chess review october 4, 2006 alexandria, va edited and presented by hybrid systems: theoretical...

Chess ReviewOctober 4, 2006Alexandria, VA

Edited and presented by

Hybrid Systems:Theoretical ContributionsPart I

Shankar SastryUC Berkeley

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

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

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

ITR Review, Oct. 4, 2006"Hybrid Systems Theory: I", S. Sastry 8

Reachability as Safety Specification

ITR Review, Oct. 4, 2006"Hybrid Systems Theory: I", S. Sastry 9

Computation of Optimal Reach Probability

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)

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

ITR Review, Oct. 4, 2006"Hybrid Systems Theory: I", S. Sastry 12

Optimal Control Actions for Room Heating Benchmark (for initial mode OFF)

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

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

Chess ReviewOctober 4, 2006Alexandria, VA

Edited and presented by

A Categorical Theory of Hybrid Systems

Aaron Ames

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

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.

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

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

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!

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

ITR Review, Oct. 4, 2006"Hybrid Systems Theory: I", S. Sastry 22

Hybrid Reduction Theorem

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

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

ITR Review, Oct. 4, 2006"Hybrid Systems Theory: I", S. Sastry 25

Goal

ITR Review, Oct. 4, 2006"Hybrid Systems Theory: I", S. Sastry 26

How to Walk in Four Easy Steps

ITR Review, Oct. 4, 2006"Hybrid Systems Theory: I", S. Sastry 27

Simulations

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

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.

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

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.

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

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