D. Ishii, K. Ueda, H. Hosobe, A. Goldsztejn: Interval-based Solving of Hybrid Constraint Systems, in Preprints of the 3rd IFAC Conference on Analysis and Design of Hybrid Systems (ADHS'09), pp. 144-149, 2009.

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

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

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

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

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

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

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

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

Example of
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

Example of
• 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

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

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

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

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

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

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

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

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)))

Overview of the Box enclosure of
Proposed Algorithm: the trajectories
1. If 0 ∉ G(YT0,Y0(T1)), computed by the
ODE solver
→ 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

Overview of the
Proposed Algorithm:
1. If 0 ∉ G(YT0,Y0(T2,l)),
→ The condition holds,
so finish the process

T2,l T2,u

T2

Overview of the
Proposed Algorithm:
1. If 0 ∉ G(YT0,Y0(T2,u)),
→ 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

Experiments (Overview)

• Implementation:
Elisa (an impl. of branch-and-prune)
[Granvilliers, 05]
+
VNODE-LP (an interval-based ODE solver)
[Nedialkov, 06]

+ Some optimizations

22

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

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

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

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

D. Ishii, K. Ueda, H. Hosobe, A. Goldsztejn: Interval-based Solving of Hybrid Constraint Systems, in Preprints of the 3rd IFAC Conference on Analysis and Design of Hybrid Systems (ADHS'09), pp. 144-149, 2009.

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D. Ishii, K. Ueda, H. Hosobe, A. Goldsztejn: Interval-based Solving of Hybrid Constraint Systems, in Preprints of the 3rd IFAC Conference on Analysis and Design of Hybrid Systems (ADHS'09), pp. 144-149, 2009.