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Set-Oriented Dimension Reduction For Reachability Computations

Bai Xue and Martin Fr¨ anzle

Universit¨ at Oldenburg, Germany {bai.xue, martin.fraenzle}@uni-oldenburg.de

November 3, 2016

Bai Xue and Martin Fr¨ anzle (Universit¨ at Oldenburg, Germany{bai.xue, martin.fraenzle}@uni-oldenburg.de) Set-Oriented Dimension Reduction For Reachability Computations November 3, 2016 1 / 32

Outline

1

Background

2

Preliminaries

3

Reachable Sets Computation

4

Examples

5

Discussions and Comparisons

6

Conclusion and Future Work

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Background

Cyber-Physical Systems: a new generation of systems with integrated computational and physical capabilities that can interact with humans through many new modalities Applications:

1

Space vehicles

2

Hybrid gas-electric vehicles

3

Fully autonomous urban driving

4

. . .

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Background

Cyber-physical systems are routinely subject to deviations coming from

1

sensor noise

2

actuation noise

3

delays over unreliable network channels

4

. . . ⇓ Requirement: Robustness ⇓ system acts safely in the context of disturbances

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Background

Reachability Analysis: the process of computing the set of reachable states for a system with uncertain initial states, parameters and inputs ⇓ Validate this robustness property of cyber-physical systems Problems inherent in Reachability Analysis: Wrapping Effects [J. Lunze and F. Lamnabhi-Lagarrigue (2009)]

Definition 1.1

The wrapping effect is the propagation and accumulation of over-approximation error through the iterative computation in the construction of reachable sets. ⇓ Poorly Pessimistic Approximation

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Background

Observations and Difficulties for Overcoming Wrapping Effects

1

Observations: The extent of wrapping effects correlates strongly with the volume of the initial set Classical Techniques: Partition the initial state space

2

Difficulties: Such partitions ⇒

1

extensive demand on computation time and memory, especially for large initial sets and/or large time horizons

2

the number of partitions: exponential increase with the dimensions of systems

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Background

Facilitating the reduction of the wrapping effect in applying existing reachability analysis techniques to perform reachability computations: A set-oriented dimension reduction based method Its Features:

1

a collection of subsets extracted from the initial set in a low dimensional space is explored for computations

2

the number of subsets increases polynomially with the dimensions of systems

Bai Xue and Martin Fr¨ anzle (Universit¨ at Oldenburg, Germany{bai.xue, martin.fraenzle}@uni-oldenburg.de) Set-Oriented Dimension Reduction For Reachability Computations November 3, 2016 7 / 32

Outline

1

Background

2

Preliminaries

3

Reachable Sets Computation

4

Examples

5

Discussions and Comparisons

6

Conclusion and Future Work

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2.1 Problem of Interest

Consider the system modelled by an ordinary differential equation: ˙ x = f(x), x(0) = x0 = (x1,0, . . . , xn,0)′ ∈ X0, (1) where

1

f(·) = (f1, . . . , fn)′ : Rn − → Rn is Lipschitz continuous;

2

the trajectory over the time interval [0, T] is defined to be φ(t; x0, 0) = x(t), in which x(t) is the solution to the system (1) with the initial condition x0 ∈ X0 at time instant t0 and φ(·) = (φ1(·), . . . , φn(·))′.

Definition 2.1 (Problem of Interest)

Compute a set Ψ(t; X0, t0) such that this set encloses all states reached by system (1) initialised in the set X0 of the interval form for the duration of t ∈ [0, T].

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2.2 Sensitivity Analysis Theory

The sensitivity to initial conditions at time t is defined by sx0(t) = ∂φ(t; x0, t0) ∂x0 , where sx0 is a square matrix of order n. Sensitivity Equation [A. Donz´ e and O. Maler (2007)]: ˙ sx0(t) = Dfsx0, sx0(0) = I ∈ Rn×n, (2) where Df is the Jacobian matrix of f along the trajectory φ(t; x0, t0) and I is the identity matrix. Remark: The ijth element of sx0 basically represents the influence of variations in the ith coordinate xi,0 of x0 on the jth coordinate xj(t) of φ(t; x0, t0).

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2.3 Interval Linear Systems

An interval linear system is of the form: ˙ y = Ay, y(t0) = y0 ∈ Y0, where y ∈ Rn, Y0 ⊂ Rn and A is a n × n matrix, the elements of which are independent intervals of real numbers.

Theorem 2.2 (M. Althoff et al. (2007))

The over-approximation of the interval matrix exponential with order p, denoted ⌈eAt

p

⌉, where A ∈ I n×n, the elements of which are independent intervals of real numbers, and t ≥ 0, is obtained by ⌈eAt

p

⌉ = I + W (t) +

p

• i=3

1 i! (At)i + E(t) with E(t) =< −1, 1 > ((A∞)t)p+1 (p + 1)! 1 1 − ǫ , ǫ = A∞t p + 2 < 1, where W (t) = At + 1

2 A2t2 can be computed exactly by the following procedure using interval arithmetic: ∀i = j : wij =

aii (t + 1

2 (aii + ajj )t2) + 1 2

• k:k=i,k=j aik akj t2, ∀i : wii = [κ(aii , t), max({aii t + 1

2 a2 ii t2, aii t + 1 2 a2 ii t2})] + 1 2

• k:k=i aik aki t2,

κ(aii , t) =

• min({aii t + 1

2 a2 ii t2, aii t + 1 2 a2 ii t2}), − 1 t /

∈ aii − 1

2 , − 1 t ∈ aii

and A∞ = max(|A|, |A|)∞, in which the absolute value and the maximum in the above expression are determined elementwise. Bai Xue and Martin Fr¨ anzle (Universit¨ at Oldenburg, Germany{bai.xue, martin.fraenzle}@uni-oldenburg.de) Set-Oriented Dimension Reduction For Reachability Computations November 3, 2016 11 / 32

Outline

1

Background

2

Preliminaries

3

Reachable Sets Computation

4

Examples

5

Discussions and Comparisons

6

Conclusion and Future Work

Bai Xue and Martin Fr¨ anzle (Universit¨ at Oldenburg, Germany{bai.xue, martin.fraenzle}@uni-oldenburg.de) Set-Oriented Dimension Reduction For Reachability Computations November 3, 2016 12 / 32

3 Reachable Sets Computation

The sensitivity matrix sx0=     

∂φ1(t;x0,0) ∂x1,0 ∂φ1(t;x0,0) ∂x2,0

. . .

∂φ1(t;x0,0) ∂xn,0 ∂φ2(t;x0,0) ∂x1,0 ∂φ2(t;x0,0) ∂x2,0

. . .

∂φ2(t;x0,0) ∂xn,0

. . .

∂φn(t;x0,0) ∂x1,0 ∂φn(t;x0,0) ∂x2,0

. . .

∂φn(t;x0,0) ∂xn,0

     Underlying Theory Behind Our Method:

Theorem 3.1

Given a system (1) with an initial set X0 = [x1,0, x1,0]×. . .×[xn,0, xn,0] ⊂ Rn and a time instant t ∈ [t0, t1], if ∂φi(t;x0,t0)

∂xj,0

, j = li, . . . , mi, i ∈ {1, . . . , n}, has an uniquely determined sign for all x0 = (x1,0, . . . , xn,0)′ ∈ X0, where {li, . . . , mi} ⊆ {1, . . . , n} and li, mi are positive integers with 1 ≤ li ≤ mi ≤ n, then there exist two sets Yi,0 and Zi,0 of dimension n − ki such that the mapping φi(t; X0, t0) can be bounded by performing computations on these two sets respectively, where ki is the number

• f the elements in the set {li, . . . , mi}.

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3 Reachable Sets Computation

The rule for choosing two sets Yi,0 and Zi,0, i ∈ {1, . . . , n}, where Yi,0 = [y i,1,0, y i,1,0] × . . . × [y i,n,0, y i,n,0] and Zi,0 = [zi,1,0, zi,1,0] × . . . × [zi,n,0, zi,n,0]: If j = li, . . . , mi, for all x0 ∈ X0 = [x1,0, x1,0] × . . . × [xn,0, xn,0], y i,j,0 = y i,j,0 =

• xj,0, if ∂φi(t;x0,t0)

∂xj,0

≥ 0 xj,0, if ∂φi(t;x0,t0)

∂xj,0

≤ 0 , zi,j,0 = zi,j,0 =

• xj,0, if ∂φi(t;x0,t0)

∂xj,0

≤ 0 xj,0, if ∂φi(t;x0,t0)

∂xj,0

≥ 0 , Otherwise, y i,j,0 = zi,j,0 = xj,0 and y i,j,0 = zi,j,0 = xj,0. ⇓ infz0∈Zi,0φi(t; z0, t0) ≤ φi(t; x0, t0) ≤ supy0∈Yi,0φi(t; y0, t0)

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3 Reachable Sets Computation

The sensitivity matrix sx0=     

∂φ1(t;x0,0) ∂x1,0 ∂φ1(t;x0,0) ∂x2,0

. . .

∂φ1(t;x0,0) ∂xn,0 ∂φ2(t;x0,0) ∂x1,0 ∂φ2(t;x0,0) ∂x2,0

. . .

∂φ2(t;x0,0) ∂xn,0

. . .

∂φn(t;x0,0) ∂x1,0 ∂φn(t;x0,0) ∂x2,0

. . .

∂φn(t;x0,0) ∂xn,0

    

Remark 1

If the monotonic property pertaining to the mapping solution φi(t; x0, t0) as presented in Theorem 3.1 is explored successfully for i = 1, . . . , n, an over-approximation of the reachable set at time instant t ∈ [t0, t1] can be constructed using some low-dimensional subsets instead of the entire initial set.

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3 Reachable Sets Computation

Remark 2

The first order Taylor series for the solution φ(t; x0, 0) with respect to the time instant 0: φ(t; x0, t0) = x0 + f(ξx0)t, where ξx0 = φ(τ; x0, 0) in which τ lies between t0 and t and x0 = (x1,0, . . . , xn,0)′ ∈ X0. The diagonal elements of the sensitivity matrix sx0(t) are of the form 1 + ∂f(x)

∂x |x=ξx0 ∂ξx0 ∂xi,0 t,

i ∈ {1, . . . , n}. ⇓ limt→t01 + ∂f(x)

∂x |x=ξx0 ∂ξx0 ∂xi,0 t = 1.

Therefore, the diagonal elements of the sensitivity matrix sx0(t) can always achieve values greater than zero if the time interval [0, t1] is sufficiently small. This corresponds to boundary reachability computations.

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3 Reachable Sets Computation

The implementation in every iteration consists of two steps mainly:

1

Solving sensitivity equation (2)

2

Performing reachability computations on subsets {Yi,0, Zi,0} chosen according to the aforementioned rule ⇓ Bound the reachable set The emphasis is on Step 1: Solving sensitivity equation (2), i.e., ˙ sx0 = Dfsx0, s(0) = I ∈ Rn×n.

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3 Reachable Sets Computation

Three steps for computing a validation solution to Equation (2), i.e. ˙ sx0 = Dfsx0 at time instant t ∈ [0, t1]:

1

Compute a prior interval enclosure X of the states traversed by system (1) originating from the initial set X0 at time instant t = t0 for the time duration t1

2

Over-approximate Df based on the enclosure X to obtain an interval matrix A by carrying out interval arithmetic operations

3

Resolve the resulting interval linear system ˙ x = Ax to produce a validated solution

• f the sensitivity equation (2) with respect to the time instant t ∈ [0, t1].

Bai Xue and Martin Fr¨ anzle (Universit¨ at Oldenburg, Germany{bai.xue, martin.fraenzle}@uni-oldenburg.de) Set-Oriented Dimension Reduction For Reachability Computations November 3, 2016 18 / 32

Outline

1

Background

2

Preliminaries

3

Reachable Sets Computation

4

Examples

5

Discussions and Comparisons

6

Conclusion and Future Work

Bai Xue and Martin Fr¨ anzle (Universit¨ at Oldenburg, Germany{bai.xue, martin.fraenzle}@uni-oldenburg.de) Set-Oriented Dimension Reduction For Reachability Computations November 3, 2016 19 / 32

4 Examples

We test our set-oriented dimension reduction method, which implemented based on vali- dated ordinary differential equation solver VNODE-LP [N. S. Nedialkov (2006)]1 , on two

• examples. The time steps in our implementations are generated by VNODE-LP automat-
• ically. (Platform: an i5-3337U 1.8GHz CPU with 4GB running Ubuntu Linux 13.10)

Example 4.1

Consider the FitzHugh-Nagumo neuron model describing the electrical activity of a neuron, ˙ x = x − x3 − y + 7

8

˙ y = 0.08(x + 0.7 − 0.8y) with X0 = [0.5, 1.5] × [2.0, 3.0] and T = 20. Computation time: 1.26 seconds.

1It is available at www.cas.mcmaster.ca/˜nedialk/vnodelp/. Bai Xue and Martin Fr¨ anzle (Universit¨ at Oldenburg, Germany{bai.xue, martin.fraenzle}@uni-oldenburg.de) Set-Oriented Dimension Reduction For Reachability Computations November 3, 2016 20 / 32

4 Examples

Example 4.2

Consider a nine-dimensional biological systema,                            ˙ x1 = 50x3 − x1x6 ˙ x2 = 100x4 − x2x6 ˙ x3 = x1x6 − 50x3 ˙ x4 = x2x6 − 100x4 ˙ x5 = 500x3 + 50x1 − 10x5 ˙ x6 = 50x5 + 50x3 + 100x4 − x6(x1 + x2 + 2x8 + 1) ˙ x7 = 50x4 + 0.01x2 − 0.5x7 ˙ x8 = 5x7 − 2x6x8 + x9 − 0.2x8 ˙ x9 = 2x6x8 − x9 with X0 is an interval consisting of the value of xi ranging in [0.09, 0.11] for 1 ≤ i ≤ 9 and T = 0.4.

ahttp://systems.cs.colorado.edu/research/cyberphysical/taylormodels/casestudies/

Computation time: 10.72 seconds.

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4 Examples

−2 −1.5 −1 −0.5 0.5 1 1.5 2 −0.5 0.5 1 1.5 2 2.5 3 3.5 x y

Figure: An over-approximation of the reachable set over the interval [0.0, 20] for Example

4.1. (Red lines – the boundary of Ψ(t0 + hi; X0, 0.0), i = 1, . . . , 310; Blue lines – the boundary of the initial set X0; Black lines – the boundary of Ψ(20; X0, 0.0).)

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4 Examples

0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.1 0.2 0.3 0.4 0.5 x8 x1

Figure: An over-approximation of the projection of the reachable set in the x8 − x1 plane

• ver the interval [0.0, 0.4] for Example 4.2. (Red lines – the boundary of the projection
• f Ψ(t0 + hi; X0, 0.0) in the x8 − x1 plane, i = 1, . . . , 37; Black lines – the boundary of

the projection of Ψ(0.4; X0, 0.0) in the x8 − x1 plane; Blue lines – the boundary of the projection of X0 in the x8 − x1 plane.)

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Outline

1

Background

2

Preliminaries

3

Reachable Sets Computation

4

Examples

5

Discussions and Comparisons

6

Conclusion and Future Work

Bai Xue and Martin Fr¨ anzle (Universit¨ at Oldenburg, Germany{bai.xue, martin.fraenzle}@uni-oldenburg.de) Set-Oriented Dimension Reduction For Reachability Computations November 3, 2016 24 / 32

5.1 Discussions by Comparing with VNODE-LP

Experiment results from VNODE-LP without subdividing initial sets: For Examples 4.1 and 4.2,

1

VNODE-LP alone cannot proceed past reachability times a 0.462964282008106[1, 2] b 0.231860158836194[7, 8]

2

The reachable sets corresponding to the above two reachability time intervals are a [−2.667 × 106, 2.667 × 106] × [1.901, 3.044] b [−1.645×108, 1.645×108]×[−3.223×108, 3.223×108]×[−1.645×108, 1.645× 108]×[−3.223×108, 3.223×108]×[−445.631, 450.805]×[−3.253×1014, 3.253× 1014]×[−61.941, 62.554]×[−3.253×1014, 3.253×1014]×[−3.253×1014, 3.253× 1014] VNODE-LP alone: TOO CONSERVATIVE!! Comparison Conclusion: Incorporating our set-oriented dimension reduction method ⇒ Reducing the wrapping effect significantly

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5.1 Discussions by Comparing with VNODE-LP

Experiment results from VNODE-LP by subdividing initial sets: For Examples 4.1 and 4.2, the corresponding minimal numbers of equal-sized subsets are

1

182

2

≥ 59 The corresponding computation times:

1

11.65 seconds for Example 4.1

2

cannot return a result due to memory exhaustion resulting from an enormous amount

• f computations for Example 4.2

Comparison Conclusion: Our set-oriented dimension reduction method ⇒ Promising in improving the scalability

• f the method implemented in VNODE-LP

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5.2 Discussions by comparing with IOLAVABE2

IOLAVABE [A. Eggers et al (2012)] encapsulates

1

the bracketing enclosure method

2

the method implemented in VNODE-LP The feature of the reachability analysis based method in IOLAVABE:

1

Constructing bracketing systems with dimension twice that of the original system

2

Exploring two corner points of the initial set to perform reachability computations Experiment results from IOLAVABE: For Examples 4.1 and 4.2, IOLAVABE cannot proceed past reachability times

1

t = 0.462964282008106[1, 2]

2

t = 0.231860158836194[7, 8]

2This package can be downloaded from

https://seshome.informatik.uni-oldenburg.de/eggers/iolavabe.php.

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5.2 Discussions by comparing with IOLAVABE

Main differences between the method implemented in IOLAVABE and our method:

1

IOLAVABE:

1

bracketing system with dimension twice that of the original system

2

two corner points

3

Basic Prerequisite: every entry in the Jacobian matrix Df has uniquely determined sign

2

Our method:

1

• riginal system

2

at most 2n subsets extracted from the initial set’s boundary

3

Basic Prerequisite: at least one entry in the associated sensitivity matrix has uniquely determined sign

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Outline

1

Background

2

Preliminaries

3

Reachable Sets Computation

4

Examples

5

Discussions and Comparisons

6

Conclusion and Future Work

Bai Xue and Martin Fr¨ anzle (Universit¨ at Oldenburg, Germany{bai.xue, martin.fraenzle}@uni-oldenburg.de) Set-Oriented Dimension Reduction For Reachability Computations November 3, 2016 29 / 32

6.1 Conclusion

Features of the set-oriented dimension reduction method:

1

subsets extracted from the initial set’s boundary was explored for computations by conducting sensitivity analysis

2

the number of subsets increases polynomially with the dimension of systems rather than exponentially Experiment Results: The experiment results show that

1

VNODE-LP integrating our set-oriented set dimension reduction method gains sig- nificant benefits: the reduction of wrapping effect and scalability improvement

2

the set-oriented dimension reduction method based on conducting sensitivity analysis is superiority over the bracketing system based method implemented in IOLAVABE for some cases

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6.2 Future Work

1 Investigate it in reachability analysis and safety verification problems

involving delay differential equations based on conducting sensitivity analysis

2 Conducting online verification of autonomous cars by combining reach-

ability analysis theory (computing over- and under-approximations), probability theory and machine learning together

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Thank you! Any Questions?

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