Hybrid Systems Theory presented by Tom Henzinger 1

Hybrid Systems Theory presented by Tom Henzinger 1. Robust hybrid systems 2. Stochastic hybrid systems 3. Compositional hybrid systems 4. Computational hybrid systems Program Review May 10, 2004 Berkeley, CA NSF UC Berkeley: Chess Vanderbilt University: ISIS University of Memphis: MSI Foundations of Hybrid and Embedded Software Systems

A Formal Foundation for Embedded Systems needs to combine Computation Theories of -composition & hierarchy -computability & complexity B + Physicality R Theories of -robustness & approximation -probabilities & discounting CHESS Program Review, May 10, 2004 2

Continuous Dynamical Systems State space: Rn Dynamics: initial condition + differential equations Room temperature: x(0) = x 0 x’(t) = -K·x(t) x x 0 t Analytic complexity. CHESS Program Review, May 10, 2004 3

Discrete Transition Systems State space: Bm Dynamics: initial condition + transition relation Heater: heat off on t Combinatorial complexity. CHESS Program Review, May 10, 2004 4

Hybrid Automata State space: Bm Rn Dynamics: initial condition + transition relation + differential equations Thermostat: off x’ = -K·x x L x 0 x l on off on x u x’ = K·(H-x) x U t CHESS Program Review, May 10, 2004 5

The Robustness Issue Hybrid Automaton Property slightly perturbed automaton CHESS Program Review, May 10, 2004 6

The Robustness Issue Hybrid Automaton Safe x=3 CHESS Program Review, May 10, 2004 7

The Robustness Issue Hybrid Automaton Unsafe x = 3+ CHESS Program Review, May 10, 2004 8

Towards Robust Hybrid Automata value(Model, Property): States B value(Model, Property): States R CHESS Program Review, May 10, 2004 9

Towards Robust Hybrid Automata value(Model, Property): States B value(m, T) = ( X) (T pre(X)) discounted. Value(Model, Property): States R discounted. Value(m, T) = ( X) max(T, pre(X)) discount factor 0< <1 CHESS Program Review, May 10, 2004 10

Reachability (co. Safety) (F Ç pre(T)) = T max(0, ¢ pre(1)) = a b T 1 c T 2 c … undiscounted property c … discounted property CHESS Program Review, May 10, 2004 11

Main Result (so far, only for discrete systems) Robustness Theorem [de Alfaro, Henzinger, Majumdar]: If discounted. Bisimilarity(m 1, m 2) > 1 - , then |discounted. Value(m 1, p) - discounted. Value(m 2, p)| < f( ). Further Advantages of Discounting: -approximability because of geometric convergence (avoids non-termination of verification algorithms) -applies also to probabilistic systems and to games (enables reasoning under uncertainty, and control) CHESS Program Review, May 10, 2004 12

Hybrid Systems Theory 1 Robust hybrid systems 2 Stochastic hybrid systems Theory of discounting. Timed and stochastic games and optimal control. CHESS Program Review, May 10, 2004 20

Hybrid Systems Theory 1 Robust hybrid systems 2 Stochastic hybrid systems Theory of discounting. Timed and stochastic games and optimal control. 3 Compositional hybrid systems 4 Computational hybrid systems Agent algebras and interface theories. Hybrid systems simulation (Ptolemy II, GME). Hybrid systems verification (polyhedral & ellipsoidal methods). Code generation from hybrid systems (Ptolemy II, Giotto). CHESS Program Review, May 10, 2004 21

The Compositionality Issue Requirements Verification automatic (model checking) Model Implementation Environment automatic (compilation) Resources CHESS Program Review, May 10, 2004 22

The Compositionality Issue Requirements no change necessary Verification Composition Component Implementation Component no change necessary Resources CHESS Program Review, May 10, 2004 23

The Compositionality Issue Requirements (time, fault tolerance, etc. ) no change necessary Verification Composition Component Implementation Component no change necessary Resources CHESS Program Review, May 10, 2004 24

The Compositionality Issue Requirements (time, fault tolerance, etc. ) Verification Agent algebras. Interface theories. no change necessary Composition Component Implementation Virtual machines. Component no change necessary Resources CHESS Program Review, May 10, 2004 25

Compositional Hybrid Systems Modeling and Simulation I: Hy. Visual (based on Ptolemy II) Hy. Visual is being used to nail down an operational semantics for hybrid systems. CHESS Program Review, May 10, 2004 26

Compositional Hybrid Systems Modeling and Simulation II: Generating SIMULINK models from Hybrid Bond Graphs GME Tank 1 Pressure Tank 2 Pressure Two Tank Example THE SIMULINK ENVIROMENT • Using Simulink blocks and subsystems we can create a block diagram that corresponds to the Hybrid Bond Graph in GME. • Simulink provides variable step solvers for the simulation of ODEs, with error control and zero crossing detection. SIMULINK CHESS Program Review, May 10, 2004 27

Hybrid Systems Theory 1 Robust hybrid systems 2 Stochastic hybrid systems Theory of discounting. Timed and stochastic games and optimal control. 3 Compositional hybrid systems 4 Computational hybrid systems Agent algebras and interface theories. Hybrid systems simulation (Ptolemy II, GME). Hybrid systems verification (polyhedral & ellipsoidal methods). Code generation from hybrid systems (Ptolemy II, Giotto). CHESS Program Review, May 10, 2004 28

Continuous System Verification: Problem Given • control system • control set • target set Consider decision problems CHESS Program Review, May 10, 2004 29

Continuous System Verification: Approach 1. Express decision problems as optimization problems 2. Derive Hamilton-Jacobi-Bellman (HJB) partial differential equation of value functions 3. Obtain support functions of using convex analysis 4. Obtain tight outer and inner ellipsoidal approximations to CHESS Program Review, May 10, 2004 30

Continuous System Verification: Implementation 1. Matlab ‘toolbox’ to calculate ellipsoidal approximations and display any 2 -d projection of reach set inner ellipsoid outer ellisoid 2. Solve control problem: find control to steer CHESS Program Review, May 10, 2004 31

From Continuous to Hybrid System Verification 1. Find states that can be reached despite disturbances in which is set of disturbances 2. Find reach set of hybrid systems in which i-th system is triggered by ‘guard’ (future work) CHESS Program Review, May 10, 2004 32

Related In-Depth Talks 3: 50 p. m. Compositional Hybrid Systems: Model Transformations on Hybrid Models (Aditya Agrawal) 4: 10 p. m. Stochastic Hybrid Systems: Hybrid Systems in Systems Biology (Wei Chung Wu, Jianghai Hu, Shankar Sastry) 4: 30 p. m. Computational Hybrid Systems: Computational Methods for Analyzing and Controlling Hybrid Systems (Claire Tomlin) CHESS Program Review, May 10, 2004 33

Related Posters Compositional Hybrid Systems: Rich Interface Theories (Arindam Chakrabarti) Compositional Metamodeling (Matthew Emerson) Stochastic Hybrid Systems: Stochastic Hybrid Systems (Alessandro Abate) Computational Hybrid Systems: Zeno Behavior in Hybrid Systems (Aaron Ames) Computation of Reach Sets (Alex Kurzhansky) CHESS Program Review, May 10, 2004 34