Georgios E. Fainekos, Savvas G. Loizou and George J. Pappas. GRASP Lab Departments of CIS, MEAM and ESE University of Pennsylvania

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1 Georgios E. Fainekos, Savvas G. Loizou and George J. Pappas CDC 2006 Math free Presentation! Lab Departments of CIS, MEAM and ESE University of Pennsylvania

2 Motivation Motion Planning π 0 π 4 x 2 30 π 2 20 π 1 π x 1

3 Motivation Motion Planning 60 SUCCESS x π 0 π 1 π 2 π 4 π x 1 Task: Task: Stay Stay always always in in π 0 and 0 and visit visit area area π 2, 2, then then area area π 3, 3, then then area area π 4 and, 4 and, finally, finally, return return to to and and stay stay in in region region π 1 while 1 while avoiding areas areasπ 2 and 2 and π 3 3 Good news RTL: π 0 D(π 0 D(π 2 D(π 2 D(π 3 D(π 3 D(π 4 (Ÿπ 4 2 Ÿπ 2 Ÿπ 3 )Uπ 3 )Uπ 1 ))) 1 )))

4 Problem Definition Temporal Logic Logic Controller Synthesis Given Givena dynamical system systemσ, Σ, a set set of of initial initial conditions X 0 and 0 and a flat flat RTL RTL formula φ over over Π, Π, construct a closed-loop system system in in the the form form of of a hybrid hybrid automaton H φ such φ such that that the theresulting system system trajectories x(t) x(t) starting at at some some point pointx(0)œx 0 satisfy 0 satisfy the the formula φ. φ. Σ:

5 In the past Approaches to synthesis of hybrid systems: Given controllers + switching rules ï verify that HS satisfies spec Given controllers + spec ï keep necessary controllers Given closed-loop dynamics + spec ï design switching rules Researchers: Alur, Henzinger, Tabuada, Davoren, Moor, Koutsoukos, Antsaklis, Stiver, Lemmon, Tomlin, Lygeros, Sastry, Habets, van Schuppen, Asarin, Bournez, Dang, Maler, Pnueli, Bemporad, Morari, Giorgetti, Lafferriere, Kloetzer, Belta, Kyriakopoulos, Kress-Gazit, Loizou, Pappas, Fainekos and many others Given spec î design controllers

Morari, Conner, Rizzi, Choset, Burridge, Koditschek, Koutsoukos, Antsaklis, Stiver, Lemmon,

6 In this presentation A top-down approach: Input: Software Specification 1 Algorithm Hybrid Automaton s 2 s 1 s 3 (Abstract) Tomlin, Lygeros, Sastry, Habets, van Schuppen, Bemporad, Morari, Conner, Rizzi, Choset, Burridge, Koditschek, Koutsoukos, Antsaklis, Stiver, Lemmon, many other Your favorite control engineer 2 List List of of Controllers {Init, Inv, Inv, Goal}

7 Hybrid Automaton A hybrid automaton is a tuple H = (X, V, E, Inv, Flow, Init, Guard, F) where X is the state space of the system Σ V is the set of control locations, E Œ V V is the set of control switches Inv : V Ø P(X) assigns an invariant set to each location Flow : V X Ø P( n ) constraints the time derivative of the continuous part of the state Init : V Ø P(X) assigns to each control location a set of initial conditions, Guard : E Ø P(X) is the guard condition that enables a control switch e in E F Œ V is the set of final locations

8 Trajectories of the Hybrid Automaton Formally, the semantics of a hybrid automaton are given in terms of timed transition systems T H = (H, H 0, Ø). H 0 Guard for edge e = (v 1, v 2 ) Inv(v 1 ) Control location v 1 Flow(v 1 ) Note: the reset map is the identity, transitions are forced

9 Trajectories of the Hybrid Automaton Formally, the semantics of a hybrid automaton are given in terms of timed transition systems T H = (H, H 0, Ø). Init(v 2 ) Control location v 2 Flow(v 2 ) Inv(v 2 ) Note: the reset map is the identity, transitions are forced

10 Language of the timed transition system The set of all trajectories η of T H starting from a state in H 0 is the language L(T H ) of the timed transition system T H. H 0 η : + Ø X

11 Linear Temporal Logic: Continuous Time Like propositional logic, but also reasoning wrt time Basic operators: Propositional: Ÿ,, ÿ Temporal:,, U, R (red) Eventually red (gray) Always gray (gray) U (red) gray Until red

12 Flat LTL with continuous time semantics Let Π be a set of atomic propositions and Π* be the set of Boolean combinations of atomic propositions, i.e. (π 1 π 2 ) Ÿπ 3. Define the denotation [[.]] : Π Ø P(X) which extends naturally over Π* as: [[π 1 π 2 ]] = [[π 1 ]] [[π 2 ]] and [[Ÿπ]] = [[π]] c Syntax in NNF: φ ::= π* φ 1 φ 2 φ 1 φ 2 π*uφ 2 π*rφ 2 Semantics: η π* iff η(0)œ[[π*]] η φ 1 φ 2 iff η φ 1 and η φ 2 η φ 1 φ 2 iff η φ 1 or η φ 2 η π*uφ iff $t 0 s.t. η t η φ 2 and "sœ[0,t] η(s)œ[[π*]] η π*rφ iff "t 0 η t η φ 2 or $sœ[0,t] s.t. η(s)œ[[π*]] Some notation: η t (s) = η(t+s) LTL(op 1,, op n ) φ = ΤUφ φ = F Rφ

13 LTL(U,, ) to HA 1 Algorithm Modular construction using using the the structure of of φ. φ. (ª (ª proof proof by by induction) Syntax in NNF: φ ::= π* φ 1 φ 2 φ 1 φ 2 π*uφ Ex. D(π 2 D(π 3 D(π 4 (Ÿπ 2 Ÿπ 3 )Uπ 1 )))

14 LTL(U,, ) to HA Proposition 2 (base (base case): case): Let Let φ = π, π, then then H π* = π* (X, (X,{v}, {v},«, «, X, X, n n,,[[π*]], [[π*]],«, «,{v}) [[π*]] v X n proof immediate from definition: η π* iff η(0)œ[[π*]]

15 Basic Building Blocks φ = φ 1 φ 2 or φ = φ 1 φ 2 Proposition 1 [Henzinger 95]: 95]: Let Let H 1 and 1 and H 2 be 2 be hybrid hybrid automata. If If H = H 1» 1 H 2, 2, then then L(T L(T H ) H ) = L(T L(T H1 ) H1 )» L(T L(T H2 ). H2 ). If If H = H 1 1 H 2, 2, then then L(T L(T H ) H ) = L(T L(T H1 ) H1 ) L(T L(T H2 ). H2 ).

16 LTL(U,, ) to HA Proposition 3: Let φ = π*uψ, then H φ = (X, V»{v }, E, Inv, Flow, Init, Guard, F) Let H ψ = (X, V, E, Inv, Flow, Init, Guard, F). Then: Inv (v ) = Init (v ) = [[π*]] and Flow(v ) = n Inv (v) = Inv(v) for all vœv Flow (v) = Flow(v) for all vœv Init (v) = Init(v) [[π*]] for all vœv E = E»{(v,v) vœv in } Guard (v, v) = Init (v) for all vœv in Guard (e) = Guard(e) for all eœe where V in = {vœv Init(v) [[π*]] «} [[π 1 ]] Ex. π 1 Uπ 2 [[π 2 ]] [[π 1 ]] H ψ v [[π 1 ]] n v X n proof immediate from definition: η π*uψ iff $t 0 s.t. η t η ψ and "sœ[0,t] η(s)œ[[π*]]

17 LTL(U,, ) to HA

18 LTL(U,, ) to HA Remarks: Fairness conditions Computational complexity Size of H1 H2: O(n 1 n 2 ) Size of final automaton O(exp( φ )) Set intersections and complementation set theoretic operations, not reachability etc. Note: graph is acyclic Final hybrid automaton might have empty guards and invariant sets Then formula might be unsatisfiable

19 LTL U, fragment Similar to LTL(U,, ) we can get an algorithm for LTL(,, ) General solution for φ 1 φ 2 or φ 1 φ 2 with φ 1 œltl(u,, ) and φ 2 œltl(,, ) More general solution for any formula: φ ::= π* φ φ 1 φ 2 φ 1 φ 2 π*uφ φ œltl(,, ) Εχ. π 0 D(π 2 D(π 3 D(π 4 (Ÿπ 2 Ÿπ 3 )Uπ 1 )))

20 Abstract Hybrid Automaton π 0 Εχ. Εχ. π 0 Dπ 0 Dπ 2 Dπ 2 Dπ 3 3 π 4 x 2 30 π π 1 π 3 Algorithm x 1 s 0 s 1 s 2 s 3

21 Abstract Hybrid Automaton Εχ. Εχ. π 0 Dπ 0 Dπ 2 Dπ 2 Dπ x Algorithm x s 0 20 x 2 15 s 1 s s x 1

22 A practical solution Backward reachability using Breadth First Search: s 0 fix initial conditions one outgoing edge from each control location s 1 s 2 s 3

23 2 Derive controller constraints s 0 s 1 s 2 Feedback control law g c : XöU Controller constraints c = {A, B, Γ} A initial conditions, i.e. x(0)œa Γ final conditions, i.e. x(t g )œγ B invariant, "tœ[0,t g ].x(t)œb s 3 A B Γ c 3 = {» eœi3 Guard(e), Inv(s 3 ), «} ï g c3 c 1 = {Guard(e I1 )»Init(s 1 ), Inv(s 1 ), Guard(e O1 )} ï g c1 c 2 = {Init(s 2 ), Inv(s 2 ), Guard(e O2 )} ï g c2 c 0 = {Init(s 0 ), Inv(s 0 ), Guard(e O1 )} ï g c0 c 0 = {Init(s 0 ), Inv(s 0 ), Guard(e O1 )} ï g c0

24 Main result Let Let φœltl φœltl U, U, be be satisfiable wrt wrtto to our our model model of of computation. Then: Then: (i) (i) φ can can be be converted into into list list of of controller specifications. (ii) (ii) The The hybrid hybrid automaton H resulting from from the the closed-loop dynamics satisfies: "ηœl(h).. η φ

25 Back to the toy example π 4 30 RTL: π 0 D(π 0 D(π 2 D(π 2 D(π 3 D(π 3 D(π 4 (Ÿπ 4 2 Ÿπ 2 Ÿπ 3 )Uπ 3 )Uπ 1 ))) 1 ))) 30 x π 0 π 1 π 2 π x 1 x 2 15 x x x x s 0 s 1 s 2 s 3 s x x 2 15 x x x 1 13 Controllers

Back to the toy example 30 25 20 π 0 π 4 x 2 15 10 π 1 π 2 π 3 5 0

26 Back to the toy example π 0 π 4 x π 1 π 2 π x 1 (a) (b) Conner et. al. IROS 2003

27 Conclusions Top-down approach for the synthesis of HA from temporal logic identified a fragment of LTL that provides a modular construction algorithm is based on the closure properties of HA Advantages of the proposed framework: Clean separation between software specification and controller design Potentially smaller number of required controllers The closed-loop dynamics can be designed using different methodology in each control location Real continuous time semantics Ability to distinguish between events that must hold at a point in time or for an interval

28 Future work Introduce robustness into the system Fainekos, Girard, Pappas: Hierarchical Synthesis of Hybrid Controllers from Temporal Logic Specifications, HSCC 2007 (to appear) Complete the framework for the full LTL i.e. add liveness Design for distributed specifications Built a complete toolbox with libraries of controllers

29 Thank You! Any Question(s)?

Hierarchical Synthesis of Hybrid Controllers from Temporal Logic Specifications

Hierarchical Synthesis of Hybrid Controllers from Temporal Logic Specifications Georgios E. Fainekos 1, Antoine Girard 2, and George J. Pappas 3 1 Department of Computer and Information Science, Univ.