An Introduction to Hybrid Systems Modeling

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1 CS620, IIT BOMBAY An Introduction to Hybrid Systems Modeling Ashutosh Trivedi Department of Computer Science and Engineering, IIT Bombay CS620: New Trends in IT: Modeling and Verification of Cyber-Physical Systems (2 August 2013) A. Trivedi 1 of 24

Dynamical Systems Dynamical System: A system

continuous) of the system time: discrete or

2 Dynamical Systems Dynamical System: A system whose state evolves with time governed by a fixed set of rules or dynamics. state: valuation to variables (discrete or continuous) of the system time: discrete or continuous dynamics: discrete, continuous, or hybrid A. Trivedi 2 of 24

3 Dynamical Systems Dynamical System: A system whose state evolves with time governed by a fixed set of rules or dynamics. state: valuation to variables (discrete or continuous) of the system time: discrete or continuous dynamics: discrete, continuous, or hybrid Discrete System Continuous System Hybrid Systems. A. Trivedi 2 of 24

4 Discrete Dynamical Systems Continuous Dynamical Systems Hybrid Dynamical Systems A. Trivedi 3 of 24

5 Abstract Model for Dynamical Systems Definition (State Transition Systems) A state transition system is a tuple T = (S, S 0, Σ, ) where: S is a (potentially infinite) set of states; S 0 S is the set of initial states; Σ is a (potentially infinite) set of actions; and S Σ S is the transition relation; A. Trivedi 4 of 24

6 Abstract Model for Dynamical Systems Definition (State Transition Systems) A state transition system is a tuple T = (S, S 0, Σ, ) where: S is a (potentially infinite) set of states; S 0 S is the set of initial states; Σ is a (potentially infinite) set of actions; and S Σ S is the transition relation; tick tick tick tick pause, 0 pause, 1 pause, 2 pause, 3 pause pause pause pause start start start start tick tick tick start count, 0 count, 1 count, 2 count, 3 tick A. Trivedi 4 of 24

7 State Transition Systems Definition (State Transition Systems) A state transition system is a tuple T = (S, S 0, Σ, ) where: S is a (potentially infinite) set of states; S 0 S is the set of initial states; Σ is a (potentially infinite) set of actions; and S Σ S is the transition relation; Finite and countable state transition systems A finite run is a sequence s 0, a 1, s 1, s 2, s 2,..., s n such that s 0 S 0 and for all 0 i < n we have that (s i, a i+1, s i+1 ). Reachability and Safe-Schedulability problems A. Trivedi 5 of 24

8 State Transition Systems Definition (State Transition Systems) A state transition system is a tuple T = (S, S 0, Σ, ) where: S is a (potentially infinite) set of states; S 0 S is the set of initial states; Σ is a (potentially infinite) set of actions; and S Σ S is the transition relation; Finite and countable state transition systems A finite run is a sequence s 0, a 1, s 1, s 2, s 2,..., s n such that s 0 S 0 and for all 0 i < n we have that (s i, a i+1, s i+1 ). Reachability and Safe-Schedulability problems We need efficient computer-readable representations of infinite systems! A. Trivedi 5 of 24

9 Extended Finite State Machines Let X be the set of variables (real-valued) of the system let X = N. A valuation ν of X is a function ν : X R. We consider a valuation as a point in R N equipped with Euclidean Norm. A predicate is defined simply as a subset of R N represented (non-linear) algebraic equations involving X. Non-linear predicates, e.g. x sin(z) = 0 Polyhedral predicates: a 1x 1 + a 2x a nx n k where a i R, x i X, and = {<,, =,, >}. Octagonal predicates x i x j k or x i k where x i, x j X, and = {<,, =,, >}. Rectangular predicates x i k where x i X, and = {<,, =,, >}. Singular Predicates x i = c. A. Trivedi 6 of 24

10 Extended Finite State Machines x<3, tick, x =x + 1, start, x =x, tick, x =x start count, pause, x =x pause x=3, tick, x =0 Extended Finite State Machines (EFSMs): Finite state-transition systems coupled with a finite set of variables The valuation remains unchanged while system stays in a mode (state) The valuation changes during a transition when it jumps to the valuation governed by a predicate over X X specified in the transition relation. Transitions are guarded by predicates over X Mode invariants Initial state and valuation A. Trivedi 7 of 24

11 Extended Finite State Machines x<3, tick, x =x + 1, start, x =x, tick, x =x start count, pause, x =x pause x=3, tick, x =0 Definition (EFSM: Syntax) An extended finite state machine is a tuple M = (M, M 0, Σ, X,, I, V 0 ) such that: M is a finite set of control modes including a distinguished initial set of control modes M 0 M, Σ is a finite set of actions, X is a finite set of real-valued variable, M pred(x) Σ pred(x X ) M is the transition relation, I : M pred(x) is the mode-invariant function, and V 0 pred(x) is the set of initial valuations. A. Trivedi 8 of 24

12 EFSM: Semantics x<3, tick, x =x + 1, start, x =x, tick, x =x start count, pause, x =x pause x=3, tick, x =0 tick tick tick tick pause, 0 pause, 1 pause, 2 pause, 3 pause pause pause pause start start start start tick tick tick start count, 0 count, 1 count, 2 count, 3 tick A. Trivedi 9 of 24

13 EFSM: Semantics The semantics of an EFSM M = (M, M 0, Σ, X,, I, V 0 ) is given as a state transition graph T M = (S M, S0 M, Σ M, M ) where S M (M R X ) is the set of configurations of M such that for all (m, ν) S M we have that ν I(m) ; S M 0 S M is the set of initial configurations such that (m, ν) S M if m M 0 and ν V 0 ; Σ M = Σ is the set of labels; M S M Σ M S M is the set of transitions such that ((m, ν), a, (m, ν )) M if there exists a transition δ = (m, g, a, j, m ) such that current valuation satisfies the guard ν g ; current and next valuations satisfy the jump constraint (ν, ν ) j ; and next valuation satisfies the invariant of the target mode ν I(m ). A. Trivedi 10 of 24

14 Discrete Dynamical Systems Continuous Dynamical Systems Hybrid Dynamical Systems A. Trivedi 11 of 24

15 Continuous Dynamical Systems A finite set of continuous variables, a set of ordinary differential equations (ODE) characterizing the flow of these variables as a function of time F : Ẋ pred(x) where ẋ is the first derivative of x. Higher-order derivatives can be written using first derivatives by introducing auxiliary variables, e.g. write θ + (g/l) sin(θ) = 0 can be written as θ = y and ẏ = (g/l) sin(θ). an initial valuation to the variables. A. Trivedi 12 of 24

16 Continuous Dynamical Systems A finite set of continuous variables, a set of ordinary differential equations (ODE) characterizing the flow of these variables as a function of time F : Ẋ pred(x) where ẋ is the first derivative of x. Higher-order derivatives can be written using first derivatives by introducing auxiliary variables, e.g. write θ + (g/l) sin(θ) = 0 can be written as θ = y and ẏ = (g/l) sin(θ). an initial valuation to the variables. Definition (Continuous Dynamical System) A continuous dynamical system is a tuple M = (X, F, ν 0 ) such that X is a finite set of real-valued variable, F : Ẋ pred(x) is the flow function, and ν 0 R X is the initial valuation. A. Trivedi 12 of 24

17 Continuous Dynamical Systems Definition (Continuous Dynamical System) A continuous dynamical system is a tuple M = (X, F, ν 0 ) such that X is a finite set of real-valued variable, F : Ẋ pred(x) is the flow function, and ν 0 R X is the initial valuation. A run or a trajectory of M = (X, F, ν 0 ) is given as a solution to the differential equations Ẋ = F (X) with initial valuation ν 0. Let a differentiable function f : R 0 R X be a solution to Ẋ = F (X) that provides the valuations of the variables as a function of time: f(0) = ν 0 f(t) = F (f(t)) for every t R 0, where f : R 0 R X is the time derivative of the function f. a run of a continuous dynamical system may not exist or may not be unique! A. Trivedi 13 of 24

18 Existence and Uniqueness Definition (Lipschitz-continuous Function) We say that a function F : R n R n is Lipschitz-continuous if there exists a constant K>0, called the Lipschitz constant, such that for all x, y R n we have that F (x) F (y) < K x y. A. Trivedi 14 of 24

19 Existence and Uniqueness Definition (Lipschitz-continuous Function) We say that a function F : R n R n is Lipschitz-continuous if there exists a constant K>0, called the Lipschitz constant, such that for all x, y R n we have that F (x) F (y) < K x y. Theorem (Picard-Lindelöf Theorem) If a function F : R X R X is Lipschitz-continuous then the differential equation Ẋ=F (X) with initial valuation ν 0 R X has a unique solution f : R 0 R X for all ν 0 R X. A. Trivedi 14 of 24

20 Existence and Uniqueness Definition (Lipschitz-continuous Function) We say that a function F : R n R n is Lipschitz-continuous if there exists a constant K>0, called the Lipschitz constant, such that for all x, y R n we have that F (x) F (y) < K x y. Theorem (Picard-Lindelöf Theorem) If a function F : R X R X is Lipschitz-continuous then the differential equation Ẋ=F (X) with initial valuation ν 0 R X has a unique solution f : R 0 R X for all ν 0 R X. Theorem (Stability wrt initial valuation) Let F be a Lipschitz-continuous function with constant K>0 and let f:r 0 R X and f :R 0 R X be solutions to the differential equation Ẋ=F (X) with initial valuation ν 0 R X and ν 0 R X, respectively. Then, for all t R 0 we have that f(t) f (t) ν ν 0 e Kt. A. Trivedi 14 of 24

21 Example: Simple Pendulum l θ mg ml 2 θ Variables y and θ flow equations: ml 2 θ = mgl sin(θ), or θ = y, ẏ = (g/l) sin(θ), initial valuations (θ, y) = (θ 0, 0). A. Trivedi 15 of 24

22 Simple Pendulum To analytically solve these equations, assume that initial angular displacement θ is small. hence sin(θ) θ. Now the equations simplify to θ = y and ẏ = (g/l)θ. The solution for these differential equations is θ(t) = A cos(kt) + B sin(kt) y(t) = AK sin(kt) + BK cos(kt), where K = g/l. Substituting θ(0) = θ 0 and y(0) = 0 from the initial valuation, we get that A = θ 0 and B = 0. The unique run of the pendulum system can be given as the function f : R 0 {θ, y} as t (θ 0 cos(kt), θ 0 K sin(kt)). A. Trivedi 16 of 24

23 Pendulum Motion θ (in degrees) and y ( in degrees/second) θ 0 cos(( g/l)t) θ 0 g/l sin(( g/l)t) t (in seconds) (b) A. Trivedi 17 of 24

24 Discrete Dynamical Systems Continuous Dynamical Systems Hybrid Dynamical Systems A. Trivedi 18 of 24

25 Another example: Bouncing Ball Bouncing Ball Consider a bouncing ball system dropped from height l and velocity 0. Is it a continuous system? A. Trivedi 19 of 24

26 Another example: Bouncing Ball Bouncing Ball Consider a bouncing ball system dropped from height l and velocity 0. Is it a continuous system? variables of interest : height of the ball x 1 and velocity of the ball x 2 flow function: A. Trivedi 19 of 24

27 Another example: Bouncing Ball Bouncing Ball Consider a bouncing ball system dropped from height l and velocity 0. Is it a continuous system? variables of interest : height of the ball x 1 and velocity of the ball x 2 flow function: What happens at impact? x 1 = x 2 and x 2 = g A. Trivedi 19 of 24

28 Another example: Bouncing Ball Bouncing Ball Consider a bouncing ball system dropped from height l and velocity 0. Is it a continuous system? variables of interest : height of the ball x 1 and velocity of the ball x 2 flow function: What happens at impact? x 1 = x 2 and x 2 = g x 1 = x 1 and x 2 = cx 2 where c is Restituition coefficient. A. Trivedi 19 of 24

29 Another example: Bouncing Ball Bouncing Ball Consider a bouncing ball system dropped from height l and velocity 0. Is it a continuous system? variables of interest : height of the ball x 1 and velocity of the ball x 2 flow function: What happens at impact? x 1 = x 2 and x 2 = g x 1 = x 1 and x 2 = cx 2 where c is Restituition coefficient. x1=0 x2 0, impact x 1=x1 x 2= cx2 start x1 = x2, x2 = g m x1 0 A. Trivedi 19 of 24

30 Discrete, Continuous, and Hybrid Systems 4 Discrete System 1 Continuous System 3 Hybrid System X X 0 X t t t A. Trivedi 20 of 24

31 Hybrid Automata: Syntax Some examples: Two leaking-water tanks systems Water-level monitor with delayed switch A leaking gas-burner Green scheduling with lower dwell-time requirements Light-bulb with three modes- dim, bright, and off. Job-shop scheduling problem A. Trivedi 21 of 24

32 Login Protocol restart, x > 60 start start user name, x:=0 valid restart, x 10 pw fail, x < 60 pw match, x < 60 x:=0 delay error conn A. Trivedi 22 of 24

33 Hybrid Automata: Syntax Definition (HA: Syntax) A hybrid automaton is a tuple H = (M, M 0, Σ, X,, I, F, V 0 ) where: M is a finite set of control modes including a distinguished initial set of control modes M 0 M, Σ is a finite set of actions, X is a finite set of real-valued variable, M pred(x) Σ pred(x X ) M is the transition relation, I : M pred(x) is the mode-invariant function, F : M (Ẋ pred(x)) is the mode-dependent flow function, and V 0 pred(x) is the set of initial valuations. A. Trivedi 23 of 24

34 Hybrid Automata: Syntax Definition (HA: Syntax) A hybrid automaton is a tuple H = (M, M 0, Σ, X,, I, F, V 0 ) where: M is a finite set of control modes including a distinguished initial set of control modes M 0 M, Σ is a finite set of actions, X is a finite set of real-valued variable, M pred(x) Σ pred(x X ) M is the transition relation, I : M pred(x) is the mode-invariant function, F : M (Ẋ pred(x)) is the mode-dependent flow function, and V 0 pred(x) is the set of initial valuations. A configuration (m, ν) and a timed action (t, a) A transition ((m, ν), (t, a), (m, ν ) solve flow ODE of mode m with ν as the starting state ν F (m) t. invariant, guard, and jump conditions. A run or execution is a sequence of transitions (m 0, ν 0 ), (t 1, a 1 ), (m 1, ν 1 ), (t 2, a 2 )... A. Trivedi 23 of 24

35 Hybrid Automata: Semantics Definition (HA: Semantics) The semantics of a HA H = (M, M 0, Σ, X,, I, F, V 0 ) is given as a state transition graph T H = (S H, S H 0, Σ H, H ) where S H (M R X ) is the set of configurations of H such that for all (m, ν) S H we have that ν I(m) ; S H 0 S H s.t. (m, ν) S H 0 if m M 0 and ν V 0 ; Σ H = R 0 Σ is the set of labels; H S H Σ H S H is the set of transitions such that ((m, ν), (t, a), (m, ν )) H if there exists a transition δ = (m, g, a, j, m ) such that (ν F (m) t) g ; (ν F (m) τ) I(m) for all τ [0, t]; ν (ν F (m) t)[j]; and ν I(m ). A. Trivedi 24 of 24

An Introduction to Hybrid Systems Modeling

CS620, IIT BOMBAY An Introduction to Hybrid Systems Modeling Ashutosh Trivedi Department of Computer Science and Engineering, IIT Bombay CS620: New Trends in IT: Modeling and Verification of Cyber-Physical