Hybrid%20Systems%20and%20Hybrid%20Automata

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Hybrid Systems and Hybrid Automata

• Lecture 20 (04/11/02)

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Schedule for Course

• Remaining work
• Presentations Jian (04/16) John, Holly,
  Yolanda, and Sree classes 04/18 and 04/23
• Project want a well-defined problem statement
  plus literature review by April 18
• my travel schedule out 04/22 (Mon), 04/26-27
  (Fri, Sat),
• 04/30-05/07 (Tues.-Tues.), and 05/12-05/16
  (Sun-Thurs)
• When do we have our final project submission and
  presentations?
• 05/08-05/10 Wed-Fri
• My suggestion Turn in final project report on
  05/08.
• Presentations on 05/09. Demos 05/09 and 05/10.
• After presentation and demo, I will allow you to
  refine project report and submit for final
  grading.
• Post grades on 05/11.

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Course Projects

• Jolan MILAN interpreter for MATLAB simulations
  (asynchronous scheduling problem)
• Jian Hierarchical modeling of Hybrid Systems
  applied to water recovery system (WRS) of Bioplex
• Hari Supervisory Control for Hybrid Systems
  Designing a controller for a three-tank testbed
• Aditya Graph transformations to aid
  task-oriented problem solving (??) look at the
  bond graph formalism
• Holly Parameter estimation methods for
  continuous dynamic systems (??)
• John Hybrid Bond Graph modeling and simulation
  methodologies applied to fuel transfer system
  (??)
• Sachin timed automata representations for (??)
• Yolanda The Hybrid Bond Graph Framework (??)
• Sree (??)

Please send me updated topic and abstract as soon
as possible e-mail will be fine.
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Project Work -- Outline

• Introduction
• Motivating scenario and what are the important
  problems in the domain
• Use this to make your problem statement
  (obviously you will not be solving every problem
  listed above)
• What you hope to achieve
• Background
• Critical literature review
• Use the above to situate where your work falls
  into in terms of the big scheme of things, why is
  it different or new?
• Project Description
• Your work, describe in one or more sections
• General solution
• Particular examples and cases you have worked on
• Experimental Results
• Conclusions and Future Work

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Lecture on Hybrid 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) -- v? V, z? Rn
• Inv ? V x Rn -- invariant condition as long as
  the discrete mode is v? V, the state of the
  system ? Inv
• Jump V x Rn ? P(V x Rn) jump condition
  defines if transition from one discrete mode to
  another is possible, and what new 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) ? Jump(v,z) guard cond. J(e,z)
z ? Rn (v ,z) ? Jump(v,z) jump map
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Hybrid Time Trajectory

• Interval Point Paradigm sequence of intervals on
  the real line, whose end points overlap.
  Endpoints are where discrete transitions occur.

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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Train gate system
Train starts atleast 2000 ft away from gate Train
travels at 40-50 ft/sec The gate is fully
raised Controller can sense train 1000 ft away At
this point controller commands gate to lower with
delay of at most ? seconds Gate lowers at
90/sec. Train slows down after sensor but still
travels at at least 30ft/sec 100 feet after
crossing second signal sensed and controller
raises gate at 90/sec Specs gate must be closed
when train within 10 feet of crossing keep gate
open for as much time as possible
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Train Gate system
Train Model
Gate Model
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Train Gate System Controller
Safety Verification, Reachability Analysis,
Simulation Controller Design
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Continuous System
Solution x(t) must be differentiable and
satisfy Unique solution requires x(0) x0
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Pathological Cases
Questions 1. Do solutions exist? 2. Do
solutions exist globally? (t ? ?) 3. Are
solutions unique Answer Lipschitz condition
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Local Existence and Uniqueness

• Lipschitz Condition

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Global Existence and Uniqueness
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Executions of Hybrid System
Execution is finite if ? is a finite sequence
ending with a compact interval Infinite if ? is
an infinite sequence or if ??(? ) ? Zeno if
??(? ) lt ?
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Blocking and Nonblocking Automata
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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