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An approach to reliable modeling, simulation and verification of hybrid systems is interval arithmetic, which guarantees that a set of intervals narrower than specified size encloses the solution. ...

An approach to reliable modeling, simulation and verification of hybrid systems is interval arithmetic, which guarantees that a set of intervals narrower than specified size encloses the solution. Interval-based computation of hybrid systems is often difficult, especially when the systems are described by nonlinear ordinary differential equations (ODEs) and nonlinear algebraic equations.We formulate the problem of detecting a discrete change in hybrid systems as a hybrid constraint system (HCS), consisting of a flow constraint on trajectories (i.e. continuous functions over time) and a guard constraint on states causing discrete changes. We also propose a technique for solving HCSs by coordinating (i) interval-based solving of nonlinear ODEs, and (ii) a constraint programming technique for reducing interval enclosures of solutions. The proposed technique reliably solves HCSs with nonlinear constraints. Our technique employs the interval Newton method to accelerate the reduction of interval enclosures, while guaranteeing that the enclosure contains a solution.

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- 1. Interval-based Solving of Hybrid Constraint Systems Sep. 17, 2009 Daisuke ISHII† Kazunori UEDA †,‡ Hiroshi HOSOBE‡ Alexandre GOLDSZTEJN * † Waseda University, Japan ‡ National Institute of Informatics, Japan * LINA, Universite’ de Nantes, France 1
- 2. Reliable Modeling, Simulation, and Verification of Hybrid Systems 1.Simple modeling of (possibly nonlinear) hybrid systems using interval constraints - [Henzinger, 00], [Hickey, 04], [Ratschan, 06], [Eggers, 08] - cf. Abstraction into piecewise linear systems Bouncing particle 3 ODE Initial constraint 2 1 Guard constraint 0 -1 1 2 3 4 5 6 7 8 9 2
- 3. Reliable Modeling, Simulation, and Verification of Hybrid Systems 2.Rigorous detection of a discrete change - Numeric techniques may compute unexpected results [Park, 96], [Esposito, 07] - Enclosing a solution by tight intervals or boxes - Guaranteeing the existence of a unique solution Bouncing particle 3 2 1 0 -1 1 2 3 4 5 6 7 8 9 3
- 4. Talk Outline 1. Hybrid constraint systems (HCSs) for formalizing the detection of discrete changes - Box-consistency for an HCS: A box enclosing a solution with a given accuracy 2. Interval-based technique for solving HCSs - Based on the branch-and-prune algorithm - Efficient domain reduction by the interval Newton method - Integration of ✴Interval-based solver for nonlinear constraints [van Hentenryck, 97] ✴Interval-based solver for nonlinear ODEs [Nedialkov, 99] 4
- 5. I = {r ∈ R | l ≤ r ≤ u}. W N (Validated) Interval Arithmetic I denotes a set of intervals. A box B is a tuple of [Moore, 66] n intervals I (I1 , . . . , In ). I n denotes a set of boxes. For an interval I, O • Extension of the lower bound, ub(I) denotes the upper lb(I) denotes numerical analysis W - Using intervals|I| denotes max{|lb(I)|, |ub(I)|}. For bound, int(I) denotes the (l, u ∈ F) or boxesdenotes the center of I, and [l, u] internal of I, m(I) (tuple a of intervals) instead of floating point numbers r ∈ R, [r] denotes an interval such that lb([r]) and ub([r]) e - Computed intervals rounded values to the their are the lower and upper enclose solutions and nearest d ﬂoating-point errors of r. round-off numbers I (over-approximation) n d For f : Rm → Rn , F : I m → I is called an f ’s interval • Let f be a function Rthe→ R , F :condition (Fi denotes extension iﬀ it satisﬁes m n m n following I → I is an t Interval extension ofvalue of F ) the i-th component of the f iff T a ∀I1 ∈ I · · · ∀Im ∈ I ∀r1 ∈ I1 · · · ∀rm ∈ Im ∀i ∈ {1, . . . , n} o (fi (r1 , . . . , rm ) ∈ Fi (I1 , . . . , Im )). V • For I1 ,constraint and1,...,xm)=0, F(X1,...,Xm) f0ais an 3 For a . . . , Im ∈ I f(x an interval extension F of , box F (I1 , . . . , Im ) is called an interval enclosure of possible interval f over I1 , . . . , Im . For a bounded set R ⊂ R, 2R values of extension denotes the smallest interval I ∈ I that encloses R. For a 5 v
- 6. Let y denotes a vector-valued continuous function over equations time R → R y(t) = f trajectory. y(t ) = y , value problem n ˙ called (t, y(t)) ∧ An initial for an ODE (IVP-ODE) is formed n+1 theODEs Interval-based Solving of 0 conjunction of 0 by where initial R, y0 ∈ R and f anR equations ∈ value problem for : ODE → R (assuming • An t0 n n Lipschitz continuity). Given an IVP-ODE, a solution de- y(t) = f (t, y(t)) ∧ y(t0 ) = y0 , ˙ noted by yt0 ,y0 is a trajectory that satisﬁes the equations. where t0 ∈ R, is0 a∈ Rn and f yt0,y0(t) : R RnR(assuming • A solution y trajectory : Rn+1 → → n Given an continuity). Given(Y0 , IVP-ODE, a, solution de- Lipschitz initial value set n+1 T0 ) ∈ I an n+1 an interval • Given aof the(Ya,trajectory , ,an interval extensionIof box solution y extension yt0 ,y0 is noted by 0 T0) I t0 ,y0 denoted by YT0 ,Y0 : that satisﬁes the equations. → I ,ysatisﬁes the following → In such that n+1 n t0,y0(t) is YT0,Y0(T) : I condition Given an initial value set (Y0 , T0 ) ∈ I , an interval ∀t0 ∈ T0 ∀y0 ∈ Y0 ∀t ∈ T (yt0 ,y0 ,i (t) ∈ YT0 ,Y0 ,i (T )), extension of the solution yt0 ,y0 , denoted by YT0 ,Y0 : I → where T is athe following condition lb(T ) ≥ ub(T0 ). I n , satisﬁes time interval such that Example 0 We employ T0 ∀y0 ∈ Y0 ∀t ∈ T (yt0VNODE T0 ,Y0 (T )), in Ne- ∀t0 ∈ an existing method ,y0 (t) ∈ Y proposed YT0,Y0(T) dialkov et a time interval such that lb(T ) ≥ ub(T0solving al. (1999)boxed value Initial and Nedialkov (2006) for -10 where T is (T0, Y0) ). IVP-ODEs based -20 interval arithmetic. Consider an IVP- on We employ an existing method ,VNODE proposedinterval ODE, an initial -30value set (T0 Y0 ) and a time in Ne- et obtain a and = YT ,YT (2006) for solving dialkov We al. (1999)box Y1Nedialkov(T1 ) using VNODE. T1 ∈ I. 0 0 IVP-ODEs based on interval arithmetic. Consider 0.0 0.5 1.0 1.5 2.0 2.5 an IVP- 6
- 7. Hybrid Constraint Systems • An hybrid constraint system (HCS) consists of: - A flow constraint ✴flow(x0, x1,..., xn) - A guard constraint ✴grd(x1,..., xn) - An initial box ✴D0=(X0,0, X0,1,..., X0,n) 7
- 8. Example of Hybrid Constraint Systems (HCSs) • Particle falling towards a sine-waved ground surface Variables X = (t, px, py, vx, vy) Trajectory time position velocity y(τ) : R → R4 3 3 2 2 1 1 0 0 -1 -1 Bouncing particle 0.0 0.2 0.4 0.6 0.8 1 2 3 4 5 8
- 9. Example of Hybrid Constraint Systems (HCSs) Flow constraint flow(t, px, py, vx, vy): y’=(yvx, yvy, 0, -9.8-0.3 yvy) ODE ∧ y(t0)=y0 Initial value Variables X = ∧ y(t)=(px, py, vx, vy) ∧ t>t0 (t, px, py, vx, vy) State causing a discrete change 3 3 2 2 Guard constraint grd(px, py, vx, vy): 1 1 sin(2 px)-py=0 0 0 -1 -1 Bouncing particle 0.0 0.2 0.4 0.6 0.8 1 2 3 4 5 9
- 10. Example of Hybrid Constraint Systems (HCSs) • Initial box D0=(T0, Y0) providing initial values t0, y0 in the flow constraint - cf. y(t0)=y0 D0 3 3 2 2 1 1 0 0 -1 -1 Bouncing particle 0.0 0.2 0.4 0.6 0.8 1 2 3 4 5 10
- 11. Solutions of an HCS • A (theoretical) solution of an HCS is a valuation of variables satisfying the flow and guard constraints • An HCS may have multiple solutions 3 3 2 2 1 1 0 0 -1 -1 Bouncing particle 0.0 0.2 0.4 0.6 0.8 1 2 3 4 5 11
- 12. Box-Consistency for HCSs • Box D is given as a rough enclosure of solutions • Consider interval extensions of the flow constraint Flow and the guard constraint Grd D 3 3 Flow 2 2 1 1 Grd 0 0 -1 -1 Bouncing particle 0.0 0.2 0.4 0.6 0.8 1 2 3 4 5 12
- 13. Box-Consistency for HCSs • (Refined) box D’=(I0,...,[l k,u k],...,In) is box-consistent [Benhamou, 1994] iff ∀k∈{0,...,n} [ Flow(I0,...,[lk,lk+),...,In) ∧ Grd(I1,...,[lk,lk+),...,In) ∧ Flow(I0,...,(uk-,uk],...,In) ∧ Grd(I1,...,(uk-,uk],...,In) ] D The smallest interval at each bound 3 3 Flow 2 D’12 Grd 1 D’21 0 D’30 -1 -1 Bouncing particle 0.0 0.2 0.4 0.6 0.8 1 2 3 4 5 13
- 14. Interval-based Technique for Solving HCSs • Computation of a set of box-consistent boxes - Each box is narrower than the specified width • Based on the branch-and-prune algorithm [van Hentenryck, 97] • Integrated with an interval-based method for solving ODEs • Efficient reduction of an input box using the interval Newton method - Proof of the existence of a unique solution within a box 14
- 15. Application of the Interval Newton Method 1. Trajectory with respect to a flow constraint yt0,y0(t) 2. Composition with a guard constraint g(yt0,y0(t)) = 0 Computed by an interval-based 3. Interval extension ODE solver H(T) = G(YT0,Y0(T)) ∋ 0 4. Interval extension of the derivative of g yt0,y0 H’(T) = Σ ( δG/δXi 1≦i≦n dYT0,Y0(T)/dT ) 15
- 16. 3.2 Interval Newton Method Application of the Interval Newton Method Given an equation h(t) = 0, where h : R → R is a 5. If a time interval T contains a solution, an interval continuously diﬀerentiable function, a solution of the equation in an interval T is Newton operator an interval obtained by the interval also included in also obtained by the following interval operator contains the solution H([m(T )]) NH,H (T ) = T ∩ [m(T )] − , H (T ) ˙ where H and H are interval of T Midpoint extensions of h and its 6. Fixpoint H,H / ˙ derivative. Nof the interval Newton 0 ∈ H(T )Nholds. The (T ) is deﬁned iﬀ operator H,H’*(T) (uni-variate) interval Newton method iteratively reﬁnes an interval enclosureto contain a unique solution taking a • T is guaranteed by the operator above. By if suﬃciently small enclosure T of a solution, iterated appli- NH,H’(T) internal(T) holds cations of NH,H (T ) will converge. The ﬁxpoint is denoted by NH,H (T ). If the condition NH,H (T ) ⊆ int(T ) holds, a ∗ ˙ unique solution t∗ ∈ NH,H (T ) exists (see Theorem 8.4 in 16
- 17. Overview of the YT0,Y0(T1) Proposed Algorithm: 1. If 0 ∉ G(YT0,Y0(T1)), Possible trajectories yy0,t0(τ) return ∅ and finish w.r.t. D0=(T0,Y0) the flow constraint → 0 ∈ G(YT0,Y0(T1)) flow 2. Else calculate T2 = NH,H’*(T1) A region possibly satisfied by the guard constraint grd T1
- 18. Overview of the Proposed Algorithm: 1. If 0 ∉ G(YT0,Y0(T1)), return ∅ and finish → 0 ∈ G(YT0,Y0(T1)) 0 G(YT0,Y0(lb(T2))) 2. Else calculate T2 = NH,H’*(T1) 3. Is the box enclosure box-consistent? → No Solve the ODE at the T2 bounds of T2 using the minimal step size 0 G(YT0,Y0(ub(T2)))
- 19. Overview of the Box enclosure of Proposed Algorithm: the trajectories 1. If 0 ∉ G(YT0,Y0(T1)), computed by the ODE solver return ∅ and finish → 0 ∈ G(YT0,Y0(T1)) 2. Else calculate T2 = NH,H’*(T1) 3. Is the box enclosure box-consistent? → No T2,l T2,u 4. Split T2 into T2,l and T2,u and process recursively T2
- 20. Overview of the Proposed Algorithm: 1. If 0 ∉ G(YT0,Y0(T2,l)), return ∅ and finish → The condition holds, so finish the process T2,l T2,u T2
- 21. Overview of the Proposed Algorithm: 1. If 0 ∉ G(YT0,Y0(T2,u)), return ∅ and finish → 0 ∈ G(YT0,Y0(T2,u)) 2. Else calculate T3 = NH,H’*(T2,u) 3. Is the box enclosure T3 box-consistent? → Yes, T2,l T2,u return D’ and finish D’ T2
- 22. Experiments (Overview) • Implementation: Elisa (an impl. of branch-and-prune) [Granvilliers, 05] + VNODE-LP (an interval-based ODE solver) [Nedialkov, 06] + Some optimizations 22
- 23. Experiments (Overview) • Efficiency of the proposed method - The number of reductions and computation time were reduced, compared to the method not applying the interval Newton method ✴ 1.5-12%, and 11-23%, respectively - The proposed method took about 200% of computation time (at least), compared to the (non-validated) numeric computation on Mathematica 23
- 24. Conclusion and Future Work 1. Hybrid constraint systems (HCSs) describe (an over-approximation of) hybrid systems using constraints 2. Interval-based technique for solving HCSs - Guarantees the existence and uniqueness of a solution in a box enclosure • Future work: Application to (bounded) model checking - Integration with SAT solvers (cf. SMT) 24
- 25. References • [van Hentenryck, 1997] P. van Hentenryck, et al.: Solving polynomial systems using a branch and prune approach. In J. on Numerical Analysis, 34(2), pp. 797-827. SIAM, 1997. • [Nedialkov, 1999] N. S. Nedialkov, et al.: Validated solutions of initial value problems for ordinary differential equations. Applied Mathematics and Computation, vol. 105 (1), pp. 21-68. Elsevier, 1999. • [Ishii, 2008] , , : . , MPS-68, pp. 133-136. 2008. 25
- 26. References (cont.) • B. Carlson and V. Gupta: Hybrid cc with Interval Constraints, In Proc. of HSCC 1998, LNCS 1386, pp. 80-95, 1998. • S. Ratschan and Z. She: Safety Verification of Hybrid Systems by Constraint Propagation Based Abstraction Refinement, In Proc. of HSCC 2005, LNCS 3414, 2005. • G. Frehse: PHAVer: Algorithmic Verification of Hybrid Systems past HyTech, In Proc. of HSCC 2005, LNCS 3414, pp. 258-273, 2005. • T. A. Henzinger, et al.: Beyond HyTech: Hybrid Systems Analysis Using Interval Numerical Methods, LNCS 1790, pp. 130-144, 2000. 26

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