Hybrid Systems: Theoretical Contributions Part I

1. Hybrid Systems:Theoretical ContributionsPart I Shankar Sastry UC Berkeley

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 "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 "Hybrid Systems Theory: I", S. Sastry

4. 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 ε "Hybrid Systems Theory: I", S. Sastry

5. Reachability as Safety Specification "Hybrid Systems Theory: I", S. Sastry

6. Computation of Optimal Reach Probability "Hybrid Systems Theory: I", S. Sastry

7. Room 2 Room 1 Temperature sensors Heater Room Heating Benchmark • 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: Two Room One Heater Example (Do Nothing, Rm 1->Rm2, Rm 2-Rm 1, Rm 1-> Rm 3, Rm 3->Rm1, Rm 2-Rm 3, Rm 3-> Rm 2) "Hybrid Systems Theory: I", S. Sastry

8. 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)0F with probability at least 0.9 for 150 minutes Temperature in Room 2 22.5 20 20 25 22.5 Temperature in Room 1 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 "Hybrid Systems Theory: I", S. Sastry

9. Optimal Control Actions for Room Heating Benchmark (for initial mode OFF) "Hybrid Systems Theory: I", S. Sastry

10. 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 "Hybrid Systems Theory: I", S. Sastry

11. 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

12. A Categorical Theory of Hybrid Systems Aaron Ames

13. 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

14. M V S S D D t e a c n : : ! ! . , T T V M t e a c n = = ; ; T D T T S D D : ! 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 "Hybrid Systems Theory: I", S. Sastry

15. 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

16. 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

17. 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

18. 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 "Hybrid Systems Theory: I", S. Sastry

19. Hybrid Reduction Theorem "Hybrid Systems Theory: I", S. Sastry

20. 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

21. 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

22. Goal "Hybrid Systems Theory: I", S. Sastry

23. How to Walk in Four Easy Steps "Hybrid Systems Theory: I", S. Sastry

24. Simulations "Hybrid Systems Theory: I", S. Sastry

25. 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

26. 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

27. 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

28. 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

29. 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 "Hybrid Systems Theory: I", S. Sastry

30. 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