Controller Synthesis for Hybrid Systems using SMT Solving

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1 Controller Synthesis for Hybrid Systems using SMT Solving Ulrich Loup AlgoSyn (GRK 1298) Hybrid Systems Group Prof. Dr. Erika Ábrahám GRK Workshop 2009, Schloss Dagstuhl

2 Our model: hybrid system Discrete controller in a continuous setting Simple example: v [0, 2] AlgoSyn (GRK 1298) Ulrich Loup 2

3 Our model: hybrid system Discrete controller in a continuous setting Simple example: v = 0 x = 0 l 0 ẋ = v v [0, 2] v v max l 1 ẋ = v v [ 2, 0] v 0 accelerate brake AlgoSyn (GRK 1298) Ulrich Loup 2

4 Our model: hybrid system Discrete controller in a continuous setting Simple example: v = 0 x = 0 l 0 ẋ = v v [0, 2] v v max l 1 ẋ = v v [ 2, 0] v 0 accelerate brake Parameter synthesis: How is v max to be chosen such that the car can always be stopped in 0.2 meters? AlgoSyn (GRK 1298) Ulrich Loup 2

5 Our model: hybrid system Discrete controller in a continuous setting Simple example: v = 0 x = 0 l 0 ẋ = v v [0, 2] v v max l 1 ẋ = v v [ 2, 0] v 0 accelerate Parameter synthesis: brake v max v 0... v 4... ( x 4 x v 4 = )? AlgoSyn (GRK 1298) Ulrich Loup 2

6 Satisfiability checking over the reals Given: First-order formula ϕ over (R, +,, <). Question: Is ϕ satisfiable? Theoretical results decidable [Tarski1948] upper bound for time complexity 2 2n variables [BrownDavenport2007] where n is the number of AlgoSyn (GRK 1298) Ulrich Loup 3

7 Satisfiability checking over the reals Given: First-order formula ϕ over (R, +,, <). Question: Is ϕ satisfiable? Theoretical results decidable [Tarski1948] upper bound for time complexity 2 2n variables [BrownDavenport2007] where n is the number of Approach exploit practical experience in satisfiability checking in propositional logic (SAT solving) AlgoSyn (GRK 1298) Ulrich Loup 3

8 Satisfiability modulo theories (SMT) SAT solver y 2 1 = 0 x 1 = 0 ( x y + 1 = 0 x + y 1 = 0 ) )? Theory solver AlgoSyn (GRK 1298) Ulrich Loup 4

9 Satisfiability modulo theories (SMT) a b c (d e)? SAT solver y 2 1 = 0 x 1 = 0 ( x y + 1 = 0 x + y 1 = 0 ) )? Theory solver AlgoSyn (GRK 1298) Ulrich Loup 4

10 Satisfiability modulo theories (SMT) a b c (d e)? a b SAT solver sat y 2 1 = 0 x 1 = 0 ( x y + 1 = 0 x + y 1 = 0 ) )? Theory solver y 2 1 = 0 )? AlgoSyn (GRK 1298) Ulrich Loup 4

11 Satisfiability modulo theories (SMT) a b c (d e)? a b c d SAT solver Theory solver unsat b c d y 2 1 = 0 x 1 = 0 x y + 1 = 0 )? y 2 1 = 0 x 1 = 0 ( x y + 1 = 0 x + y 1 = 0 ) )? incremental minimal infeasible subset AlgoSyn (GRK 1298) Ulrich Loup 4

12 Satisfiability modulo theories (SMT) a b c (d e)? a b c e SAT solver Theory solver unsat b c e y 2 1 = 0 x 1 = 0 x + y 1 = 0 )? y 2 1 = 0 x 1 = 0 ( x y + 1 = 0 x + y 1 = 0 ) )? incremental minimal infeasible subset backtracking AlgoSyn (GRK 1298) Ulrich Loup 4

13 Satisfiability modulo theories (SMT) a b c (d e)? SAT solver y 2 1 = 0 x 1 = 0 ( x y + 1 = 0 x + y 1 = 0 ) )? Theory solver incremental minimal infeasible subset backtracking unsat, unsatisfiability proof AlgoSyn (GRK 1298) Ulrich Loup 4

14 Theory solver: existing implementations Cylindrical algebraic decomposition (CAD) QEPCAD, Reduce,... Gröbner bases Maple, Mathematica, Singular, Maxima, CoCoA, Reduce,... Other methods Virtual substitution (Reduce) Interval arithmetic (Ariadne) More on computer algebra: [Kaplan2005] AlgoSyn (GRK 1298) Ulrich Loup 5

15 Theory solver: CAD y y 2 1 = 0 1 )? 1 1 x 1 AlgoSyn (GRK 1298) Ulrich Loup 6

16 Theory solver: CAD y y 2 1 = 0 x 1 = 0 x y + 1 = 0 )? 1 incremental 1 1 x 1 AlgoSyn (GRK 1298) Ulrich Loup 6

17 Theory solver: CAD y y 2 1 = 0 x 1 = 0 x + y 1 = 0 )? 1 incremental minimal infeasible subset? backtracking 1 1 x 1 AlgoSyn (GRK 1298) Ulrich Loup 6

18 Theory solver: CAD y y 2 1 = 0 x 1 = 0 x + y 1 = 0 )? 1 incremental minimal infeasible subset? backtracking 1 1 x 1 AlgoSyn (GRK 1298) Ulrich Loup 6

19 Theory solver: Gröbner bases Mathematical background p 1 = 0... p k = 0 has no solution q 1p q k p k = 1 for some polynomials q 1,..., q k AlgoSyn (GRK 1298) Ulrich Loup 7

20 Theory solver: Gröbner bases Mathematical background p 1 = 0... p k = 0 has no solution q 1p q k p k = 1 for some polynomials q 1,..., q k {x 2 1} )? AlgoSyn (GRK 1298) Ulrich Loup 7

21 Theory solver: Gröbner bases Mathematical background p 1 = 0... p k = 0 has no solution q 1p q k p k = 1 for some polynomials q 1,..., q k y 2 1 = 0 {x 2 1, y 2 1} )? incremental AlgoSyn (GRK 1298) Ulrich Loup 7

22 Theory solver: Gröbner bases Mathematical background p 1 = 0... p k = 0 has no solution q 1p q k p k = 1 for some polynomials q 1,..., q k y 2 1 = 0 x 1 = 0 incremental )? {x 1, y 2 1} x 2 1 = (x + 1)(x 1) AlgoSyn (GRK 1298) Ulrich Loup 7

23 Theory solver: Gröbner bases Mathematical background p 1 = 0... p k = 0 has no solution q 1p q k p k = 1 for some polynomials q 1,..., q k y 2 1 = 0 x 1 = 0 x y + 1 = 0 )? incremental minimal infeasible subset {1} 1 = 1 3 (y 2 1) 1 3 (y + 2)(x 1) (y + 2)(x y + 1) AlgoSyn (GRK 1298) Ulrich Loup 7

24 Theory solver: Gröbner bases Mathematical background p 1 = 0... p k = 0 has no solution q 1p q k p k = 1 for some polynomials q 1,..., q k y 2 1 = 0 x 1 = 0 x + y 1 = 0 )? {x 1, y 2 1} incremental minimal infeasible subset backtracking AlgoSyn (GRK 1298) Ulrich Loup 7

25 Theory solver: Gröbner bases Mathematical background p 1 = 0... p k = 0 has no solution q 1p q k p k = 1 for some polynomials q 1,..., q k y 2 1 = 0 x 1 = 0 x + y 1 = 0 )? incremental minimal infeasible subset backtracking {1} 1 = (y 2 1) y(x 1) +y(x + y 1) AlgoSyn (GRK 1298) Ulrich Loup 7

26 Future work Enhance theory solver: Adjust the methods to the SMT framework (Redlog, Thomas Sturm, Universidad de Cantabria; Singular, Viktor Levandovskyy, RWTH Aachen). Manage differential equations. Generate unsatisfiability proofs. Integrate interval arithmetic (Ariadne, Pieter Collins, CWI Amsterdam). Work on the applications: Termination analysis (Computer science 2, RWTH Aachen) SFB 686 Niedertemperaturverbrennung AlgoSyn (GRK 1298) Ulrich Loup 8

27 References Alfred Tarski. A Decision Method for Elementary Algebra and Geometry. University of California Press, Christopher W. Brown and James H. Davenport. The complexity of quantifier elimination and cylindrical algebraic decomposition. In Dongming Wang, editor, ISSAC, pages ACM, Michael Kaplan. Computeralgebra AlgoSyn (GRK 1298) Ulrich Loup 9

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