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Using Taylor Models in the Reachability Analysis of Non-linear Hybrid Systems Xin Chen 1 Erika ´ Abrah´ am 1 Sriram Sankaranarayanan 2 1 RWTH Aachen University, Germany 2 University of Colorado, Boulder, CO. SWIM 2012 Xin Chen Using Taylor Models in the Reachability Analysis of Non-linear Hybrid Systems 1 / 43

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Using Taylor Models in the Reachability Analysisof Non-linear Hybrid Systems

Xin Chen1 Erika Abraham1 Sriram Sankaranarayanan2

1RWTH Aachen University, Germany

2University of Colorado, Boulder, CO.

SWIM 2012

Xin Chen Using Taylor Models in the Reachability Analysis of Non-linear Hybrid Systems 1 / 43

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Outline

1 Verification of hybrid systems

2 Taylor model method

3 Intersections of Taylor models and guards

4 Experimental results

5 Future work

Xin Chen Using Taylor Models in the Reachability Analysis of Non-linear Hybrid Systems 2 / 43

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Outline

1 Verification of hybrid systems

2 Taylor model method

3 Intersections of Taylor models and guards

4 Experimental results

5 Future work

Xin Chen Using Taylor Models in the Reachability Analysis of Non-linear Hybrid Systems 3 / 43

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Motivations

Given a hybrid system on which we would like to verifiy

1 Reachability − A given state set is reachable.

2 Safety − No unsafe state is reachable.

3 Liveness − Some good properties keep happening.

4 ...

Many important properties can be reduced to reachability.

Xin Chen Using Taylor Models in the Reachability Analysis of Non-linear Hybrid Systems 4 / 43

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Motivations

Given a hybrid system on which we would like to verifiy

1 Reachability − A given state set is reachable.

2 Safety − No unsafe state is reachable.

3 Liveness − Some good properties keep happening.

4 ...

Many important properties can be reduced to reachability.

Xin Chen Using Taylor Models in the Reachability Analysis of Non-linear Hybrid Systems 4 / 43

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Hybrid automata

· · ·

· · ·

· · ·

`

d~xdt = f (~x , t)

~x ∈ Inv x ∈ G→ ~x ′ = P(~x) + I

f (~x , t) is a polynomial

Continuous dynamics

Inv is defined by a constraint set

Mode invariant

G is defined by a constraintset, P is a polynomial, andI is an interval

Discrete transition

Xin Chen Using Taylor Models in the Reachability Analysis of Non-linear Hybrid Systems 5 / 43

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Reachability computation by flowpipe construction

Xin Chen Using Taylor Models in the Reachability Analysis of Non-linear Hybrid Systems 6 / 43

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Reachability computation by flowpipe construction

Xin Chen Using Taylor Models in the Reachability Analysis of Non-linear Hybrid Systems 6 / 43

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Reachability computation by flowpipe construction

Xin Chen Using Taylor Models in the Reachability Analysis of Non-linear Hybrid Systems 6 / 43

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Reachability computation by flowpipe construction

Xin Chen Using Taylor Models in the Reachability Analysis of Non-linear Hybrid Systems 6 / 43

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Reachability computation by flowpipe construction

Xin Chen Using Taylor Models in the Reachability Analysis of Non-linear Hybrid Systems 6 / 43

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Reachability computation by flowpipe construction

Xin Chen Using Taylor Models in the Reachability Analysis of Non-linear Hybrid Systems 6 / 43

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Reachability computation by flowpipe construction

Xin Chen Using Taylor Models in the Reachability Analysis of Non-linear Hybrid Systems 6 / 43

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Difficulties

Continuous part:

1 Overestimation is heavily accumulated.

2 Global accuracy improvement is inefficient.

Discrete part:

1 Flowpipe/guard and flowpipe/invariant intersections aredifficult to compute.

2 Reducing the complexity of intersections is also hard.

We try to find new trade-off between accuracy and efficiency.

Xin Chen Using Taylor Models in the Reachability Analysis of Non-linear Hybrid Systems 7 / 43

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Difficulties

Continuous part:

1 Overestimation is heavily accumulated.

2 Global accuracy improvement is inefficient.

Discrete part:

1 Flowpipe/guard and flowpipe/invariant intersections aredifficult to compute.

2 Reducing the complexity of intersections is also hard.

We try to find new trade-off between accuracy and efficiency.

Xin Chen Using Taylor Models in the Reachability Analysis of Non-linear Hybrid Systems 7 / 43

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Outline

1 Verification of hybrid systems

2 Taylor model method

3 Intersections of Taylor models and guards

4 Experimental results

5 Future work

Xin Chen Using Taylor Models in the Reachability Analysis of Non-linear Hybrid Systems 8 / 43

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High order approximations

Polynomial p is a k-order approximation of a smooth functionf : D → R iff

(a) f ∈ C k

(b) f (~c) = p(~c) for the center point ~c of D and for each0 < m ≤ k :

∂mf

∂~xm

∣∣∣∣~x=~c

=∂mp

∂~xm

∣∣∣∣~x=~c

.

Xin Chen Using Taylor Models in the Reachability Analysis of Non-linear Hybrid Systems 9 / 43

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Examples

Several high order approximations of f (x) = ex with x ∈ [−1, 1].

x

f (x)

-1 -0.5 0 0.5 10

1

2

3f (x) = ex

p(x) = 1

0-order approximation

Xin Chen Using Taylor Models in the Reachability Analysis of Non-linear Hybrid Systems 10 / 43

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Examples

Several high order approximations of f (x) = ex with x ∈ [−1, 1].

x

f (x)

-1 -0.5 0 0.5 10

1

2

3f (x) = ex

p(x) = 1 + x

1-order approximation

Xin Chen Using Taylor Models in the Reachability Analysis of Non-linear Hybrid Systems 10 / 43

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Examples

Several high order approximations of f (x) = ex with x ∈ [−1, 1].

x

f (x)

-1 -0.5 0 0.5 10

1

2

3f (x) = ex

p(x) = 1 + x + x2

2

2-order approximation

Xin Chen Using Taylor Models in the Reachability Analysis of Non-linear Hybrid Systems 10 / 43

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Taylor models

Introduced by Berz and Makino.

Taylor model: a pair (p, I ) over an interval domain D. Itdefines the set

p + I = {~x = p(~x0) + ~y | ~x0 ∈ D ∧ ~y ∈ I}

Closed under many basic operations.

Berz. Modern Map Methods in Particle Beam Physics. ser. Advances in Imaging andElectron Physics. Academic Press, 1999, vol. 108.

Xin Chen Using Taylor Models in the Reachability Analysis of Non-linear Hybrid Systems 11 / 43

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High order over-approximations by Taylor models

x

f (x)

-1 -0.5 0 0.5 1

-1

0

1

2

3f (x) = ex

p(x) = 1

p(x) + [−1.8, 1.8]

0-order over-approximation

Xin Chen Using Taylor Models in the Reachability Analysis of Non-linear Hybrid Systems 12 / 43

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High order over-approximations by Taylor models

x

f (x)

-1 -0.5 0 0.5 1

-1

0

1

2

3f (x) = ex

p(x) = 1 + x

p(x) + [−0.8, 0.8]

1-order over-approximation

Xin Chen Using Taylor Models in the Reachability Analysis of Non-linear Hybrid Systems 12 / 43

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High order over-approximations by Taylor models

x

f (x)

-1 -0.5 0 0.5 1

-1

0

1

2

3f (x) = ex

p(x) = 1 + x + x2

2

p(x) + [−0.3, 0.3]

2-order over-approximation

Xin Chen Using Taylor Models in the Reachability Analysis of Non-linear Hybrid Systems 12 / 43

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Taylor model arithmetic

Addition:(p1, I1) + (p2, I2) = (p1 + p2, I1 ⊕ I2)

Multiplication:

(p1, I1) ∗ (p2, I2) = (p1 ∗ p2, (B(p1)⊗ I2)⊕ (I1 ⊗ B(p2))⊕ (I1 ⊗ I2))

Truncation:

Trunck((pn, I )) = (pk , I ⊕ B(pn − pk))

More operations: anti-derivation, Lie derivation, ...

Makino and Berz. Taylor models and other validated functional inclusion methods.J. Pure and Applied Mathematics, vol. 4, no. 4, 2003.

Xin Chen Using Taylor Models in the Reachability Analysis of Non-linear Hybrid Systems 13 / 43

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Verified integration by Taylor models

Assume the ODE is d~xdt = f (~x , t) and the initial set is given by a

Taylor model X0 over domain D0. In the i th integration step,

1 Compute a k-order approximation pk(~x0, t) over~x0 ∈ Xi−1, t ∈ [0, ti − ti−1] for the flow starting from theTaylor model Xi−1.

2 Evaluate a proper remainder Ik such that (pk(~x0, t), Ik) is anover-approximation of the flow in [0, ti − ti−1].

3 Compute the i th flowpipe X[ti−1,ti ] = (pk(Xi−1, t), Ik), andXi = (pk(Xi−1, ti − ti−1), Ik).

Berz and Makino. Verified integration of ODEs and flows using differential algebraicmethods on high-order Taylor models. Reliable Computing, vol. 4, 1998.Berz and Makino. Rigorous integration of flows and ODEs using Taylor models. InProc. of SNC’09.

Xin Chen Using Taylor Models in the Reachability Analysis of Non-linear Hybrid Systems 14 / 43

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Verified integration by Taylor models

Assume the ODE is d~xdt = f (~x , t) and the initial set is given by a

Taylor model X0 over domain D0. In the i th integration step,

1 Compute a k-order approximation pk(~x0, t) over~x0 ∈ Xi−1, t ∈ [0, ti − ti−1] for the flow starting from theTaylor model Xi−1.

2 Evaluate a proper remainder Ik such that (pk(~x0, t), Ik) is anover-approximation of the flow in [0, ti − ti−1].

3 Compute the i th flowpipe X[ti−1,ti ] = (pk(Xi−1, t), Ik), andXi = (pk(Xi−1, ti − ti−1), Ik).

Berz and Makino. Verified integration of ODEs and flows using differential algebraicmethods on high-order Taylor models. Reliable Computing, vol. 4, 1998.Berz and Makino. Rigorous integration of flows and ODEs using Taylor models. InProc. of SNC’09.

Xin Chen Using Taylor Models in the Reachability Analysis of Non-linear Hybrid Systems 14 / 43

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Verified integration by Taylor models

Assume the ODE is d~xdt = f (~x , t) and the initial set is given by a

Taylor model X0 over domain D0. In the i th integration step,

1 Compute a k-order approximation pk(~x0, t) over~x0 ∈ Xi−1, t ∈ [0, ti − ti−1] for the flow starting from theTaylor model Xi−1.

2 Evaluate a proper remainder Ik such that (pk(~x0, t), Ik) is anover-approximation of the flow in [0, ti − ti−1].

3 Compute the i th flowpipe X[ti−1,ti ] = (pk(Xi−1, t), Ik), andXi = (pk(Xi−1, ti − ti−1), Ik).

Berz and Makino. Verified integration of ODEs and flows using differential algebraicmethods on high-order Taylor models. Reliable Computing, vol. 4, 1998.Berz and Makino. Rigorous integration of flows and ODEs using Taylor models. InProc. of SNC’09.

Xin Chen Using Taylor Models in the Reachability Analysis of Non-linear Hybrid Systems 14 / 43

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Verified integration by Taylor models

Assume the ODE is d~xdt = f (~x , t) and the initial set is given by a

Taylor model X0 over domain D0. In the i th integration step,

1 Compute a k-order approximation pk(~x0, t) over~x0 ∈ Xi−1, t ∈ [0, ti − ti−1] for the flow starting from theTaylor model Xi−1.

2 Evaluate a proper remainder Ik such that (pk(~x0, t), Ik) is anover-approximation of the flow in [0, ti − ti−1].

3 Compute the i th flowpipe X[ti−1,ti ] = (pk(Xi−1, t), Ik), andXi = (pk(Xi−1, ti − ti−1), Ik).

Berz and Makino. Verified integration of ODEs and flows using differential algebraicmethods on high-order Taylor models. Reliable Computing, vol. 4, 1998.Berz and Makino. Rigorous integration of flows and ODEs using Taylor models. InProc. of SNC’09.

Xin Chen Using Taylor Models in the Reachability Analysis of Non-linear Hybrid Systems 14 / 43

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Performance

Complexity: If p is a polynomial with n variables and k degree,then it could have

(n+kk

)monomials in the worst case.

Accuracy:

The initial set in every step is represented by a high ordermodel.

By Taylor model arithmetic, overestimation is effectivelyrestricted.

Auxiliary methods can be used to further improve theaccuracy, such as shrink wrapping, preconditioning, ...

Xin Chen Using Taylor Models in the Reachability Analysis of Non-linear Hybrid Systems 15 / 43

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Performance

Complexity: If p is a polynomial with n variables and k degree,then it could have

(n+kk

)monomials in the worst case.

Accuracy:

The initial set in every step is represented by a high ordermodel.

By Taylor model arithmetic, overestimation is effectivelyrestricted.

Auxiliary methods can be used to further improve theaccuracy, such as shrink wrapping, preconditioning, ...

Xin Chen Using Taylor Models in the Reachability Analysis of Non-linear Hybrid Systems 15 / 43

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Performance

Complexity: If p is a polynomial with n variables and k degree,then it could have

(n+kk

)monomials in the worst case.

Accuracy:

The initial set in every step is represented by a high ordermodel.

By Taylor model arithmetic, overestimation is effectivelyrestricted.

Auxiliary methods can be used to further improve theaccuracy, such as shrink wrapping, preconditioning, ...

Xin Chen Using Taylor Models in the Reachability Analysis of Non-linear Hybrid Systems 15 / 43

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Continuous systems:

Van-der-Pol oscillator Brusselator Rossler attractor

How can we apply Taylor models to hybrid systems?

Xin Chen Using Taylor Models in the Reachability Analysis of Non-linear Hybrid Systems 16 / 43

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Continuous systems:

Van-der-Pol oscillator Brusselator Rossler attractor

How can we apply Taylor models to hybrid systems?

Xin Chen Using Taylor Models in the Reachability Analysis of Non-linear Hybrid Systems 16 / 43

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Outline

1 Verification of hybrid systems

2 Taylor model method

3 Intersections of Taylor models and guards

4 Experimental results

5 Future work

Xin Chen Using Taylor Models in the Reachability Analysis of Non-linear Hybrid Systems 17 / 43

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Taylor model flowpipe construction for hybrid systems

In a mode:

Verified integration by using Taylor models.Compute flowpipe/invariant intersections.

For a discrete transition:

Compute flowpipe/guard intersections.Compute the image of the reset mapping.

Xin Chen Using Taylor Models in the Reachability Analysis of Non-linear Hybrid Systems 18 / 43

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Taylor model flowpipe construction for hybrid systems

In a mode:

Verified integration by using Taylor models.Compute flowpipe/invariant intersections.

For a discrete transition:

Compute flowpipe/guard intersections.Compute the image of the reset mapping.

Xin Chen Using Taylor Models in the Reachability Analysis of Non-linear Hybrid Systems 18 / 43

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Over-approximate an intersection

Intersection of a Taylor model (p, I ) and a guard:

~x = p(~x0, t) + ~y ∧ ~x0 ∈ D0 ∧ t ∈ [0,∆]︸ ︷︷ ︸Taylor model domain

∧ ~y ∈ I

︸ ︷︷ ︸Taylor model

∧ γ(~x)︸︷︷︸guard predicate

Xin Chen Using Taylor Models in the Reachability Analysis of Non-linear Hybrid Systems 19 / 43

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Over-approximate an intersection

We propose the following techniques to over-approximate theintersection (p, I ) ∩ G .

Domain contraction -Contract the domain of p such that (p, I ) with the newdomain is the over-approximation.

Range over-approximation -Over-approximate the Taylor model (p, I ) by a convex set S ,then compute S ∩ G .

Template method -Compute a Taylor model over-approximation (p∗, 0) based ona given template.

They can be used in a combination.

Xin Chen Using Taylor Models in the Reachability Analysis of Non-linear Hybrid Systems 20 / 43

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Over-approximate an intersection

We propose the following techniques to over-approximate theintersection (p, I ) ∩ G .

Domain contraction -Contract the domain of p such that (p, I ) with the newdomain is the over-approximation.

Range over-approximation -Over-approximate the Taylor model (p, I ) by a convex set S ,then compute S ∩ G .

Template method -Compute a Taylor model over-approximation (p∗, 0) based ona given template.

They can be used in a combination.

Xin Chen Using Taylor Models in the Reachability Analysis of Non-linear Hybrid Systems 20 / 43

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Domain contraction - An example

t

D0

TM

guard

(a)

t

D0

new TMguard

(b)

Xin Chen Using Taylor Models in the Reachability Analysis of Non-linear Hybrid Systems 21 / 43

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Domain contraction - Main procedure

Compute an interval contraction by interval constraintpropagation:

1 Contract the interval in one dimension.

2 Propagate the result to the next dimension.

Accuracy can be further improved by

Perform the contraction through all dimensions for severaltimes.

Determine an order on the dimensions and perform thecontraction for one time.

Xin Chen Using Taylor Models in the Reachability Analysis of Non-linear Hybrid Systems 22 / 43

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Domain contraction - Main procedure

Compute an interval contraction by interval constraintpropagation:

1 Contract the interval in one dimension.

2 Propagate the result to the next dimension.

Accuracy can be further improved by

Perform the contraction through all dimensions for severaltimes.

Determine an order on the dimensions and perform thecontraction for one time.

Xin Chen Using Taylor Models in the Reachability Analysis of Non-linear Hybrid Systems 22 / 43

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Domain contraction - Contraction in one dimension

Compute the lower bound for the contracted domain in the i th

dimension.

1: while the size of [lo, up] is larger than ε do2: Split [lo, up] into [lo, a] and [a, up] wherein a = lo+up

2 ;3: if (p(~y), I ) with ~y ∈ D and (~y)i ∈ [lo, a] intersects the guard

then4: up← a;5: else6: lo← a;7: end if8: end while9: return lo;

Xin Chen Using Taylor Models in the Reachability Analysis of Non-linear Hybrid Systems 23 / 43

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Domain contraction - Contraction in one dimension

Compute the lower bound for the contracted domain in the i th

dimension.

1: while the size of [lo, up] is larger than ε do2: Split [lo, up] into [lo, a] and [a, up] wherein a = lo+up

2 ;3: if (p(~y), I ) with ~y ∈ D and (~y)i ∈ [lo, a] intersects the guard

then4: up← a;5: else6: lo← a;7: end if8: end while9: return lo;

Xin Chen Using Taylor Models in the Reachability Analysis of Non-linear Hybrid Systems 23 / 43

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Domain contraction - Emptiness checking

Check the emptiness of

{~x ∈ Rn | ~x = p(~y) + ~z ∧ ~y ∈ D ∧ ~z ∈ I ∧ γ(~x)}

Approximation methods: Conservative simplification on p and γ.

Xin Chen Using Taylor Models in the Reachability Analysis of Non-linear Hybrid Systems 24 / 43

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Domain contraction - An example

Xin Chen Using Taylor Models in the Reachability Analysis of Non-linear Hybrid Systems 25 / 43

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Domain contraction

Advantages:

Polynomial computation time.

The result is still a Taylor model.

Disadvantages:

The contracted domain is always an interval.

Large overestimation might be introduced when either p or γis of high degree.

Xin Chen Using Taylor Models in the Reachability Analysis of Non-linear Hybrid Systems 26 / 43

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Range over-approximation

1 Over-approximate (p, I ) by a set S which can be a

support function -Conservative over-approximation of a polynomial.zonotope -Zonotopes are order 1 Taylor models and vise versa.

2 Over-approximate S ∩ G by a Taylor model.

Sankaranarayanan, Dang and Ivancic. Symbolic Model Checking of Hybrid SystemsUsing Template Polyhedra. In Proc. of TACAS’08.Le Guernic and Girard. Reachability Analysis of Hybrid Systems Using SupportFunctions. In Proc. of CAV’09.

Xin Chen Using Taylor Models in the Reachability Analysis of Non-linear Hybrid Systems 27 / 43

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Range over-approximation

Advantages:

Available algorithms can be used.

The orientation of the over-approximation can be chosen.

Disadvantages:

Multi-time over-approximation.

Best orientation is not easy to find.

Xin Chen Using Taylor Models in the Reachability Analysis of Non-linear Hybrid Systems 28 / 43

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Template method

We call p∗(~x0) a template polynomial if its domain is given by

~x0 ∈ Du = [`1, u1]× [`2, u2]× · · · × [`n, un]

wherein `1, . . . , `n, u1, . . . , un are unknown parameters.

We find the parameters such that the Taylor model (p∗, 0)contains the intersection (p, I ) ∩ G :

∀~x .((~x = p(~y) + ~z ∧ ~y ∈ D ∧ ~z ∈ I ∧ ~x ∈ G )

→ ∃~x0.(~x = p∗(~x0) ∧ ~x0 ∈ Du))

We may also add more constraints to limit the overestimation.

Sankaranarayanan, Sipma and Manna. Constructing invariants for hybrid systems.Formal Methods in System Design. 2008.Gulwani and Tiwari. Constraint-Based Approach for Analysis of Hybrid Systems. InProc. of CAV’08.

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Template method

We call p∗(~x0) a template polynomial if its domain is given by

~x0 ∈ Du = [`1, u1]× [`2, u2]× · · · × [`n, un]

wherein `1, . . . , `n, u1, . . . , un are unknown parameters.

We find the parameters such that the Taylor model (p∗, 0)contains the intersection (p, I ) ∩ G :

∀~x .((~x = p(~y) + ~z ∧ ~y ∈ D ∧ ~z ∈ I ∧ ~x ∈ G )

→ ∃~x0.(~x = p∗(~x0) ∧ ~x0 ∈ Du))

We may also add more constraints to limit the overestimation.

Sankaranarayanan, Sipma and Manna. Constructing invariants for hybrid systems.Formal Methods in System Design. 2008.Gulwani and Tiwari. Constraint-Based Approach for Analysis of Hybrid Systems. InProc. of CAV’08.

Xin Chen Using Taylor Models in the Reachability Analysis of Non-linear Hybrid Systems 29 / 43

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Template method

We call p∗(~x0) a template polynomial if its domain is given by

~x0 ∈ Du = [`1, u1]× [`2, u2]× · · · × [`n, un]

wherein `1, . . . , `n, u1, . . . , un are unknown parameters.

We find the parameters such that the Taylor model (p∗, 0)contains the intersection (p, I ) ∩ G :

∀~x .((~x = p(~y) + ~z ∧ ~y ∈ D ∧ ~z ∈ I ∧ ~x ∈ G )

→ ∃~x0.(~x = p∗(~x0) ∧ ~x0 ∈ Du))

We may also add more constraints to limit the overestimation.

Sankaranarayanan, Sipma and Manna. Constructing invariants for hybrid systems.Formal Methods in System Design. 2008.Gulwani and Tiwari. Constraint-Based Approach for Analysis of Hybrid Systems. InProc. of CAV’08.

Xin Chen Using Taylor Models in the Reachability Analysis of Non-linear Hybrid Systems 29 / 43

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Template method

Advantages:

Flexibility.

The unknown parameters can be found by SMT solving.

Disadvantages:

Bad scalability because of the quantifier elimination.

Best template is not easy to find.

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Outline

1 Verification of hybrid systems

2 Taylor model method

3 Intersections of Taylor models and guards

4 Experimental results

5 Future work

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Glycemic Control in Diabetic Patients

dGdt = −p1G − X (G + GB) + g(t)dXdt = −p2X + p3IdIdt = −n(I + Ib) + 1

VIi(t)

Typical parameters:

p1 = 0.01, p2 = 0.025, p3 = 1.3× 10−5, VI = 12,n = 0.093, GB = 4.5, Ib = 15.

G plasma glucose concentration above the basal value GB

I plasma insulin concentration above the basal value IBX insulin concentration in an interstitial chamber

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Glycemic Control in Diabetic Patients

g(t) infusion of glucose into the bloodstreami(t) infusion of insulin into the bloodstream

The infusion of glucose after a meal:

g(t) =

t

60 t ≤ 30120−t

180 t ∈ [30, 120]0 t ≥ 120

Two control schemes:

i1(t) =

253 G (t) ≤ 4

253 (G (t)− 3) G (t) ∈ [4, 8]

1253 G (t) ≥ 8

i2(t) =

{1 + 2G(t)

9 G (t) < 6503 G (t) ≥ 6

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Glycemic Control in Diabetic Patients

Our platform: CPU i7-860 2.8 GHz Memory: 4 GBOrder of the Taylor models: 9 Time step: 0.02 Time horizon: [0,360]

Total time: 2138 s Time of intersection: 663 s Memory: 430 MB

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Glycemic Control in Diabetic Patients

Order of the Taylor models: 9 Time step: 0.02 Time horizon: [0,360]

Total time: 1804 s Time of intersection: 443 s Memory: 410 MB

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Vehicle model

The dynamics of a non-holonomic vehicle is given as follows,

dxdt = vct

dydt = vst

dvdt = u1

dctdt = σv 2st

dstdt = −σv 2ct

dσdt = u2

m1 m2 m3

st > 0.7

st < 0.65

st > 0.8

st < 0.75

Initial set:

x ∈ [1, 1.2] y ∈ [1, 1.2] v ∈ [0.8, 0.81]st ∈ [0.7, 0.71] ct ∈ [0.7, 0.71] σ ∈ [0, 0.05] .

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Vehicle model

x

y

Xin Chen Using Taylor Models in the Reachability Analysis of Non-linear Hybrid Systems 37 / 43

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Vehicle model

x

y

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Vehicle model

x

y

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Vehicle model

Order of the Taylor models: 9 Time step: 0.1 Time horizon: [0,10]

Total time: 85 s Time of intersection: 40 s Memory: 4 MB

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Scalability tests

D = 6deg = 2 deg = 4

size order T mem size order T mem

34 8 4

34 18 4

9 211 12 9 449 12

54 13 4

54 27 4

9 298 12 9 587 12

74 18 4

74 36 4

9 373 12 9 687 12D = 8

deg = 2 deg = 4size order T mem size order T mem

34 14 4

34 23 4

9 1389 28 9 2430 28

54 18 4

54 27 4

9 1433 28 9 2446 28

74 22 4

74 28 4

9 2074 28 9 2474 28D = 10

deg = 2 deg = 4size order T mem size order T mem

34 16 4

34 27 8

9 t.o. - 9 t.o. -

54 22 8

54 46 8

9 t.o. - 9 t.o. -

74 35 8

74 62 8

9 t.o. - 9 t.o. -

Dang and Testylier. Hybridization domain construction using curvature estimation.In Proc. of HSCC’11.

Xin Chen Using Taylor Models in the Reachability Analysis of Non-linear Hybrid Systems 39 / 43

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More examples

http://systems.cs.colorado.edu/research/cyberphysical/taylormodels/

Xin Chen Using Taylor Models in the Reachability Analysis of Non-linear Hybrid Systems 40 / 43

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Outline

1 Verification of hybrid systems

2 Taylor model method

3 Intersections of Taylor models and guards

4 Experimental results

5 Future work

Xin Chen Using Taylor Models in the Reachability Analysis of Non-linear Hybrid Systems 41 / 43

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Future work

Taylor model flowpipe construction with varying orders andtime steps.

Split Taylor models according to equilibriums.

Taylor model contraction for domains.

Heuristics for choosing templates.

Xin Chen Using Taylor Models in the Reachability Analysis of Non-linear Hybrid Systems 42 / 43

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Thank you!Questions?

Xin Chen Using Taylor Models in the Reachability Analysis of Non-linear Hybrid Systems 43 / 43