![Page 1: Chess Review October 4, 2006 Alexandria, VA Edited and presented by Hybrid Systems: Theoretical Contributions Part I Shankar Sastry UC Berkeley](https://reader030.vdocuments.us/reader030/viewer/2022032801/56649d555503460f94a325e8/html5/thumbnails/1.jpg)
Chess ReviewOctober 4, 2006Alexandria, VA
Edited and presented by
Hybrid Systems:Theoretical ContributionsPart I
Shankar SastryUC Berkeley
![Page 2: Chess Review October 4, 2006 Alexandria, VA Edited and presented by Hybrid Systems: Theoretical Contributions Part I Shankar Sastry UC Berkeley](https://reader030.vdocuments.us/reader030/viewer/2022032801/56649d555503460f94a325e8/html5/thumbnails/2.jpg)
ITR Review, Oct. 4, 2006"Hybrid Systems Theory: I", S. Sastry 2
Broad Theory Contributions: Samples
• Sastry’s 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
• Tomlin’s group: Developed new mathematics for– Safe set calculations and approximations,– Estimation of hybrid systems
• Sangiovanni’s 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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ITR Review, Oct. 4, 2006"Hybrid Systems Theory: I", S. Sastry 3
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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ITR Review, Oct. 4, 2006"Hybrid Systems Theory: I", S. Sastry 4
Qualitative vs. Quantitative Verification
System is safe System is unsafe
System is safe with probability 1.0
System is unsafe with probability ε
Qualitative Verification
Quantitative Verification
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ITR Review, Oct. 4, 2006"Hybrid Systems Theory: I", S. Sastry 8
Reachability as Safety Specification
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ITR Review, Oct. 4, 2006"Hybrid Systems Theory: I", S. Sastry 9
Computation of Optimal Reach Probability
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ITR Review, Oct. 4, 2006"Hybrid Systems Theory: I", S. Sastry 10
Room Heating Benchmark
38'-6"
17
'-6"
4'-3"
6'-4
3/8
"
22'-6"
5'-1
1 3
/16"11'-9"
17'-1
1 3
/16"
23
'-1
0 3
/8"
5'-0"
2'-6
"
6 ft. x 3 ft.
5'-0
"
Temperature sensors
Room 1 Room 2
Heater
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:
)())}()(())(({)()1(
)(}))()(())(({)()1(
2212122
11121111
kntkxkxkxxkxkx
kntkkxkxkxxkxkx
ca
ca
(Do Nothing, Rm 1->Rm2, Rm 2-Rm 1, Rm 1-> Rm 3, Rm 3->Rm1, Rm 2-Rm 3, Rm 3-> Rm 2)
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ITR Review, Oct. 4, 2006"Hybrid Systems Theory: I", S. Sastry 11
Probabilistic Maximal Safe Sets for Room Heating Benchmark (for initial mode OFF)
202522.520
25
22.5
Temperature in Room 1
Tem
pera
ture
in
Room
2
Starting from this initial condition in OFF mode and following optimal control law, it is guaranteed that system will remain in the safe set (20,25)×(20,25)0F with probability at least 0.9 for 150 minutes
Note: The spatial discretization is 0.250F, temporal discretization is 1 min and time horizon is 150 minutes
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ITR Review, Oct. 4, 2006"Hybrid Systems Theory: I", S. Sastry 12
Optimal Control Actions for Room Heating Benchmark (for initial mode OFF)
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ITR Review, Oct. 4, 2006"Hybrid Systems Theory: I", S. Sastry 13
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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ITR Review, Oct. 4, 2006"Hybrid Systems Theory: I", S. Sastry 14
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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Chess ReviewOctober 4, 2006Alexandria, VA
Edited and presented by
A Categorical Theory of Hybrid Systems
Aaron Ames
![Page 13: Chess Review October 4, 2006 Alexandria, VA Edited and presented by Hybrid Systems: Theoretical Contributions Part I Shankar Sastry UC Berkeley](https://reader030.vdocuments.us/reader030/viewer/2022032801/56649d555503460f94a325e8/html5/thumbnails/13.jpg)
ITR Review, Oct. 4, 2006"Hybrid Systems Theory: I", S. Sastry 16
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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ITR Review, Oct. 4, 2006"Hybrid Systems Theory: I", S. Sastry 17
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
TT = Vect; T = Man;
TD
D
S : D ! T
S : D ! Vect,S : D ! Man.
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ITR Review, Oct. 4, 2006"Hybrid Systems Theory: I", S. Sastry 18
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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ITR Review, Oct. 4, 2006"Hybrid Systems Theory: I", S. Sastry 19
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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ITR Review, Oct. 4, 2006"Hybrid Systems Theory: I", S. Sastry 20
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 hard—reduction is more important!
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ITR Review, Oct. 4, 2006"Hybrid Systems Theory: I", S. Sastry 21
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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ITR Review, Oct. 4, 2006"Hybrid Systems Theory: I", S. Sastry 22
Hybrid Reduction Theorem
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ITR Review, Oct. 4, 2006"Hybrid Systems Theory: I", S. Sastry 23
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
![Page 21: Chess Review October 4, 2006 Alexandria, VA Edited and presented by Hybrid Systems: Theoretical Contributions Part I Shankar Sastry UC Berkeley](https://reader030.vdocuments.us/reader030/viewer/2022032801/56649d555503460f94a325e8/html5/thumbnails/21.jpg)
ITR Review, Oct. 4, 2006"Hybrid Systems Theory: I", S. Sastry 24
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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ITR Review, Oct. 4, 2006"Hybrid Systems Theory: I", S. Sastry 25
Goal
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ITR Review, Oct. 4, 2006"Hybrid Systems Theory: I", S. Sastry 26
How to Walk in Four Easy Steps
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ITR Review, Oct. 4, 2006"Hybrid Systems Theory: I", S. Sastry 27
Simulations
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ITR Review, Oct. 4, 2006"Hybrid Systems Theory: I", S. Sastry 28
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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ITR Review, Oct. 4, 2006"Hybrid Systems Theory: I", S. Sastry 29
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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ITR Review, Oct. 4, 2006"Hybrid Systems Theory: I", S. Sastry 30
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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ITR Review, Oct. 4, 2006"Hybrid Systems Theory: I", S. Sastry 31
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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ITR Review, Oct. 4, 2006"Hybrid Systems Theory: I", S. Sastry 32
Overview of Main Result
• The categorical approach to hybrid systems allows us to decompose the study of Zeno equilibria into two steps:1. We identify a sufficiently rich, yet simple, class of
hybrid systems that display the desired stability properties: first quadrant hybrid systems
2. 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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ITR Review, Oct. 4, 2006"Hybrid Systems Theory: I", S. Sastry 33
Some closing thoughts
• Key new areas of research initiated• Some important new results• Additional theory needed especially for
networked embedded systems