Hybrid Systems Theoretical Contributions Part I Edited and
Hybrid Systems: Theoretical Contributions Part I Edited and presented by Shankar Sastry UC Berkeley Chess Review October 4, 2006 Alexandria, VA
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 "Hybrid Systems Theory: I", S. Sastry ITR Review, Oct. 4, 2006 2
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 "Hybrid Systems Theory: I", S. Sastry ITR Review, Oct. 4, 2006 3
Qualitative vs. Quantitative Verification Qualitative Verification System is safe System is unsafe Quantitative Verification System is safe with probability 1. 0 "Hybrid Systems Theory: I", S. Sastry System is unsafe with probability ε ITR Review, Oct. 4, 2006 4
Reachability as Safety Specification "Hybrid Systems Theory: I", S. Sastry ITR Review, Oct. 4, 2006 8
Computation of Optimal Reach Probability "Hybrid Systems Theory: I", S. Sastry ITR Review, Oct. 4, 2006 9
Room Heating Benchmark • • • Temperature in two rooms is controlled Two Room One Heater Example by one heater. Safe set for both rooms Room 2 Room 1 is 20 – 25 (0 F) Goal is to keep the temperatures within corresponding safe sets with a high Temperature probability sensors SHS model – Two continuous states: – Three modes: OFF, ON (Room 1), ON (Room 2) Heater – Continuous evolution in mode ON (Room 1) – Mode switches defined by controlled Markov chain with seven discrete actions: (Do Nothing, Rm 1 ->Rm 2, Rm 2 -Rm 1, Rm 1 -> Rm 3, Rm 3 ->Rm 1, Rm 2 -Rm 3, Rm 3 -> Rm 2) "Hybrid Systems Theory: I", S. Sastry ITR Review, Oct. 4, 2006 10
Probabilistic Maximal Safe Sets for Room Heating Benchmark (for initial mode OFF) Temperature in Room 2 25 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)0 F with probability at least 0. 9 for 150 minutes 22. 5 20 20 22. 5 Temperature in Room 1 25 Note: The spatial discretization is 0. 250 F, temporal discretization is 1 min and time horizon is 150 minutes "Hybrid Systems Theory: I", S. Sastry ITR Review, Oct. 4, 2006 11
Optimal Control Actions for Room Heating Benchmark (for initial mode OFF) "Hybrid Systems Theory: I", S. Sastry ITR Review, Oct. 4, 2006 12
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] CDC 2006 "Hybrid Systems Theory: I", S. Sastry ITR Review, Oct. 4, 2006 13
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 "Hybrid Systems Theory: I", S. Sastry ITR Review, Oct. 4, 2006 14
A Categorical Theory of Hybrid Systems Edited and presented by Aaron Ames Chess Review October 4, 2006 Alexandria, VA
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 "Hybrid Systems Theory: I", S. Sastry ITR Review, Oct. 4, 2006 16
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, T • Example: T = Vect; T = Man; • The objects being considered can be “hybridized” D by considering a small category (or “graph”) together with a functor (or “function”): S: D ! T D – is the “discrete” component of the hybrid system – T is the “continuous” component • Example: hybrid vector space S : D ! Man. "Hybrid Systems Theory: I", S. Sastry S : D ! Vect, hybrid manifold ITR Review, Oct. 4, 2006 17
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 "Hybrid Systems Theory: I", S. Sastry ITR Review, Oct. 4, 2006 18
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 "Hybrid Systems Theory: I", S. Sastry ITR Review, Oct. 4, 2006 19
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! "Hybrid Systems Theory: I", S. Sastry ITR Review, Oct. 4, 2006 20
Hybrid Reduction: Motivation • Problem: – There 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 "Hybrid Systems Theory: I", S. Sastry ITR Review, Oct. 4, 2006 21
Hybrid Reduction Theorem "Hybrid Systems Theory: I", S. Sastry ITR Review, Oct. 4, 2006 22
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 "Hybrid Systems Theory: I", S. Sastry ITR Review, Oct. 4, 2006 23
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 "Hybrid Systems Theory: I", S. Sastry ITR Review, Oct. 4, 2006 24
Goal "Hybrid Systems Theory: I", S. Sastry ITR Review, Oct. 4, 2006 25
How to Walk in Four Easy Steps "Hybrid Systems Theory: I", S. Sastry ITR Review, Oct. 4, 2006 26
Simulations "Hybrid Systems Theory: I", S. Sastry ITR Review, Oct. 4, 2006 27
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 "Hybrid Systems Theory: I", S. Sastry ITR Review, Oct. 4, 2006 28
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. "Hybrid Systems Theory: I", S. Sastry ITR Review, Oct. 4, 2006 29
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 "Hybrid Systems Theory: I", S. Sastry ITR Review, Oct. 4, 2006 30
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. "Hybrid Systems Theory: I", S. Sastry ITR Review, Oct. 4, 2006 31
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 "Hybrid Systems Theory: I", S. Sastry ITR Review, Oct. 4, 2006 32
Some closing thoughts • Key new areas of research initiated • Some important new results • Additional theory needed especially for networked embedded systems "Hybrid Systems Theory: I", S. Sastry ITR Review, Oct. 4, 2006 33
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