enclosing temporal evolution of dynamical systems us ......enclosing temporal evolution of dynamical...

Enclosing Temporal Evolution of Dynamical Systems Us-ing Numerical Methods.

Olivier Bouissou (CEA LIST)Alexandre Chapoutot and Adel Djoudi (ENSTA Paristech)

NFM 2013May 14-16, 2013

Problem motivationHybrid systems verification

I We consider Hybrid Systems given as a state space system:

x = fx(t, x) Problem dynamic

x = fz(t, x) when z(t, x) Reset function

I Note that fx is not necessarily continuous but piecewise continuous.

I We want to compute bounds on the system trajectory, i.e. a piecewisecontinuous function x∞ such that:

∀t ∈ R+, x ′∞(t) = fx(t, x∞(t)) and x+∞(t) = fz(t, x−∞(t)) if z(t, x−∞(t)) .

with x ′ the right-derivative, x+ the right-limit and x− the left-limit.

I Two main issues:1.2. detecting and solving the discrete jumps

2 / 27NFM 2013, May 14-16, 2013

Problem motivationHybrid systems verification

I We consider Hybrid Systems given as a state space system:

x = fx(t, x) Problem dynamic

x = fz(t, x) when z(t, x) Reset function

I Note that fx is not necessarily continuous but piecewise continuous.

I We want to compute bounds on the system trajectory, i.e. a piecewisecontinuous function x∞ such that:

∀t ∈ R+, x ′∞(t) = fx(t, x∞(t)) and x+∞(t) = fz(t, x−∞(t)) if z(t, x−∞(t)) .

with x ′ the right-derivative, x+ the right-limit and x− the left-limit.

I Two main issues:1. solving the differential equations2. detecting and solving the discrete jumps

2 / 27NFM 2013, May 14-16, 2013

Problem motivationTime ellapse operator

Definition: given an ODE x = f (x), an initial state x0 ⊆ Rn and a duration h,compute the set

Rf (x0, h) ={x(t) | t ∈ [0, h], x = f (x) and x(0) ∈ x0

}I Basis for hybrid systems reachability algorithm.

I Generally not exactly computable, depending on the nature of f and therepresentation for x0:

1. if f is linear and x0 is a convex polyhedra, Rf (x0, h) can be exactlycomputed using matrix exponential.

2. if f is not linear, over-approximation must be performed.

3 / 27NFM 2013, May 14-16, 2013

Problem motivationLinear case

The problem is:

x = Ax x(0) ∈ x0

with A ∈ Rn×n and x0 ⊆ Rn.

I Can be resolved exactly.

I x(t) = x0 · eAt .I Solution from SpaceEx [Frehse et al. 2011]:

1. Get an eclosure R0 of x on [0, h] using an Euler like extrapolation.

2. Compute Ri = Ri−1 · eAh.

3. Ri is an enclosure of x on [ih, (i + 1)h].

4. Must compute the matrix exponential eAh.

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The GRKLib method[Bouissou Martel 2007]

I More general method to compute over-approximations of Rf (x0, h).

I Depends on interval arithmetics: x0 must be given as an interval.

I Modifies the numerical method RK4 that computes approximations ofx∞(h):

k1 = f (0, x0)k2 = f

(h2, x0 + h

2k1

)k3 = f

(h2, x0 + h

2k2

)k4 = f (h, x0 + hk3)x = x0 + h

6

(k1 + 2k2 + 2k3 + k4

)

h

I Picard iteration

I Taylor expansion

5 / 27NFM 2013, May 14-16, 2013





k1 = f (0, x0)k2 = f

(h2, x0 + h

2k1

)k3 = f

(h2, x0 + h

2k2

)k4 = f (h, x0 + hk3)x = x0 + h

6

(k1 + 2k2 + 2k3 + k4

)

h

I Picard iteration∀t ∈ [0, h], x∞(t) ∈ R

I Taylor expansion

5 / 27NFM 2013, May 14-16, 2013





k1 = f (0, x0)k2 = f

(h2, x0 + h

2k1

)k3 = f

(h2, x0 + h

2k2

)k4 = f (h, x0 + hk3)x = x0 + h

6

(k1 + 2k2 + 2k3 + k4

)

h

I Picard iteration

I Taylor expansionx∞(t) ∈ x + e

5 / 27NFM 2013, May 14-16, 2013

The GRKLib methodPicard operator

I Given an interval x0 such that x(0) ∈ x0, we look for R such that∀t ∈ [0, h], x(t) ∈ R.

I Iterative algorithm:1. Set R0 = x0 t x where x is the result of RK4 method using interval

arithmetic.2. n = 03. while(true)4. Rn+1 = x0 + [0, h]f (Rn)5. if Rn+1 ⊆ Rn break;6. else n = n + 1

I This process ends if f is globally Lipschitz.

I The post fixpoint R can be used as an over-approximation of Rf (t, x0).

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The GRKLib methodEnclosure at h

I The enclosure R obtained previously is generally too large to continue theintegration on [h, 2h].

I We use a different method to compute a tight approximation xh at h, fromwhich we start the same process to reach 2h.

I Computation of xh: xh = Φ(x0) + e where

Φ(x0, h) = x0 + h6

(k1 + 2k2 + 2k3 + k4)

e = h5

120

(dΦ5

dt5 (x0, [0, h])− df 4

dt4 ([0, h],R))

7 / 27NFM 2013, May 14-16, 2013

The GRKLib methodLimitations of GRKLib

xh = x0 +h

6(k1 + 2k2 + 2k3 + k4) +

h5

120

(dΦ5

dt5(x0, [0, h])− df 4

dt4([0, h],R)

)1. Use of interval arithmetics

I This formula makes the width of x increase.I Solution: use of interval center-ray form and computation of the Jacobian

of Φ (same as for VNODE).

I Wrapping effect that makes the error increase.

I One solution: use Lohner QR decomposition, but complexity is n3 where nis the dimension of the ODE.

8 / 27NFM 2013, May 14-16, 2013

The GRKLib methodLimitations of GRKLib

xh = x0 +h

6(k1 + 2k2 + 2k3 + k4) +

h5

120

(dΦ5

dt5(x0, [0, h])− df 4

dt4([0, h],R)

)1. Use of interval arithmetics

2. Use of only RK4 method

I There exists many other numerical methods.I In Simulink, more than 6 Runge-Kutta like methods.

I Each performs well on different problemsI See experimentations.

I We can bound the error for each method, using a method similar toGRKLib.

I More on that later.

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Main contribution of this work

Solution to these problems1. Use of affine arithmetics.

2. Use of a generic guaranteed solver.

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Affine sets

I Extension of interval arithmetics to reduce the dependency issue.

[a, a] + [b, b] = [a + b, a + b]

x = [0, 1] ⇒ x − x = [−1, 1]

I Main idea: parametrize variables w.r.t. a set of noise symbols εi withεi ∈ [−1, 1].

x = x0 + x1ε1 + x3ε3

y = y0 + y1ε1 + y2ε2

I This representation encodes the linear relation between noise symbols andvariables and allows for precise linear transformation.

I What is a noise symbol ?I Initial uncertainty: [a, b]→ a+b

2+ b−a

2ε

I Non-linear operations

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Affine setsGeometric representation

Given m variables x1, x2, . . . , xm, an affine set

x j = x j0 +

n∑i=1

x ji εi

is a zonotope Z = γ(x1, . . . , xm).

Example: x = 20− 4ε1 + 2ε3 + 3ε4 y = 10− 2ε1 + ε2 − ε4

x

y

12 / 27NFM 2013, May 14-16, 2013

Affine setsArithmetics

Given two affine sets

x = x0 +n∑

i=1

xiεi y = y0 +n∑

i=1

yiεi

I Addition, subtraction and scalar multiplication (κ ∈ R)

κ · x ± y = κ · x0 ± y0 +n∑

i=1

(κ · xi ± yi )εi

I Multiplication

x × y = x0y0 +n∑

i=1

(y0xi + x0yi )εi +n∑

i=1

xiεi

n∑i=1

yiεi

= x0y0 +n∑

i=1

(y0xi + x0yi )εi + rn+1εn+1

I Other operations (sin, cos, . . . ) are computed using Taylor series.

Remark: each non-linear operation generates a new noise symbol.

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Runge-Kutta numerical scheme

Any explicit Runge-Kutta method can be described by a “Butcher table”:

, e.g.ODE23 method :

d2 a21

d3 a31 a32

......

. . .

ds as1 as2 · · · as,s−1

w1 w2 · · · ws

w ′1 w ′2 · · · w ′s

For some metods, the second extrapolation zn+1 allows for an approximation ofthe integration error and thus a step-size control.

15 / 27NFM 2013, May 14-16, 2013

Runge-Kutta numerical scheme

Any explicit Runge-Kutta method can be described by a “Butcher table”, e.g.ODE23 method :

012

12

34

0 34

1 29

13

49

29

13

49

724

14

13

18

k1 = f (tn, xn)

k2 = f (tn +1

2hn, xn +

1

2hk1)

k3 = f (tn +3

4hn, xn +

3

4hk2)

xn+1 = xn + h

(2

9k1 +

1

3k2 +

4

9k3

)k4 = f (tn + 1hn, xn+1)

zn+1 = xn + h

(7

24k1 +

1

4k2 +

1

3k3 +

1

8k4

)

For some metods, the second extrapolation zn+1 allows for an approximation ofthe integration error and thus a step-size control.

15 / 27NFM 2013, May 14-16, 2013

(Non-guaranteed) Set-based integration usingaffine arithmetics

[Bouissou et al., RSP 2012]

We consider a numerical integration scheme, e.g. ODE23:

k1 = f (tn, xn) k2 = f (tn +1

2hn, xn +

1

2hk1) k3 = f (tn +

3

4hn, xn +

3

4hk2)

xn+1 = xn + h

(2

9k1 +

1

3k2 +

4

9k3

)I Write x = Φ(x0, h).

I Suppose x0 ∈ x0 where x0 is an affine set.

I Suppose we evaluate all above equations using affine arithmetics,computing an affine set x .

I Then clearly x ∈ x and x is a tight enclosure of {Φ(x0, h) | x0 ∈ x0}.

16 / 27NFM 2013, May 14-16, 2013

(Non-guaranteed) Set-based integration usingaffine arithmetics

Example: car.ode

init x = [0,1];init y = [0,1];init theta = [0,0.1];init v = 7;init delta = 0.0;

u1 = 0;u2 = 0.2;

x’ = v*cos(delta)*cos(theta);y’ = v*cos(delta)*sin(theta);theta’ = v*sin(delta)/5;v’ = u1;delta’ = u2;

17 / 27NFM 2013, May 14-16, 2013

Runge-Kutta numerical schemeComputing the error

x0

Φ(x0, h)

I How can we perturb Φ(x0, h) so that itcontains x∞(x0, h) for all x0 ∈ x0 ?

I We use the order condition ofRunge-Kutta methods.

Order condition: a method is of order p iff

∀k ∈ [1, p]dkΦ

dtk(x0, 0) =

dkx∞dtk

(0)

Example:

I Euler method is of order 1

I ODE23 method is of order 2

I RK4 method is of order 4

I ODE78 method is of order 7

18 / 27NFM 2013, May 14-16, 2013

Runge-Kutta numerical schemeComputing the error

x0

Φ(x0, h)

I How can we perturb Φ(x0, h) so that itcontains x∞(x0, h) for all x0 ∈ x0 ?

I We use the order condition ofRunge-Kutta methods.

Order condition: a method is of order p iff

∀k ∈ [1, p]dkΦ

dtk(x0, 0) =

dkx∞dtk

(0)

Consequence: the error e = x∞(x0, h)− Φ(x0, h) is

e =hp+1

(p + 1)!

(f (p) (ξ, x∞(ξ))− dp+1Φ

dtp+1(x0, η)

)with ξ, η ∈ [0, h].

18 / 27NFM 2013, May 14-16, 2013

Guaranteed integration using affine arithmeticsComputing the error

e =hp+1

(p + 1)!

(f (p) (ξ, x∞(ξ))− dp+1Φ

dtp+1(x0, η)

)We need to evaluate this.

Two steps:

1. compute the apriori enclosure R such that ∀ξ ∈ [0, h], x∞(ξ) ∈ R.

2. compute the error e.

19 / 27NFM 2013, May 14-16, 2013


e =hp+1

(p + 1)!

(f (p)

([0, h], R

)− dp+1Φ

dtp+1(x0, η)

)We need to evaluate this. Two steps:



19 / 27NFM 2013, May 14-16, 2013


e =hp+1

(p + 1)!

(f (p)

([0, h], R

)− dp+1Φ

dtp+1(x0, [0, h])




19 / 27NFM 2013, May 14-16, 2013


e =hp+1

(p + 1)!

(f (p)

([0, h], R

)− dp+1Φ

dtp+1(x0, [0, h])




Step 1.

I we use again the Picard operator, except this time with affine arithmetic.

I comparison of affine form: we can use the order relation as in [Goubault,Putot, 2011].

19 / 27NFM 2013, May 14-16, 2013


e =hp+1

(p + 1)!

(f (p)

([0, h], R

)− dp+1Φ

dtp+1(x0, [0, h])




Step 2.

I a direct evaluation of the formula in affine arithmetics gives anover-approximation of the error.

I no need to perform QR decomposition as for intervals as affine arithmeticsdoes not suffer from wrapping effect.

19 / 27NFM 2013, May 14-16, 2013

Guaranteed integration using affine arithmeticsControlling the step-size

From the Picard iteration:

I it may happen that h is too large for Picard iteration to terminate.

I after a fixed number of iterations, we reject the step-size and start againfrom h/2.

Next we can adapt the step-size to reach a user-defined tolerance.Two ways to define the error:

I from the embedded numerical scheme (ODE23 for instance);

I from the guaranteed error.

The second option is more efficient as it reduces the number of rejected Picarditeration

20 / 27NFM 2013, May 14-16, 2013

Implementation

We developped a tool named RKN:

I contains 12 methods;

I parses a simple language for defining ODEs;

I uses a formal differentiation tool to compute the derivatives.

We compared the results of RKN with VNODE:

I VNODE is the state of the art tool for guaranteed integration;

I it is based on interval arithmetics and Taylor series expansion;

I results show that RKN is more “adaptable” than VNODE.

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Implementation issues

Problem with affine arithmetics: creation of one noise symbol per non-linearoperation.

I Even an addition creates a noise symbol when computing with floatingpoint numbers !

I Use a sparse representation for the list of noise symbols.

I Truncate the list of noise symbols by keeping only the ones bigger than auser-defined threshold.

Avoiding the creation of noise symbols:

I we compute the error term using interval arithmetics.

I we convert it to affine arithmetics and add it to the current value.

I similar to following the numerical scheme and perturbing it by intervalerror terms.

22 / 27NFM 2013, May 14-16, 2013

Experimental results1. Oil-reservoir problem

init y = 10

init z = 0

y’ = z

z’ = (z * z) - (3 / (0.001 + (y * y)))

I Non-linear ODE with a very stiff behaviour around t = 35.I Different numerical methods give very different approximations.

30 32 34 36 38 40

−5

0

5

10Euler method

Fixed step ODE23

Variable step ODE23

23 / 27NFM 2013, May 14-16, 2013


init y = 10

init z = 0

y’ = z

z’ = (z * z) - (3 / (0.001 + (y * y)))


Results with RKN:

34.6 34.8 35 35.2 35.4

−20

−10

23 / 27NFM 2013, May 14-16, 2013


init y = 10

init z = 0

y’ = z

z’ = (z * z) - (3 / (0.001 + (y * y)))


Results with RKN:

I using the guaranteed Heun method, we could enclose the solution.

I computation time is ∼10s.

I the step-size adaptation is necessary to reduce h when the problembecomes stiff.

I VNODE could not compute enclosures of it (problem with the Taylorseries expansion of 1/(0.001 + y 2)).

23 / 27NFM 2013, May 14-16, 2013

Experimental results2. High dimensional problems

y1

y2

y3

...yn

=

−2 1 0 0 · · · 01 −2 1 0 · · · 00 1 −2 1 · · · 0

. . .0 0 · · · 0 1 −2

y1

y2

y3

...yn

with y(0) =

100...0

.

I C3 problem from the DETEST suite.

I We study the computation time of RKN with increasing dimension.

I We use the RK4 numerical scheme and a tolerance of 10−6.

RKN VNODEDim T Prec T Prec

40 1.064 1.284 · 10−5 7.46 2.175 · 10−07

80 2.551 1.432 · 10−5 58.89 2.175 · 10−07

120 4.297 1.386 · 10−5 196.56 2.175 · 10−07

140 5.285 1.440 · 10−5 334.08 2.175 · 10−07

24 / 27NFM 2013, May 14-16, 2013

Experimental results3. Stiff problem

init y1 = 2

init y2 = 0

y1’ = y2

y2’ = 5 * (1 - (y1 * y1)) * y2 - y1

25 / 27NFM 2013, May 14-16, 2013

Experimental results3. Stiff problem

init y1 = 2

init y2 = 0

y1’ = y2

y2’ = 5 * (1 - (y1 * y1)) * y2 - y1

I We compute over-approximations of the trajectories.

I Computation time: 0.12s with ODE23

I On this example, VNODE cannot reach t = 8 with order 4, requires atleast order 8

I Computation time for VNODE, order 10: 0.08s

25 / 27NFM 2013, May 14-16, 2013

Experimental results4. Some pictures

init x = [0,1];

init theta = [0,0.1];

init y = [0,1];

init v = 7;

init delta = 0.0;

x’ = v*cos(delta)*cos(theta);

theta’ = v*sin(delta)/5;

y’ = v*cos(delta)*sin(theta);

delta’ = 0.2;

26 / 27NFM 2013, May 14-16, 2013


init x1 = [0.8,1]

init x2 = 0

tmp = x1*x1-1

x1’ = x2

x2’ = -0.2 * tmp * x2 - x1

26 / 27NFM 2013, May 14-16, 2013


init u = [0.95,1.05];

init v = [-1.05,-0.95];

u’ = v;

v’ = (u^2);

26 / 27NFM 2013, May 14-16, 2013

Conclusion

We presented:

I a generic framework for making explicit numerical methods guaranteedODE solvers.

I an interesting use of affine arithmetics which shows to be a good tradeoffbetween complexity and accuracy for problems with many numericalcomputations.

I our method compares very well to VNODE and scales well tohigh-dimensional problems. It even outperforms VNODE on stiff problems.

Perspectives:

I study implicit ODE solvers which are known to work better for stiffproblems.

I extend this work to handle DAEs (basic equation in Modelica) and PDEs.

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enclosing temporal evolution of dynamical systems us ......enclosing temporal evolution of dynamical...

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