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.

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