Hybrid Systems: Theoretical Contributions Part I

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Hybrid SystemsTheoretical ContributionsPart I

• Shankar Sastry
• UC Berkeley

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Broad Theory Contributions Samples

• Sastrys group Defined and set the agenda of
  the following sub-fields
• Stochastic Hybrid Systems
• Category Theoretic View of Hybrid Systems,
• State Estimation of Partially Observable Hybrid
  Systems
• Tomlins group Developed new mathematics for
• Safe set calculations and approximations,
• Estimation of hybrid systems
• Sangiovannis group defined
• Intersection based composition-model as common
  fabric for metamodeling,
• Contracts and contract algebra refinement
  relation for assumptions/promises-based design in
  metamodel

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Quantitative Verification for Discrete-Time
Stochastic Hybrid Systems (DTSHS)

• Stochastic hybrid systems (SHS) can model
  uncertain dynamics and stochastic interactions
  that arise in many systems
• Quantitative verification problem
• What is the probability with which the system can
  reach a set during some finite time horizon?
• (If possible), select a control input to ensure
  that the system remains outside the set with
  sufficiently high probability
• When the set is unsafe, find the maximal safe
  sets corresponding to different safety levels

Abate, Amin, Prandini, Lygeros, Sastry HSCC 2006
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Qualitative vs. Quantitative Verification
Qualitative Verification
System is safe
System is unsafe
Quantitative Verification
System is safe with probability 1.0
System is unsafe with probability e
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Discrete-Time Stochastic Hybrid Systems
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Entities
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Definition of Reach Probability
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Reachability as Safety Specification
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Computation of Optimal Reach Probability
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Room Heating Benchmark
Two Room One Heater Example

• Temperature in two rooms is controlled by one
  heater. Safe set for both rooms is 20 25 (0F)
• Goal is to keep the temperatures within
  corresponding safe sets with a high probability
• SHS model
• Two continuous states
• Three modes OFF, ON (Room 1), ON (Room 2)
• Continuous evolution in mode ON (Room 1)
• Mode switches defined by controlled Markov chain
  with seven discrete actions

(Do Nothing, Rm 1-gtRm2, Rm 2-Rm 1, Rm 1-gt Rm 3,
Rm 3-gtRm1, Rm 2-Rm 3, Rm 3-gt Rm 2)
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Probabilistic Maximal Safe Sets for Room Heating
Benchmark (for initial mode OFF)
Note The spatial discretization is 0.250F,
temporal discretization is 1 min and time horizon
is 150 minutes
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Optimal Control Actions for Room Heating
Benchmark (for initial mode OFF)
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More Results

• Alternative interpretation
• Problem of keeping the state of DTSHS outside
  some pre-specified unsafe set by selecting
  suitable feedback control law can be formulated
  as a optimal control problem with max-cost
  function
• Value functions for max-cost case can be
  expressed in terms of value functions for
  multiplicative-cost case
• Time varying safe set specification can be
  incorporated within the current framework
• Extension to infinite-horizon setting and
  convergence of optimal control law to stationary
  policy is also addressed

Abate, Amin, Prandini, Lygeros, Sastry CDC2006
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Future Work

• Within the current setup
• Sufficiency of Markov policies
• Randomized policies, partial information case
• Interpretation as killed Markov chain
• Distributed dynamic programming techniques
• Extensions to continuous time setup
• Discrete time controlled SHS as stochastic
  approx. of general continuous time controlled SHS
• Embedding performance in the problem setup
• Extensions to game theoretic setting

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A Categorical Theory of Hybrid Systems

• Aaron Ames

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Motivation and Goal

• Hybrid systems represent a great increase in
  complexity over their continuous and discrete
  counterparts
• A new and more sophisticated theory is needed to
  describe these systems categorical hybrid
  systems theory
• Reformulates hybrid systems categorically so that
  they can be more easily reasoned about
• Unifies, but clearly separates, the discrete and
  continuous components of a hybrid system
• Arbitrary non-hybrid objects can be generalized
  to a hybrid setting
• Novel results can be established

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Hybrid Category Theory Framework

• One begins with
• A collection of non-hybrid mathematical
  objects
• A notion of how these objects are related to one
  another (morphisms between the objects)
• Example vector spaces, manifolds
• Therefore, the non-hybrid objects of interest
  form a category,
• Example
• The objects being considered can be hybridized
  by considering a small category (or graph)
  together with a functor (or function)
• is the discrete component of the hybrid
  system
• is the continuous component
• Example hybrid vector space
  hybrid manifold

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Applications

• The categorical framework for hybrid systems has
  been applied to
• Geometric Reduction
• Generalizing to a hybrid setting
• Bipedal robotic walkers
• Constructing control laws that result in walking
  in three-dimensions
• Zeno detection
• Sufficient conditions for the existence of Zeno
  behavior

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Applications

• Geometric Reduction
• Generalizing to a hybrid setting
• Bipedal robotic walkers
• Constructing control laws that result in walking
  in three-dimensions
• Zeno detection
• Sufficient conditions for the existence of Zeno
  behavior

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Hybrid Reduction Motivation

• Reduction decreases the dimensionality of a
  system with symmetries
• Circumvents the curse of dimensionality
• Aids in the design, analysis and control of
  systems
• Hybrid systems are hardreduction is more
  important!

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Hybrid Reduction Motivation

• Problem
• There are a multitude of mathematical objects
  needed to carry out classical (continuous)
  reduction
• How can we possibly generalization?
• Using the notion of a hybrid object over a
  category, all of these objects can be easily
  hybridized
• Reduction can be generalized to a hybrid setting

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Hybrid Reduction Theorem
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Applications

• Geometric Reduction
• Generalizing to a hybrid setting
• Bipedal robotic walkers
• Constructing control laws that result in walking
  in three-dimensions
• Zeno detection
• Sufficient conditions for the existence of Zeno
  behavior

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Bipedal Robots and Geometric Reduction

• Bipedal robotic walkers are naturally modeled as
  hybrid systems
• The hybrid geometric reduction theorem is used to
  construct walking gaits in three dimensions given
  walking gaits in two dimensions

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Goal
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How to Walk in Four Easy Steps
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Simulations
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Applications

• Geometric Reduction
• Generalizing to a hybrid setting
• Bipedal robotic walkers
• Constructing control laws that result in walking
  in three-dimensions
• Zeno detection
• Sufficient conditions for the existence of Zeno
  behavior

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Zeno Behavior and Mechanical Systems

• Mechanical systems undergoing impacts are
  naturally modeled as hybrid systems
• The convergent behavior of these systems is often
  of interest
• This convergence may not be to classical''
  notions of equilibrium points
• Even so, the convergence can be important
• Simulating these systems may not be possible due
  to the relationship between Zeno equilibria and
  Zeno behavior.

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Zeno Behavior at Work

• Zeno behavior is famous for its ability to halt
  simulations
• To prevent this outcome
• A priori conditions on the existence of Zeno
  behavior are needed
• Noticeable lack of such conditions

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Zeno Equilibria

• Hybrid models admit a kind of Equilibria that is
  not found in continuous or discrete dynamical
  systems Zeno Equilibria.

• A collection of points invariant under the
  discrete dynamics
• Can be stable in many cases of interest.
• The stability of Zeno equilibria implies the
  existence of Zeno behavior.

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Overview of Main Result

• The categorical approach to hybrid systems allows
  us to decompose the study of Zeno equilibria into
  two steps
• We identify a sufficiently rich, yet simple,
  class of hybrid systems that display the desired
  stability properties first quadrant hybrid
  systems
• We relate the stability of general hybrid systems
  to the stability of these systems through a
  special class of hybrid morphisms hybrid
  Lyapunov functions

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Some closing thoughts

• Key new areas of research initiated
• Some important new results
• Additional theory needed especially for networked
  embedded systems