Result

• Freewheeling Diode

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Parameter Abstractions

• Principle of Invariance of State
• Switching transition for parameter abstraction
  depends on a posteriori state vector value

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.
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Time Scale Abstraction

• Perfect Elastic Collision
• elasticity effects
• condensed to a point
• in time
• conservation of state
• conservation of energy
• Collision Chain
• energy state
• changes

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Time Scale Abstraction

• State Vector change governed by the principle of
  Conservation of State.
• Mode change from interval to point to interval.
  Point is called a pinnacle.

E.g., colliding bodies Newtons Collision rule
v2 - v1 ? (v2 - v1 ) ? - coefficient
of restitution and equate Forces m1 (v1 - v1 )
m2 (v2 - v2 )
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Hybrid Systems Issues and Challenges

• Building Hybrid Models of Complex Systems
• Systematic introduction of abstraction phenomena
• Composing hybrid automata
• Design of Hybrid Systems
• Verification and Validation of Hybrid
  Trajectories
• Monitoring and Control of Hybrid Systems
• Issues of switching transients
• Fault Detection and Isolation
• Combining discrete-event and continuous paradigms
• Fault Adaptive Control
• Fault detection, isolation, study of
  consequences, controller selection, transient
  management